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Free energy and phase transition of the matrix model on a plane-wave Hadizadeh, Shirin 2006

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Free Energy and Phase Transition of the Matrix Model on a Plane-Wave by Shirin Hadizadeh B.Sc , Shahid Beheshti University, 2000 M . S c , Shahid Beheshti University, 2003 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia February 2006 © Shirin Hadizadeh 2006 11 A b s t r a c t It has recently been observed that the weakly coupled plane wave matrix model has a density of states which grows exponentially at high energy. This implies that the model has a phase transition. The transition appears to be of first order. However, its exact nature is sensitive to interactions. In this thesis, we analyze the effect of interaction by computing the relevant parts of the effective potential for the Polyakov loop operator in the finite temperature plane-wave matrix model to three loop order. We also compute correction to the Hagedorn temperature to order two loops. iii Contents Abs t rac t i i Contents i i i Lis t of Figures v Acknowledgements vii 1 In t roduct ion 1 2 Hagedorn Behavior of M a t r i x M o d e l 10 2.1 A Toy Model 10 2.2 Polya Style of Counting States 14 2.2.1 The d = 1 Case 15 2.2.2 The d > 1 Case 15 2.2.3 Extensions 18 2.2.4 Hagedorn Behavior in Free Second Quantized Theories 23 3 Thermodynamics of the M a t r i x M o d e l 25 3.1 The Toy Model 25 3.2 Confinement-Deconflnement Transition 30 3.3 The Matrix Model at One Loop 32 3.3.1 Gauge fixing 33 3.3.2 Classical ground states 34 3.3.3 Semiclassical expansion 38 4 D i s t r i b u t i o n of the Angles f3Aa 42 4.1 Systematic improvement of the semicircle 46 4.2 High temperature limit 48 4.3 Free energy 48 4.4 Symmetry restoration 49 Contents iv 5 Higher loop order 51 5.1 Phase transition 52 5.2 Free Energy to Two Loop Order 53 5.2.1 Propagators 54 5.2.2 Fermion Contributions 57 5.2.3 The Theta Diagram 62 5.2.4 The Figure-Eight Diagram 65 6 Conclus ion 68 6.1 The Free Energy at Three Loops 68 6.1.1 Tripple Bubble Diagram 71 6.1.2 Theta-Bubble Diagram 73 6.1.3 Circle-T Diagram 75 6.1.4 Two-Rung Ladder Diagram 77 6.2 Thermal Green functions 80 Appendices 82 A Super symmetry of the Massive M a t r i x M o d e l 83 A . l Symmetry Algebra 83 A.2 Plane wave limit of the lOd IIB AdS5 x S 5 action 84 A.3 Matrix theory action 86 B F i x i n g the Static Diagonal Gauge 91 C D e r i v i n g u>sc(z) i n pp-wave M a t r i x M o d e l 95 Bib l iography 100 V List of Figures 5.1 Two loop fermionic diagram 57 5.2 Theta diagram 63 5.3 Eight-shape diagram 65 6.1 P-cat's eye diagram 70 6.2 Q-cat's eye diagram 70 6.3 PQ-cat's eye diagram 71 6.4 PQP-tripple bubble diagram 71 6.5 QPQ-tripple bubble diagram 71 6.6 PPQ-tripple bubble diagram 72 6.7 PQQ-tripple bubble diagram 72 6.8 PPP-tripple bubble diagram 72 6.9 QQQ-tripple bubble diagram 73 6.10 Simly scalar diagram 73 6.11 P-theta bubble diagram 73 6.12 Q-theta bubble diagram 74 6.13 PP-theta bubble diagram 74 6.14 PQ-theta bubble diagram 74 6.15 QP-theta bubble diagram 75 6.16 QQ-theta bubble diagram 75 6.17 PP-circle-T diagram 75 6.18 PQ-circle-T diagram 76 6.19 QQ-circle-T diagram 76 6.20 Fermion-scalar circle-T diagram 76 6.21 Scalar-scalar circle-T diagram 77 6.22 PP-two-rung ladder diagram 77 6.23 QQ-two-rung ladder diagram 77 6.24 Q-two-rung ladder diagram 78 6.25 P-two-rung ladder diagram 78 6.26 PQ-two-rung ladder diagram 79 List of Figures vi 6.27 Fermion-scalar two-rung ladder diagram 79 6.28 Scalar-scalar two-rung ladder diagram 79 V l l Acknowledgements I feel most fortunate to have had the opportunity to study and broaden my knowledge in the University of British Columbia and to carry out my Master's thesis at the Department of Physics and Astronomy. The writing of this thesis would not have been possible without the help and encouragement of many people. It is a great pleasure to extend my sincere gratitude toward all of them. First and foremost, I am highly indebted to my supervisor, Dr. G. W. Semenoff, as this thesis borrows heavily from his unlimited help, stimulat-ing effort and unrestricted patience. He guided me not to get lost during the development of this thesis and provided a motivating and enthusiastic atmosphere during the discussions we had. Dr. Semenoff was so devoting and patient in correcting my draft. It was a great pleasure to do this thesis under his supervision. I am deeply greateful to my other supervisors, Dr. D. Witt and Dr. K . Schleich, excellent teachers, whose suport and advise I have relied on during my masters program. I have enjoied my collaboration with Donovan Young and Bojan Ra-madanovic and would like to acknowledge them for making the process of writing this thesis a managable task. A special thanks goes to Fereydoun Hormozdiari for assisting me in type-setting and also being a main source of hope and encouragement. Without his help the completion of this thesis would have been an unachievable goal. At last but not least, I would like to express my heartfelt and everlasting appreciation to my family for their endless love and support. I specially owe a great debt to my mother and father for sacrificing their own joy and comfort, and giving me moral support throughout all years of my study. Nothing would have ever been accomplished without their countinous love and encouragement. This thesis is dedicated to my parents. Chapter 1 i Introduction It is believed that a maximally super/symmetric matrix quantum mechanics called BFSS matrix model describes the full dynamics of M-theory. In 1997, T. Banks, W. Fichler, S.H. Shenker and L. Susskind suggested an equivalence between the large iV limit of the supersymmetric matrix quantum mechan-ics which describes .DO-branes and uncompactified eleven dimensional M -Theory. The evidence for the conjecture consists of several correspondences between the two theories. This was the first nonperturbative formulation of a quantum theory which includes gravity. M-theory is the strongly coupled limit of type IIA string theory. In the limit of infinite coupling it becomes an eleven dimensional theory in a back-ground infinite flat space. Our knowledge about this theory is that: 1) at low energy and large distances it is described by eleven dimensional supergravity and 2) it possesses membrane degrees of freedom with membrane tension ^ where lp is the elevenh dimensional Planck length. There was a problem in believing that we are actually able to consider M-theory as any kind of con-ventional quantum field theory. The degrees of freedom describing the short distance behavior were simply unknown. BFSS has put forward a conjecture about these degrees of freedom and about the Hamiltonian governing them. The conjecture is that M-theory in the light cone frame is exactly described by N—> oo limit of a particular supersymmetric matrix quantum mechanics. The system is the same one that has been used even before their work to study the small distance behavior of DO-branes [l]-[3]. They tried to de-velop a precise realization of P. K . Townsend's suggestion [4] stating that the supersymmetric formulation of membrane theory proposes that membranes could be viewed as composites of DO-branes. In what follows we will try to present the BFSS model in short and some evidence of it. Their strongest evidence for the conjecture is a demonstration that their model contains the excitations which are widely believed to exist in M-theory, supergravitons and large metastable classical membranes. They also presented a calculation of supergraviton scattering in a very special Chapter 1. Introduction 2 kinematic region, and argued that their model reproduces the expected result of low energy supergravity. As the last evidence they discussed that their model may also satisfy the Holographic Principle [5],[6]. Open strings which connect the branes when they are practically on top of each other introduce a new kind of coordinate space in which the nine spatial coordinate of a system of N DO-branes become nine N x N matrices, Xlab. The matrices X are accompanied by 16 fermionic superpartners 9afi which transform as spinors under the SO(9) group of the transverse rotation. The matrices may be thought of as the spatial components of 10 dimensional Super Yang Mills (SYM) fields after dimensional reduction to zero space di-rections. These Yang Mills fields describe the open strings which are attached to the DO-branes. The Yang Mills quantum mechanics has U(N) symmetry and is described by the Lagrangian Here conventions have been used in which the fermionic variables are 16 component nine dimensional spinors. BFSS propose this Lagrangian as the most general infinite momentum frame Lagrangian with at most two derivatives which is invariant under the gauge symmetry and the Supergalilean group [7]-[9]. Apart from simplicity their main reason for suggesting this Lagrangian is that there are some partial evidence that the large N limit of the quantum theory it defines is indeed Lorentz invariant. Let us rewrite the action in units in which the eleven dimensional Planck length is 1. According to one of the correspondences between M-theory and type IIA string theory, the compactification radius R is related to the string coupling constant by r = gilp = gls (1.2) where ls is the string length scale t r X - X - + 29T6 - -tr[X\ Xj}2 - 26T '^[0,Xl] (1.1) •p (1.3) and therefore the change of units is easily made and one finds (1.4) Chapter 1. Introduction 3 where Y = \ and Dt = dt + iA. The units of time have been also changed to eleven dimensional Planck units. We emphasize that the SUSY transformation laws (e and e are two inde-pendent 16 component anticommuting SUSY parameters), 5Xi = -2eTf6 50 = | [DtX^i + 7 - + \ [X\ XJ]ltj\ e + e (1.5) 5A = -2eT9 involve a gauge transformation. As a result, the SUSY algebra closes on the gauge generators. The Hamiltonian has the form H = R T r 1 ^ + I r K i ) yif + Y^] j ( L 6 ) where II is the canonical conjugate to Y. Note that in the R —> oo, all finite energy states of this Hamiltonian have infinite energy. We are only interested in states whose energy vanishes like jr in the large iV limit, so that this factor becomes the inverse power of longitudinal momentum which we expect for the eigenstates of a longitudinal boost invariant system. Thus the correct infinite momentum frame limit, the only relevant asymptotic states of the Hamiltonian should be those whose energy is of order jj. Their conjecture is thus that M-theory formulated in the infinite momen-tum frame is exactly equivalent to the N —> oo limit of the supersymmetric quantum mechanics described by the Hamiltonian (1.6). The calculation of any physical quantity in M-theory can be reduces to a calculation in matrix quantum mechanics followed by an extrapolation to large N. Since the work of BFSS, matrix theory has passed numerous tests [10], however an honest quantum mechanical study of the model seems extremely difficult due to the flat directions in the potential (resulting in the continuous spectrum), the difficulty of distinguishing single-particle and multi-particle states, the lack of a tunable coupling constant (interactions are of order one), and the large number of degrees of freedom in the large N limit. Indeed, even determining the normalizable ground state wavefunction for the case N = 2 (proven to exist in [11]) is an unsolved and notoriously difficult problem. In 2002, D. Berenstein, J. Maldacena and H. Nastase studied a mas-sive deformation of the BFSS matrix model appropriate to M-theory on a pp-wave background. In fact they presented a matrix model associated to the discrete light cone quantization (DLCQ) description of the maximally Chapter 1. Introduction 4 supersymmetric eleven dimensional pp-wave. Before we start on a brief ex-planation of their work we shortly show how pp-wave geometries arise as a limit of AdSp x Sq. We first consider the case of AdS5 x S5. The idea is to consider the trajectory of a particle that is moving very fast along the 5 s and to focus on the geometry that this particle sees. The AdS5 x S5 metric is ds2 = R2 -dt2 cosh2 p + dp2 + sinh 2 pdtij + dxp2 cos2 0 + dO2 + sin 2 6d(l\ (1.7) We want to consider a particle moving along the ip direction and sitting at p = 0 and 8 = 0. We will focus on the geometry near this trajectory. In order to do this systematically we introduce the coordinates and then performing the rescaling x+ = f + , x~ = R2x~, p = —, 9 — R —> oo (1.9) R R In this limit the metric (1.7) becomes ds2 = -4dx+dx~ - (r 2 + f)dx+2 + dy1 + dr1 (1.10) where f and y parameterize points on R4. We can also see that only the components of F with a plus index survive the limit. We see that this metric is of the form of a plane wave metric ds2 = -Adx+dx- - p?z2dx+2 + dz2 F+1234 = F+5678 = const x p, where z parameterizes a point in R8. The mass parameter p can be intro-duced by rescaling (1.9) as x" —> ^ - and x+ —> px+. These solutions were studied in [12]. There is a nice, simple matrix model associated to these backgrounds. The M-theory pp-wave background is Chapter 1. Introduction 5 ds2 = -Adx~dx+ - [(§ f{x\ + xl + xj) + (f )2(x2 + ... + x2)dx+2 + dx2 -P+123 = t1 (1-12) This metric arises as a limit of AdS*\ x S7 or AdS7 x S 4 (both cases give the same metric) and has a large symmetry group with 32 supersymmetries. The algebra is a contraction of the AdS^j x S7,4 superalgebras. In analogy to the discussion in BFSS case we do D L C Q along the direction x~ ~ x" + 2uR, and we consider the sector of the theory with momentum 2p+ = —p_ = ^ . Then the dynamics of the theory in this sector is given by the U(JV) matrix model S — S0 + Smass where S0 = J dtTr 9 9 § 2 M ( w ' ) 2 + *T j D o* + ^  E ^ > 2(2R) i = l , 2 , 3 j=4 + ^(2 i?) (^f [* ,^] ) Smass = J dtTr 3 - f i TrW<l>k<f>l)ejkl j,k,i=i (1.13) where we have set lp = 1. We also have that £ = x+ and 4> = ^ where r is the physical distance in eleven dimensions. S0 is the usual matrix theory of BFSS (To compare with BFSS note that due to the form of the metric and the way R is defined in here 2R^eTe = ^ B F S S - ^ e n o r m a l i z e lp so that \[&, = lvg~^ when we go to type IIA string theory). SVnass adds mass to the scalar fields and fermion fields. Chapter 1. Introduction 6 The action (1.13) has the transformation rules 3 9 5 * = + 67J7^)I>V7l23 - 3$*jJ>V7l23 + i=l i=4 (1-14) 5A0 = * T e( i ) e(t) = e - i 5 T 1 2 3 t e 0 In appendix A we show that the action (1.13) is determined by the supersym-metry algebra of the plane wave metric [13]. The matrix model Hamiltonian associated to this action is equal to H = — p+. Note that the bosons and fermions have different masses, three of the bosons have mass ^ while six of them have mass On the other hand all the 3 6 fermions have mass ^. This is possible because the supersymmetries (1.14) are time dependent and therefore do not commute with the Hamiltonian. This is in agreement with the SUSY algebra of plane waves [13], see appendix A. The pp-wave matrix model for non-zero p avoids a number of difficulties associated with the p = 0 case. Firstly the quadratic terms in the potential remove all flat directions and yield a potential that becomes infinite in all directions. This implies a discrete spectrum for the massive matrix model and an isolated set of classical supersymmetry vacua. These vacua may be associated with all possible ways of dividing up the momentum into different numbers of membranes from a single membrane carrying N units of momenta to N membranes each carrying a single unit of momentum. Since these vacua are separated by potential barriers we see that the massive matrix model (at low energies) also has a natural distinction between single and multi-membrane states. The main difference between B M N model and BFSS model is their su-persymmetry which in the latter case is that of a maximally supersymmetric plane wave spacetime rather than eleven dimensional Minkowski space. A great advantage of the plane wave matrix model is that it has a weak coupling regime where it can be studied systematically using perturbation theory[14]-[22], particularly in [14], the massive matrix model has been studied and they explore that the existence of a tunable parameter in the matrix model opens up the possibility of a weakly coupled regime which can be studied Chapter 1. Introduction 7 in perturbation theory. It also has a powerful supersymmetry algebra which allows the extrapolation of some perturbative results to the strong coupling regime. Recently the variant of the matrix model which is conjectured to describe a discrete light cone quantization of M-theory on a pp-wave background has been formulated it will most convenient to write the action in form that makes SO(3) x 50(6) symmetries manifest, so we divide the scalars in to X1 with i — 4, ...,9 and Xa with a — 1,2,3, and write the fermions as tpja where / is a fundamental index of 5(7(4) ~ SO(6) and a is a fundamental index of 5(7(2) ~ 50(3). The relation between these fermions and the real 16 components spinor \I/ is the following. Under the decomposition 50(9) -> 50(6) x 50(3) (1.15) the spinor representation spilts up as i6^, (4'2^ (J'2) ( L i e ) where we take I as a fundamental SU(4) index an a as a fundamental SU(2) index. our spinors obey a reality condition which in the reduced notation be-comes simply (^)Ia = fta (1.17) To decompose the terms in the action we introduce the matrices gau which relate the antisymmetric product of two 4 repersentations of the 5£/(4) to the vector of 50(6). Then, we may choose and l a = ( ° t 1 X 5 a " l l * = ( ~ a l X l ° ) ( H 9 ) 7 ^ 1 x ( 5 °) t 0 J ^ ^ ^ 0 a*xl J U ' i y j These Dirac matrices satisfy the Clifford algebra as long as we take normal-Chapter 1. Introduction 8 ization so that ga(gby + gb(ga) (1.20) with these conventions the matrix model describing the D L C Q of M-theory in the maximally supersymmetirc pp-wave background is given by where all of the variables are NxN matrices. The U (N) symmetry is gauged. A l l the variables transform in the adjoint representation of the gauge group X1 —> UXlW , etc. The time derivatives are covariant, DX = dTX — i[A,X] with an N x N Hermition gauge field A. eap and e&i5 are antisymmetric tensors. This matrix model reduces to the BFSS model if we set p — 0 and can be considered a one parameter deformation of it. The main difference between the two is that in the former case the action has (super)symmetries identical to those of the residual inverse of eleven dimensional Minkowski space in light cone quantization whereas the latter has symmetries appropriate to pp-wave spacetime. As we mentioned before, in the action of the BFSS model the classical potential (—Tr([Xl,Xj})) has flat directions any set of matrices which mutually commute have zero energy. In (1.21) these flat directions are cancelled by the mass terms. Perturbation theory is accurate in the large mass gap p limit. Just as the BFSS model (1.21) could also be regarded as the effective ac-tion describing the low energy dynamics of a collection of iV £)0-branes of type IIA string theory on Minkowski space with the appropriate re-interpretation of the parameters, (1.21) could also describe D0-branes in a background of type IIA string theory. The plane wave matrix model (1.21) can be systematically analyzed in perturbation theory. When iV is large, the expansion parameter is [23] Note that all dimensional quantities are in units of the eleven dimensional 5 = IdrTr (DXiDXi + DX~aDX'a + i^IaD^Ia + f[X\ Xi}2+ +f[X*,X1]2 + i ? 2 [ X a , X f V - R^Iaa%3[Xa,ipIp} + +lea^Iag}J[Xi,^J] - ^^ai{g^)IJ{X\^j}-_ ( | ) 2 ( X S ) 2 _ ( | ) 2 ( X i ) 2 _ l ^ t / c * ^ _ ^fe^X'-X^) (1.21) (1.22) Chapter 1. Introduction 9 Planck length which we have set to one. Aside from being a model of M-theory on a plane-wave background, there is an interesting connection between the matrix model (1.21) and four dimen-sional M = 4 supersymmetric Yang-Mills theory. At the classical level, the degrees freedom and dynamics described by (1.21) are a consistent truncation of the Yang-Mills theory which keeps only those modes which are invariant under a certain SU{2) subalgebra of the full £7/(2, 2|4) superconformal alge-bra. In the planar limit, this truncation seems to also hold at the one-loop level [20]. It is then natural to speculate about whether it would inherit some of the properties of M = 4 supersymmetric Yang-Mills theory, such as integrability of the large JV limit, at least in the leading orders of per-turbation theory. This has been studied in detail in recent work [22]. It has been shown that the form of the dilatation operator for supersymmet-ric Yang-Mills theory which has been conjectured using integrability as an input, when restricted to the appropriate sector of the theory, is identical to the Hamiltonian of the plane wave matrix model, at least for computing energy levels to three loop order. 10 Chapter 2 Hagedorn Behavior of Matrix Model In this section, we will compute the Hagedorn temperature of matrix models by counting their gauge invariant states or equivalently the numbers of gauge invariant operators. It will be shown that when the dimension iV of the matrices is infinite they have a Hagedorn density of states. 2.1 A Toy Model we will begin with a toy model which captures the salient features of the physics involved. Consider a set of d bosonic N x N Hermitian matrices, Xj(t), where j — 1, ...,d and t is the time. These are coupled to an N x N "gauge field" A(t). The action is S = f * E ^ ( ( | ^ i + » [ A ^ ] ) a - ^ a ) (2-1) If it were not for the gauge field appearing in the covariant derivative this would be a theory of d independent matrix oscillators. Introducing the gauge field makes the model invariant under the gauge transformation X3{t) C/(£)X,(£)[/t(i) A{t) U{t)A(t)U\t) - iU(t)U\t) (2.2) The equation of motion which is obtained by varying A(t) has no time deriva-tives and is an equation of constraint Chapter 2. Hagedorn Behavior of Matrix Model 11 0 = £ [ * , ( * ) , n ^ - O (2.3) where nj(t) = ftXj(t) + i[A(t),Xj(t)} (2.4) is the canonical momentum, with Poisson bracket {X?(t), nckd(t)} = Sjk5ad5bc (2.5) The left-hand-side of the constraint equation is the generator of an infinites-imal time-independent gauge transformation, 6Xj = {Tr(AG),Xj} = -[A,Xj] (2.6) When we quantize, this toy model is simply described by d decoupled matrix oscillators with the constraint (2.3) on the physical states. C l a i m . In the limit N —> oo this toy model has a Hagedorn density of states and Hagedorn temperature T » = <2-7> The case d = 1 should be interpreted as TH = oo. We Wi l l also use the parameter f3 = ^ for the inverse temperature and the notation /?# = T~ = Ind LJ To substantiate this claim we will count the number of physical states which have a given asymptotically large energy and extract @H from the exponential growth (if there is any). The Hamiltonian which follows from canonical quantization of the action (2.1) is H = Y.\Tr^ + "2X*) (2-i = i where the momentum obeys the canonical commutation relation {Xf,Uckd} = i5jk5ad5bc (2.9) Chapter 2. Hagedorn Behavior of Matrix Model 12 It is easy to see that the constraint (2.3) commutes with the Hamiltonian. We shall take the strategy of quantizing the model of free oscillators with Hamiltonian (2.8) and commutators (2.9). From the resulting quantum states we then isolate a subspace of physical states which are annihilated by the constraint operator in equation (2.3). We define creation and annihilation operators 1 = (iL, - iuXj) , a] = - L =(n , + iuXj) (2.10) '1wv J J J V2LO so that [af,alcd} = 5jk5ad5bc (2.11) and after dropping the ground state energy the Hamiltonian d H = J2uTr(alaj) (2.12) i = i The vacuum state is annihilated by all of annihilation operators af |0) = 0 Va, b = 1 , N ; j = 1 , d (2.13) A basis of quantum states is found by operating creation operators on the vacuum, |0) , o f |0) , a f af< |0> ... (2.14) These basis vectors are eigenstates of the Hamiltonian. From these states we must choose a subspace which is annihilated by the constraint operator in (2.3). That operator generates infinitesimal time independent gauge transformations. Quantum states will be invariant if they are invariant under the finite transform a j -> UajU* , a ] Ua]W , withUW = 1 (2.15) Such invariant states are generally obtained from the general basis states by taking traces over the matrix indices, (If TV were finite there would also be the possibility of other invariants such as the determinant of aj. The energy of the state created by this operator is Nu>) Chapter 2. Hagedorn Behavior of Matrix Model 13 |0) , Tr (aj) |0) , Tr (a]a{) |0> , Tr (aj) Tr (a\) |0) ... (2.16) The operators which create gauge invariant states are thus of the form and can be thought of as "words" which are made from the "letters" a\. The length of a word is the number of letters which it contains. There are d different letters at our disposal. Inside a word, the ordering of the letters is important, since they are matrices and do not commute with each other. However, because of the cyclicity of the trace, words which are related by cyclic permutations of the letters are equivalent. Note that for finite N the words are not independent. Generally, a very long word can be expressed in terms of linear combinations of sentences made from shorter words: in the U(N) theory, the operators Tr{a[)k for k > N can be expressed in terms of those with 0 < k < N. However, the long words do become independent when the limit N —> oo is taken. Here, we shall always assume that the large N limit is to be taken. For this reason, we shall treat all distinct words as independent operators. This is valid if there is only one kind of matrix (d = 1) since then only the eigenvalues are invariant and there are N degrees of freedom. If there are more than one matrices then the number of independent traces is of order iV2since the matrices can not be simultaneously diagonalaized. Finally, we can create a state by applying several "words" to the vacuum. Such multi-trace operators, Tr 4 ) Tr ( a ^ . . . ^ ) ...Tr ( < . . . < ) (2.17) will be called "sentences". The length of a sentence is the sum of the lengths of its constituent words, or the total number of letters in the sentence. As each creation operator, or letter, creates an excitation of energy u, the energy is the length of the sentence in units of u. The energy of the above expression is E = Lu(n + m + p+ ...) (2.18) The number of these traces with a fixed energy, E, does not scale like A^2 Chapter 2. Hagedorn Behavior of Matrix Model 14 as N —> oo, instead it approaches a constant as iV is taken large. Thus at low enough temperatures, the free energy should approach an iV-independent constant as N is taken large. However, the number of independent traces does increase rapidly with the energy E.1 Obviously, words or sentences should be considered equal if they produce the same quantum state. As operators, words commute with each other. Therefore, the order of the words in a sentence does not make any difference. In order to find the Hagedorn temperature we should find the density of states at asymptotically large energy. If the theory has a Hagedorn spectrum, the increase in the density of states will be exponential, p(E) ~ ep"B (2.19) This formula can be corrected by prefactors; we shall discuss them later. We will concentrate for the time being on extracting the coefficient in the expo-nent which we shall identify with the inverse, of the Hagedorn temperature in the next section. 2.2 P o l y a Style of Count ing States To find the density of states, we are essentially looking for the number sn of sentences of length n. In order to compute this number, we will need to consider also the number u>k of words of length k. It is convenient to define s0 = <jjQ = 1 (2.20) As is common in combinatorial problems, it is helpful to define the generating functions of those numbers, oo W(x) = Y^wkxk (2.21) k=Q 1 At temperatures higher than TH, the thermodynamic canonical ensemble does not exist. It could be made to exist by keeping ./V large but finite. That would cut off the exponential growth in the asymptotic density of states at some large energy, of order TV 2 . Then we could consider a temperature that is greater than TH- Both the energy and entropy would be dominated by states at and above the cutoff scale and would be of order N2. The divergence of the free energy ~ iV 2 occurs as we take the limit N —> oo at constant temperature (noting that the Hagedorn temperature does not depend on JV). Chapter 2. Hagedorn Behavior of Matrix Model 15 S(x) = ] T S n x " (2.22) 71=0 2.2.1 The d = 1 Case For the case d = 1 of an alphabet with a single letter o) = a\, there is obviously only one word Tr(a))k of each length k, or Wk = 1. A sentence of length n is nothing but a collection of such words, where their order is immaterial. Hence, the number of sentences of length n is just the number of partitions of n into natural numbers, sn = p(n). It is well known from the days of Euler that p ( x ) = En=oP(n)xn = (I + x + x2 + + x2 + x4 + + x3 + x6 + ...). = n ~ i ( i - ^ r l (2.23) and from the celebrated work of Hardy and Ramanujan that p(n) ~ e c ^ (2.24) with c = ^/|vr « 2.5651. The number of sentences of length n is therefore sub-exponential, and the partition function converges for all finite temperatures; consequently we conclude that /?# = 0 and in this case Ty = oo as claimed. 2.2.2 The d > 1 Case An upper bound on the number of words of length k is wk < dk as there are d independent choices for each letter. The cyclic property of the trace makes some of those words equivalent. Obviously, each word can have at most k equivalent representations; therefore, a lower bound on the number of inequivalent words is wk > Tdk Chapter 2. Hagedorn Behavior of Matrix Model 16 We claim that the latter bound is in fact a good approximation 1 ,k wk « -d* To substantiate this claim, we will utilize the famous Polya theorem. Theorem (Polya) . Let G be a group acting on a set X. Then the number of colorings of the elements of X with d colors, distinct under the action of G, where X(g, X) is the number of cycles in the action of g on X. In our case, X — is the string of letters in a word, and the group of rotations is also G = acting on X by addition. The colors are the different kinds of letters, so their number is indeed d. The number of colorings is just the number of inequivalent words. Let us begin with the case when k = p is a prime number. Then, the identity has \X\ — p orbits, and each other rotation has only one; therefore, This is indeed very close, for large p, to ^dp. For the case where k is not a prime, a similar formula can be written in terms of Euler's totient function, but it suffices to note that apart from the identity, again having | X | = A; orbits, all other rotations can have at most k/2 orbits (in fact, there can be at most one such orbit, for g = k/2 when k is even). Therefore (2.25) wp = -{d? + (p-l)d) (2.26) wk < Udk + (k- l)dk'2) (2.27) which is again close to ^dk. Accordingly, we shall henceforth use the approximation for k > 1. This results in Chapter 2. Hagedorn Behavior of Matrix Model 17 oo oo 1 W{x) = ]T wkxk « 1 + = 1 - l o g ( ! - d x ) ( 2 - 2 8 ) k=0 k=l A sentence can be composed of only one word; The contribution to sn is then wn, and to S(x) is W(x). It can be composed of two words; the contribution to sn is then naively the convolution (w * w)n = YlZi=o w m W n - m and to S(x) is W(x2), but we should remember that the order of the words is immaterial. As for large n the sentences composed of two identical words are negligible (the generating function of those is W{x2)), we should approx-imately divide by two. In general we approximate that the number of words is so large that generically no two will be identical in a sentence; if there are a words in a sentence, we should divide the naive approximation by a!. Consequently, O O -. .. o o S(x) « ^ ( X ) ° = e W ( x ) ~ e-7~T~fe = e ' ^ ( 2 " 2 9 ) a=0 ' n = 0 SO sn « e.dn ~ e ( l o g d ) " (2.30) and the Hagedorn temperature, in units of LO, is indeed as claimed. Note that if x = e-P" (2.31) then o o Z(r3) = S(x) = Y,Sne->3nu (2.32) 71=0 is exactly the partition function, so in light of (2.29) we get The partition function clearly diverges at the inverse Hagedorn temperature PH s u c n that d.e-pHU = l (2.34) Chapter 2. Hagedorn Behavior of Matrix Model 18 which gives indeed /3# = In fact, a more careful use of Polya's theorem can give an exact expression [24] for Z((3) or S(x), but this is irrelevant for our purposes. 2.2.3 Extensions The Traceless Case The matrices we have encountered thus far were assumed to be Hermitian. They can equivalently be characterized as matrices in the adjoint represen-tation of the U(N) algebra. It is natural to explore the variant where the matrices are in the adjoint of SU(N) (we will denote the various magnitudes for the SU(N) version with a tilde). We will now show that our results are essentially unchanged if we have the gauge group SU(N) instead of U(N). A matrix in the adjoint representation of SU(N) is Hermitian, but with the extra constraint of being traceless. In particular Tra) — 0 and so there are no single-letter words: cJi = 0 (2.35) In the d = 1 case, therefore, the partitions of n should not contain ones. As any partition without ones of any m <n can be uniquely completed, with the addition of n — m ones, to a partition of n, and vice versa, obviously and so s n — s n ~ s n - l ~ ~7^~sn ~ ~ ~ = S n (2.37) on yjn We see that the exponential behavior, and hence the Hagedorn temperature, remains as in the U(N) case, and the change is only in the irrelevant poly-nomial prefactors. A l l this can also be seen from the fact that the relevant partition function is oo p{x) = JJ(1 - x ^ - i = (1 _ x)P(x) 2=2 (2.38) Chapter 2. Hagedorn Behavior of Matrix Model 19 In the d > 1 case, (2.35) implies W(x) = W(x) - dx (2.39) and S{x) » eW^ = e-dxew^ « e~dxS{x) . , « e(l - dx + \d2x2 - ±d3x3 ± ...)(1 + ^ + d V + d3x3 + ...) 1 J so the coefficient of xn is now 3 „ « e r f * ( l - l + i - i + ... + (-iri) but the last factor is approximately e _ 1 , so sn = « d" (2.41) We get the same Hagedorn temperature as in the U(N) case, the only change being in an irrelevant constant factor. Fermions Fermionic matrix degrees of freedom behave quite similarly to bosonic ones of the same mass. We can form the fermionic creation operators b\. Each entry of such a matrix is a Grassmannian number which squares to zero; however, there are so many of those, (i.e. TV2) that when forming words of finite length, none vanishes, and more generally, the words remain independent. However, each word squares to zero, so we can not use the same word more than once in a sentence. For the d = 1 case, this means that the partitions of n should be to distinct numbers. The number of such partitions, q(n), behaves also as q(n) ~ ec'^" (2.42) when now d = \J\^- The corresponding generating function for the sen-tences is easily seen to be 2 incidentally, the number q(n) of partitions of n into distinct parts is equal to the the number of partitions into odd parts; this is an amusing exercise in generating functions, as formally Chapter 2. Hagedorn Behavior of Matrix Model 20 OO OO S(x) = Q(x) = Y, q(n)xn = + (2.43) For the d > 1 case, we should again take into account the restriction that no two equal words appear inside a sentence. However, we have already argued that such cases are rare and used the approximation that they are negligible, when we argued that the "a words in a sentence" term W(x)a should be divided by a! (so that S(x) is the exponent of W(x)). We conclude that the Hagedorn temperature of the d > 1 fermionic matrix model is equal to its bosonic counterpart. Different Masses When there are several matrices with different masses, there is obviously still a Hagedorn density of states, with the Hagedorn temperature between two bounds: the Hagedorn temperature of the same number of matrices but all with the same mass, which is either the minimal or maximal one. However, it is quite easy to generalize (2.34) to get an exact result. C l a i m . For a model with d > 1 matrices, bosonic or fermionic, of masses LOi > 0, i = l,...,d, the inverse Hagedorn temperature /?# is given by the solution of The idea of the proof in the equal masses case carries over when we define d (2.44) To see that, we first pick a mass scale to and write LOi — UVi (2.45) OO OO n(i+^i)=n(i — x . 2 j - l \ - l i=l j = l (To prove this it is useful to use Jacobi triple product identity). Chapter 2. Hagedorn Behavior of Matrix Model 21 i = l F(x) = YxVi (2-46) as a generalization of dx. We still have oo -. W(x) « 1 + £ 7 :^(^) F C = 1 - log(l - ^(x)) (2.47) and k k=l S{x)*ew^*e.—*-— (2.48) 1 - t (re) Where now sra should be interpreted as the number of states with energy nui and not as the number of sentences of length n; indeed, n doesn't even have to be integer. Clearly, S'(rc)diverges at xu = e-P"" (2.49) where F(xH) = 1 (2.50) Still using (2.31), we get Z((3)^e. 1 (2.51) which diverges at the claimed inverse Hagedorn temperature. As F is a monotone decreasing function of /?, where F = d > lior /3 — 0 and F — 0 < 1 for P = oo, there is exactly one root of equation (2.44). If LOi — 0 for some i, then there is an infinite number of zero modes, the Hagedorn temperature is formally zero, and we see that the relation (2.44) continues to hold formally. It is also easy to see that this reasoning, properly interpreted, does not rely upon the number of matrices being finite. Absence of Prefactor We would also like to refine our analysis and find any energy power prefactor multiplying the leading exponential in the density of states. Stated otherwise, Chapter 2. Hagedorn Behavior of Matrix Model 22 assuming that the density of states behaves as p{E) ~ CEae?»E (2.52) we wish to find a. C l a i m . There is no power-like prefactor in matrix models, i.e. a = 0. In order to see this, assume (2.52), and compute the partition function, Z{B) = J P(E)e-f>EdE = j f CE°e-V-™EdE = J j ~ ^ (2-53) or equivalently by (2.31,2.32,2.49) S(x) CT{a + l)ma+1 (\og(xH/x)) a+l It is easv to see that S(x)S"(x) a + 2- log(xH/x) (2.54) (2.55) S'(x)2 a + 1 so that a can be extracted from the behavior of the latter expression at the divergence, S(x)S"(x) a + 2 a + l S'(x)2 On the other hand, it is immediate from (2.48) that S(x)S"(x) 2F'(x)2 - {F{x) - l)F"(x) S'(x)2 and therefore, from (2.50), S(x)S"(x) S'(x) F'{xf = 2 (2.56) (2.57) (2.58) Equating (2.56) and (2.58) we get indeed that a = 0 as claimed. Essentially, this behavior is dictated by the fact that F(x), a monotonous function, has a non zero derivative at x — XH-Chapter 2. Hagedorn Behavior of Matrix Model 23 2.2.4 Hagedorn Behavior in Free Second Quantized Theories Let us study a free quantum theory of harmonic oscillators of frequencies u>i > 0. Their number may be finite or infinite, and they might be bosonic or fermionic. The corresponding Hamiltonian, after discarding the zero-point energy, is H = ^2itoia\ai. The Fock space states of the second quantized the-ory (the "sentences") are generated directly by applying the creation operators (the "letters") on the vacuum: as the creation operators are not matrices, and no gauge invariance condition is imposed, there is no need for "words" in this context. C l a i m . The inverse Hagedorn temperature PH in this model is given by the solution of More accurately, it is the maximal value of P where the left hand side diverges. Note that this implies that the Hagedorn temperature for a system with a finite number of oscillators is always infinite, so there is no Hagedorn tran-sition (for that, it is essential that there are no zero modes u>i = 0). Even for systems with an infinite number of oscillators the Hagedorn temperature might be infinite. It is indeed infinite in the sensible cases where the model comes from a field theory in a finite volume. To understand the claim, we will begin with the example of d bosonic oscillators of equal frequency to. As in the the corresponding matrix case of subsection 2.1.1.2, sn counts the number of sentences of length n made out of d letters. The order of the letters does not matter, as the creation operators commute, and therefore sn = ^ " ^ ^ ^ ^ (think of arranging n circles and d — 1 dividers in a line). As the density of states sn « const.nd~l behaves polynomially and not exponentially, there is no Hagedorn behavior. This can also be easily seen from the generating function: (2.59) n + d-l d-l d Y[(l + x + x2 + x3 + ...) (2.60) The term x J in the i — th multiplicand represents j excitations of the i — th Chapter 2. Hagedorn Behavior of Matrix Model 24 oscillator. The generalization to different frequencies is immediate: using (2.31,2.45), a bosonic oscillator a) contributes a factor of j^p, while a fermionic one can be excited at most once, as (tf)2 — 0, and so contributes a factor of 1 + x". Accordingly, s ^ = n T - r ^ - n ( 2 - 6 1 ) Bosons Fermions Now, take the logarithm: logS(x) = Y - l o g ( l - ^ ' ) + Y ^(1 + ^) (2-62) Bosons Fermions If there is a subsequence of the going to zero, then log Six) diverges for all x > 0, so by (2.31), PH — oo; this is clearly true also from (2.59). Assume, on the other hand, that v* = inf > 0. Then for 1 > x > 0 we have xVi <xv* < 1, so there is some 77 < 1 such that (1 — rj)xVi < — log(l — xVi), log(l + xVi) < (1 + r\)xVi. Consequently, (2.62) and the right hand of (2.59) converge or diverge together. This concludes the proof of the claim. As an example, let us take a classical (non-relativistic) string tied in its two ends and free to oscillate in one transverse dimension. The string's fre-quencies are harmonics of some natural frequency, u>i — i.to for i = 1,2,3,.... The number of states of energy n.u is again p(n), the number of parti-tions of n, which is subexponential (2.24), so there is no Hagedorn behavior. S(x) = P(x) is given by (2.23) that has a radius of convergence one around the origin. In this case, The right hand side of (2.59), V \ e - ^ = £V xl = ^ has indeed the same radius of convergence. If the string lives in D spacetime dimensions, there are D — 2 transverse directions, so S(x) — P(x)D~2, which still displays no Hagedorn behavior. The lack of Hagedorn behavior is generic in field theories. In the free case in a finite volume, the number of oscillators grows like the phase space, or as a power of the energy. The result (2.59) implies that we should look at expressions of the form ^So^ 2 - 1 ~ ( I - ^ + I > a s w a s shown, in a different context, in (2.60). The divergence is still at H — 1, corresponding to PH — 0 or TH = 0 0 . 25 Chapter 3 Thermodynamics of the Matrix Model In this Section, we will demonstrate that this is indeed the case using a direct analysis of the thermodynamics of the matrix model. We will be interested in the thermodynamic partition function, where we take the Hamiltonian, H, for the model corresponding to the matrix model in pp-wave background and form the partition function Z[N,R,P] = Tre-l}H = e-WN>R-ft where /? = ^ is the inverse temperature and F[N, R, /?] is the free energy. Before we discuss this, we shall again illustrate the method by using a simple toy model. 3.1 The Toy M o d e l Consider again the toy model of section 1. The thermodynamic partition function has a functional integral representation Z = J [dA)[dX,\e-^dT^M^)^l) (3.1) where the fields have periodic boundary conditions in Euclidean time A(T + f3) = A(T),XJ(T + (3) = XJ(T) (3.2) To proceed, we must fix a gauge. The periodic boundary condition prevents us from choosing the A — 0 gauge. The best we can do is to fix a gauge where the gauge field is static and diagonal, •%-A(T) = 0 , Aab = Aa6ab (3.3) Chapter 3. Thermodynamics of the Matrix Model 26 In that case, (DXj)ab = ^-Xf + i(A* - Ab)Xf (3.4) The Faddeev-Popov determinant for fixing this gauge is computed in the Appendix B. It turns out to be similar to the Vandermonde determinant which appears in the measure of unitary matrix integrals, detDdet-^- = IT s i n - \ A a - Ab\ (3.5) dr 2 a=£b where det' is a determinant restricted to the space of non-constant modes. Then, the integral over the oscillator fields in (3.1) is Gaussian and can be performed explicitly Z= mdA a r j s in^ |A a -A 6 | r jde t (- (A. + i'A.-A,,)) + uA a=l a+b a,b \ ^ ' / (3.6) The determinant can be evaluated explicitly (see Appendix B, also [25]). This model with d — 1, and a different critical behavior from the one which we shall be discussing in the following, has been studied extensively in the context of lower dimensional string theories. For example, see [26],[27]. We obtain JV / \ ^ Z = [f\dAa]Jsm^\Aa-Ab\]ji—-7ir l— —- (3.7) J th ii 2 ii V s m h ( f ( w + < A « - A » » ) ) The effective action for the eigenvalues is then 5 eff = \ d l n f s i n h (f ^ + ~ A b ^ ) ) ~ l n s i n f \Aa ~ AA (3-8) We have dropped an A-independent ground state energy. There are now N remaining degrees of freedom, the variables Aa, and because there are two jV-fold summations, the action is of order N2. For this reason, in the large N Chapter 3. Thermodynamics of the Matrix Model 27 limit the integration in (3.6) can be done in the saddle point approximation. In that case there is a well-defined expansion of the integral as a power series in To begin, we must find the saddle point by minimizing the effective action. The equation for the minimum, for each a, is Y* J 1 1 1 ^ : / ^ As = Y<** £ (A - Ab) (3.9) j-f cosh/?w - cos/?(A a - Ab) {-f 2 V b > x ' It is useful to introduce the eigenvalue density, 1 N a=l In the large ./V limit, it becomes a continuous function of 9 with support on some or all of the interval—n < 9 = (5A < u, and it is normalized so that d9p{9) = 1 (3.11) Obviously, the density is constrained to be positive, p{9) > 0. Now, the action can be written as 5 e f f = N*fd9d9>P(9)p(e>) {din (sinh ( f + \{9 - 9'))) -- l n ( s i n i | 6 l - 0 ' | ) } { 6 A l ) We see there is a sort of competition between the eigenvalue repulsion of the second term, coming from the Vandermonde determinant, and the eigenvalue attraction of the first term, coming from the action. The attraction is more pronounced in the high temperature regime, 0 —» 0. the equation for the eigenvalue density is given by where it is understood that the principal value around the singularity should be taken in the right hand side integral. Note that this equation should be satisfied only in the support of p{6). This equation should be solved together with the normalization condition (3.11) and the positivity constraint. Chapter 3. Thermodynamics of the Matrix Model 28 First, we observe that the constant eigenvalue density P. = I (3-14) is always a solution of (3.13) where both sides of the equation vanish. How-ever, this solution is stable only for a range of values of (3. To see this, we Fourier expand inside (3.12),1 d9d9'p(9)p(9') Y cos(n(0 - 9')) (3.15) 7 1=1 n From this, we can see that if (5 is large enough, then the phase where the density is constant (so that the Fourier modes vanish, J d9p{9)em9 = 0 for n 7^  0) is stable. As the temperature is raised, and (3 is decreased, the first instability sets in at = = 1/TH where T~ = is <3-16' This coincides with the Hagedorn temperature in (2.7) which we found by estimating the high energy density of states in section 1. Note that the approximations of section 1 amount to keeping the n = 1 term only in (3.15), and that the arguments there are invalid in the deconfmed phase above the Hagedorn temperature. The straightforward generalization of this model where we have different frequencies for the bosonic oscillators and possibly some fermionic oscillators gives the general equation for the critical temperature of the matrix model, Y^e~0HUli = l (3.17) i This also coincides with the equation for the Hagedorn temperature which was found in section 1. The phase transition that we have identified is the usual Gross-Witten 1Here we use the identity ln [ sin - 1 = - 2^ — — + const. n = l Chapter 3. Thermodynamics of the Matrix Model 29 [28] phase transition of unitary matrix models. In that phase transition the distribution of eigenvalues of the unitary matrix rearranges itself from one which has compact support near the identity matrix to one which is distributed on the whole unit circle. Above the phase transition the eigenvalue distribution is not uniform. Without loss of generality, we may assume it is an even function concentrated around 9 = 0. The distribution can be approximated by the semi-circle one, ^ = ^ ^ 2 2 ( 3 - 1 8 ) This distribution has support in the region | sin | | < y | and the parameter £ is restricted to the range £ < 2. This approximation is exact when we truncate the low temperature expansion of the matrix contribution to the effective action at its first nontrivial, p(9) dependent, term d ln( l - e-/*"--(*-9'>) « _ d e - ^ e - * ( 9 - e ' ) (3.19) and the saddle point equation is approximately J d9'p{9') (^de-^ sin(0 - 9') - cot ^(9- 0')) = 0 (3.20) This equation is solved by (3.18) when2 f = 2 ( l - y/1 - l/de-P^j (3.21) This solution exists only when de~^" > 1, that is when the temperature is greater than the Hagedorn temperature. In this region £ < 2 and the semi-circle distribution is well-defined. Note, however, that the expansion parameter in the approximation (3.19), e~@u, is at least e " f e u = 1. 2 To see this, we need the integrals / de1 p{e') cothe -e1) = ^smt where again the principal value is implied, and / d8p(6) cos 0= ( 1 - | Chapter 3. Thermodynamics of the Matrix Model 30 This approximation is not too bad, however, even in the high temperature limit. For (3 = 0 we get a narrow distribution, as £ ~ j. It is easy, though, to solve exactly for the density in this limit. Equation (3.13) reduces, for d > 1, to J d8'p(0') cot 1 (0 - 0') = 0 Because of the implicit principle value we immediately see that in this limit, p{6) = 5(9) is a solution (with support only at 0 = 0); the eigenvalue attrac-tion totally wins over. 3.2 Connnement-Deconfinement Transi t ion The matrix model phase transition that we have found is characteristic of a deconfining phase transition in a gauge theory which is heated to a sufficiently high temperature. The matrix quantum mechanics has the gauge symmetry (2.2) where, since all variables transform under the adjoint representation, the unitary matrices are periodic up to an element of the center of the group, U(T + (3) = ZU(T) (3.22) and where z is an element of the center of the group. If the gauge group is U(N), the center is U(l). If it is SU(N), the center is Z^, the multiplicative group of the N-th roots of unity. No local operators are sensitive to the center element in (3.22). However, there is a nonlocal operator which transforms under the center, the Polyakov loop operator P = TrPexp (i d rA( r )^ (3.23) where V denotes path ordering. Under a gauge transform with boundary condition (3.22), P -> zP (3.24) The center symmetry is often treated as a global symmetry of the finite temperature gauge theory. As a global symmetry one can ask the question as to whether or not it is spontaneously broken. An order parameter is the expectation value of the Polyakov loop, (P). If the center symmetry is unbroken, this expectation value must vanish. If it is broken, then (P) can Chapter 3. Thermodynamics of the Matrix Model 31 be non-zero. The vanishing or non-vanishing of (P) has an interpretation in terms of confinement. The expectation value, = J[dA] [dX}e- X H T r « D X ) W ) T r V e l Sg drA{r) _ J[dA][dX]e-IoP*TkTT«DX)>+^x>) P [ P { °» (3.25) is the ratio of partition functions for the system where one classical quark source is inserted to the partition function in the absence of the source. F—F0 is interpreted as the difference of the free energies with and without the quark. When the expectation value vanishes, this is interpreted as taking an infinite amount of energy to insert the quark. Thus, the phase with unbroken center symmetry is confining. When the symmetry is broken, the free energy of the extra quark is finite and that phase is deconfined. The use of the word "deconfinement" in a theory where there is no spatial extent over which particles can be separated must be justified carefully. It is possible to distinguish the two phases by the behavior of the free energy, F[T]. In a weakly coupled theory, the free energy should be proportional to the number of degrees of freedom. In the confining phase the number of degrees of freedom at a given energy are of order the number of color singlets, which in the lower energy parts of the spectrum does not grow with the rank of the gauge group, N, but is of order one. In a deconfined phase, the number of degrees of freedom is the number of elements of the matrices, which is of order N2. Thus, we would expect F lim —- = 0 confined JV->oo N2 lim — ^ 0 deconfined (3.26) When interactions are turned on, the large iV limit which we are using is the ' t Hooft limit [23], where the coupling constants are also scaled as N is taken large so that A defined as A = (^)3N is held constant. Note that this limit is different from the large N limit with N/R held constant which would decompactify the null direction in M-theory. In a gauge theory such as the plane wave matrix model, where all vari-ables transform in the adjoint representation, there is an order parameter for Chapter 3. Thermodynamics of the Matrix Model 32 confinement called the Polyakov loop [29],[30]. It is the trace of the holonomy of the gauge field around the finite temperature Euclidean time circle, p=±-Tr(ei$A) (3.27) It is related to the difference of free energies of the state where an additional fundamental representation quark is introduced and the state without the quark, Fq[T\-Fn[T\ = -T ln (P) The free energy of a free quark in the confined phase should be large compared to that in the deconfined phase. In fact, (P) = 0 confined , , (P) ^ 0 deconfined 1 ' is interpreted as requiring infinite energy to insert a fundamental represen-tation quark into the system when it is in the confining phase, whereas it is finite in the deconfined phase. The Polyakov loop operator is widely used to study finite temperature de-confining phase transitions in higher dimensional gauge theories, [31]-[40]. A phase transition where both criteria for deconfinement (3.26) and (3.28) occur has been found in the large N limit of the matrix model . It has been argued to be generic to the large N limit of matrix quantum mechanics when the matrices have a gapped spectrum. The transition temperature is of order of the mass gap. 3.3 The M a t r i x M o d e l at One Loop Now, let us consider a perturbative expansion of the plane wave matrix model at finite temperature. The partition function has Euclidean path integral rep-resentation similar to those of finite temperature quantum field theories. We consider the theory in Euclidean space where time is compact and identified as r ~ r + /? (3.29) with P = l/T (3.30) Chapter 3. Thermodynamics of the Matrix Model 33 T is the temperature. The partition function is given by the functional integral Z = J[dA][dX'}[dip]e~So drL[A^M (3_31) where L is the Euclidean time Lagrangian L = ±Tr (DXiDXi + DX~aDX~a - ^IaD^Ia) + + 2liTr ( ( f ) 2 (X&? + ( £ ) 2 ( ^ ) 2 + l^ia + infemXaXbXt + +R^Iaof[X\^\ - ^ea^aIg)j[X\^J] + f e«^aI{g*)IJ[X\ x/,aJ]--^[X\X^}2 - f[Xa,Xb}2 - i ? 2 [ X 5 , X f ) The bosonic and fermionic variables have periodic and antiperiodic boundary conditions, respectively A{T + (5)=A{T) , Xi(r + (3) = Xi(r) , ^ ( r + (5) = -ip{r) (3.32) Since the boundary conditions for fermions and bosons are different, super-symmetry is broken explicitly. Of course this is expected at finite temperature where bosons and fermions have different thermal distributions. Supersym-metry is restored in the zero temperature limit. We will see the results of this explicitly in the following. A parallel discussion of the BFSS matrix model at finite temperature can be found in ref. [25]-[41]. 3.3.1 Gauge fixing To begin, we must fix the gauge. It is most convenient to use the gauge freedom to make the variable A static and diagonal, -~^Aab = 0 , Aab = Aa5ab (3.33) Once this is done, the remaining degrees of freedom of A are the time-independent diagonal components, AA. We shall see that they eventually appear in the form exp (i/3Aa). Chapter 3. Thermodynamics of the Matrix Model 34 The Faddeev-Popov determinant for the first of these gauge fixings is 3 d e t ' { - T r { ~ T r + ~ *>) ) = d e t ' ("£) {-If + * * * " (3.34) where the boundary conditions are periodic with period (3. The prime means that the zero mode of time derivative operating on periodic functions is omitted from the determinant. Once the gauge field is time-independent, we do the further gauge fixing which makes it diagonal. The Faddeev-Popov determinant for diagonalizing it is the familiar Vandermonde determinant, ni^-^i (3-35) This is also just the factor that the time independent zero mode would con-tribute to the second of the determinants in (3.34). Including it gives the determinant f jde t / det + i{Aa - Ab)\ (3.36) where there is now no prime on the second factor. These determinants can be found explicitly. We will do this shortly. 3.3.2 Classical ground states We will perform a semiclassical expansion of the free energy in the sector of the theory with the classical ground state Xa = 0 = X1 m the large N, 't Hooft limit. This is a double expansion. First, it keeps only planar Feynman diagrams, which is the leading order in an expansion in 1/N2. Secondly, it is "\perturbative in that it keeps those diagrams which are of low orders in the coupling constant. Since N is large, the appropriate coupling constant is A defined as A = (^fN. The configuration Xa = 0 = X1 has zero classical energy. In fact, because of supersymmetry, the ground state energy of the theory quantized beginning 3Using zeta-function regularization, Chapter 3. Thermodynamics of the Matrix Model 35 with this vacuum is zero to all orders in perturbation theory. This will provide us with an important check of our finite temperature computations. Of course, temperature breaks supersymmetry and the free energy of the thermodynamic state, which is what we compute, is non-zero. However, we will always be able to check whether the zero temperature limit of the free energy, which is the ground state energy, vanishes. We find that its expansion in the coupling constant A indeed does so to order A 2 . In this subsection, for completeness, we comment on the fact that there are a large number (—> oo as N —> oo) of other ground states with zero energy. To see this, observe that the classical potential in (1.21) can be written in the form V = *Tr 2 ( ^ r + ^ l ' l j V ^ (i[X\Xj})2 + + (}[x\x«}f+(£-)\xr \6RJ It has isolated global classical minima (V = 0) where (3.37) Xd = ° ' Xd = ^RJa (3-38) and Ja form an N-dimensional representation of the SU(2) algebra, [Ja, Jb] — ieabcJc, and could be either irreducible or reducible representations of SU(2). The representations of SU(2) are interpreted as fuzzy spheres which are spherically symmetric states of membranes [42] (see also [43],[44]). They approach a classical membrane when the spin of an irreducible representation is large, the state with a single membrane being described by a single N-dimensional irreducible representation of SU(2) with spin j — ^f^. Recently, an alternative interpretation of these solutions as transverse five-branes has been given [19]. In principle, any of the classical vacua corre-sponds to a five-brane state. A single classical five-brane, with its five spatial dimensions forming a five-sphere and its time direction lying along the light-cone, is believed to correspond to the vacuum where the representation of SU(2) is highly reducible, consisting of N repetitions of the singlet J1 = 0 representation. A detailed comparison of the fluctuation spectrum of the matrix model about these states with the fluctuations of the classical five-brane on a plane-wave background was given in ref. [19] and good agreement was found. A stack of k coincident five-branes is thought to correspond to Chapter 3. Thermodynamics of the Matrix Model 36 the state with n = N/k repetitions of the fc-dimensional representation of SU(2). If the representations have differing dimensionalities, with maximum dimension k, this still corresponds to k spherical five-brane states where the branes have different radii. The main difference between a five-brane state and a membrane state is that, for the five-brane, as iV is taken to infinity, the number of repetitions of irreducible representations of SU(2) is taken to infinity with their dimen-sionalities held fixed, whereas for the membrane, the number or repetitions of an irreducible representation (number of membranes) is held fixed and the dimensionalities are taken to infinity. The distinction between these two cases that will be important to us in the following occurs in the way that the gauge symmetry of (1.21) is realized. Vacua with non-trivial representations of SU(2) break the gauge symmetry, realizing it in a Higgs phase. There is a residual gauge symmetry which interchanges representations of the same dimensionality. If there are rik representations of dimension k, the gauge symmetry breaking pattern is the true gauge group is the group modulo its center. For a membrane state, as we take the limit JV —> oo, the rank of the residual gauge group remains finite, whereas for a five-brane, it becomes infinite. The coupling constant which governs the size of corrections to perturba-tion theory depends on the classical vacuum about which one is expanding. For membrane vacua, the effective coupling constant is For the five-brane, where the multiplicity of representations is n, the effective coupling is the' t Hooft coupling (3.39) Here, because all degrees of freedom transform in the adjoint representation, (3.40) The difference comes from the residual U(N) symmetry of the five-brane so-Chapter 3. Thermodynamics of the Matrix Model 37 lution and the fact that index loops in perturbation theory contribute factors of n which can only be controlled in the' t Hooft limit, n —> oo with A fixed (3-41) The decoupling limit, which isolates the theory living on the five-brane, is the same as the 't Hooft limit in (3.41). Intuitively, this is the weakest coupling limit that can be taken that retains any of the interactions in the matrix model. If it were taken for a membrane state, the matrix model would be non-interacting. It is only the five-brane states which retain interactions. The ' t Hooft coupling is the radius of the five-brane squared in units of the string scale r2 — ~ A (3.42) a In addition, the classical solution for gauge field in (3.37) must obey the equation Acb J c l = 0 (3.43) If Ja are an irreducible representation of SU(2), by Schur's Lemma, A c j must be proportional to the unit matrix Ac\ = c.T and a symmetry of the theory allows us to set c = 0. The gauge symmetry is realized by the Higgs mecha-nism and fluctuations of the gauge field are all massive. On the other hand, when the representation is reducible, there are gauge fields which commute with the condensate. The parts of A c ] which commute with the condensate remain undetermined by the classical equations. The volume of the space of all possible such A c ] forms a moduli space of the classical solutions which must still be integrated over, even to obtain the leading order in the semi-classical approximation to the partition function. The implications of the solutions (3.38) have been discussed in detail in ref. [19]. They were interpreted in terms of the spherical membrane and transverse spherical five-branes which exist on the 11-dimensional plane wave background. In refs. [45],[46] it was argued that the phase transition that we shall study here occurs only when there is a residual gauge symmetry and only in the limit where the rank of the residual gauge group goes to infinity. According to ref. [19], this is the limit of the theory which describes five-branes. Perturbation theory in that limit is governed by a ' t Hooft coupling similar to (1-22) with N replaced by the rank of the residual gauge group. Chapter 3. Thermodynamics of the Matrix Model 38 Ref. [19] argued that, in the't Hooft limit, the barrier between the degenerate vacua becomes infinitely high. They also argued that the limit decouples the five-brane from other degrees of freedom and focuses on its internal dynamics. The results of refs. [45],[46] can be interpreted as saying that the phase transition of the matrix model occurs only in five-brane states and not in membrane states, and it seems to be associated with internal dynamics of the five-branes. Of course, refs. [45],[46] analyzed only the weak coupling limit of the matrix model which is far from the limits which are conjectured to describe supergravity of an 11-dimensional spacetime continuum where the five-brane would live. In order to apply it directly to any known behavior of 5-branes, it would have to be extrapolated to strong't Hooft coupling. This is of course a difficult problem. The perturbative expansion in the present paper perhaps gives an indication that the phase transition is of first order but much more would have to be done to answer the question of persistence or nature of the phase transition in the 5-brane regime. One interesting extension of the present work would be to examine the dependence of the nature of the phase transition on the number of 5-branes. For k coincident five-branes, Xa contains N/k rc-dimensional representations of SU(2), with k being held fixed in the large N limit, /c-dependence of the phase transition temperature was investigated in ref. [45]. 3.3.3 Semiclassical expansion If we expand about the classical vacuum = 0 = X * , . we find the partition function in the 1-loop approximation is where Dab = — i(Aa — Ab). We discuss how to evaluate the determinants explicitly in the appendix B. The first two terms in the numerator are the Faddeev-Popov determinant. The third term comes from fermions whereas the denominator is from bosons. Using the formula Z= / dAaJ{ det' ( -d/dr) det (-Dab) det8 (-Dab + (3.44) det 3/ 2 {-Dlb + f) det 3 (-Dlb + g) (3.45) Chapter 3. Thermodynamics of the Matrix Model 39 with periodic boundary conditions and det + ivj = 2 c o s h ^ (3.46) with antiperiodic boundary conditions, we can write 4 [v y r d {(3Aa) j - r [1 - e ^ - ^ ) ] [ l + e - M 4 + i f 3 ( A a - A b ) } 8 / l l On ^ [1 - e-Pn/3+iP(Aa-Ab)]3\l _ e-[3ii/6+ip(Aa-Ab)]6 J ~ * a=l a,y£b 1 J 1 J (3.47) Note that, because of supersymmetry, the zero temperature (/? —> oo) limit of the partition function is one. It also has a symmetry under replacing e-^ by l/e-Pi1. We must now do the remaining integral when N —> oo. There are N integration variables Aa and the action, which is the logarithm of the inte-grand is generically of order iV 2 which is large in the large N limit. For this reason, the integral can be done by saddle point integration. This amounts to finding the configuration of the variables Aa which minimize the effective action: 5 eff = J2 ( ~ ~ eiP{Aa~Ab)} ~ 8 ln[l + e - ^ / 4 + i ^ A ° - A b ) } + +3 ln[l - e - / W 3 + i / 3 ( ^ - / U ) ] + 6 L N [ ! _ e-M6+if3(Aa-Ab)^ ( 3 _ 4 8 ^ To study the minima, it is illuminating to Taylor expand the logarithms in the phases (this requires some assumptions of convergence for the first log) ^ 1 - 8 ( - ) n + 1 r 3 n - 3 r 4 n - 6r2n I ± SeS = }2 — 0 - n & » (3.49) n = l Here, r = exp (-/3///12) (3.50) 4 Because the matrix model action in pp-wave background is invariant under replacing A by A plus a constant times the unit matrix, we see that the integrand in equation (3.47) is indeed invariant under translating all values of Aa by the same constant. Chapter 3. Thermodynamics of the Matrix Model 40 and ^ = J j ^ e M ' ( 3 ' 5 1 ) a = l Recalling (3.27), we note that <f)n are multiply wound Polyakov loop operators evaluated in the static, diagonal . The zeroth moment is normalized <po = 1 (3.52) The other elements are constrained by sum rules. The density defined by a = l n is a non-negative function. For example, if only <f>o and <f>±i are nonzero, (3.53) implies that |<£i| < 1/2. In this one-loop approximation, the action is quadratic in the Polyakov loops. When all coefficients of the quadratic terms are positive, the action is minimized by cj)n = 0 for n ^ 0. This is the confining phase. When a coefficient becomes negative, the effective action is minimized with one of the loops nonzero. The result is a condensation of the loops. As we raise the temperature from zero (and lower j3 from infinity), the first mode to condense is n = 1. This occurs when r c = 1/3 - v Tc = ^ • 0 7 5 8 5 3 3 ^ and 4>\ 7^ 0 when T > Tc-Note that this condensation breaks a U(l) symmetry. This is associated with the center of the gauge group U(l) € U(N). It arises from the fact that all variables are in the adjoint representation. In the Euclidean path integral, gauge transformations X(T) —> U(T)X(T)U^(T) must preserve the periodicity of the dynamical variables. They therefore must be periodic up to an element of the center, U((3) — el9U(0). The Polyakov loop, on the other hand, being the holonomy on the time circle, does transform as P —• eIEP. Even once the static, diagonal gauge is fixed, there is a vestige of this symmetry where (3Aa —> (3Aa + 6 or <f>n —> ein6<j)n. This symmetry restricts Chapter 3. Thermodynamics of the Matrix Model 41 the form of the effective action for Polyakov loops, so that the term with 4>ki • • • <i>kn must have ki = 0. It is a good symmetry of the confined phase and it is spontaneously broken in the deconfined phase. The Polyakov loop operator is an order parameter for this symmetry breaking. 42 Chapter 4 Distribution of the Angles / 3 A a At temperatures greater than Tc the eigenvalues Aa distribute themselves so that they are clustered near a particular point on the unit circle. To examine the possibility, we consider the equation of motion for the eigenvalues, uj(e0+z) + aj(e°-z) + 8w(-r 3 z) + 8( j ( - r" 3 z) = = 3u(r4z) + 3to(r-4z) + 6cv(r2z) + 6w(r" 2z) (4.1) where r = e^/ 1 2 and the resolvent is defined as 0 = 1 w(z) is holomorphic for z away from the unit circle and has asymptotic behavior, a ; ( 0 0 ) = 1 and u(0) — — 1. In the large limit the poles in u>(z), which occur at the location of the elements elf3Aa, coalesce to form a cut singularity on a part or perhaps all of the unit circle. u(z) remains holomorphic elsewhere in the complex plane. We must remember that equation (4.1) is valid only when z is one of the gauge field elements el/3Aa. In that case, the sum in (4.2), which turns into an integral in the large N limit, must be defined as a principal value. In (4.1) this is gotten by averaging over approaching the unit circle from the inside and from the outside. It is easy to find one exact solution of (4.1). If we consider the case where ei/3Aa a r e u n i f o r m i y distributed over the unit circle, so that the sum in (4.2) is symmetric under z —> el6z we can average over the symmetry orbit to get The result is certainly holomorphic everywhere away from the unit circle and is discontinuous on the entire unit circle. Chapter 4. Distribution of the Angles f3Aa 43 The resolvent (4.3) is always a solution of (4.1) for any value of r. This is the symmetric, confining solution of the matrix model, where the Polyakov loop operator, whose expectation value is a particular moment of to(z) for large z, vanishes. We would expect that this confining solution is only stable if the temperature is low enough. At some critical temperature it becomes an unstable solution and there should be other solutions which have lower free energy. The confining phase which we discuss above is stable when r > 3 or r < 1/3. When r = 3 or r = 1/3, we can find a 1-parameter family of solutions, / \ ( —I — az \z\ < 1 . . . . ^ = { l + a/z \z\>l <4-4> This is an acceptable solution when \a\ < 2.1 If we plug it into equation (4.1) and assume that r > 1, we obtain (3r~4 + 8r~ 3 + 6r~2 - l)(z - 1/z) = 0 (4.5) which is solved when r = 3. If we assume r < 1 we find an equation which is solved by r — 1/3. To examine the phase transition further, we expand about r = oo. We expect the transition to occur at r = 3 which is not really large, but we will see that corrections are of order 1/r4, at the 1-percent level. The asymptotic expansions of the resolvent is 7 1=1 O O w(*) = - l - 2 £ V - 7 _ n (4.7) 7 1=1 where a = l are the expectation values of the Polyakov loop operator for n windings. If 'Here, a/2 is the expectation value of the Polyakov loop operator which must be less than one. Chapter 4. Distribution of the Angles (3Aa 44 we assume that r > 1, an asymptotic expansion of the equation (4.1) is o,(e°+z) + - ( e ° "z ) = 2 £ ( A + i _ + _ , _ „ / ) ( 4 . 8 ) n = l ^ ' Remember that this equation is valid only when z is inside the cut disconti-nuity of to(z) which is assumed to occur on a segment of the unit circle. In the large r limit, the right-hand-side of this equation can be approx-imated by the leading terms. It is then similar to the equations for the eigenvalue distributions in adjoint unitary matrix models which have been solved in the literature [47]-[49]. It is easy to find a solution of (4.8) if we truncate the right-hand-side by retaining only the n = 1 term. Consider the semi-circle distribution of Gross and Witten [50]2 1 I \ I I W ~W = i + t\z~z)~ (T+t) y-+ z) M (More details of solving the loop equation for the resolvent which leads to the equation (4.9) can be found in appendix C). This function has a cut singularity on the unit circle between branch points z± = — t ± i\/l — t2 where we take t in the range — 1 < t < 1. When t —> 1 the endpoints of the cut touch each other and the cut covers the whole unit circle. This is the where the Gross-Witten phase transition occurs in their unitary matrix model [50]. In their case, it is a third order phase transition. In the present case it is a first order phase transition. The solution that we found above, when r = 3, is just a special case of the semicircle (4.9) when t = 1. 2 A n example of the eigenvalue distribution p(8) is the semicircle distribution, which is given by p g c W = f cos § ^ 2 ( 1 + ^ - 4 s i n 2 f 0 < s i n I < ^ ¥ SC I 0 \[W < sin f < 1 To get (4.9) we integrate Chapter 4. Distribution of the Angles (3Aa 45 Let, us explore u>sc(z) a little more. First we note that it obeys UJsc(1/Z) = -UJSC(Z) (with the appropriate change in the sign of the square root). This means that the r]n = rj_n = rj* for all n. We can expand for small z, usc{z) = - 1 - —z - 2 z 2 - ^ ^ J-z3 + ... (4.10) from which we identify 3 - * ( I - * ) 2 ( l - £ ) 2 ( 5 i + l ) % = 1 , »7i = — , V2 = '-^J~ , Vs = " ^ - (4-11) We see that, there is a critical point at t = 1. At that point, 770 — 1, i]±i = 1/2 and »7|fc|>x = 0- This is precisely the value of t for which the edges of the cut meet, so that the cut covers the entire unit circle. This is also precisely the exact solution (4.4) which we found when r — 3, here with the special value a = 1/2. In fact, the semicircle distribution gives a good approximation to the solution when r is slightly less than 3. For z in the cut, 2 / 1 ujsc(e0+z) +ojsc(e°-z) = ^ f - - z If, for the moment, we truncate the right-hand-side of (4.8) to the term with n = 1, we see that the equation is solved by the semi-circle distribution when (8r~ 3 + 3r~ 4 + 6r" 2 ) (4.12) ( l + t ) ( 3 - t ) Also, remembering that t falls in the range—1 < t < 1, we get the critical value of r, r c r j t = 3 by setting t = 1. (4.12) has a solution only when r < 7~ c r^ = 3 (and, here we have assumed r > 1). When r — 3, the n = 2 term on the right-hand-side of (4.8), which we have ignored, contains (8r~3 + 3r~ 4 + 6r-~2) — .086. Thus, we see that, to an accuracy of about ten percent, the semicircle distribution is an approximate solution of the model for temperatures just above the critical temperature. Chapter 4. Distribution of the Angles [3Aa 46 4.1 Systematic improvement of the semicircle It is clear what has to be done to improve this approximation. We can begin with an Ansatz for the resolvent which has a single cut singularity placed on the unit circle K uj(z) = Y (a™ (z~n - zn) - 6 « (2™ - z~n~l) V l + 2tz + z 2 ) (4.13) n=l To get the general solution, we should consider all orders by putting K —* oo. An approximate solution is found by truncating at some order K. We will see below that this approximate solution is good near the phase transition. The coefficients in (4.13) must be arranged so that, in an asymptotic expansion in small z, 1. all of the poles of order l/zK,1/z cancel and co(0) — —1 so, the asymptotic series then has the form LO = — 1 — 2r\\Z — 2r)2Z2 — . . . This gives K +1 conditions that the 2K + 1 parameters (a n , bn, t) must obey. 2. From the above expansion, we determine the moments in terms of the parameters r/i(an,bn,t) , ri2(an,bn,t) , ... , ijK(an,bn,t) 3. Then we use the equation (4.8), with the right-hand-side truncated to order K to get K conditions oi = f(r)vi(an,bn,t) , (4.14) a-2 = j'{r2)r}2(an, bn, t) , (4.15) aK = f{rK)nK{an,bn,t) (4.16) where f(r) — + ^ + ^ ) . This gives K further equations which completely determine the 2K + 1 parameters of the solution. Chapter 4. Distribution of the Angles /3Aa 47 For example, if we choose K = 2, requiring that the poles cancel and u(0) = — 1 yields the three conditions a2-b2 = 0, a1-b1-tb2 = 0, h(l + t) + b2^(l - t2) = 1 We can use these equations to eliminate bi, 0-1,0,2 t 1. 1 3* - 1, 1 b\ = o 2 H , a-i — b2 H , ao = Oo 1 2 1 + t ' 2 1+t ' Then, we can calculate the first and second moments by considering an asymptotic expansion oito(z). We get [t + lfb2 - 2(t - 3) m = 8 — % = - ^ ( 3 t - 5 ) ( l + i ) 3 + ^ ( l - i ) 2 and finally, using (4.14) and (4.15), we get the equations 3 t - l 1 = f(t + 1)362 - 2(t - 3) 2 1 + t M b2 = f(r2) ( - ^ ( 3 i - 5 ) ( l + i ) 3 + l ( l - t ) 2 Of course, we already know that these equations are solved at the critical point by 62 = 0,£ = l , r = 3 —»• / ( r ) = 1. If we consider a value of r somewhat less than the critical value, f(r) = l + e In this case, / ( r > ) _ i ! Z . + 25i M ; 2187 162 We get i = 1 - 2^1 and 187 62 = e 2000 Chapter 4. Distribution of the Angles (SAa 48 This demonstrates that, close to the phase transition, the semicircle dis-tribution gives an accurate description of the de-confined phase and this de-scription can be systematically corrected. It would be interesting to explore this further to determine precise thermodynamic properties of that phase. The next term in the series on the right-hand-side of (4.8) which we have ignored, since we truncated to order 2, is proportional to f{rz) with r « 3, which suggests that in the vicinity of r = 3, the error is less than one percent. However, we caution that this is the case only for r close to 3. When r = 2, /V) = . i i . 4.2 H i g h temperature l imi t Another limit we could consider is the high temperature limit where r —> 1. In that case, we expect that the values of elliA" are concentrated near a point. In fact, the case where they are at a single point u{z) = Z-T) is a saddle point for all values of r. However, for r ^ 1 it has infinite positive energy, crossing over to infinite negative energy when r — 1. To see that it is a solution, we note that, in this case the eigenvalue support is at z = r\ and therefore the variable in (4.1) is rj. Then tu(e0+r]) + bj{eQ~r)) = 0. Also u){rsrf) +uj{^rj) = 0 and (4.1) is solved. It is easy to see from the action that this solution is unstable for all values of r except r — 1. 4.3 Free energy This shows the nature of the phase transition. At the critical point, it will turn out that the free energy is continuous, but the expectation of the Polyakov loop is equal to a and is ambiguous. Just below the transition, the theory is approximately described by the semi-circle distribution for which the Polyakov loop is 1/4. So we see that it jumps in value from 0 to 1/4 at the phase transition. It is for this reason that we expect the transition to be of first order. Indeed, by examining the free energy, we see that it is given by Chapter 4. Distribution of the Angles (3Aa 49 7 m(z a -zb)-8 ln(za - r 3 z 6 ) + 3 ln(z a - r 4 z 6 ) + 6 ln(z a - r 2 z h )) (4.17) N2 a^b For the symmetric solution, where po(0) — ^ 7o = 0 When r is large, we can expand to get 7 N2 a^b In 11 — z;aMI 6 8 3 \ z a + - ^ + - 7 ) - + r2 r 3 r 4 / zb (4.18) If we keep only the first term in the large r expansion (note that there is another term which competes with 1/r 4 which we ignore for now), this is approximately an adjoint unitary matrix model of the kind solved in the literature [51]. It is solved by the semicircle distribution or the symmetric distribution. The free energy is 7 0 The value of the Polyakov loop is —Tr e^A N 0 ,r > 3 3 ( 1 + v / r r 7 k , ' r < 3 , r > 3 ,r < 3 (4.19) (4.20) 4.4 Symmet ry restoration Because of the low dimensionality of the system that we are discussing, the symmetry breaking which occurs in the de-confined phase could be destroyed by quantum fluctuations. In fact, it would generally be the case in theories with local interactions. For example, if N is finite, symmetry breaking is not possible. The phase transition that we have discussed here can only occur when ./V is infinite. Mathematically, we can think of large iV as the analog of a large volume Chapter 4. Distribution of the Angles (3Aa 50 limit in a statistical mechanical system. If iV is large but not infinite, the symmetry is not broken in a mathematical sense but the decay rate of a non-symmetric state is exponentially suppressed in the volume, in this case ~e~N-. The deconfined solution has a spectral density p{9). Because of symme-try of the problem under replacing 6 by 0+constant, there would be a zero mode of the linear equation for the fluctuations of p(0), with wave-function ip{9) ~ This mode would provide the motion which would restore the symmetry. However, in the case of the semi-circle distribution, because of the square-root singularity at the edge of the distribution, this function is not square-integrable, and therefore not normalizable. Chapter 5 Higher loop order 51 1 In the previous Section, we have computed the effective action as a function of the variables 4>n in the one-loop approximation. The result is the quadratic potential in equation (3.49). To find the stable phase, it is necessary to find those values of 4>n which minimize (3.49), subject to the constraint (3.53). The phase transition occurs when the curvature of the Gaussian potential for <pi develops a vanishing moment. This is a first order phase transition [47]. Unlike generic first order phase transitions, it occurs at the point where the curvature of the potential first goes to zero. As a result, there is no energy barrier separating the two phases and no coexistence region of the ordered and disordered phases. These features of the phase transition are very sensitive to higher order corrections. In the following, we shall take into account the leading effect of higher order corrections. Since, as the temperature is raised from zero, the variable <j>i is the first to condense, we focus on its effective action. We have computed the relevant parts of the effective action up to three loop order. First of all, the ^-independent part of the effective action (what one obtains by putting <j>n — 0 for n ^ 0) vanishes up to order A 2 . The relevant parts which depend on <f)n have the form ^ 5 e f f = A i ( r ) \4>i\2 + A 2 ( r ) \<f>2\2 + XP^r) {4>iM-2 + c.c.) + , . +A 2 P 2 ( r ) |<H4 + ... The coefficients in (5.1) are : A version of this chapter has been published. Hadizadeh, S., Ramadanovic, B., Se-menoff, G. W. , Young, D. (2005) Free Energy and Phase Transition of the Matrix Model on a Plane-Wave. Phys. Rev. D 71 065016 Chapter 5. Higher loop order 52 A i ( r ) = [1 - 8r 3 - 3r 4 - 6r2] - 24A [ln(r) r 2 ( r 2 + l )(r + l) 4 ] -- 3 A 2 r 2 [ ln(r) 2 (68 r 1 0 + 352 r 9 + 904 r 8 + 1536 r 7 + 2256 r 6 + +3104r 5 + 4120 r 4 + 2304r3 + 928 r 2 + 192 r + 16) - (5.2) - ln(r) (27 r 1 0 + 152 r 9 + 390 r 8 + 640 r 7 + 915 r6 + 1232 r 5 + + 1748 r 4 + 1184 r 3 + 466 r 2 + 440 r + 102)] + ... A 2 ( r ) = ^ (1 + 8r 6 - 3r 8 - 6r 4) + . . . (5.3) Pi(r) = - 1 2 1 n ( r ) r 4 ( 2 r 4 - 4 r 3 + 3 r 2 - 4 r + 5)(r + l ) 4 (5.4) P 2 (r) = 3r 4 [ - ln(r) 2 ( 136 r 1 2 + 512 r 1 1 + 704 r 1 0 - 1308 r 8 - 1376 r 7 + +1560 r 6 + 6400 r 5 + 10896 r 4 + 8096 r 3 + 2136 r 2 + 1536 r + 240 ) + + ln(r) ( 27 r 1 2 + 120 r 1 1 + 166 r 1 0 - 32 r 9 - 271 r 8 - 16 r7 + 1044 r 6 + +2624 r 5 + 4036 r 4 + 3256 r 3 + 774 r 2 + 768 r + 944 ) ] + .. . (5.5) Eliminating <f>2 using its equation of motion, we obtain the effective action for (pi, in the large iV limit, and to order A 2 : - ^ 5 e / / = A 1 ( r ) | ^ | 2 + A 2 P 2 (r) A 2 ( r ) l ^ i | 4 + (5.6) 5.1 Phase transi t ion As we raise the temperature from zero, the quadratic term in 4>i m t-n e effective action vanishes at the critical value of r, 1 9 6 • 5 I + A y i n ( 3 ) - A 2 2 2 • 3 9 which translates to the critical temperature h-11-53-3061 ^ , 132 • 1867 2 ln(3) + 2 4 _ 3 8 ln(3) + . 121n(3) ^ 2 6 • 5 x , 1 + A ^ - A 23 • 19927 1765769 , , + ^ „ B ln(3) + 2 2 - 3 7 2 4 • 3 8 (5.7) (5.8) Chapter 5. Higher loop order 53 The zeroth order term in the critical temperature is the one found in [45]. The term of first order in A agrees with the result quoted in ref. [48]. Also, from (5.6) we see that the quartic term in 4>i is negative over the entire range 0 < r < 1. This means that the phase transition is of first order. When r is just less than the critical TQ = 1/3, the extremum of the effective action at <pi = 0 is only a local minimum. The effective action has a second zero when 1 9 1 1 A 2 P 2 ( r ) - P 2 ( r ) / A 2 ( r - ) Higher order terms in the effective action are individually small at this value of \d>t I 2 when — n , . n i f L , . << 1. We are further constrained by the fact that |0!| < 1. This requires that ~ p2^r)Jpf(r)/A2(r) < ^ 2 << 1- The number — „ , , p ^ r w . , . is less than 0.10 in the range 0.2555 < r < 1/3 and is less P2{r)-P{(r)/A2(r) ° — ' than 0.001 in the range 0.3174 < r < 1/3. If r is sufficiently close to rc, we can reliably say that the absolute min-imum of the potential is not at <j>i — 0 but is elsewhere. This sets an upper bound on the transition temperature T . <r T — ^ 'cnt. < J c 1 2 1 n ( 3 ) The tunneling barrier for bubble nucleation during the first order phase tran-sition is of order 1/A 2. 5.2 Free Energy to Two Loop Order We have that Z = f[dX}[di()}[dip^}e~s, as gauge fixing has already taken place and ghosts do not contribute. The free energy is F = —SQfe, and thus contributes another sign to the results. At two loops and above, SQQ is simply the sum of connected vacuum diagrams. Chapter 5. Higher loop order 54 5.2.1 Propagators Our strategy here will be to construct a Euclidean Green function which obeys the equation ^ - i A a b ^ + LO2^J 9AB(T) = 8{T) (5.9) which has the periodicity 0(T + /3) = S ( T ) (5.10) We will begin by constructing g(r) in the domain — (3 < r < (3 and then con-tinuing it periodically outside of this domain. For this we use the heavyside function 1 0 < r < (3 0 -{3 < T < 0 (5.11) Then our ansatz for the Green function is 5 ( r ) = (9+(T)9(T) + g-(r)0(-r)) (5.13) where ^ = &-iPAab = eif)Ab lei(3Aa = z ^ Z a ( 5 J 4 ) The Green function equation is obeyed if ( " J ^ + ^ ) 9±(T) = 0 — 9±{T) = a±e"T + b±e~"T (5.15) and 9+{T)=g-(T) , ±.g+(r) - ±g_(T) = -I (5.16) And the green function is periodic within the domain —f3 < r < (3 if r < 0 vg^r) = g+(r + (3) (5.17) The unique solution of these equations is Chapter 5. Higher loop order 55 If needed, this Green function should be extended periodically to all values of T . We note that this Green function is a sum of two green functions for linear differential operators, M = T O M " S ; ( T | D T ^ + ^ I ' ' » < 5 ' 1 9 ) which implies 9(r) = ^(9i(r)+g2(r)) (5.20) where (D + LV)9I(T) = 5(T) , (-D + U)92(T) = 5(T) (5.21) with the same periodic boundary condition that is satisfied by g(r). ^ = ^ + ( ^ T ^ J 0 ( T ) + ^ ( ^ ^ J (5-23) Similarly, a fermionic propagator obeys (-D + U)S(T) = S(T) (5.24) with the anti-periodic boundary condition S(T + (3) = S ( T ) (5.25) We can similarly construct it in the interval —j3 < r < (3 and continue it anti-periodically to the real line. The Fermionic propagator is We begin by deriving the propagator for the scalar Xa and X1 fields. To Chapter 5. Higher loop order 56 do so we must find the Green's function for the operator — D2 + to, where is the mass of the field. First we note that: + (dT - iAab)2 2LO [to - (dT - iAab) co + (dT- iA ab) (5.27) where Aab = Aa — Ab, and thus we begin by finding the Green's function g(r' — T) for the operator — D+to, enforcing the periodic boundary conditions g{i3/2) = g(-(3/2): / ( r ' - r ) = e e-(iAab+u)(3 1 _ e-(iAab+u)f3 8{r' - T ) + 1 1 _ e-{iAab+u)P where 8(T' — r) is the standard Heaviside function defined so that: while [8(s) + f?(—s)]s=0 = 1. This ensures: (r - r') (5.28)" 0 s < 0 1 s > 0 [-(&V - iAab) + to] g{r' -T) = S(T' - r). (5.29) According to (5.27), we then have that the Green's function for —D2 + io2 is and so the full propagator for the scalars is: (5.30) (XUr) XUr')) = ^ S^5ad8bc [g(r' - r) + g*(r - r')}ab (5.31) where, for this expression only, we can take i,j to be either flavor of scalar. The fermion propagator comes from an inversion of — D + to alone, how-ever this time with antiperiodic boundary conditions so that gj(f3/2) — —gf(—/3/2). The result is that the Green's function is: Chapter 5. Higher loop order 57 5 / ( T ' - T ) = e ( i A « » + « ) ( T ' - T ) (5.32) Resulting in the fermion propagators: <W7a)a*(r) { ^ J p ) c d ( r ' ) ) = 2R5ad5bc5i5J9fab{T' - r) < ( ^ t / a ) a b ( r ) (Vj/?)cd(r')> = - 2 / 2 ^ * j ^ 3 / a * ( r - r') while (V'V;) = (^V) = 0- The negative sign in the second propagator comes from the Grassman nature of the fermions, the conjugation from the order of the matrix indices, and the time reversal from just that - the time reversal. 5.2.2 Fermion Contributions The fermion diagrams are the following, with two flavors of scalar propagat-ing: Figure 5.1: Two loop fermionic diagram The relevant term in the action is: l-Tr [+^Iaaf [Xs^jp] - \ea^Ia9\j[X\^JP\ + \i<# rpIa(g^)IJ[X\^ (5.33) The diagram comes from expanding exp(—S) to second order in the path integral. Since (tpip) = (V^ V'O = 0, the surviving terms are: Chapter 5. Higher loop order 58 ( i i / dr dr' {Tr (^aa[Xa, rb]) (r) Tr ^ab[X\ V]) (r') +Tr ( - i eV ' t^[A :« ) V t ] ) (T) Tr (§ ^ p P , (V) (5-3 4) +Tr (§ e V ^ ^ 4 , V < ] ) (r) Tr ( - | e ^ s W ^ ] ) (/)}> Considering the first term first, and writing it in terms of matrix indices we have: 1 J dr dr' ( V l ^ a ( * ^ < » - ^ c ^ J (r) 4eal (x\f*Pfd - rpefXlfd) (r ')) (5.35) Keeping only the planar contributions, and noting again that (ipip) = (V^V^) — 0 this becomes: i Jdrdr'cr^ {(x?aXbd) U b c ^ d e ) (^V-/) + where the first field in each expectation value is evaluated at r and the second at T'. Recalling the form of the propagators (5.31) and (5.33) we have: [(9f)bc{g}-)ab + (gf)ab(g}-)bc] (5.37) where the subscript " —" indicates time reversal. The factor of 8 comes from SjTraa. The factor of 3 from the fact that there are three scalars of the first flavor. Now, noting that R3 = g2, and that u>i — /J/3 we have: - — E / d T d T ' [9(9 + g*-)c«] [(gf)bc(g}-)ab + (9f)ab(g}-)bc} (5.38) ^ abc Now we attack the fermion propagator terms: ,(2RY dr dr' 3R 2o7i (9+9*-\ Chapter 5. Higher loop order 59 where <pab = eiA°b+_u', 8 = 6{T' - r ) , and 8 = 9(T - r ') . Using the fact that 62 = 6 and that 88 = 0, and that Aab = Aa — Ab, we have: Therefore: (9f)bc(9*f-U + (9f)ab(9f-)bc = -eiA<^'-r) { + -(i+Cb3)(i+C"^)J while for the scalar propagators we have: 6 + [ce/ (l+Cf)(l+^c ) + r i . 1 " 9 + 9- A-ca'Pca ^ + BcayCa *(r—r') + 8+ where. Aca = ( i - p-^-J Now it can be seen that, Bca = (1 -(5.40) (5.41) (5.42) (5.43) [ ( S / W s J - U + (9f)ab(9}-)bc} [g + g*Jca = G8 + H8 (5.44) and by changing variables in the second term such that r and r ' are inter-changed, one notes that H —> G*, so that: [{gs)bc{g}-)ab + (gj)ab{g)J)bc] [g + gt]"ca = (G + G*)8 = 2Re(G) 8 (5.45) Now notice that the elAac^T'~T) term from the fermion propagators kills the gauge field dependence of the scalar propagators: eiAac(r'-T) ^-T) _ eu,{T'-T) eiAac{T'-T)^*(r-T') _ e - o j ( r ' - r ) (5 46) Thus yielding the following form for G: Chapter 5. Higher loop order 60 G {Acae^T'-^ + Bcae-^T'-^) (AbcBab + AabBbc) GcaCabCbc (5.47) where: Cat = |1 - C f e ) ! 2 ^ = 11 + O / ) ! " 1 - fc'H) B„6 = l " a t ''fa) B, *ab 1 + The integrations over r and r ' are performed using: /3/2 /J /2 dr / dr' 0(r' - r) e" ( r '-T ) = /J /2 J-P/2 - 1 - /?. (5.48) The subsequent reduction to the following form is performed using Maple. The result is: C e o . C a i t C b c L1 X {± [cos(3Aab + cos(3Abc\ '[e-M* + e - 3 / 3 M / 4 + e - l l / W 1 2 _|_ g-17/3^/12 _ 2 e - 7 / W 1 2 _ 2 e - 1 3 ^ / 1 2 ] + [2 + 2cos /M c a ] [e~^/ 2 + e - 7 / W 6 - 2 e - 5 ^ / 6 ] where the negative sign comes from (5.47), the factor of 2 from (5.45), the factor of 3/3/// from the Maple reduction, and the final factor of —2 • 9 g 2 / / i from (5.38). The form of the C"s is: Cab = l-2e-(3ujcos(3Aab + e -2/3w Cab = l + 2e-fiucosPAab + e - 2 / J a (5.49) In the case of periodic boundary conditions for the fermions, the propagator term gf picks up an overall sign, while all barred quantities should be replaced by their unbarred equivalents. The final result is that the [cos (3Aab+cos [3Abc\ term switches sign and the fermionic C"s are replaced by C ' s (with LU = / i /4 , of course). Chapter 5. Higher loop order 61 Now for the second and third terms in (5.34), starting with the second term: - ± f d T d T > ( e ^ S « ( i y L - f e ) W x Q ) x e^i{Xifrp}d-^.fX)d) (r')> There are more planar contributions here than for the first term of (5.34), we have: -IJdrdr' e eg^j \(X*aX}d) (^de) (^e/) + + (XlcXif) <V4^ e> {^fd) + (XLX'ef) ( ^ f d ) ( t i b ^ e ) + (5.51) + (XicX}d) ( ^ e f ) ( ^ d e ) } Not surprisingly each term contributes the same quantity, and a factor of four is gained, the result is: + ^ (2R)2 • 6 ^ j"dr dr' abc R 2lO~2 (9 + 9-Tc [(g}-U(g*f be (5.52) Here we encounter the structure Tre2Trgg^ — — 8 • 6, where the sign comes from the fact that e = ia2. The factor of 6 counts the six scalars of the second flavor. The third term in (5.34) is identical except that (tp^ip) —»• ( V ^ ) , and thus the full expression is: 8 • 4 ~32~ ( 2 i ? ) 2 - 6 ^ jdrdr' abc or more concisely, [(g}-)ab(g}-)bc + [gf)ab{gf)bc] (5.53) 21 h-J dr dr' [36 (g + g*_)%] [(g}_)ab(g}_)bc + (<?/Ws/k] (5.54) abc We continue as before by evaluating the fermion propagators: Chapter 5. Higher loop order 62 (l + ^ ) ( l + 0aT) (9f-U(9f-)bc-e (1 + ^ ( 1 + ^ ) Now the same integration variable switch employed in (5.44) and (5.45) can be used here, yielding: _ (Ame^'-^ + B n e - ^ ' - T ' ) { A a b A b c e 2 ^ + BabBbce-2^T'w>) ^ c a C ^ C t , , ; (5.55) Maple is then used to obtain the final result: c^C ,<?, fH^ [cosPAab + cospAbc] xC ' j-g-V /4 + e -5 /3M/12 + e - l l ^ / 1 2 + g-13/3^/12 _ 2e- 3 i<W 4 - 2e" 7 / 3^ / 1 2] + C O S / M c a [ e - 7 ^ / 6 + e - / W 6 _ e - /3 / i /2 _ e -5/3/x/6] +1 + e - 4 / 3 M / 3 + 2 e - 2 ^ / 3 _ 2 e - / W 3 _ 2 e -^} (36 2 £ ' where the negative sign comes from (5.45), the factor of 3/3/2/z from the Maple reduction, and the final factor of 2 • 36g 2//j from (5.54). For periodic boundary conditions, the final effect is as before. That is, only the [cos PAab + cos PAbc] term switches sign, and C"s are replaced by regular C's. 5.2.3 The Theta Diagram The "Theta Figure" is given by the combination'of the two three-vertices for the scalars of the first kind: Chapter 5. Higher loop order 63 Figure 5.2: Theta diagram From the action we get the vertex as: Tr[e-al-XaXhXc} (5.56) so we can write the diagram as: ™/2 2 3 2 dr / 6 ^ 6 J e - / ^ ( r ) X b b c ( r ) ^ 0 ( r ) X i ( r ' ) X f / ( r ' ) ^ ( r ' ) (5.57) There is therefore three propagators between X(T) and X(r ' ) ' s the a,b,c and d, e, f all range over 1,2,3 and the e limits them to the totally symmetric and totally antisymmetric combinations. That makes up for 6 on each side. Furthermore the requirement that the diagram be planar makes sure that if the d, b, c combination is symmetric then d, e, / can not be anti-symmetric and vice versa. The planar contractions also introduce a sign, due to their mixed symmetry. There is therefore 6 x 3 combinations of the propagators with the summation over the indices: 2 R 3 8wJ abc 2 dr d T ' [ 5 ( r - r ' ) + 3 V - T ) U 2 bc\ (5.58) where Diagram can then be written as: 2R3 8ui-T ^ a b c l ^ d r f ^ d r ' r1 W l = 3 + + hih 1_ ab 6c J6c I r + (5.59) (5.60) Which can also be written as: Chapter 5. Higher loop order 64 - ( T ' - T ) _ Cab • • be .. ca^ r - r ' (5.61) It is possible to interchange r and r ' in the second line of the integral in order to get everything multiplied by the same Heaviside function. The second line is then just a complex conjugate of the first because Cab is a real quantity. It is then possible to write the diagram as: ^ £ a 6 c Z^tcZ A d T fr ^ ' R e [ ( A 6 ^ b c A a ) ( 0 a 6 0 6 c 0 c a ) ( T ' - T ) + + ( A b A c £ c a ) ( 0 a 6 0 6 c 0 : - V ' - T ) + - + (BabBbcBca) ( # T 1 <f>£1P'1) ^ This can be simplified by noticing that: (5.62) (5.63) Then diagram is given by: ^ J2abc alcbcz, A dr JTf dT>Re(AabAbcAcay^-r) + 4uf L^iabc CabCbcCca +Re(AabAbcBca + ...)e^T'-T^ + Re{AabBbcBca + ...)e-«i('-'-r)+ +Re(BabBbcBca)e-^T'-^ Performing the integration: z±!^y^ i \Re(At,AhA )ktM!di=^l+ 4wf (5.64) +R.e(AabAbcBca + ...y+^y0"1 + R,e(AabBbcBca + -+ +Re( /3 a f c £ b c £ C ( I 1—3/3oJi—e" 1 9w? (5.65) Using the given definitions of A and B from (5.43) it is possible to simplify the above, using Maple: x I1 E a b c Z ^ f e r {1 + e" 2 ^ - 9 e - W + 1 6 e - A , _ 9 e - * W 3 [cos(/?Aab) + cos(/?A c) + cos(/M c a)] 2 7 / V V -+ 12e-^j} Chapter 5. Higher loop order 65 5.2.4 The Figure-Eight Diagram Figure 5.3: Eight-shape diagram This comes from expanding the action to first order, exp(—S) ~ 1 — 5, and so we pick up a sign: f J% {Tr[X\ X*]*) dr = f f% (Tr[X\ X>f) dr = f 1% (Tr[X*,Xr) d r + f J% E i < 3 (Tr[X\ X']*) dr +]lf%2Za2Zl(Tr[X*,Xr)dT (5.66) Let us consider one of the above terms Tr([Xa,Xb}[Xa,Xb}) = Tr({XaXb - XbX~a){XaXb - XbXa)) = 2Tr(XaXbXaXb - XaXaXbX b V a vb y a y « y 6 vb\ (5.67) While the first term has a non-planar contribution, hence the whole expres-sion for the d-flavor bosonic field would be R E a < 6 Yjabcd 1-0%(~ (XabXbc) i X c d X d a ) ) d r = ~R E a < b Eabcd f% 2,2 (&) 2 S^SM^Pat^P^r) 5 ' 6 8 where Pab(cv) = (g + g*_rab (5.69) as per (5.37) and (5.31). Therefore what we get is Chapter 5. Higher loop order 66 ( D3 \ o o fP/2 1772 H r E / dr (5.70) The only difference for the i-flavor would be the value of the Sj<j, hence putting everything together we find the following result for (5.66): ~ X ^abd J-l3/2\^Pab(UJl)Pad{ui) + + | | P a b ( c 2 ) P a d ( c 2 ) + ^ P a b ( W l ) P a d ( ^ ) ] ^ ^ ' Considering that u>\ = p,/3 and CJ 2 = p,/6, then the final expression would be P 3 ^ j-*1'2 abd J-?/2 Now, consider the product of two P's R3 rPI2 - T ^ E / [ 2 7 P a 6 ( - i ) P a d ( - i ) + 5 4 0 P a 6 (a ; 2 ) P a d ( W 2 ) + 3 2 4 P Q b (w 1 ) P a d ( - 2 ) ]dT (5.72) PabPad = {Htf + G19)(H2d + G26) = HYH2Q + GXG2Q (5.73) It so happens that, when r is set equal to r ' tf1 = ff2 = G 1 = G 2 = ^ ( l - e - 2 ' J ' ' ' ) (5.74) Using the fact that [0(s) + 6{—s)]3-o = 1, we arrive at the final form: 2 7 / 3 g 2 r [ i _ e - ^ M / 3 ] 2 r 1 _ e - ^ / 3 ] 2 [ 1 _ e - 2 f ) / x / 3 ] [ 1 _ e - p M / 3 ] The final equation can be taylor expanded in r. The result is o2-loops _ _ 2 7 f t £ V / (1-r 8 ) o f , ( 1 - r " ) (l- r 8 )(l-r 4 ) ^ e / / - 4 M 2 2^abc\ c^ctf 1 1 C ^ C ^ + ( r 8 + 4 r 4 + l ) ( r 4 - l ) 4 + [ c o 8 / 3 / l < , t + c o J » / 3 A | , e + c o a i 3 A c a 1 2 r 4 ( r 4 - l ) 4 , | 1 6 r V 4 - r 2 + l ) ( r 4 - l ) V * + l ) [ c o g / 3 . 4 ^ ( O . / D J ! 3 2r> 4 - l ) 2 ( r 2 + l)[coS ; 9.4 a b+c 0s/3.4^^ Where we have: C2 = l-2r4cos{3Aab + r8 Chapter 5. Higher loop order 67 C^ = l-2r2cosPAab + r4 Cab = l + 2r3cos(3Aab + r6 This expression can be restated to exhibit its <j>n dependence: + ^ %r;Xl^l) ( - l ) " r 3 H + 4 H - 16^ gi ( _ 1 ) n + m r 3 | n | + 3 H + +32 ^ a 1 ^ (-1)" r 3 | " l + 2 | r T l 1 - 1 2 r 4 M + 2 H - 2 0 r 2 | n | + 2 | m | - (5.76) _ 4 ( r * - l " ) ( r 8 + r * + l ) p f j ^ | 1 6 ( r - 2 + l ) ( r 1 0 - l ) / ^ m p , 3 . ^ ( r 4 + l ) 3 r ™ ^ ! ^) ' l u ( r 4 + 1 ) ( r 4 + r 2 + 1 ) 2 V J-j ^mu^,1*) - l e ^ ^ x x y - ^ ( - i ) m ^(3 ,2 ) ; Where we define the function Fmn(a, b) in the following manner: { F^n(a, b) m,n>0 or m,n<0 F^n(a,b) n<0,m>-n or n>0,m<-n F^n(a, b) m < 0, m > —n or m > 0, m < —n And where we have: Fin(a,b) =ra(2+n+m)+b ' r 6 ( n + r n ) ~ a n r ~ b — 2 a + a n r — 2a — an+b(ri+m) J _ r 2 a + b ~T~ ^ _ r 2 a + 6 ~T~ r b —1 r - 2 a - a n + b n r ~ 2 a - f a n ^ .b ^ rb^_r2a ~T~ rb_i_r2a — 2 a — an-\- b n F 2 n ( a , 6 ) = r a ( " + m ) + f c a — a n - f - b ( n - f - m ) ^, — b — a n — b n ~ + l - r 2 » + 6 1" — a r t 4 - b ( n + m ) _ | _ y , — b — a n T > — a n r — bn — 6 — a n "| H 1 - r 6 (5.77) (5.78) F 3 n ( a , 6 ) = r ^ - m ) + 6 „ „ + 6 ( „ + m ) + 2 a + 6 n ^ , - 6 + o , n + 2 a m — a n + b n + 2 ^ + T -1 - r r — an-r-b(n+m) ^ j , a n + 2 a m b + r 2 a (5.79) In the zero temperature limit, r —> 0. We then see that the free energy is 1 + 20 + 12 — 1 — 32 = 0, which is what is expected from supersymmetry. Chapter 6 68 Conclusion We have found that the phase transition in the weakly coupled plane wave matrix model is indeed of first order. As the temperature is raised from zero, the curvature contained in the quadratic term in the effective action still vanishes at some critical temperature. However, before that point is reached, when there is still an energy barrier between the two phases, the deconfined phase becomes the lower energy state. This is the generic behavior at a first order phase transition. In fact, this behavior is seen in other adjoint matrix models [50],[51],[52]. It is also the behavior that is seen in the collapse of Anti de Sitter space to a black hole, which is thought to be the analog of this phase transition in supergravity of a similar deconfinement in M = 4 supersymmetric Yang-Mills theory [53]. Our analysis does not allow us to compute the first order phase transition temperature accurately, only to deduce that it is of first order. It does, how-ever, allow us to compute corrections to the Hagedorn temperature. This is the temperature at which, if the confining phase is superheated beyond where it is a global minimum of the free energy, it eventually becomes per-turbatively unstable. It is just the place where the corrected curvature of the effective action vanishes. 6.1 The Free Energy at Three Loops To do perturbation theory, we split the Euclidean action into three parts, a free action, an interaction action with three-point vertices and an interaction action with four-point vertices, S0 = / drTri-X' (-D2 + (u,/6)2) X1 + -Xa (-D2 + (/^/3)2) Xa + J-0/2 2 2 + ^ j Q ( - D + /i/4)V/aX6.1) Chapter 6. Conclusion 69 Z ^ / 2 / 1 5 3 = / drRiTr i^-e-alcXaXbXE + tfIaaf[X&, ^ ] -•7-/3/2 \ -3 - ^ Q ^ t Q / 5 b [ ^ , V t /3J] + \ea^aI{gi])IJ[X\ M ) (6-2) /•/3/2 D3 . SA = - dr—Tr ([X\ Xj]2 + [Xa, Xb}2 + 2[Xa, X'}2 7 - / 3 / 2 4 V (6.3) respectively. The effective action S^frJA] is obtained by a perturbative expansion of the partition function. e~SeS[A] = J[dX}[diP}e-s°-S3-s* , SeS[A] = 1 - < e ^ 3 ^ 4 > (6.4) where the bracket indicates the connected part of the expectation value in the free theory. Taylor expanding the exponential of interactions and retaining those terms which have non-zero contributions to two loop order gives >eff =2^ In det{-Dah)de1*(-Dab + *) + ( < S4 > - - < S2 > 1-<s2>-1-<s2si>+±<sl>) + The first line on the right-hand-side is one-loop, the bracket after it contains two-loop and the next bracket the three-loop contributions. At three loops there are many more diagrams. The three loop contribu-tions are naturally more complicated, so much so that a computer code was written to produce the results. The following figures show the three loop dia-grams and their associated zero temperature limits. We introduce some new notation to simplify the presentation. The letter "P" denotes the circulation of the m = /j,/3 scalar, while "Q" denotes the m = fi/6 scalar. Where there is no indication, it is obvious from the allowed interactions dictated by the Chapter 6. Conclusion 70 action. Pab(t21) = ^-\g(t2-t1) + g*(t1-t2)ral Qabihi) = irMt2 - * i ) + / ( t l - * 2 ) ] ^ (6.6) Fab(hl) — 2Rgfab(h — tl) Gabihl) — —2Rg*fab{h — tl) Cat's Eye Diagram Results should be multiplied by R2. The general form of the propagator for these diagrams is 3 \Pab(tw)Pbc{tw)Pcd{tw)Pda(tio) + 15 7iQab{tlo)Qbc(tw)Qcd{tw)Qda{tlo) +9 Pab(tlo)Pbc(tlo)Qcd(tlo)Qda{tlo) + 18 Pab{t\o)Qbc{tw)Pcd{tw)Qda{tw) and the numerical result of the share of each diagram in free energy is as the following: Figure 6.1: P-cat's eye diagram _ 2187 pR6 ~~ 64 [i5 Figure 6.2: Q-cat's eye diagram _ 10935 PR6 ~ ~~2 ~yT~ Chapter 6. Conclusion 71 Figure 6.3: PQ-cat's eye diagram _ 2187 pR6 6.1.1 Tripple Bubble Diagram Results should be multiplied by R2. Figure 6.4: PQP-tripple bubble diagram PR6 27{Pab(0)Pcd(0) + Pab(0)Paa(0)]Qac(tW)Qca(tW) = 6561' Q P I Q Figure 6.5: QPQ-tripple bubble diagram PR6 54[Q a f c(0)Q c d(0) + Qab(0)Qad(0)}Pac(tw)Pca(tw) = 656V M 5 Chapter 6. Conclusion 72 Figure 6.6: PPQ-tripple bubble diagram = 36[Pab(0)Qcd(Q) + Pab{0)Qad(0)]Pac(tw)PCa{tw) = 2187 PR* / i 5 P O O Figure 6.7: PQQ-tripple bubble diagram 90[P o t(0)Q c d(0) + Pab(0)Qad(0)]Qac(tw)Qca(tw) = 4 3 7 4 0 ^ f I1 p p > Figure 6.8: PPP-tripple bubble diagram l2Pab(Q)Pcd{0)Pac{tw)Pca{t 10J 729 PR6 ~4~~pT Chapter 6. Conclusion 73 Figure 6.9: QQQ-tripple bubble diagram 150Q a f e (0)Q c d (0)Q a c ( i 1 0 )Q c o ( i 1 0 ) = 7 2 9 0 0 ^ J t1 6.1.2 Theta-Bubble Diagram Results should be multiplied by R. Figure 6.10: Simly scalar diagram - 9 / J 2 Pab(tw)Pbc(tlo)Pac(t2l)Pcd(t2o)Pda(t2o) — ~ 10935 PR6 ~32 ]F Figure 6.11: P-theta bubble diagram -12/ i 2 P a b (0)P c a ( i 1 o)P a c ( t 2 o)P c d ( i 2 i )P d a ( t 2 i ) = - 7 2 9 ^ Chapter 6. Conclusion 74 Figure 6.13: PP-theta bubble diagram = -12Pab(0)Pca(tw)Pac(t20)[Fcd(t21)Gda(t21) + F «-> G] = 0 Figure 6.14: PQ-theta bubble diagram = -S6Pab(0)Qca(t10)Qac(t20)[Fcd(t2i)Fda(t21) + F^G] = Chapter 6. Conclusion 75 Figure 6.15: QP-theta bubble diagram - -36Qab(0)Pca(tw)Pac(t2o)[Fcd(t21)Gda(t21) + F <- G] = 0 Figure 6.16: QQ-theta bubble diagram -60Qab(0)Qca(tw)Qac(t20)[Gcd(t21)Gda(t21) + F <-» G] = -145800 (3R6 6.1.3 Circle-T Diagram The results do not need to be rescaled. Figure 6.17: PP-circle-T diagram Pa 6 ( i 2 0)Pcd ( i 3 l ) [ ^ ( i l 0 ) / ; i 6 c ( i 2 l ) i ?ac ( ^ 2)G o a ( t 3 0 ) + F <- G] = 0 Chapter 6. Conclusion 76 Figure 6.18: PQ-circle-T diagram ~ P a 6 ( i 2 o ) Q c d ( i 3 i ) [ ^ ( t i o ) G 6 c ( i 2 1 ) G a c ( t 3 2 ) G d a ( i 3 o ) + F ~ G] = 2 9 1 6 ^ -Figure 6.19: QQ-circle-T diagram = - 3 Qa6 ( i2o )Qc d ( t3 i ) [G 6 d ( i io )F b c ( i2 i )G a c ( i 3 2 )G d a ( i3o) + F <-> G] = 7776 /3R6 Figure 6.20: Fermion-scalar circle-T diagram -2/iP a 6(t 3o)i 'ca(t3l)nc(t32)[FM(t2o)G ( f c(t 2 1)G d o(t 1o) - F <- G] = 0 Chapter 6. Conclusion 77 Figure 6.21: Scalar-scalar circle-T diagram 1 6561 (3R6 — 2"/^  Pab(tlo)Pbc(ho)Pca(t3o)Pdb{t2l)Pad{t3l)Pdc{z32) = 6.1.4 Two-Rung Ladder Diagram The results do not need to be rescaled. I = 1 Figure 6.22: PP-two-rung ladder diagram = | Pab(tw)[Pcd(t32)Gbc(h2) + Pbc{h2)Gcd{t32)]Gbd{t2o)Gbd(tl3)Fda(tw) + + (F «-> G) = 0 Figure 6.23: QQ-two-rung ladder diagram Chapter 6. Conclusion 78 9 Qab(tw) \Qcd{h2)Gbc{t32) + Qbc(t32)Gcd(t32)]Fbd(t20)Fbd(ti3)Fda(t10) + (F G) = -34992 0R6 Figure 6.24: Q-two-rung ladder diagram = 6Qab(tl0)Qba{t32)Fbc{t20)Fca(t20){Fad(t31)Fdb(t31) + Gdb(t31)Gad(t31)] + +(F <-• G) = 7 3 8 7 2 ^ Figure 6.25: P-two-rung ladder diagram = 3 Pab(tio)Pba(t32)Fbc(t2o)Gca(t20)[Fad(t3i)Gdb(t3i) + Fdb(t3i)Gad(t31)}+ +(F G) = 0 Chapter 6. Conclusion 79 Figure 6.26: PQ-two-rung ladder diagram = -9Pa6(iio)arf(i32)[i?6d(^2o)Gdo(t1o)Gdb(i31)Gbc(i32) + (F «- G)] -9P„ 6 ( t 1 o )g c o ( t 3 2 ) [ i ; da(t2o )G M ( t 1 o )G a d ( t 3 1 )G d c ( t32) + (F <-> G)] = 5 8 3 2 ^ Figure 6.27: Fermion-scalar two-rung ladder diagram = 6/i 2 P o b (t 2 o)P 6 a(t3 1 )P a c (t 3 2 )P c b (t 32)[F6 d (t io)G d o (i 1 o) + (P <- G)] = 0 Figure 6.28: Scalar-scalar two-rung ladder diagram Chapter 6. Conclusion 80 One can check that the sum of the above factors is zero, as guaranteed by SUSY. The vertices for these diagrams can be deduced from the interaction terms in the action. The propagators depend on temperature and are discussed in the next section. The technique that we use is to work in time, rather than momentum representation of the propagators. The integration over times is elementary and most of the work involves extracting the moments in r?0 = ell3A". Lib-eral use of computer algebra systems was used to unravel the expansion of diagrams in this object. 6.2 Thermal Green functions The free field correlation function of the scalar field is where the Green function is M - ) L = M _ ^ + W 6 ) a l ° ) - T / / 3 41nr l _ I r 2 L .21 ^ + rjr „2T ~T/P rjr* I R 2 (6.7) 9{r) nr where r] — e i/3Aal, Va/ilb and r = e P^l12. This Green function is defined on the interval r//3 € (-1/2,1/2), with Si( l /2) = P i ( - 1 / 2 ) a n d i* m u s t be extended periodically to all real values r//3+integers. An alternative Chapter 6. Conclusion 81 expression which is sometimes useful is 0 i I [ r2]M/P + j y n v-nrr2y/P + rin •2-ir/P 41nr „ 2 1 - r / ^ ' n = l The first term is the zero temperature Euclidean Green function and the last two terms are the homogeneous solutions of the Euclidean wave equa-tion which must be added to the first term in order to satisfy the periodic boundary conditions. Note that both green functions have a coefficient rfRIP. This factor cancels identically in all vacuum diagrams. It has been separated explicitly since some loop integrations are easier once it is canceled. Similarly, 'x:b(r)xlM)Q = ^5ad5bc[g2(r)}ab b2(r)] ab -D2ab + (M/3) : r|0) (6.8) prj -T/P 81nr t [ r 4 l r / / 3 + - rjr rj'T 4 1 -T/P r + l _ r 4 + i _ M r J + 1 r\r • [r4r IP - T prf r/P 41T/P 81nr I \ The Fermion Green function is 1 „ 4 i -T/P' n=l [9f(r)}ab = (7 '-Dab + ifi/A) |0) = - r r r / / 3 1 + r3/rj —r V -T) (6.9) It is antiperiodic #/(—1/2) = —gf (1/2). Note that the expressions in (6.7),(6.8) are unchanged by the replacement r 1/r. To conclude we summarize our results: • We obtain parts of the free energy to three loop order • We check our three-loop computation by taking the zero temperature Chapter 6. Conclusion 82 limit of it and finding that the free energy vanishes as the temperature is taken to zero. This is expected as a result of the supersymmetry of the model which is restored at zero temperature. • We also find that the ^-independent part of the free energy vanishes to this order. Note that this is not the same as the zero temperature limit. • We find the full shift in Tc to order A 2 . • The form of the effective action that we find confirms the first order nature of the phase transition at weak coupling. 83 A p p e n d i x A S u p e r s y m m e t r y o f t h e M a s s i v e M a t r i x M o d e l A . l Symmetry Algebra In this subsection we make some remarks about the super symmetry algebra. We will consider the l i d wave [10] but similar remarks apply for the 10 dimensional waves [54]. We define a generator e = —p_ and a generator h — —p+. The generator e commutes with all the other operators. Some of the (anti)commutation relations are [a;,aj] = e6ij, i,j = 1,...,9 [K 4} = | a l . [K ai] = i = 1, 2,3 [h,al}=Qah [h,ai] = --ai, i , j = 4, ...,9 tl ^ t [u 1 I1 1i\=Qal, [h,ai]  --c {Kp, *>f 7*} = ePJl, 0 = {b, b'} = {b\ b*} = [b} a] = [b, at] M i = 4 ^ , [hXp\ = --AK? {Q,Q} = {S,S} = 0, S[.=Q^ where the undoted greek indices indicate spinor indices of SU(2) an the doted ones denote spinor indices of SO(6) (the ones downstairs are in the 4 of SU(4 ) and the upstairs one are in the 4 of S U ( 4 ) ) and 7* and P are Appendix A. Supersymmetry of the Massive Matrix Model 84 three and six dimensional gamma matrices respectively1 In addition we have (anti)commutators of the S and Q with 6s or as which give as or 6s. We will not write those since we will give them implicitly below when we discuss the superparticle. The main observation we want to make is that the structure of the representations of this algebra is very simple. Since e commutes with everything we can diagonalize it. Then the commutation relations of the as and 6s (and their adjoints) become bosonic and fermionic harmonic oscilla-tors. Then the rest of the symmetries acts linearly on these oscillators. We can identify h with the light cone Hamiltonian, so we see that the o) and 6^  oscillators describe the center of mass motion of the state. In fact we could subtract from Q,S,h, Mi3 an expression bilinear in these oscillators (which is a realization of Q, S, h, Mio- in terms of oscillators) so than then Q, S, h, Mi3-act on the relative state. Note that Q, S are supersymmetries that do not commute with the Hamiltonian. In the matrix model, the oscillators a and 6 are going to result from quantizing the U(l) degree of freedom and the shift of Q, S, h, Mi3 that we mentioned above amounts to separating the U(l) degree of freedom to leave the SU(N) degrees of freedom. A .2 Plane wave l imi t of the lOd IIB AdS5 x S5 action Here we prove that the GS action of Metsaev [47] can be obtained as a limit of the AdS5 x S5 action of [53]. There is a general formalism one can use in both cases. Indeed, as shown in [49], for D branes propagating in supercoset manifolds, one can write down an action in terms of supervielbeins (vielbeins of the target superspace realized as a coset manifold). The kinetic term is always of the type where Lf are the bosonic components of the supervielbein 1-forms pulled back on the worldsheet. In general there can be also a W Z term, defined as (A.2) ^ h e relation of the generators in (B.l) and those in [10] is schematically as follows ai e J + ie*' and similarly for at, the 6s and b^s are linear combinations of Q+ in [10] and similarly S and Q are linear combinations of Q_ in [10]. Appendix A. Supersymmetry of the Massive Matrix Model 85 the integral of a form on a n + 1 dimensional manifold with M as boundary. The supervielbeins are found from the general procedure in [49] as LA = LA + 29y^S-^^)?(Dey (A.3) and where the matrix M is defined by (M2)p = -er>f°Aosf!A (A.4) the coefficients fAp are the structure constants of the fermi-fermi part of the superalgebra {Fa,Fp} = jA^BA- If 0 is constant, one gets the W Z parametrization of superspace. Here (D8)a = d0a + (LABA0)a (A.5) is the Kil l ing spinor operator acting on the 0's (the Kil l ing spinor equation would be De(x) = 0). The GS string action in a general supergravity background was given in [55] and is S = - \ [ d2o^g~gijLdLj +i [ sIJLd A L r j d A LJ (A.6) 2 JdM3 JM3 where La are the bosonic supervielbeins and L1 the fermionic ones. In the case of AdS5 x S5 the simple form of the action based on the above approach has been found in [53],[56], S=~\J d2a(V=ggljLdLd + 4ie« J' dssIJLiQJTdLi) (A.7) where j l _ fsinhjsM) n Q ^ Ls ~ {—M—VU> , (A R) L% = e%dX™ - 4 i 6 7 r a ( 5 ^ M / 2 ) D 6 ) 7 V ' ] The fermionic light-cone gauge was fixed in [57], and is the same as in flat space, namely F+0 = 0. With this fermionic light-cone gauge, one gets that the matrix M2 = 0, and so the only nontrivial information is encoded in DQ. Appendix A. Supersymmetry of the Massive Matrix Model 86 But that has the general form DQ1 = {5IJ(d + \^^) + i e " F m . . . w F ' 1 - ' V J ) e J (A.9) 4 4o and consequently it has the correct limit from the AdS$ x 55 case to the pp wave case. The last step is the fixing of the bosonic light-cone gauge, which for the AdS5 x S5 case was done in [58]. Metsaev [47], using the gauge y/ggab = Vab X+(T,O) = T (A.10) finds then the action 1 u2 L = --daxIdaxI - y x | - iip^-padaip + ifitpy-Ilip ( A . l l ) A . 3 M a t r i x t h e o r y a c t i o n The action for a single DO-brane can be obtained as the superparticle action moving in (1.21) in the Green-Schwarz formulation. Indeed, for a Z)0-brane in fiat space, the light-cone gauge superparticle action gives the free massless bosons X1 and fermions 9 (spinors of SO(9)), which is the free DO action. As we mentioned in the case of the GS string, the super-brane action has a kinetic and a W Z term. But in the case of the superparticle, there is no 2d form one can write down (except for the target space AdS2 x S2 where one has the target space invariants eab). So the superparticle action has only the kinetic term. The supervielbeins for the l i d supersymmetric pp-wave can be obtained as a limit from the AdS7 x 5 4 supervielbeins, just as above for the lOd wave as a limit of the AdS$ x 55. Indeed, from the above formalism, the supervielbeins can be written in a universal form depending only on the structure constants f£p of the superalgebra, and in terms of the Kil l ing spinor operator. But we know that the wave space symmetry algebras are a contraction of the AdS x S ones, and that the Kill ing spinor operators are also a similar limit (they only depend on F). The supervielbeins for the AdS7 x S4 case have been given in [59]. If one takes the general formulas there and substitutes F + 1 2 3 = p and the fact that \ Appendix A. Supersymmetry of the Massive Matrix Model 87 LO~1 are the only nonzero components of u^" one obtains D6 = d6 + ^ ( e T , + 1 2 3 - 8e^T12^)8 - ^w~T_j0 (A.12) and also (M 2)% = % [(r>+123 - s ^ r 1 2 3 ] ) ^ ^ ( A 1 3 ) - i | [ ( r , ^ ) « ( 0 r « + 1 2 3 ) / 3 + 24(r [_ 10)«(^r 2 3]) / 3] 1 ' ' The superparticle action dte~lLALA (A.14) will have a symmetry similar to the one of the free superparticle with L f = i M — iO AY^9. This fc symmetry needs to be gauge fixed by choosing the fermionic light-cone gauge. The procedure is exactly similar to the su-perstring in AdS5 x S5 and its limit the 10 d wave (see [47]). As there, one can choose the gauge r+e = 0 er + = o (A.15) which we can see from the expression of the AdS7 x 5 4 M 2 above that makes M 2 = 0, and so LA = dx^eA + ^0F AD9 (A.16) and where D 9 = dQ+^uTf+™-Re^T12^)6--Lo- iT-ie = a ' 0 + ^ e + r - + r 1 2 3 0 - ^ e + r 1 2 3 £ 12K ' 2 12 6 (A.17) where we have used the gauge condition to kill the terms with T and T+i. One can then see that we get (in spacetime light cone parametrization) L+ = e+ = dx+ U = ei = dx1 (A.18) and 9 L~=e- + \eT-D6; e~ = dx~ - \ { ^ ) 2 £ ( x * ) 2 ^ - \ { ^ f E ^ ) 2 ^ i=l,2,3 i=4 (A.19) Appendix A. Supersymmetry of the Massive Matrix Model 88 to be used in the action S = J dt{2LtL~ + L\Ut) (A.20) Then fixing the bosonic light cone gauge e = 1, x~{t) = t one gets the action r 9 s = / dt^x'f - ( £ ) 2 ( x * ) 2 - & £ ( x i ) 2 + dT~e - 7#r-r123#] ^ i=l,2,3 i=4 (A .21) We now rewrite the l i d fermions and gamma matrices in terms of 9d ones. We choose the representation P* = Y ® CT3 r° = 1 ® ia2 (A.22) r 1 1 = I <g> o i And we also choose a real (Majorana) representation for the spinors and gamma matrices: C — T0, 9 = 9TC = 9^T0. Then we have r - = >/2(o) r+ = V2(o) r 0 r - = - v / 2 ( i ) r + - = i ® a 3 (A .23) Then, take '-(JO^-U)-0**-0 (a-24) So, take 0 and so the fermion terms in the action sum up to V 2 ( V ^ + ^ V * V ) (A.25) We can now absorb the y/2 in front of this expression in the definition of the fermions. We turn to proving SUSY of this action, and generalizing it to the non-abelian case. We will leave the coefficient of the fermion mass term free, since we will find another solution for it in the abelian case. Appendix A. Supersymmetry of the Massive Matrix Model 89 Let us then start with the lagrangian 9 9 i = l 1=1,2,3 i=4 (A.26) and look for a SUSY transformation of the type 5Xi = <bTfe(t) = x^eit) + nX^M^it) (A.27) e(t) = e^Mte0 Then the terms of order 1 in the susy transformation cancel , the terms of order u, give the equation M[ = ± a / 4 7 l 2 3 - M (A.28) where the two values are for i = 1, 2, 3 and i = 4 , 9 respectively, and the terms of order a? give (A.29) (A.30) M[M - 1/9 - a / 4 7 l 2 3 M / = 0 i = 1, 2, 3 M / M - 1/36 + a / 4 7 l 2 3 M " ; = 0 i = 4 , . . , 9 We then obtain M = 6 7 l 2 3 Ml = ( ± a / 4 - 6 ) 7 l 2 3 and a = 1 or 1/3 (2 solutions) and b — -1/12 or —1/4 (the 2 corresponding solutions). There are also solutions where we change the sign of both a and b, but these correspond to the symmetry y, —> —u,. The extension to the nonabelian theory is obvious; besides the usual commutator terms which are present in the lagrangian and SUSY rules in flat space, we have an extra coupling of order \i. Indeed, Myers [15] has found a term FtijkTr(XlX^Xk) in the action for N DO-branes in constant R R field. In our case, after the limit to the plane wave geometry (infinite boost), the coupling is F+ijkTr(XlX^Xk) ~ yTr(XlXjXk)eijk (A.31) So the lagrangian is Appendix A. Supersymmetry of the Massive Matrix Model 90 L = El i (#) a - W 3 ) 2 El=1,2,3(^)2 - W 6 ) 2 EL(*T + *T*-- a ( A i / 4 ) * T 7 i 2 3 * + d/xff EL *=i Tr(XiXiXk)eijk+ +2g2Tr([Xi, X^}2) + 2 i f f T r ( * T 7 i [ * , X']) (A.32) and the SUSY rules are SXi = $>Tfe(t) 5$ = (x*f + fiX'fi+a/i - 6 ) 7 l 2 3 + ig[X\ ^ ] 7 i j ) e(t) (A.33) e(t) = e»Mte0 The terms of order g° in the SUSY transformation of L work the same way as for one .DO-brane, since they are bilinear in fields. The terms of order g cancel (they would fix the coefficient of the ^tpX term in the action). The terms of order pg are proportional to Tr(^TYj 1i23^(t)[Xl, Xj]) and split into i,j both = 4,..., 9, one of i,j — 1,2,3 and the other = 4,..., 9 which both give the equation 3o + a/4 = 0 (A.34) and the case when both i,j are 1,2,3 which gives d = 2(6 - a / 4 ) (A.35) So now a and b are restricted to just a = 1, b = —1/12. This solution is the one we found from the general formalism. The terms of order g2 cancel (they would fix the coefficient of the [X, X}2 term in the action). The action has the almost the same nonlinearly realized SUSY as in flat space. In flat space, the nonlinear SUSY is 5$ = e (constant), and the X's constant. In our case, we have 8^ = €(t) = e^ 7 l 2 3 t e 0 (A.36) 91 A p p e n d i x B F i x i n g t h e S t a t i c D i a g o n a l G a u g e The thermodynamic partition function is defined by the Euclidean path in-tegral Z = J[dA]...exp(- dr^Tv {(DX)2 + .. (B.l) where T is the temperature (we use units where Boltzmann's constant = 1) the variables have periodic boundary conditions, A ( r ) = A ( r + l / r ) , X(r) = X{r + l/T) , . . . This model has the gauge invariance, X^U(t)X(t)U\t) , A(t) - U(t) (A(t) - ij^j U\t) where U(T) = ZU(T + l / T ) is a periodic function of time and z is constant element of the center of the group. It is present here since the gauge transform is entirely in the adjoint. In a sense, the true gauge group is the factor group U(N)/ZN. There is a quantity, the Polyakov loop operator P = T r ( P e * ' o / T * ^ M ) which transforms non-trivially, P^zP This is interpreted as a global symmetry of the theory. Its realization is re-lated to the gauge interaction in the theory and can be used to analyze con-finement. There are two phases. When the symmetry is good, the Polyakov loop operator averages to zero: (P) = 0. When the symmetry is sponta-neously broken, the Polyakov loop operator can have non-vanishing expecta-Appendix B. Fixing the Static Diagonal Gauge 92 tion value, (P) ^ 0. Of course in this very low dimensional system, a discrete symmetry can only be broken in the large N limit. The expectation value of P is interpreted as the energy of this system with one additional external fundamental representation source. Indeed, the path integral quantization of the system with such a source would replace the partition function (B.l) by (P) and with n such sources the partition function would be replaces by (Pn). Thus, we interpret F[T] = - T l n « P » as the free energy that it would take to introduce a fundamental representa-tion source into the system. When (P) = 0 this free energy is infinite and we say that the system is confining, whereas when (P) ^ 0 this symmetry is finite and we say that the system is deconfined. Let us analyze the partition function (B.l) in more detail. We can fix a static gauge by enforcing the gauge condition Then, taking into account the Faddeev-Popov ghost determinant, the parti-tion function becomes [<LA(T)] ...5 (dA/dr) det Tr D' exp -( 1 /T ^ - j f ovlTr ({DX? + ...) The measure in the path integral can be defined using the Fourier transform, oo A(T) = Vr A n e 2 * i n T r The integration measure can be defined in this basis, \dA(r)\ = \{dAn Appendix B. Fixing the Static Diagonal Gauge 93 The Faddeev-Popov determinant is a determinant over the non-zero modes, 5{dA(T)/dT) det Y[Y[8(2irinTAn)- {2irinT)(2mnT + iA" a,b n^O b ) = Hl{S(An)-[(2mnT + zAaii)_ a,b n^O where A^ is the adjoint. Integrating the modes of An with n ^ 0 and decomposing the measure into an integral over the eigenvalues of A and the unitary matrices necessary to diagonalize yields the measure [dA] = fj dAa l[(Aa - Ab)2[dn] a = l b^a where Q, is the unitary matrix which diagonalizes A. Also, when A is diagonal A^ = Aa — Ab. Then, the total measure becomes JV JV oo / \ n dAa n n (27rinT+^ - a^ n27r*nr iV W O J - n y l l l l (2mnT + z(Aa - Ab)) a = l n=—oo where we have used the zeta-function regularization to evaluate the infinite product oo .. rj 2mnT = H(2TTTIT)2 = (2nT)2^e^'^ = ± n^O n = l Riemann's zeta function is defined by OO -. — n s n = l and has the values C(o) \ , C(0) = -l-ln(2rr) Appendix B. Fixing the Static Diagonal Gauge 94 To take the remaining product, we use the formula Y[ {2-KinT + icp) = exp I ln(2ninT + icp) J n=—oo \ n / = e X P ( - / ^ / ^ ^ ( 2 7 r i n r 4-^ )0 = exp ( - / dcp/ ^  £ ^ M t h ( , / 2 T ) _ ^ _ ) where we have taken two derivatives in order to get a convergent series and the contour C encircles all of the singularities of the cothz/2T function located at z = 2-KinT. Doing the contour integral by collapsing the contour onto the singularity at z = —icp gives ir=-oo (2™^ + icp) = exp (-JdcpJ ^ c o t h ( - ^ / 2 T ) ) = exp ( / dcp (± cot(cp/2T) + P)) = exp (lnsin(<£/2T) + a + Pep) = ea+^ sin(<^/2T) where a and P are arbitrary constants parameterizing the subtractions that are needed to renormalize the product. If we arbitrarily set both of them to zero, we get the gauge-fixed measure for integrating over gauge fields as The absolute value sign is there because the Faddeev-Popov determinants should always have absolute value signs. 9 5 Appendix C Deriving CJSC (^) in pp-wave Matrix Model In the pp-wave matrix model after the guage fixing static diagonal gauge and expanding about the classical vacuum corresponding to the single five-brane, the effective actin is oo 5 ef f^ ] = E A " l T r e ^ l ( a i ) n=l and the partition function is (for n = 1) Z = ( dAe-SeS[A] (C.2) In the static diagonal gauge, it is convenient to define uab = e^Saj, (C.3) and du = d(PAa)[dv] fl \ei/3A° - eipA"\2 (C.4) a<b where v is a matrix which diagonalizes u. The symmetries of this model are: S[u] = S[vuv^} . S[Zu] = S[u}; \Z\ = 1 { ^ 0 > Note that in Gross-Witten model we only have the first symmetry. The partition function in terms of u is Z= I dueXTruTru] (C.6) Appendix C. Deriving LOSC{z) m pp-wave Matrix Model 96 or N Z= f f[d(pAa)]J\e^ - e x ^ l P A ^ P A b (C.7) J 1 L a—I a<b using the following | e ^ „ _ e i ^ „ | 2 = 4 s i n 2 ( ^ - ^ ) (C.8) we get Z= f n d ^ e ^ ^ H - ^ 2 ^ ) ! ^ ^ ^ - ' ^ 6 (C.9) ^ a = l Since both terms inside the summation are of order N2, as iVgets large we can use the saddle point integration, ^ c o t ( ^ a ~ ^ b ) + iA ^ - ( ^ e - ^ ^ - e - ^ ^ e ^ ) ^ = 0 a#6 \ 6 b / (CIO) To simplify this equation we rewrite the cot(^ ' 4 ° 2 / 3 / 4 ' ' ) in terms of the expo-nential functions. DAq-JiA, f3A . - M T -e-^"(E6ei/3^)) = 0 or a f^a \ 6 b / and if we define: C E E y v The equation of motion is finally given by: e i / 3 A „ _|_ gi /3 / l ^ + i A ( e ^ - c * - e - ^ - c ) = 0 (0.14) a^b Appendix C. Deriving tosc(z) in pp-wave Matrix Model 97 Now we consider to2(z) in which to(z) has been defined in (4.2) 1 to [z = N2 \ Let us define z _ &ipAa ei(3Aa _|_ eif3Ab pipAa _ pif3Ab pipAb pif3Aa pifiAb eif3Aa _|_ ei(3Ab Z + e ipAa z — e iPAa pifiAa _ eif3Ab Using equation of motion one gets A = -NX{eil3Aac* - e-i/3A» Replacing (C.17) back into (C.15) one gets to-, _ J _ f I V - , / z - e i 0 A a _ i p A a + e i / 3 A b \ / z + eH3Aq \ Z ) — \ [ l ^ a ^ b \y eif>Aa - e i 0 A b e i p A a _ e i 0 A b J y z - e i f 3 A a + {a^ b}} + + [a b] (C.15) (C.16) (C17) (C.18) The combination of the first term in summation and its counterpart in [a <-> b] gives a factor of 2~2a^b which is -^ 2- Therefore what remains is 2A eipA„c* _ e-if3Aac) x ) z + e iPAa z — e i(3Aa (C.19) Now let us add and subtract the following to (C.19) 2 A V - , * <=. fz + e^A"' — > (ZC ) X - r - r -N ^ y z! Xz-e1^" a x Using definition (4.2) we get (C.20) to \z) = \ - 2X(zc* - f )to(z) + f E a izc* - f - e i M " c * + e " i / 3 A " c ) x e i 0 A a (C.21) Appendix C. Deriving wsc{z) in pp-wave Matrix Model 98 or LO2 W =1 - 2 X ^ - + £ £(<* + )(* - ( f r $ £ ) a ^ ' (C.22) Canceling the (z — e l / 3 j4") term from numerator and denumerator in the sum-mation, we have UJ2(Z) = 1 - 2\{zc* - -z)UJ{Z) + ^ {ZC* + ~Z + EI/3A"C* + e" i / 3 / 1"c) (C.23) a and finally u>2{z) = l - 2 A ( z c * - - Z ) L O { Z ) + 2 \ { Z C * + ^ ) + ^ (e i / 3 A oc* + e'^c) (C.24) a Using the definition (C.13), the last term simply reduces to -i2\cc* By rewriting (C.24) we get a quadratic equation for UJ(Z) u)2{z) + 2\(zc* - -)LO(Z) - 2\(zc* + -)- ±Xcc* - 1 = 0 (C.25) z z Solutions of the above equation are u(z) = -X(zc* - f )± ±y/\2(zc* + \)2 + 2X(zc* + §) 4- 4Acc* - 4A2cc* + 1 (C.26) Now for one phase we have p(0) = ~ and therefore c = 0. 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Sugawara, "Thermal Amplitudes in D L C Q Superstrings on pp-waves", Nucl. Phys. B 650, 75 (2003), hep-th/0209145. [57] R. C. Brower, D. A . Lowe and C. I. Tan, "Hagedorn Transition for Strings on pp-waves and Tori with Chemical Potentials", Nucl. Phys. B 652, 127 (2003), hep-th/0211201. [58] G. Grignani, M . Orselli, G. W. Semenoff and D. Trancanelli, "The Super-string Hagedorn Temperature in a pp-wave Background", JHEP 0306, 006 (2003), hep-th/0301186]. [59] S. J. Hyun, J. D. Park and S. H. Y i , "Thermodynamic Behavior of IIA String Theory on a pp-wave", JHEP 0311, 006 (2003), hep-th/0304239. 105 Index free energy, 14 adjoint representation, 8, 18, 30, 31, 36, 40 annihilation operator, 12 BFSS, 1-3, 5 BFSS matrix model, 1, 3, 33 BFSS model, 1, 6, 8 B M N model, 6 Boltzmann's constant, 91 bosonic light cone gauge, 86, 88 canonical quantization, 11 center symmetry, 30, 31 classical five-brane, 35 classical ground state, 34 classical level, 9 classical membrane, 1, 35 classical minima, 35 classical potential, 8, 35 classical quark source, 31 classical solution, 37 classical supersymmetry vacua, 6 classical vacua, 35 classical vacuum, 36, 38, 95 Clifford algebra, 7 condensation, 40 confining phase, 31, 32, 40, 43, 68 confining solution, 43 connected vacuum diagram, 53 coupling constant, 2, 3, 31, 34-36 creation operator, 12, 13, 19, 23 critical point, 45, 47, 48 critical temperature, 28, 43, 45, 52, 53, 68 critical value, 45, 47, 52 cyclicity, 13 D brane, 84 DO-brane, 1, 2, 8, 86, 89, 90 deconfining phase transition, 30, 32 dilatation operator, 9 dimensional reduction, 2 Dirac matrices, 7 discrete light cone quantization (DLCQ), 3, 5, 7, 8 discrete symmetry, 92 effective action, 8, 26, 27, 29, 39-41, 51-53, 68, 69, 82 eigenvalue attraction, 27, 30 eigenvalue density, 27, 28 eigenvalue distribution, 29, 44, 99 eigenvalue repulsion, 27 eigenvalue support, 48 Euclidean action, 68 Euclidean Green function, 54, 81 Euclidean path integral, 40, 91 Euclidean path integral representa-tion, 32 Euclidean space, 32 Euclidean time, 25 Euclidean time circle, 32 Euclidean time Lagrangian, 33 Euclidean wave equation, 81 Index 106 Faddeev-Popov determinant, 26, 34, 38, 93, 94 Faddeev-Popov ghost determinant, 92 fermion propagator, 55-59, 61 fermionic light cone gauge, 85, 87 finite temperature gauge theory, 30 five-brane, 35-38, 95 five-brane solution, 37 five-brane state, 37 flat directions, 3, 6, 8 flat space, 1, 85, 86, 89, 90 fluctuation spectrum, 35 Fock space, 23 free energy, i , 25, 31, 32, 34, 35, 43, 48, 49, 53, 67, 68, 70, 81, 92 fuzzy sphere, 35 gauge field, 8, 10, 32, 34, 37, 42, 59 gauge fixing, 33, 34, 53 gauge generator, 3 gauge group, 8, 18, 30, 31, 36, 37, 40, 91 gauge interaction, 91 gauge invariance, 91 gauge invariance condition, 23 gauge invariant operator, 10 gauge invariant state, 10, 13 gauge symmetry, 2, 30, 36, 37 gauge theory, 30-32 gauge transformation, 3, 10, 12, 30, 40 gauge transformation , 91 global symmetry, 30, 91 Grassmannian number, 19 Gross-Witten model, 95 Gross-Witten phase transition, 29, 44 GS string, 85, 86 Hagedorn behavior, 10, 23, 24 Hagedorn density, 10, 11, 20 Hagedorn spectrum, 14 Hagedorn temperature, 10, 11, 14, 17-21, 23, 28, 29, 68 Hagedorn transition, 23 Higgs phase, 36 holographic principle, 2 infinite momentum frame, 3 infinite momentum frame Lagrangi; 2 infinite momentum frame limit, 3 instability, 28 integrability, 9 Kil l ing spinor equation, 85 large N limit, 2, 3, 9, 13, 27, 31, 32, 38, 39, 42, 52, 92 light cone frame, 1 light cone Hamiltonian, 84 light cone parametrization, 87 light cone quantization, 8 light-cone gauge, 86 Lorentz invariant, 2 M-theory, 1-4, 7-9, 31 matrix oscillator, 10, 11 membrane, 1, 6, 35-37 membrane state, 36, 37 membrane vacua, 36 multi-membrane state, 6 non-zero mode, 93 • Index 107 open string, 2 partition function, 15, 17, 18, 22, 25, 31, 32, 37-39, 69, 91, 92, 95 phase transition, i , 28, 30, 37, 38, 43, 46, 48, 49, 51-53, 68, 82 planar limit, 9 Planck length, 1, 2, 9 plane wave, i , 6 plane wave background, 37 plane wave geometry, 89 plane wave limit, 84 plane wave matrix model, 6, 8, 9, 31, 32, 68 plane wave metric, 4, 6 plane wave spacetime, 6 Poisson bracket, 11 Polya style of counting states, 14 Polya theorem, 16 Polyakov loop, 30 Polyakov loop operator, 30 Polyakov loop, 32, 40, 48, 49 Polyakov loop operator, 32, 40, 41, 43, 91 positivity constraint, 27 pp-wave, 4, 86 pp-wave background, 3, 4, 7, 8, 25 pp-wave geometry, 4 pp-wave matrix model, 6, 95 pp-wave spacetime, 8 principal value, 27, 42 propagator, 54-58, 60, 63, 70, 80 reality condition, 7 residual gauge symmetry, 37 R R field, 89 saddle point, 27, 48 saddle point approximation, 27 saddle point equation, 29 saddle point integration, 39, 96 scalar propagator, 59 Schur's lemma, 37 semi-circle distribution, 29, 44, 45, 48, 50 static diagonal gauge, 40, 91, 95 super Yang Mills field, 2 super-brane action, 86 superalgebra, 5, 85, 86 supergalilean group, 2 supergraviton, 1 supergravity, 1, 2, 38, 68, 85 superparticle, 84, 86, 87 supersymmetry, 6, 33, 34, 39, 67, 82, 83 supersymmetry algebra, 6, 7 SUSY, 3, 80, 88-90 SUSY transformation law, 89 SUSY algebra, 3, 6 SUSY transformation law, 3, 89, 90 symmetry algebra, 83, 86 t' Hooft coupling, 36-38 t' Hooft limit, 31, 34, 37, 38 time-independent gauge transforma-tion, 11 U(l ) symmetry, 40 U(N) symmetry, 2, 8, 36 unit circle, 29, 42, 44-46 Vandermonde determinant, 26, 27, 34 weak coupling, 6, 38, 82 worldsheet, 84 Index 108 Yang Mills field, 2 Yang Mills quantum mechanics, 2 zero classical energy, 34 zero mode, 21, 23, 34, 50 zero-point energy, 23 

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