oo, so that null and timelike geodesies are never closed in finite affine parameter. 2.5 Holographic Screens For simplicity we keep P t 2 = 0. As explained earlier (Section 2.2.5) the screen is found as follows. Consider an observer fixed at the origin of our coordinate system. At some moment he emits lightrays in all spatial directions; hence he lives along the caustic of the resulting congruence. The equations for these geodesies were found in the previous section. The expansion of this congruence is calculated, and the place where it vanishes is found. This defines the location of the surface as a function of time. The screen is then the union of the surfaces along time. : Of particular interest here is the region bounded by the screen. We find that this "holographic region" includes the entire spacetime, as a result of the compactness of z. Now that I have found the geodesic equations y^ (2.21) I can easily con-struct the generator of the geodesic; it is given by E + Pz + e + P? Pz + E = 0. (2.24) (2.25) Chapter 2. Compactified Plane Waves 39 So y gives the equation of the curve and k is the tangent vector. Then the geodesic expansion is 6 = V • k = 3„fc" + T»aka (2.26) where the T are the Christoffel symbols (Appendix A). It is convenient to rescale A, A -» A/(P Z + E). So we find 0 = 3/?cot(/?A) (2.27) so that the screen is located at affine parameter = P.28) The screen is parametrized by angles and t0. The angle between the p and z direction is given by 77, but it will be useful to continue to express this as pm. The screen is located at p = pm = ^ t a n | , and along x = Xo and = , * j £ ~ J ? f ° V ( 3 - 3 5 ) \zi2\ \zn\ F12I 3 where h, h are the conformal weights, and Xi2 = x\ — x2. The x basis is related to the m basis used earlier via r = / J^xi~mxJ~™$i(x)- (3-36) Chapter 3. Little String Theory 66 This gives [56] ird2(m + m')5(j - j')B(j) T(j + m)r(j - ra) <**»**>'mw> - \z12\2>>\zl2\2^(2j) T( l - j + m ) r ( l - j - m)' (3.37) The functions B(j) and 7(2.7) are defined in [48, 47]: B{j) = (3-38) and T M = ^ . (3.39, The constant v and the function B(j) both equal 1 in the large k limit we are taking. In calculating the spacetime two point function, one has to divide by the volume of the conformal group. This normally makes the two point function vanish, but in this context it is compensated by the volume of target space [58, 14, 56]. This leaves a finite piece, which was calculated in [56]. In making the correlation function of our vertex operators the addi-tional ingredients make little difference. They modify the conformal weights in the denominator of (3.37), and multiply the correlator by a polarization dependent tensor in the flat spatial directions. The two point function of the vertex operators gij(qi,p) is then given by (9ij{Qi,p)gM,p')) oc (8ikSji + 6u5jk)55(q + q')S(p + p') x x ^ 1 r ( j + m ) r ( j - m ) 2j - 1 7 ( 2 j ) T( l - j + m ) r ( l -j-fh) v ' ' Here ra, ra, and j are regarded as functions of the quantum numbers c/j, p. We now proceed to comment on the structure of this result and its analytic continuation to real frequencies. Chapter 3. Little String Theory 67 3.4.1 Euclidean Space The temperature Green's function can be written as a function of the Mat-subara frequencies p as i r2(| + | y i W T ? + | ) ( 3 4 1 ) * 7(1 + V/ T W + ? ) r 2 ( i - ! v / l + 2fcg2+p2 + £)-This is found using the conditions m = p/2, m = —p/2 (3.30) and the solution • 3 = \ + \ v / l + 2fcg 2+p 2 (3.42) to the mass-shell condition (3.34). A second solution to the mass-shell condition is discarded as it violates the lower bound on j. The upper bound imposes a constraint on a combination of the Euclidean energy .p and the transverse momentum q, y/l + 2kq2 + p2 < k - 2. (3.43) There is an additional condition m = j — n = p/2, for n an integer of definite sign, since those are the allowed values in the discrete representations. Positive n corresponds to a highest-weight representation, while negative n corresponds to a lowest-weight representation. This translates, using the mass-shell condition, to ka2 ^ - + n 2 = j ( 2 n - l ) . (3.44) Since j is positive it is clear that n must be positive as well, so only the highest-weight representations are allowed. So we see that only a discrete set of points in the frequency-momentum space is obtained by the Euclidean string theory calcualtion. Our approach Chapter 3. Little String Theory 68 is to analytically continue the result as needed, and attempt interpretation of the correlator as a function of Lorentzian frequencies8. 3.4.2 Minkowski Space The temperature Green's function Q{p) can be analytically continued to the retarded Green's function GR(LO) using p —> —iio + 5. The analytic continu-ation of Q is not unique since it can, for example, be multiplied by a phase which vanishes at the poles. Typically one requires a fall-off condition for large Euclidean frequencies in order to fix the analytic continuation uniquely. These fall-off conditions require knowledge of the correlators at arbitrar-ily high Euclidean frequencies. As our frequencies are a priori bounded for any finite k, we are unable to discuss those fall-off conditions. Since LST is only defined for finite k, taking k —> oo would not give reliable results. Fur-thermore, because LST is non-local, it is not clear what the fall-off constraint should be. Even without modifying the correlator found by Teschner, which is ex-act in tree level string theory, it is possible that small modifications to the mass-shell condition (3.34) will result in satisfying the asymptotic fall-off conditions. We assume this to be the CcLSGj clS mentioned a real time calcu-lation of the retarded Green function will be useful to confirm or refute this point. We assume therefore that the form of the correlator (3.41) applies to 8 We note that the original derivation of the correlators [48] utilizes analytic continuation as well. It would be interesting to explore alternative Lorentzian continuations, along the lines of [59]. Chapter 3. Little String Theory 69 Lorentzian frequencies as well. So the retarded Green's function is taken to be R ~ T ( l + .y/l + 2kq2 - w 2 ) r 2 ( | - + 2fcg2 - w 2 + f ) ' In a retarded Green's function an instabilty will manifest itself as a pole (or other type of singularity, for more complex instabilities) below the real axis. This will be a process which grows, rather than decays, as a function of time. As we are looking for modes which may destabilize the system upon including arbitrarily small corrections, the interesting structure for our purposes occurs on the real axis. This structure represents modes that are stable in the thermal ensemble, an unusual feature. The retarded Green's function found above has only one set of singu-larities, coming from the factor T(—y/\ + 2kq2 — u2). The singularities are located at u = ±y/l + 2kq2 - a2, (3.46) where a is a non-negative integer. The positive root is discussed below; the negative root corresponds to the existence of anti-matter. The singularities are located along the real and imaginary axes. The points along the real axis are bounded (for generic9 spatial momenta u < y/l + 2kq2. (3.47) We see that a large number of possible instabilities is present for non-zero spatial momenta q2, a number of order kq2. We note that LST contains a 9There are some more possibilities at special kinematic points, but as those will be sensitive to small corrections, we concentrate on generic kinematics. Chapter 3. Little String Theory 70 continuum of modes above a gap, whose role in the theory is unclear. A typical momentum of those modes, just above the gap, is q2 ~ Therefore there are order k possible "nearly destabilizing" modes, and nearly all of them are above the gap. The frequencies of those modes are much smaller than k, so we expect them to be reliable. For zero spatial momentum and no winding there is one possible mode, at oo = l . 1 0 The retarded Green's function also has a set of zeros. These are on the imaginary axis, so in particular they can eliminate a possible divergence only for special values of qi, the spatial momentum. Thus we find evidence that LST could indeed be unstable at the Hagedorn temperature. We see a similar picture to the one suggested in [14], where the instability depends on the spatial volume, or the spatial momentum in our case. This picture is one of being more likely to be unstable at small volume, or large momenta. However the locations of the poles we find could be sensitive to our choice of analytic continuation, as discussed earlier. It would be useful to study the choice of continuation in more detail. It would also be interesting to study the corrections in 1/p to see if they shift the poles below the real axis and confirm that LST really is unstable. An additional potential pole at w = 0 is eliminated by a zero at that frequency. Chapter 4. Inhomogeneous D-brane Decay 71 Chapter 4 Inhomogeneous D-brane Decay 4.1 Introduction As described in the introductory chapter, one of the outstanding questions in string theory is whether open strings and closed strings are independent degrees of freedom. If we start with closed strings, we can build D-branes as solitonic solutions of gravity — so D-branes appear as non-perturbative states built from closed strings. The D-branes are dynamical, and their low-energy excitations are open strings. But how can we see these open strings explicitly in the closed-string sector? Similarly, we can start with the open-string worldsheet. The D-brane living on the boundary of the worldsheet decays, leaving behind only closed strings — how do these closed strings appear on the open-string worldsheet? This question is the counterpart to the one in the previous paragraph. D-branes seem to provide the connection between open and closed strings. D-branes were first understood as sets of endpoints of open strings. Strings ending on a D-brane have Dirichlet boundary conditions in the directions transverse to the brane, so that the strings cannot leave the brane. They have Neumann boundary conditions in the directions parallel to the brane, so that they are free to move along the brane. The special case of Neumann Chapter 4. Inhomogeneous D-brane Decay 72 boundary conditions in all directions is given by a space-filling brane. Open strings only appear in this way, and they are the low-energy excitations of the D-brane. Although one can use the GSO projection to remove tachyonic modes from the superstring, it is not possible to remove all tachyonic modes. Since open strings always end on D-branes, one can say this another way: the GSO projection can be used to stabilize only some types of D-brane. So the question of understanding the open strings becomes one of understanding decaying D-branes. D-brane decay is perhaps most clearly understood in the boundary C F T (BCFT) formalism. The open string is described by a C F T on a two-dimensional worldsheet with a boundary, unlike the closed-string worldsheet, which has no boundary. The Dirichlet boundary conditions can be imposed through a boundary interaction term. The changing state of a D-brane as it decays is then understood as a time-dependent boundary interaction. We can also try a field analysis. The string field is a wave-function ex-panded in the free-string Hilbert space. However there are an infinite number of fields so it is not clear that perturbative string field theory is well-defined. There are contributions from an infinite number of Feynman diagrams even at first order in the string coupling, so one expects this term to be large even at small string coupling. In the language of string field theory, the tachyonic character of the low-est string mode is interpreted as saying that this spacetime field is being expanded about a local maximum in its potential. Based on this analysis, Sen proposed several computations that test vthe reliability of string field Chapter 4. Inhomogeneous D-brane Decay 73 theory, given some basic assumptions of the tachyon potential. Sen conjectured [5] that the height of the local maximum of the tachyon potential is given by the D-brane tension, and that the minimum of the tachyon potential is the closed-string vacuum. He suggested that lower-dimensional D-branes be thought of as solitonic solutions of string theory on the background of a space-filling brane. More recently, he proposed an explicit form for the boundary interaction. This chapter considers a more general interaction than the simple case that Sen studied. Before delving into the details, it is useful to review what is already known about the dynamics of stable D-branes. Then I introduce unstable branes and discuss Sen's conjectures. I review the radiation emitted during the homogeneous decay, and lastly I calculate the radiation emitted during an inhomogeneous decay. 4.2 Dynamical D-branes D-branes are dynamical objects that act as sources for closed strings [60]. This can be seen as follows. The one-loop open-string diagram is equivalent to a closed-string tree-level diagram. The one-loop open-string diagram is a cylinder, with the length corresponding to the length of the open string. Since an open string ends on a D-brane it is clear that the cylinder can also be attached to a D-brane. In the time direction one sees the loop as the cross-section of the cylinder. On the other hand, viewing the length of the cylinder as time, one sees the tree-level propagation of a closed string. But then this represents the emission of a closed-string from the brane. Chapter 4. Inhomogeneous D-brane Decay 74 time space 1 •space open string 1 •time closed string Figure 4.1: Closed-string radiation from D-brane. D-branes appear in this context as boundary states, which are sources for closed strings. A finite tension, or mass per unit volume, can be defined for a D-brane; it is calculated to be inversely proportional to the closed-string coupling. Thus at weak string coupling the D-branes are essentially static objects. The open strings ending on the branes are the quantum excitations of the brane. Since the scalar mode of the open string is tachyonic, D-branes are generically unstable. In superstring theory, the GSO projection eliminates strings that couple to certain RR fields. This stabilizes some D-branes in Type II superstring theory by removing the tachyonic fluctuations. There is another important way to see the existence of stable D-branes. Closed strings couple to the RR field strength, and thus carry off multipole moments from the brane. Polchinski recognized that this shows that stable D-branes in superstring theory must be sources of RR charge. The RR charge must be conserved, and since other objects in string theory do not carry RR charge one would not expect these D-branes to decay. Chapter 4. Inhomogeneous D-brane Decay 75 In fact, stable D-branes are BPS states. In the presence of a supersym-metric Lagrangian, all states are subject to the BPS bound, which restricts the charge of the state to be less than its mass. States that preserve some su-persymmetry saturate this bound and are known as BPS states; their charges and masses are equal. Stable D-branes and their bound states are important examples of BPS states. BPS states are special because their masses are protected by supersymmetry; the spectrum calculated at weak coupling is also present at strong coupling. One way to understand that a stable D-brane is a BPS state is as follows. The D-brane relates to each other the left- and right-moving supersymmetry generators on the string worldsheets, so there are fewer independent genera-tors — an isolated D-brane breaks half the supersymmetries. If the unbroken supersymmetries are not projected out by the GSO projection, then the D-brane is stable and the configuration preserves half the maximum number of supersymmetries. Since the D-brane is massive and carries only a RR charge, the RR charge must be equal to the mass, which as mentioned above goes like the inverse closed-string coupling. If, on the other hand, the unbroken supersymmetries are removed by the GSO projection, then there are no remaining supersymmetries to stabilize the D-brane, and it does decay. The BPS bound is trivially satisfied but not saturated, since unstable branes are massive and neutral. In this case one can also see the decay by noting that open-string tachyonic modes exist. Stable D-branes are an important part of the string spectrum. They are required by many dualities, including T-duality and S-duality. At large string coupling there is no clear sense in which D-branes are the endpoints Chapter 4. Inhomogeneous D-brane Decay 76 of open strings. Of course, open strings themselves are not defined at strong string coupling — they are the weak-coupling quanta. But we still know the D-brane masses and their low-energy excitations, or dynamics. 4.3 Unstable D-branes Now that we know of the existence of stable D-branes, we can ask what there is to learn about unstable D-branes. Unstable D-branes certainly exist in superstring theory, since pairs of oppositely charged branes are not protected by the BPS bound. There are also uncharged branes, which are intrinsically unstable — the bosonic D-branes are also in this category and serve as useful toy models. Depending on the dimensionality of a D-brane in superstring theory, there may be no RR form for it to couple to, since the GSO projection removes either the even- or the odd-form RR potentials. A Dp-brane1 couples to a (p + l)-dimensional RR potential and thus a (p+ 2)-form field strength. Thus in Type IIA theory odd D-branes must be neutral, and their masses cannot be protected by the BPS bound; likewise in Type IIB string theory the even D-branes are neutral. The tachyonic open-string mode living on these D-branes is not projected out by the GSO projection. These D-branes are unstable and do decay. DD pairs2 are always unstable when the distance between the branes is 1 This is a D-brane with p spatial direcions, along which the string satisfies Neumann boundary conditions. The open string is free to move along the D-brane but cannot move transverse to it; the Dirichlet boundary conditions appear in the remaining 9— p directions. 2 A D-brane is simply a D-brane with total charge opposite to that of the other D-brane Chapter 4. Inhomogeneous D-brane Decay 77 small. We can see this immediately, because the pair is massive but neutral. The instability is realized by the open string stretched between the D-branes. The scalar modes that live on just one brane are projected out, but the scalar mode of the string stretched between the two branes is not. The zero-point energy of an open string is the unstretched tachyon mass. When the D-branes are far apart the contribution to the mass of the stretched strings from the tension dominates and the D-branes are stable, although they attract one another. But when they are close together the negative zero-point energy dominates, and the lowest open-string mode is tachyonic. In all cases the unstable D-branes do not satisfy the BPS condition.. This is because there is no RR flux for these branes to couple to, and thus they do not carry any RR charge — but they still have a nonzero mass. One can see that the tachyonic open-string mode coupling to these D-branes is not projected out by the GSO projection. Thus generically one should expect them to decay to the lowest energy state having the same quantum numbers. Once the D-brane has decayed we do not expect to find open-string excitations, so we expect the unstable D-brane to decay to an excited state of the closed-string vacuum. 4.4 Sen's Conjectures Sen conjectured that the result of D-brane decay will be the closed-string vacuum [5, 61, 6, 62, 7]. As a corollary he noted that the potential difference in question. This is an oriented string theory, so opposite charge is realized by opposite orientation in spacetime. Chapter 4; Inhomogeneous D-brane Decay 78 between the open-string and closed-string vacuum energies must then be the D-brane tension. He also proposed that lower-dimensional D-branes should be thought of as solitonic solutions on the background of a space-filling brane. These conjectures are straightforward, but their importance lies in the sug-gestion that there is a field-theoretic formulation of string theory that can test these conjectures. We can also use open-string field theory to see how much energy is ra-diated away. Since the mass of the D-brane is 1/g the available energy is infinite to zeroth order in g. Hence a finite answer would imply that not all energy is being radiated away. Using compact D-branes, Sen showed that the height of the tachyon ef-fective potential is universal [5] — it is independent of the precise form of the boundary interaction describing the decay. The effective potential of T is a function whose minima gives the exact value of (T) [9] (Fig. 4.2). The height of the potential is determined strictly by zero-momentum correlation func-tions. Such correlation functions on a space-filling D-brane can only probe the volume of spacetime, which is certainly independent of the interactions in the theory. It would be useful to see this from the perspective of open-string field theory. The worldsheet action is in terms of quantized oscillators living on the worldsheet. This gives a quantum mechanical single-string theory. From this we can try to construct a string field theory, based on a field \f/ whose modes are strings, * = j d26p [