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Fusion algebras and cohomology of toroidal orbifolds Duman, Ali Nabi 2010

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Fusion Algebras and Cohomology of Toroidal Orbifolds by Ali Nabi Duman  B.Sc., Bilkent University, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics)  The University of British Columbia (Vancouver) April, 2010 c Ali Nabi Duman 2010  Abstract In this thesis we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the group G = (Z/2)3 via the pairing  θg (φ) R(G)  ⊗θh (φ) R(G) →θgh (φ) R(G) where  θ : H 4 (G, Z) → H 3 (G, Z) is the inverse transgression map and φ is a carefully chosen cocycle class. We find the rank of the fusion algebra X(G) = θg (φ) R(G)  g∈G  as well as the relation between its basis elements. We also give  some applications to topological gauge theories. We next show that the twisted fusion algebra of the group (Z/p)3 is isomorphic to the non-twisted fusion algebra of the extraspecial p-group of order p3 and exponent p. The final point of my thesis is to explicitly compute the cohomology groups H ∗ (X/G; Z) where X/G is a toroidal orbifold and G = Z/p for a prime number p. We compute the particular case where X is induced by the ZG-module (IG)n , where IG is the augmentation ideal.  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Abstract  Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 A Twisted Fusion Algebra  . . . . . . . . . . . . . . . . . . . .  6  2.1  Projective representations . . . . . . . . . . . . . . . . . . . .  8  2.2  The twisted fusion product for finite groups . . . . . . . . . .  10  2.3  Calculations  14  2.4  . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.3.1  2-cocycles in G with values in U (1)  . . . . . . . . . .  14  2.3.2  The projective representations . . . . . . . . . . . . .  18  2.3.3  The relations.  26  . . . . . . . . . . . . . . . . . . . . . .  Topological gauge theories  . . . . . . . . . . . . . . . . . . .  29  3 The Fusion Algebra of an Extraspecial p-group . . . . . . .  32  3.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  3.2  Finite group modular data  . . . . . . . . . . . . . . . . . . .  34  3.3  Twisted example G = (Z/p)3 . . . . . . . . . . . . . . . . . .  36  3.4  Modular data for extraspecial p-group . . . . . . . . . . . . .  39  iii  Table of Contents 4 Cohomology of Toroidal Orbifolds  . . . . . . . . . . . . . . .  48  4.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  4.2  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .  49  4.3  The case L = IGm . . . . . . . . . . . . . . . . . . . . . . . .  52  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59  iv  Acknowledgments I am thankful to the most gracious, most merciful God who has given to me numerous bounties. If I want to count these bounties I cannot manage to complete. I sincerely acknowledge my supervisor Alejandro Adem for his support, patience and guidence during my research. I am also grateful to my colleagues Jose Cantarero, German Combariza and Jose Manuel Gomez. FinalIy I would like to thank my family for their tremendous support.  v  Chapter 1  Introduction In the past twenty years there has been a great influence of physical ideas on mathematics. One such example is the subject of string theory which increased the interest on some topological objects such as orbifolds, a very natural generalization of manifolds, and also on the invariants related to these objects, for example, the Euler characteristic, homotopy groups, homology groups and cohomology rings. The study of orbifolds in physics began with the introduction of the conformal field theory model on singular spaces by Dixon-Harvey-Vafa-Witten in 1985. Orbifolds are spaces that locally look like a quotient of an open set of a vector space by the action of a group such that the isotropy group at each point is finite. Chen and Ruan [19] discovered a cohomology theory for these inspired by their ideas in quantum cohomology and orbifold string theory. The interest in K-theory is also introduced from the consideration of a Dbrane charge on a smooth manifold and the notion of discrete torsion on an orbifold by Vafa. K-theory relates to equivariant theories if the orbifolds are the quotient of a smooth manifold M by a compact Lie group acting on M . Adem and Ruan [7] defined the twisted orbifold K-theory to study the resulting Chern isomorphism. In [8], Adem, Ruan and Zhang give an associative stringy product for the twisted orbifold K-theory of a compact, almost complex orbifold. This product is defined on the twisted K-theory  1  Chapter 1. Introduction of the inertia orbifold ∧X where the twisting gerbe τ is assumed to be in the image of the inverse transgression map H 4 (BX) −→ H 3 (B ∧ X). If the orbifold is X = ∧[∗/G] and G is a group then the resulting ring τK  orb (∧[∗/G])  is called fusion algebra.  One way to approach the fusion algebra is the finite group modular data which is mainly explored in [20]. In our context, when we mention the term modular data one should understand two symmetric matrices associated to a finite group G. This modular data was originally introduced, in Lusztig’s determination of the irreducible characters of the finite groups of Lie type [31], [32]. To describe the unipotent characters, he considered the modular data for some particular finite groups. The primary fields of the fusion algebra parametrize the unipotent characters associated to a given 2-sided cell in the Weyl group. Lusztig interprets this fusion algebra as the Grothendieck ring for G-equivariant vector bundles; in other words, the equivariant K-theory. The most physical application of this modular data is in (2+1)-dimensional quantum field theories where a continuous gauge group has been spontaneously broken into a finite group [13]. Non-abelian anyons (i.e. particles whose statistics are governed by the braid group rather than the symmetric group) arise as topological objects. The effective field theory describing the long distance physics is governed by the quantum group of [23]. A set of modular data (i.e. matrices S and T ) may be obtained for any choice of finite group G. Much information about a group can be recovered easily from its character table including whether it is abelian, simple, solvable, nilpotent, etc. For instance, G is simple if and only if for all irreducible χ ≡ 1, χ(a) = χ(e) only for a = e. Thus it may be expected that finite group modular data, which probably includes the character table, should provide more information about the group, i.e. be sensitive to a 2  Chapter 1. Introduction lot of the group-theoretic properties of G. One way to generalize this data is to twist with a cocycle from the cohomology group. One can obtain topological (e.g. oriented knot) invariants from this twisted data, as explained in [9]. These invariants are functions of the knot group (i.e., the fundamental group of the complement of the knot). Although non-isomorphic knots can have the same invariant, these invariants can distinguish a knot from its inverse (i.e., the knot with opposite orientation), unlike the more familiar topological invariants arising from affine algebras. Another important example of orbifolds is obtained by group actions on tori induced by the integral representations of finite groups. These orbifolds appear as examples of interest in physics. For this reason we are interested in computing the cohomology groups H ∗ (X/G; Z) where X/G is a toroidal orbifold. One can give an integral presentation ϕ : G → GLn (Z) for finite group G. In this way G acts linearly on Rn preserving the integral lattice Zn , thus inducing a G-action on the torus Xϕ = X := Rn /Zn . The quotient X → X/G naturally has the structure of an orbifold as a global quotient, and these kind of orbifolds are usually referred to as toroidal orbifolds. The goal of this chapter is to compute the cohomology groups H ∗ (X/G; Z) for the particular case where G = Z/p for a prime number p. The quotients of the form X/G appear naturally in different contexts. For example, given a topological space Y , the m-th cyclic product of Y is defined to be the quotient CP m (Y ) := Y m /Z/m,  3  Chapter 1. Introduction where Z/m acts by cyclically permuting the product Y m . In the particular case where the representation ϕ : G → GLn (Z) induces the ZG-module (ZG)n , the associated torus X is (S 1 )p )n , where G acts cyclically on each (S 1 )p and diagonally on the product ((S 1 )p )n . In this case X/G = (((S 1 )p )n )/Z/p ∼ = ((S 1 )n )p /Z/p where now Z/p acts cyclically on the ((S 1 )n )p . Therefore X/G ∼ = CP p ((S 1 )n ). The homology groups of quotient spaces of the form X m /K, where K ⊂ Σm , have long been studied. In particular, in [38] Swan formulated a method for the computation of cyclic products of topological spaces. In Chapter 2, the stringy product on twisted orbifold K-theory in [8] is reviewed, and an example for non-trivial twistings is calculated. For a finite group G this reduces to the twisted fusion algebra. Here we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the group G = (Z/2)3 via the pairing θg (φ) R(G)  ⊗θh (φ) R(G) →θgh (φ) R(G) where θ : H 4 (G, Z) → H 3 (G, Z) is the  inverse transgression map and φ is a carefully chosen cocycle class. We find the rank of the fusion algebra X(G) =  g∈G  θg (φ) R(G)  as well as the  relation between its basis elements. We also provide some applications to topological gauge theories. In Chapter 3, we study the twisted fusion algebra for the finite group (Z/p)3 . One way to calculate the coefficients in the fusion algebra is to use conformal field theory. The modular data in conformal field theory provides  4  Chapter 1. Introduction the Verlinde formula, which gives the fusion coefficients for fusion algebras. The modular data associated to finite groups is introduced in the work of Coste, Gannon and Ruelle [20]. They also consider the data arising from a cohomological twist, and write down the modular data for a general twist in terms of quantities associated directly with the finite group. We utilise the results from the conformal field theory, namely finite group modular data, to prove this algebra is isomorphic to the non-twisted fusion algebra of extraspecial p-group with exponent p and of order p3 . Chapter 4 is devoted to computing the cohomology of orbit space X/G where G = Z/p and X = ((S 1 )p−1 )s and the action is induced by the action on the ZG-module H 1 (X) = L = IGs , where IG is the augmentation ideal of ZG.  5  Chapter 2  A Twisted Fusion Algebra Inspired by the Chen-Ruan cohomology for orbifolds, it has been shown by Adem, Ruan, Zhang [8] that there is an internal product in twisted orbifold K-theory α Korb (X). The information determining this stringy product lies in H 4 (BX, Z) instead of H 3 (BX, Z): Given a class φ ∈ H 4 (BX, Z), it induces a class θ(φ) ∈ H 3 (B ∧ X, Z) where ∧X is the inertia stack. As a result, one can define a twisted K-theory on  θ(φ) K(∧X).  The map θ can be thought of  the inverse of the classical transgression map. The construction of this internal product is motivated by the so-called Pontryagin product on KG (G) for a finite group G. Indeed, if the orbifold is X = ∧[∗/G], one obtains the same product for the orbifold K-theory in the untwisted case. There is also an explicit calculation of the inverse transgression map θ for the cohomology of finite groups (see [8]). Using these results, we exhibit a non-trivial product structure in the case of G = (Z/2)3 . We use an integral cohomology class φ ∈ H 4 (G, Z) such that under the inverse transgression it maps non-trivially for every twisted sector, yielding a product structure on the algebra  θ(φ) K  G (G)  = X(G) =  g∈G  θg (φ) R(G)  defined via the pairing θg (φ) R(G) ⊗θh (φ) R(G) →θgh (φ) R(G). In this chapter, we derive the relations between the basis elements of the algebra X(G), and we prove the uniqueness of this product in this particular case. G = (Z/2)3 is indeed the abelian group of smallest rank such that it has non-trivial  6  Chapter 2. A Twisted Fusion Algebra transgressions (see [8], section 5). These twisted rings have also been worked out in the conformal field theory literature. In [35], the modular invariant (i.e. S and T matrices) of this group G = (Z/2)3 is calculated. As a result, one can calculate the relations between basis elements of X(G) using the Verlinde formula. Moreover, the same example is considered in [15], where a decomposition formula for twisted K-theory is given and the product is calculated after tensoring by rational numbers. There is also a physical counterpart of this theory. In our case this map is the inverse transgression map, which is actually the map coming from the correspondence between the Chern-Simons action and the WessZumino terms that arise in connecting a specific three-dimensional quantum field theory to its related two-dimensional quantum field theory. One can see that the Chern-Simons theory associates to each group element gi ∈ G a 2-cocycle βi of the stabilizer group Ngi , which is G = (Z/2)3 in our abelian case. We use the formulations in [25] to calculate the partition function Z(S 1 × S 1 × S 1 ). It is also worth mentioning that the algebra X(G) corresponds to a fusion algebra in this physical context. In this chapter, we first introduce projective representation and its basic properties. We next give some preliminaries and the definition of our fusion algebra. In the third section, we calculate the rank and the uniqueness of this algebra as well as the relation between the basis elements which are the projective representations. Finally, we present the application to topological gauge theories by using the formulation in [25].  7  Chapter 2. A Twisted Fusion Algebra  2.1  Projective representations  For the entirety of this section, G is a finite group. More information on projective representations can be found in [30]. Definition 2.1.1. Let V be a complex vector space. A projective representation of G is a map ρ : G → GL(V ) such that there exists α : G × G → C∗ with • ρ(x)ρ(y) = α(x, y)ρ(xy) for all x, y ∈ G, and • ρ(1) = I. One can easily see that α is a 2-cocycle of G. The associativity of multiplication in GL(V ) combined with the first condition gives the cocycle condition, and α(g, 1) = α(1, g) = 1 for all g ∈ G by the second condition. Any projective representation associated to α is called an α-twisted representation of G. As in the case of complex representations of G, we have a notion of isomorphism between two representations. Definition 2.1.2. Two α-twisted representations ρi : G → GL(Vi ), i = 1, 2, are called linearly equivalent if there is a vector space isomorphism f : V1 → V2 with ρ2 (g) = f ρ1 (g)f −1 for all g ∈ G. Clearly, the direct sum of two α-twisted representations is an α-twisted representation. Thus we can form the monoid of isomorphism classes of αtwisted representations of G. Rα (G) is the associated Grothendieck group. Note that if α is the trivial cocycle, then Rα (G) is R(G), the complex representation ring of G.  8  Chapter 2. A Twisted Fusion Algebra It is not hard to see that the tensor product of two α-twisted representations is no longer an α-twisted representation. Note however that if α and β are two cocycles, then the tensor product of an α-twisted representation and a β-twisted representation is an (α + β)-twisted representation. This can be extended to a pairing  Rα (G) ⊗ Rβ (G) → Rα+β (G). In order to study Rα (G), we introduce the α-twisted group algebra Cα G. We denote by Cα G the vector space over C with basis {¯ g }g∈G and with ¯ = α(g, h)gh extended by linearity. This makes Cα G into a Cproduct g¯h algebra with ¯ 1 as the identity. Definition 2.1.3. If α and β are cocycles of G, then we say that Cα G and Cβ G are equivalent if there exists a C-algebra isomorphism φ : Cα G → Cβ G and a mapping t : G → C∗ such that φ(¯ g ) = t(g)˜ g for all g ∈ G, where g¯ and g˜ are the bases for the two twisted group algebras. This defines an equivalence relation on such twisted algebras, and we have the following result. Lemma 2.1.4. There is an equivalence of twisted group algebras, Cα G Cβ G, if and only if α is cohomologous to β. In fact, α → Cα G induces a bijective correspondence between H 2 (G, C∗ ) and the set of equivalence classes of twisted group algebras of G over C. In [30], it is proved that these twisted group algebras play the same role in determing Rα (G) that CG plays in determining R(G): Theorem 2.1.5. There is a bijective correspondence between α-twisted representations of G and Cα G-modules. This correspondence preserves sums 9  Chapter 2. A Twisted Fusion Algebra and bijectively maps linearly equivalent representations into isomorphic modules.  2.2  The twisted fusion product for finite groups  In this section, we review a special case for the product in twisted orbifold Ktheory which is formalised by Adem, Ruan and Zhang in [8]. We consider the inertia orbifold ∧[∗/G] where G is a finite group. In this case, the untwisted orbifold K-theory of ∧[∗/G] is simply KG (G), which is additively isomorphic to  (g) R(ZG (g)),  where ZG (g) denotes the centraliser of g in  G, and the sum is taken over conjugacy classes. The product in KG (G) is defined as follows. An equivariant vector bundle over G can be thought of as a collection of finite dimensional vector spaces Vg with a G-module structure on  g∈G Vg  such that gVh = Vghg−1 . The product is defined as (V  Vg1 ⊗ Wg2 .  W )g = g1 g2 =g  One can give an alternative definition. We first define the maps e1 : G × G → G, e2 : G × G → G and e12 : G × G → G as e1 (g, h) = g , e2 (g, h) = h and e12 (g, h) = gh respectively, which are G-equivariant up to the conjugation action. If α, β are elements in KG (G) the product is defined as α β = e12∗ (e∗1 (α)e∗2 (β)). We now need to review the inverse transgression map for finite groups to extend the latter definition to twisted K-theory. In order to define the product in twisted K-theory, Adem, Ruan and Zhang [8] define a map to match up the levels which appear in the twistings. This cochain map θ is 10  Chapter 2. A Twisted Fusion Algebra called inverse transgression map, and it induces the homomorphism θ∗ : H k+1 (BG, Z) → H k (B ∧ G, Z). If the orbifold G is [∗/G], where G is a finite group, the inverse transgression has a classical interpretation in terms of the shuffle product. Recall that ∧[∗/G] is equivalent to  (g) [∗/ZG (g)]  (see [8]). Hence we would  like to focus on the map θg : C k (G, U (1)) → C k−1 (ZG (g), U (1)). If G is a finite group then the cochain complex C ∗ (G, U (1)) is in fact equal to HomG (B∗ (G), U (1)), where B∗ (G) is the bar resolution for G (see [18], page 18). If t is the generator of Z, the shuffle product is Bk (ZG (g)) ⊗ B1 (Z) → Bk+1 (G) given by [g1 |g2 | . . . |gk ] [ti ] =  σ[g1 |g2 | . . . |gk |gk+1 ], σ  where gk+1 = g i , σ ranges over all (k, 1)–shuffles and σ[g1 |g2 | . . . |gk+1 ] = (−1)sgn(σ) [gσ(1) |gσ(2) | . . . |gσ(k+1) ].  (2.1)  A (k, 1)–shuffle is an element σin the symmetric group Sk+1 such that σ(i) < σ(j) for 1 ≤ i < j ≤ k. We can dualize this using integral coefficients. Given a cocycle φ ∈ C k+1 (G, Z), one can see that the inverse transgression θg (φ) ∈ C k (ZG (G), Z) can be defined as θg (φ)([g1 |g2 | . . . |gk ]) = φ([g1 |g2 | . . . |gk ] [g])  (2.2)  where g1 , g2 , . . . , gk ∈ G. Hence it induces a map in integral cohomology. 11  Chapter 2. A Twisted Fusion Algebra We can now induce the inverse transgression map for H ∗ (G, F2 ) for G = (Z/2)3 using Bockstein homomorphism. We want to find a non-trivial cocycle in the image of the inverse transgression map. Notice that H ∗ (G, F2 ) is a polynomial algebra on three degree one generators x, y and z. In general, for an elementary abelian 2-group, the modulo 2 reduction map for k > 0 is a monomorphism H k (G, Z) → H k (G, F2 ) which is the kernel of the Bockstein homomorphism Sq 1 : H k (G, F2 ) → H k+1 (G, F2 ). In order to get nontrivial cocycle in the image of the inverse transgression map, we choose α = Sq 1 (xyz) = x2 yz + xy 2 z + xyz 2 , which represents a non-square element in H 4 (G, Z). The following lemma is proved in [8] by analyzing the multiplication map in the cohomology. Lemma 2.2.1. Let g = xa y b z c be an element in G = (Z/2)3 , where we are writing in terms of the standard basis (identified with its dual). Let us consider α = Sq 1 (xyz) = x2 yz + xy 2 z + xyz 2 which represents a non-square element in H 4 (G, Z). Then θg∗ (α) = a(y 2 z + z 2 y) + b(x2 z + xz 2 ) + c(x2 y + xy 2 ), and so θg∗ (α) is non-zero on every component except the one corresponding to the trivial element in G. Proof.  See [8], Lemma 5.2.  ∗ in the cohomology up to This implies that for all g, h ∈ G, θg∗ + θh∗ = θgh  coboundaries. This also implies that the correspondence g → θg (α) defines a homomorphism. In the case of G = (Z/2)3 , we have the isomorphism θ? (α) : G → H 3 (G, Z) = G. We now define the product as follows:  12  Chapter 2. A Twisted Fusion Algebra Definition 2.2.2. Let τ be a 2-cocycle for the orbifold defined by the conjugation action of a finite group G on itself which is in the image of the inverse transgression. The product on τ KG (G) is defined by the following formula: if α, β ∈τ KG (G), then α β = e12∗ (e∗1 (α)e∗2 (β)). If τ = θ(φ) then we have the following formula proved in [8]:  e∗1 τ + e∗2 τ = e∗12 τ up to coboundary. Hence the product e1 (α)e2 (β) lies in e∗1 τ +e∗2 τ  ∗  KG (G) =e12 τ KG (G).  Applying e12∗ , this is mapped to τ KG (G), which gives the product in the twisted K-theory. ∗ , the following product is defined Using the identification θg∗ + θh∗ = θgh  on the algebra θ(φ)  KG (G) = X(G) =  θg (φ)  R(G)  g∈G  via the pairing θg (φ)  R(G) ⊗  θh (φ)  R(G) →  θgh (φ)  R(G).  In the next section, we investigate the properties of this algebra while calculating its rank and the relations between the irreducible projective representation.  13  Chapter 2. A Twisted Fusion Algebra  2.3 2.3.1  Calculations 2-cocycles in G with values in U (1)  We will assume that G = (Z/2)3 for the remainder of this chapter. We recall that, for a finite dimensional complex vector space, a mapping ρ : G → GL(V ) is called a projective representation of G if there exists a U (1)valued 2-cocycle α ∈ Z 2 (G; U (1)) such that ρ(x)ρ(y) = α(x, y)ρ(x, y) for all x, y ∈ G and ρ(1) = IdV . Hence in order to compute  θg (φ) R(G)  we  first need to find the 2-cocycles in H 2 (G, U (1)) corresponding to θg (φ) in H 3 (G, Z) where both cohomology groups are isomorphic to G. For this purpose, we consider the isomorphism  H 2 (G, U (1)) → H 3 (G, Z) induced by the natural coefficient sequence 0 → Z → R → U (1) → 1. As H 3 (G, Z) ∼ = G, we need to find the eight non-cohomologous 2-cocycles in H 2 (G, U (1)) corresponding to each θg (φ) for all g ∈ G. We now determine the relations in order to obtain the 2-cocycles in C 2 (G, U (1)). Any 2-cocycle β in C 2 (G, U (1)) should satisfy: δβ = 1. By the boundary formula of the bar resolution of G, we derive:  β(g2 , g3 )β(g1 g2 , g3 )−1 β(g1 , g2 g3 )β(g1 , g2 )−1 = 1 for all gi ∈ G, i = 1, 2, 3. Some interesting relations result when we plug in  14  Chapter 2. A Twisted Fusion Algebra g1 = g3 = g and g2 = 1 into this formula and we obtain β(g, 1) = β(1, g).  (2.3)  Moreover, for g1 = g2 = g, we have β(1, g3 )β(g, g) = β(g, g3 )β(g, gg3 ).  (2.4)  As β is defining a projective representation, say ρ, it should satisfy ρ(1)ρ(g) = β(1, g)ρ(g). This implies β(1, g) = 1 for all g ∈ G. Now we consider the following tables for 2-cocycles βi : G × G → U (1) which satisfies the identities (3.3) and (2.4). We call these cocycles fundamental cocycles.We choose β1 as the trivial co-cycle. Here, xi , yi and zi ’s are in U (1), and they will be determined later. β2  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  1  x1  x2  x1  x2  x3  x3  g3  1  x1  −1  x4  −x1  x6  −x4  −x6  g4  1  x2  −x4  1  x5  x2  −x4  x5  g5  1  x1  −x1  −x5  −1  x8  −x8  x5  g6  1  x2  −x6  x2  −x8  1  −x8  −x6  g7  1  x3  x4  x4  x8  x8  1  x3  g8  1  x3  x6  −x5  −x5  x6  x3  1  15  Chapter 2. A Twisted Fusion Algebra  β3  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  y1  y2  −y1  −y2  y3  −y3  g3  1  y1  1  y4  y1  y6  y4  y6  g4  1  −y2  y4  1  y5  −y2  y4  y5  g5  1  −y1  y1  −y5  −1  y8  −y8  y5  g6  1  y2  y6  y2  −y8  1  −y8  y6  g7  1  −y3  y4  y4  y8  y8  1  −y3  g8  1  y3  y6  −y5  −y5  y6  y3  1  β4  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  z1  z2  −z1  −z2  z3  −z3  g3  1  −z1  1  z4  z1  z6  z4  z6  g4  1  z2  z4  −1  z5  −z2  −z4  −z5  g5  1  z1  z1  z5  1  z8  z8  z5  g6  1  −z2  z6  −z2  −z8  1  −z8  z6  g7  1  −z3  z4  −z4  −z8  z8  −1  z3  g8  1  z3  z6  −z5  z5  −z6  −z3  −1  As the multiplication of two cocycles gives us another cocycle, one can construct five more cocycles, one of which is the trivial cocycle. We recall that for β ∈ Z 2 (G; U (1)), an element g ∈ G is called β-regular if β(g, x) = β(x, g) for all x ∈ CG (g) (see [30], page 107). Thus all of the 2-cocycles have two 16  Chapter 2. A Twisted Fusion Algebra β-regular elements, one of which is 1, the other one is different for each cocycle. For example, one can immediately see that the β-regular elements for β2 , β3 and β4 are g2 , g3 and g4 , respectively. On the other hand, from the boundary formula we note that a 2-coboundary should satisfy β(g1 , g2 ) = σ(g1 )σ(g2 )σ(g1 g2 )−1 where σ is in C 1 (G, Z). As G is abelian, we have β(gi , gj ) = β(gj , gi ) for all gi and gj in G. This implies that all of these eight cocycles represent different cohomology classes, as multiplication with a coboundary does not change the β-regular elements. Thus, we have following proposition. Proposition 2.3.1. The rank of Proof.  θg (φ) R(G)  is 2 if θg (φ) is nontrivial.  A basic result of projective representations states that  α R(G)  is a free abelian group of rank equal to the number of distinct α-regular conjugacy classes of G (see [30], theorem 6.7). So, the ranks of  βi R(G)  is 2  for any non-trivial βi . On the other hand, we obtained eight non-cohomologous cocycles which should correspond to θg (φ)’s because H 2 (G, U (1)) and H 3 (G, Z) are isomorphic to G. The result follows. We can therefore conclude: Corollary 2.3.2. The rank of X(G) is equal to 22. Proof. and  The ranks of  beta1 R(G)  βi R(G)  are 2 for any non-trivial βi accounting for 14  has rank 8.  17  Chapter 2. A Twisted Fusion Algebra  2.3.2  The projective representations  In order to compute the irreducible projective representations of G, it is helpful to determine the xi , yi and zi ’s. From the boundary formula, we have the following relations in β2 : −1 = x1 x3 x4 x5 −1 = x2 x3 x5 x8 1 = x6 x8 x3 x1 . By a routine calculation, one can check that the other relations depend on these three relations. We can choose x1 = x2 = x3 = x4 = −x5 = x6 = x7 = x8 = 1 that obviously satisfy these relations. Similarly, we find yi ’s and zi ’s. The other cocycles are computed by multiplying β2 , β3 and β4 . We will later show that the choice of xi , yj and zk from the set {±1} does not change our representations. Here are our eight cocycles. β2  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  1  1  1  1  1  1  1  g3  1  1  −1  1  −1  1  −1  −1  g4  1  1  −1  1  −1  1  −1  −1  g5  1  1  −1  1  −1  1  −1  −1  g6  1  1  −1  1  −1  1  −1  −1  g7  1  1  1  1  1  1  1  1  g8  1  1  1  1  1  1  1  1  18  Chapter 2. A Twisted Fusion Algebra  β3  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  1  1  −1  −1  1  −1  g3  1  1  1  1  1  1  1  1  g4  1  −1  1  1  −1  −1  1  −1  g5  1  −1  1  1  −1  −1  1  −1  g6  1  1  1  1  1  1  1  1  g7  1  −1  1  1  −1  −1  1  −1  g8  1  1  1  1  1  1  1  1  β4  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  1  1  −1  −1  1  −1  g3  1  −1  1  1  −1  −1  1  −1  g4  1  1  1  −1  1  −1  −1  −1  g5  1  1  1  1  1  1  1  1  g6  1  1  1  1  1  1  1  1  g7  1  −1  1  −1  −1  1  −1  1  g8  1  1  1  −1  1  −1  −1  −1  19  Chapter 2. A Twisted Fusion Algebra  β5  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  1  1  −1  −1  1  −1  g3  1  1  −1  1  −1  1  −1  −1  g4  1  −1  −1  1  1  −1  −1  1  g5  1  −1  −1  1  1  −1  −1  1  g6  1  1  −1  1  −1  1  −1  −1  g7  1  −1  1  1  −1  −1  1  −1  g8  1  1  1  1  1  1  1  1  β6  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  −1  1  1  −1  −1  1  −1  g3  1  −1  −1  1  1  −1  −1  1  g4  1  1  −1  −1  −1  −1  1  1  g5  1  1  −1  1  −1  1  −1  −1  g6  1  −1  −1  −1  1  1  1  −1  g7  1  −1  1  −1  −1  1  −1  1  g8  1  1  1  −1  1  −1  −1  −1  20  Chapter 2. A Twisted Fusion Algebra  β7  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  1  1  1  1  1  1  1  g3  1  −1  1  1  −1  −1  1  −1  g4  1  −1  1  −1  −1  1  −1  1  g5  1  −1  1  1  −1  −1  1  −1  g6  1  −1  1  −1  −1  1  −1  1  g7  1  1  1  −1  1  −1  −1  −1  g8  1  1  1  −1  1  −1  −1  −1  β8  1  g2  g3  g4  g5  g6  g7  g8  1  1  1  1  1  1  1  1  1  g2  1  1  1  1  1  1  1  1  g3  1  −1  −1  1  1  −1  −1  1  g4  1  −1  −1  −1  1  1  1  −1  g5  1  −1  −1  1  1  −1  −1  1  g6  1  −1  −1  −1  1  1  1  −1  g7  1  1  1  −1  1  −1  −1  −1  g8  1  1  1  −1  1  −1  −1  −1  By considering the 2-cocycles that we obtained, it is obvious that there is no 1-dimensional projective representation whenever the 2-cocycle is not trivial. For the trivial cocycle, we have eight 1-dimensional representations which are just the irreducible linear representations of G. For the other cases, we find two 2-dimensional irreducible representations for each of the 2-cocycles (see [30], Theorem 6.7). Let ρi1 and ρi2 be the irreducible representations corresponding to the cocycle βi . It is also enough to give the 21  Chapter 2. A Twisted Fusion Algebra matrices corresponding to the generators of G. Without loss of generality, we assume g2 = (1, 0, 0), g3 = (0, 1, 0), g4 = (0, 0, 1), g5 = (1, 1, 0), g6 = (1, 0, 1), g7 = (0, 1, 1), g8 = (1, 1, 1). Here,  βi R(G)  is the projective  representation for 2-cocycle βi . We give only the matrices corresponding to three elements the rest can be obtained from these. β2 R(G)  :   1 0  ρ21 : g2 →    ρ22 : g2 →   β3 R(G)  −1  0 −1  0     g3 →        g3 →   0  1  −1 0  0 −1 1  0       g4 →       1 0     g4 →     0 1  −1  0 −1     0  :   ρ31 : g2 →    ρ32 : g2 →  β4 R(G)  0 1    0  1  −1 0  0 −1 1  0       g3 →        g3 →   −1  0  0  −1  1 0 0 1         g4 →      g4 →   0 −1  0 1 1 0  −1 0        :   ρ41 : g3 →   0 1 1 0       g5 →   0  i  −i 0       g6 →   1  0  0 −1     22  Chapter 2. A Twisted Fusion Algebra   ρ42 : g3 →   β5 R(G)  0  −1  −1  0  ρ51 : g2 →    ρ52 : g2 →   0   g5 →   1  1  −1 0  0 −1  0 −i i  0      −1 0   g6 →   0  1     0       g4 →        g4 →   0 1 1 0  0  −1  −1  0       g5 →        g4 →     1 0    0 1  −1  0  0  −1     :   ρ61 : g2 →    ρ62 : g2 →   β7 R(G)    :   β6 R(G)    0  −1 0  0 −1 1  1  0         g3 →      g3 →   0    0 i i 0  −i  −i  0  0  i     g6 →        g6 →     1 0    0 1  −1  0 −1  0     :   ρ71 : g2 →   0 1 1 0       g3 →   −i 0       g6 →   1  0  0 −1     23  Chapter 2. A Twisted Fusion Algebra   ρ72 : g2 →   β8 R(G)  0  −1  −1  0        0 −i   g3 →   i     g6 →   0  −1 0 0  1     :   ρ81 : g2 →    ρ82 : g2 →   0 1 1 0       g5 →   0  −1  −1  0    0    i  −i 0     g6 →     0 −i   g5 →   i    1  0 −1     g6 →   0  0     −1 0 0  1     The projective representations ρ1 and ρ2 are not linearly isomorphic to each other. Indeed, for the 2-cocycles β2 , β3 , β5 and β6 , there exist gi such that   ρ1 (gi ) =   while   ρ2 (gi ) =   1 0 0 1     −1  0  0  −1     . Thus there is no M ∈ GL2 (Z) such that M ρ1 (gi )M −1 = ρ2 (gi ). The other representations of β4 , β7 and β8 are obtained from the book of Karpilovski [30] (see page 120 and 124 Theorem 7.1 and 7.2). Thus none of them are linearly isomorphic to each other. One question that needs to be answered is whether these representations depend on the choice of xi , yi and zi . One can check by calculation that  24  Chapter 2. A Twisted Fusion Algebra the representations only depend on the values of βi (g, g). More precisely, ρ(g)ρ(g) should be equal to β(g, g) for all g ∈ G. For example,  ρ(g3 )ρ(g8 ) = β(g3 , g8 )ρ(g7 ) ⇔ ρ(g3 )β(g2 , g7 )ρ(g2 )ρ(g7 ) = β(g3 , g8 )β(g2 , g4 )ρ(g2 )ρ(g4 ) ⇔ ρ(g3 )β(g2 , g7 )ρ(g2 )β(g3 , g4 )ρ(g3 )ρ(g4 ) = β(g3 , g8 )β(g2 , g4 )ρ(g2 )ρ(g4 ) ⇔ ρ(g3 )ρ(g3 )β(g2 , g7 )β(g3 , g4 ) = β(g3 , g8 )β(g2 , g4 )I which is true if and only if ρ(g3 )ρ(g3 ) = β(g3 , g3 )I by the relations we get from the boundary formulas. The other elements can be checked in a similar manner. Thus we found the basis of our algebra. We will show that this product is unique up to coboundary. First we prove the following result. Proposition 2.3.3. If φ and φ are cohomologous cocycles in H 4 (G, Z) then the fusion algebras corresponding to these cocycles are isomorphic. Proof.  If φ and φ are cohomologous cocycles then θg (φ) and θg (φ ) rep-  resent the same cohomology class in H 3 (G, Z). In order to compute the 2-cocycle corresponding to θg (φ) ∈ H 3 (G, Z) ∼ = G, we will use the isomorphism induced from the short exact sequence  0 → Z → R → U (1) → 1. Thus θg (φ) is mapped to a certain class of 2-cocycles in H 2 (G, U (1)) ∼ = G. As we found eight non-cohomologous 2-cocycles in G, it is enough to check how the representations change if we multiply our fundamental 2-cocycles by a 2-coboundary. This is a basic result of projective representation theory 25  Chapter 2. A Twisted Fusion Algebra (see page 72 in [30]). After multiplying our fundamental 2-cocycles by a 2-coboundary, the new projective representation of this cocycle becomes linearly isomorphic to the former one. The result follows from the above argument.  2.3.3  The relations.  Now we are able to calculate the relation of this basis using the pairing θ(φ)g R(G)  ⊗  θ(φ)h R(G)  →  θ(φ)gh R(G).  The calculations are nothing but  solving linear equations. Namely one should prove that ρki ⊗ ρli ’s are linearly isomorphic to a sum of some basis elements. Here ρ1i denotes the irreducible regular representations of G for i = 1, 2, . . . , 8. Let us start with ρ1i ⊗ ρji which is linearly isomorphic to ρjk for some k ∈ {1, 2} where j = 1 because ρ1i ⊗ρji should be 2 dimensional βj representation. Here is the results of these multiplications:  26  ρ11  ρ11  ρ12  ρ13  ρ14  ρ15  ρ16  ρ17  ρ18  ⊗  ρ11  ρ12  ρ13  ρ14  ρ15  ρ16  ρ17  ρ18  ρ17  ρ18  ρ14  ρ13  ρ16  ρ15  ρ11  ρ12  ρ12  ρ16  ρ14  ρ18  ρ12  ρ17  ρ11  ρ15  ρ13  ρ13  ρ15  ρ13  ρ12  ρ18  ρ11  ρ17  ρ16  ρ14  ρ14  ρ14  ρ16  ρ17  ρ11  ρ18  ρ12  ρ13  ρ15  ρ15  ρ13  ρ15  ρ11  ρ17  ρ12  ρ18  ρ14  ρ16  ρ16  ρ12  ρ11  ρ15  ρ16  ρ13  ρ14  ρ18  ρ17  ρ17  ρ11  ρ12  ρ13  ρ14  ρ15  ρ16  ρ17  ρ18  ρ18  ρ22  ρ21  ρ22  ρ22  ρ21  ρ21  ρ22  ρ21  ρ21  ρ21  ρ22  ρ21  ρ21  ρ22  ρ22  ρ21  ρ22  ρ22  ρ32  ρ32  ρ31  ρ32  ρ31  ρ32  ρ31  ρ31  ρ31  ρ31  ρ31  ρ32  ρ31  ρ32  ρ31  ρ32  ρ32  ρ32  ρ42  ρ42  ρ42  ρ41  ρ42  ρ41  ρ41  ρ41  ρ41  ρ41  ρ41  ρ41  ρ42  ρ41  ρ42  ρ42  ρ42  ρ42  ρ51  ρ52  ρ52  ρ51  ρ51  ρ52  ρ52  ρ51  ρ51  ρ52  ρ51  ρ51  ρ52  ρ52  ρ51  ρ51  ρ52  ρ52  ρ61  ρ62  ρ61  ρ62  ρ62  ρ61  ρ62  ρ61  ρ61  ρ62  ρ61  ρ62  ρ61  ρ61  ρ62  ρ61  ρ62  ρ62  ρ71  ρ71  ρ72  ρ72  ρ72  ρ72  ρ71  ρ71  ρ71  ρ72  ρ72  ρ71  ρ71  ρ71  ρ71  ρ72  ρ72  ρ72  ρ82  ρ81  ρ81  ρ81  ρ82  ρ82  ρ82  ρ81  ρ81  ρ81  ρ82  ρ82  ρ82  ρ81  ρ81  ρ81  ρ82  ρ82  Chapter 2. A Twisted Fusion Algebra  27  Chapter 2. A Twisted Fusion Algebra Another type of multiplication is ρji ⊗ ρji , which is linearly isomorphic to a sum of four irreducible regular representations ρ1k as βj (g, h)2 = 1 for all j. We calculate all of these using the associativity of our algebra X(G) and investigating the eigenvalues of the matrices. Of course, one can also calculate them by defining the linear isomorphism explicitly. As ρi1 (gj ) = −ρi2 (gj ) for three gj ’s in the definitions of representations, we have ρi1 ⊗ ρi1 = ρi2 ⊗ ρi2 as well as ρi2 ⊗ ρi1 = ρi1 ⊗ ρi2 . ⊗  ρ21  ρ22  ρ21  ρ11 + ρ13 + ρ14 + ρ17  ρ12 + ρ15 + ρ16 + ρ18  ρ22  ρ12 + ρ15 + ρ16 + ρ18  ρ11 + ρ13 + ρ14 + ρ17  ⊗  ρ31  ρ32  ρ31  ρ11 + ρ12 + ρ14 + ρ16  ρ13 + ρ15 + ρ17 + ρ18  ρ32  ρ13 + ρ15 + ρ17 + ρ18  ρ11 + ρ12 + ρ14 + ρ16  ⊗  ρ41  ρ42  ρ41  ρ14 + ρ16 + ρ17 + ρ18  ρ11 + ρ12 + ρ13 + ρ15  ρ42  ρ11 + ρ12 + ρ13 + ρ15  ρ14 + ρ16 + ρ17 + ρ18  ⊗  ρ51  ρ52  ρ51  ρ11 + ρ15 + ρ16 + ρ18  ρ12 + ρ13 + ρ14 + ρ17  ρ52  ρ12 + ρ13 + ρ14 + ρ17  ρ11 + ρ15 + ρ16 + ρ18  ⊗  ρ61  ρ62  ρ61  ρ11 + ρ13 + ρ16 + ρ18  ρ12 + ρ14 + ρ15 + ρ17  ρ62  ρ12 + ρ14 + ρ15 + ρ17  ρ11 + ρ13 + ρ16 + ρ18  28  Chapter 2. A Twisted Fusion Algebra ⊗  ρ71  ρ72  ρ71  ρ13 + ρ14 + ρ15 + ρ16  ρ11 + ρ12 + ρ17 + ρ18  ρ72  ρ11 + ρ12 + ρ17 + ρ18  ρ13 + ρ14 + ρ15 + ρ16  ⊗  ρ81  ρ82  ρ81  ρ12 + ρ13 + ρ14 + ρ18  ρ11 + ρ15 + ρ16 + ρ17  ρ82  ρ11 + ρ15 + ρ16 + ρ17  ρ12 + ρ13 + ρ14 + ρ18  The last type of multiplication that we have to consider is ρji ⊗ρnm , where distinct i, m are in {1, 2} and j and n are in {2, 3, . . . , 8}. As ρji ⊗ ρnm is fourdimensional, it should be linearly isomorphic to 2ρl1 , 2ρl2 or ρl1 + ρl2 . None of −2ρl1 , −2ρl2 nor −ρl1 − ρl2 are possible as they are not βl representations as indicated by the list of our representations in the previous section. The representations 2ρl1 and 2ρl2 are also impossible by the following associativity argument. Suppose ρji ⊗ ρnm = 2ρl1 by the table above we can always find ρ1k such that ρ1k ⊗ ρji = ρji and ρ1k ⊗ ρl1 = ρl2 . This gives a contradiction if we multiply each side of ρji ⊗ ρnm = 2ρl1 by ρ1k . We can conclude ρji ⊗ ρnm = ρl1 + ρl2 . We have finished calculating all the relations.  2.4  Topological gauge theories  In [25], Dijkgraaf and Witten show that three dimensional Chern-Simons gauge theories with a compact gauge group can be classified by the integer cohomology group H 4 (BG, Z). Wess-Zumino interactions of such groups G are classified by H 3 (G, Z). The relation between three dimensional sigma models involves a certain natural map of H 4 (BG, Z) to H 3 (G, Z) which is 29  Chapter 2. A Twisted Fusion Algebra the inverse transgression map defined in the second section. Our calculations provide an example of three-dimensional topological theories with finite gauge group. In this context, our algebra X(G) is a fusion algebra. In QFT (Quantum Field Theory) one can associate to a (d + 1)-dimensional manifold M a certain number Z(M ), the partition function. A detailed discussion is provided in [25]. We consider the partition function of the 3-torus S 1 × S 1 × S 1 . If g, h and k are three commuting gauge fields, the partition function is evaluated to give Z(S 1 × S 1 × S 1 ) =  1 |G|  W (g, h, k), g,h,k∈G  where [g, h] = [h, k] = [k, g] = 1. We define W as  W (g, h, k) =  α(g, h, k)α(h, k, g)α(k, g, h) . α(g, k, h)α(h, g, k)α(k, h, g)  for α ∈ H 3 (BG, U (1)). The Chern-Simons theory associates to each group element gi ∈ G a 2-cocycle βi , which we calculated above. Again, by the result of Witten and Dijkgraaf [25], we may express βi in terms of a 3-cocycle α ∈ H 3 (G, U (1)):  βi (h1 , h2 ) =  α(gi , h1 , h2 )α(h1 , h2 , gi ) . α(h1 , gi , h2 )  This may also be obtained by the formula for inverse transgression map (2.2): βi (h1 , h2 ) = θgi (α)([h1 |h2 ]) = α([h1 |h2 ] [gi ]) =  α([gi |h1 |h2 ])α([h1 |h2 |gi ]) . α([h1 |gi |h2 ])  Note that the shuffle product (2.1) is defined via additive notation.  30  Chapter 2. A Twisted Fusion Algebra Thus the action W can be written in terms of 2-cocycles: W (gi , h, k) = βi (h, k)βi (k, h)−1 . We define  g (h)  = β(g, h)β(h, g)−1 for fixed g which is a one dimensional  representation of G. Thus an element g is β-regular if and only if  g  = 1.  This implies r(G, β) =  1 |G|  β(g, h)β(h, g)−1 g,h∈G  where r(G; β) denotes the number of irreducible projective representations of β R(G).  Our 2-cocycles defined in the second section satisfies this condition.  Comparing all these results we obtain the following result for the partition function of the 3-torus where G = (Z/2)3 : Proposition 2.4.1. Z(S 1 × S 1 × S 1 ) =  r(G; βi ) = 22. i  Using our representations, we find that the basis elements να of Hilbert space corresponds to 3-torus in QFT. These basis elements are given in [25] as  να (gi , h) = T rρi (h). In this context, our algebra X(G) can be regarded as the smallest twisted non-trivial fusion algebra for abelian groups.  31  Chapter 3  The Fusion Algebra of an Extraspecial p-group 3.1  Introduction  In this chapter, we give an application of the finite group modular data which is mainly explored in [20]. This modular data was originally introduced, in Lusztig’s determination of the irreducible characters of the finite groups of Lie type [31], [32]. To describe the unipotent characters, he considered the modular data for some particular finite groups. The primary fields of the fusion algebra parametrize the unipotent characters associated to a given 2-sided cell in the Weyl group. Lusztig interprets this fusion algebra as the Grothendieck ring for G-equivariant vector bundles; in other words, the equivariant K-theory. The most physical application of this modular data is in (2+1)-dimensional quantum field theories where a continuous gauge group has been spontaneously broken into a finite group [13]. Non-abelian anyons (i.e. particles whose statistics are governed by the braid group rather than the symmetric group) arise as topological objects. The effective field theory describing the long distance physics is governed by the quantum group of [23]. A set of modular data (i.e. matrices S and T ) may be obtained for any  32  Chapter 3. The Fusion Algebra of an Extraspecial p-group choice of finite group G. Much information about a group can be recovered easily from its character table including whether it is abelian, simple, solvable, nilpotent, etc. For instance, G is simple if and only if for all irreducible χ ≡ 1, χ(a) = χ(e) only for a = e. Thus it may be expected that finite group modular data, which probably includes the character table, should provide more information about the group, i.e. be sensitive to a lot of the group-theoretic properties of G. One way to generalize this data is to twist with a cocycle from the cohomology group. One can obtain topological (e.g. oriented knot) invariants from this twisted data, as explained in [9]. These invariants are functions of the knot group (i.e., the fundamental group of the complement of the knot). Although non-isomorphic knots can have the same invariant, these invariants can distinguish a knot from its inverse (i.e., the knot with opposite orientation), unlike the more familiar topological invariants arising from affine algebras. In this chapter, we consider the internal product in twisted orbifold Ktheory α Korb (X) formalised by Adem, Ruan and Zhang [8] employing results from the associated finite group modular data. Our aim is to exhibit that the non-trivial product structure for α KG (G) for G = (Z/p)3 is the same as the product structure for KH (H) where H is the extraspecial p-group of order p3 and exponent p. The product structure in these theories is the same as in the fusion algebras associated to the same finite group. We first give some preliminaries and the definition of finite group modular data, then mention the results of Coste, Gannon and Ruelle [20] who calculated the S matrix for (Z/p)3 with the particular twist. After this, we calculate the S matrix of the extraspecial p-group and show that it gives the same fusion algebra as the twisted case of (Z/p)3 . 33  Chapter 3. The Fusion Algebra of an Extraspecial p-group  3.2  Finite group modular data  In this section, we concentrate on the modular data associated with any finite group G. We first fix a set R of representatives of each conjugacy class of G. The identity e of G is in R, and more generally the center Z(G) of G is a subset of R. For any a ∈ G, let Ka denote the conjugacy class of a in G and CG (a) be the centralizer of a in G. We have |G| = |Ka ||CG (a)|. The primary fields of the G modular data are labeled by pairs (a, χ), where a ∈ R, and where χ is an irreducible character of CG (a). We will write Φ for the set of these pairs. We define the S matrix and T matrix as  S(a,χ),(b,χ ) =  =  1 |G|  1 |CG (a)||CG (b)|  χ(gbg −1 )∗ χ (g −1 ag)∗ g∈G(a,b)  χ(xhx−1 )∗ χ (ygy −1 )∗ , g∈Ka ,h∈Kb ∩CG (g)  T(a,χ)(a ,χ ) = δa,a δχ,χ  χ(a) χ(e)  where G(a, b) = {g ∈ G|agbg −1 = gbg −1 a} and where x and y are any solutions to g = x−1 ax and h = y −1 by. If G(a, b) is empty then the sum is equal to zero. The matrix S is symmetric and unitary, and gives rise (via the Verlinde (c,χ )  formula) to non-negative integer fusion coefficients N(a,χ13)(b,χ2 ) , where c Nab = d∈Φ  ∗ Sad Sbd Scd . S0d  One way to generalize the group data is by introducing some twisting. The twisting of the modular data is described in [20], where the explicit  34  Chapter 3. The Fusion Algebra of an Extraspecial p-group expressions for the modular matrix S appear. The primary fields Φα in the model twisted by a given 3-cocycle α consist of all pairs (g, χ) where g ∈ R and χ is a βg twisted irreducible character of CG (g), where βg is defined by βg (a, b) = α(g, a, b)α(a, a−1 ga, b)−1 α(a, b, (ab)−1 gab) from a normalized element α of H 3 (G, U (1)). The S matrix is calculated as α S(a,χ)(b,χ ) =  1 = |G|  1 |G|  g∈Ka ,g ∈Kb ∩CG (g)  g∈Ka ,g ∈Kb ∩CG (g)  ∗  βg (g , x−1 )βg (g, y −1 ) βg (x−1 , h)βg (y −1 , h )  χ(h)∗ χ (h )∗ ,  βa (x, g )βa (xg , x−1 )βb (y, g)βb (yg, y −1 ) βa (x, x−1 )βb (y, y −1 )  ∗  χ(h)∗ χ (h )∗ ,  where g = x−1 ax = y −1 h y, g = y −1 by = x−1 hx, h ∈ CG (a), h ∈ CG (b). In order to compute the T matrix, in [20]the 1-cochain  a  : CG (a) → U (1)  is introduced. It is determined by the following equalities: a (e)  βa (h, g) = x−1 ax  = 1,  −1 a (h) a (g) a (hg) ,  (x−1 hx) =  βa (x, x−1 hx) a (h), βa (h, x)  for all g, h ∈ CG (a) and x ∈ G. Then the T matrix is α T(a,χ)(a ,χ ) = δa,a δχ,χ  χ(a) a (a). χ(e)  35  Chapter 3. The Fusion Algebra of an Extraspecial p-group  Twisted example G = (Z/p)3  3.3  The simplest non-cyclic group is G = (Z/n)2 for a positive integer n, but it does not lead to something new (see [8]). The cohomology group H 3 ((Z/n)2 , U (1)) = (Z/n)3 has three generators, but all 3-cocycles give β’s which are all coboundaries. Thus all twistings are cohomologically trivial. The more interesting case of G = (Z/n)3 is given in [20]. Here, we recall that work in order to compare it with our calculation in the next section: The cohomology H 3 (G, U (1)) = (Z/n)7 is generated by the following cocycles:  (j)  αI (a, b, c) = exp 2iπaj (bj + cj − bj + cj )/n2 , (jk)  αII (a, b, c) = exp 2iπaj (bk + ck − bk + ck )/n2 ,  1 ≤ j ≤ 3, 1 ≤ j < k ≤ 3,  αIII (a, b, c) = exp 2iπa1 b2 c3 /n , where the group elements are the triplets a = (a1 , a2 , a3 ). The cocycles which involve a non-trivial power of αIII define non-trivial twistings. The 2-cocycles are of the form  βa (b, c) = exp 2iπq(a1 b2 c3 − b1 a2 c3 + b1 c2 a3 )/n . Given a, we wish to count the number of classes b (elements here) which are βa -regular, i.e., which satisfy βa (b, c) = βa (c, b) for all c. Tak-  36  Chapter 3. The Fusion Algebra of an Extraspecial p-group ing c = (1, 0, 0), (0, 1, 0) and (0, 0, 1), the βa -regular elements b are those which satisfy a2 b3 − a3 b2 ≡ a1 b3 − a3 b1 ≡ a1 b2 − a2 b1 ≡ 0 (mod f ), where f = n/gcd(q, n). The number of solutions (b1 , b2 , b3 ) ∈ (Z/n)3 to this system is equal to n3 [gcd(a1 , a2 , a3 , f )/f ]2 . It remains to sum those numbers for all a to obtain the number of primaries: |Φα | =  n6 f3  (pkp − 1)(1 + p−1 + p−2 ) + 1 . p|f p prime  We now consider the case when n is an odd prime number p, and when the 3-cocycle is αIII . When G is abelian, all factors in the formula for S α that involve the cocycles drop out, and one is left with the simple expression: α S(a, χ),(b, ˜ χ ˜) =  1 ∗ χ (b) χ ∗ (a), |G|  where χ and χ are respectively βa and βb -projective characters, for the cocycles given above with q = 1. It remains to compute the projective characters. One then finds p distinct, irreducible βa -projective representations of dimension p if a is not the identity, while there are of course p3 representations of dimension 1 if a = e. Depending on the value of a = (a1 , a2 , a3 ), the characters are given in the following table, where it is implicit that the element g = (g1 , g2 , g3 ) must be βa -regular for the character not to vanish. In the first three cases, the character label u runs over Z/p, and in the last column, u takes all values  37  Chapter 3. The Fusion Algebra of an Extraspecial p-group in (Z/p)3 .  χ(g)  a1 = 0 −1 −1 2 /2 a1 ug1 −a1 a2 a3 g1 p ξp  a1 = 0, a2 = 0  a1 = a2 = 0, a3 = 0  a1 = a2 = a3 = 0  −1 a ug2 p ξp 2  −1 a ug3 p ξp 3  u·g ξp  −1 If for instance a2 is also invertible, then a−1 1 g1 = a2 g2 , so that the first 2 a−1 ug2 −a1 a−1 2 a3 g2 /2  character value is also equal to χ(g) ˜ = p ξp 2  . The primary  fields are thus (e, χu ) and (a, χ ˜u ), for a total of |Φα | = p3 + (p3 − 1)p = p4 + p3 − p. The matrices S α and T α are now straightforward to establish. Taking α the condition of β-regularity into account, one finds that S(a,χ u )(b,χ  u  is  )  almost block-diagonal:    1  p   1 ξ −u1 b1 −u2 b2 −u3 b3 p p −1 −1 −ua1 b1 −u b1 a1 +(a2 a3 b1 +b2 b3 a1 )/2 1 ξ −u1 a1 −u2 a2 −u3 a3 ξp −1 −1 p p ×δ(b2 −a1 a2 b1 ) δ(b3 −a1 a3 b1 ) 1 p2  1 ξ −u2 a2 −u3 a3 p p 1 ξ −u3 a3 p p  0 0  1 ξ −u2 b2 −u3 b3 p p  1 ξ −u3 b3 p p    0  0        −1 −1 −ua2 b2 −u b2 a2 ξp −1 ×δ(b3 −a2 a3 b2 )  0  0 −1 −1 −ua3 b3 −u b3 a3 ξp  where the blocks correspond to the subsets {a = e}, {a1 = 0} , {a1 = 0, a2 = 0}, and {a1 = a2 = 0, a3 = 0}. u−a1 a2 a3 /2  α The entries T(a,χ)(a,χ) are 1 for a = e = (0, 0, 0) and ξp  in any  other case. In the next section, we observe that the modular data of the extraspecial p-group gives the same fusion coefficients with this twisted case. Hence we can conclude that these two fusion algebras are the same.  38  Chapter 3. The Fusion Algebra of an Extraspecial p-group  3.4  Modular data for extraspecial p-group  When discussing what he called electric/magnetic duality in [35], Propitius observed that the modular data for (Z/2)3 with a particular twist equals that of the untwisted dihedral group D8 (for an appropriate identification of primary fields). In the more recent work of [20] it is claimed that there are many more such examples; when calculating the modular data for an appropriately twisted (Z/p)3 , it is noted that the quantum dimensions and number of primaries suggest that this twisted data would yield the modular data for the extraspecial p-group H of order p3 , which is the central extension of the cyclic subgroup Z/p × Z/p. This group may be represented as A, B, C|Ap = B p = C p = [A, C] = [B, C] = 1, [A, B] = C . In this section we prove that this is indeed the case. It is convenient to use an isomorphic realization of H via (r, s, t) ∈ (Z/p)3 with the following defining relations. (r, s, t)−1 = (−r, −s, −t − rs), (r, s, t)(r , s , t ) = (r + r , s + s , t + t − sr ). The irreducible characters are calculated in [29]. For fixed x, y, z in {1, 2, . . . , p− 1}, we define the character as  χ(r, s, t) =     0,   pξ tz+rx+sy , p  if  p d  s ∨ ( dp r ∧ z = 0); otherwise;  where d = gcd(z, p). It may be rewritten in a more compact form using the  39  Chapter 3. The Fusion Algebra of an Extraspecial p-group Kronecker delta function:  χ(r, s, t) =     ξprx+sy ,  if z=0;   δ(p|r)δ(p|s)pξ rx+sy+tz , if z = 0; p  where δ(p|r) = 1 if p = r and 0 otherwise. Using this triple realization, the conjugacy class of the element (a, b, c) can be given as C(a,b,c) = {(a, b, c + bx − ay) | x, y ∈ Z/p}. Thus two elements, (a, b, c) and (a , b , c ), belong to the same conjugacy class if and only if a = a , b = b and there exist integers x, y and z such that c = c − ay + bx + pz. This equation is solvable if and only if c ≡ c ( mod gcd(a, b, p)). Therefore every conjugacy class contains exactly one element of the set L = {(a, b, c) ∈ {0. . . . , p − 1}3 |c < gcd(a, b, p)}. If neither a nor b is equal to 0 then gcd(a, b, p) = 0. Hence c should be equal to 0. If a = b = 0 then c can be any number in {0, . . . , p − 1} as gcd(0, 0, p) = p in this case. So the representative of the conjugacy classes are either in the form of (a, b, 0) = (0, 0, 0) or (0, 0, c). We now calculate the centralizer for each element. The elements (r, s, t) of CG (a, b, 0) should satisfy (r, s, t)(a, b, 0) = (a, b, 0)(r, s, t), (a + r, s + b, t − as) = (a + r, s + b, t − rb). Therefore as = rb and (r, s, t) = (a, asr−1 , c). CG (a, b, 0) is of order p2 . From the basic results of finite group theory there are two groups of order p2 up to isomorphism. CG (a, b, 0) cannot be the cyclic one as it has at least two elements of order p, namely (a, 0, 0) and (0, 0, c). Thus, CG (a, b, 0) is 40  Chapter 3. The Fusion Algebra of an Extraspecial p-group isomorphic to Z/p × Z/p. The elements (r, s, t) of CG (0, 0, c) should satisfy (r, s, t)(0, 0, c) = (0, 0, c)(r, s, t), (r, s, c + t) = (r, s, c + t). Hence CG (0, 0, c) = G. 2  2  Let us denote the irreducible characters by Γi |pi=1 and χi |pi=1+p+1 for Z/p× Z/p and G, respectively. The primary fields of the modular data may be written as (0, 0, c), χi for all c ∈ G and (a, b, 0), Γi where (a, b, 0) = (0, 0, 0). We find that the number of the primary fields is p(p2 + p − 1) + (p2 − 1)p2 = p4 + p3 − p which is equal to the number of primary fields of twisted modular data of (Z/p)3 . We next calculate the entries of the S matrix. We first consider the primary fields (a, χi ) and (b, χj ) where a = (0, 0, r) and b = (0, 0, t). We use the formula given in the second section: S(a,χi )(b,χj ) =  1 |CG (a)||CG (b)|  i −1 ∗ j −1 ∗ g∈G(a,b) χ (gbg ) χ (g ag) .  We know |CG (a)||CG (b)| = p6 , G(a, b) = {g ∈ G|agbg −1 = gbg −1 a} = {g ∈ G|(0, 0, r + t) = (0, 0, r + t)} = G as well as χi (gbg −1 ) = χi (b) for g ∈ G. Hence S(a,χi )(b,χj ) =  1 i ∗ j ∗ χ (b) χ (a) . p3  By the above formula for χ, if z in the formula is equal to 0 then there are p2 characters such that χ(0, 0, t) = 0. We denote these characters by χi,j where i and j are in {0, . . . , p − 1}. If z = 0 then there are p − 1 characters of the form as z ∈ {1, . . . , p − 1}. We denote these characters by χzi after indexing the (p − 1) z’s. Hence, by the formula for the S matrix, we have 41  Chapter 3. The Fusion Algebra of an Extraspecial p-group  S((0,0,r),χi,j )((0,0,t),χi ,j ) = 1/p3 , −rzj −tzi  S((0,0,r),χzi )((0,0,t),χzj ) = 1/pξp  ,  S((0,0,r),χi,j )((0,0,t),χz ) = 1/p2 ξp−rz . One may compare these results with the S α matrix of the twisted modular data for (Z/p)3 . In order to get the same entries as S α , we change ((0, 0, t), χzi ) to ((0, 0, zi−1 t), χzi ), and notice that zi corresponds to a3 in the notation of the twisted matrix. We basically proved that the blocks at the corners of each matrix are the same. It is trivial to see that the dimension of these blocks is also the same. We now need to check the other blocks. For the other blocks we need to consider the primary fields ((r, s, 0), Γi ) where s or t are not equal to 0. We have the following formula S((r,s,0),Γi )((0,0,t),χz ) =  1 |CG (a)||CG (b)|  Γi (gbg −1 )∗ χz (g −1 ag)∗ g∈G(a,b)  where a = (0, 0, t) and b = (r, s, 0). By the Kronecker delta functions δ(p|r) and δ(p|s) in the definition of χ, the character χz (g −1 ag) should be equal to zero as at least one of r or s is nonzero. These zeros corresponds to the (4, 2), (4, 3), (2, 4) and (3, 4) block entries of the S α matrix of the twisted case if we index the blocks starting from the left upper corner which is the (1, 1) block. Another kind of entry that we need to consider is S((0,0,t),χi,j )((r,s,0),Γi ) =  1 |CG (a)||CG (b)|  χi,j (gbg −1 )∗ Γi (g −1 ag)∗ g∈G(a,b)  42  Chapter 3. The Fusion Algebra of an Extraspecial p-group where a = (0, 0, t) and b = (r, s, 0). Here, we know |CG (a)| = p3 , |CG (b)| = p2 and G(a, b) = {g ∈ G|agbg −1 = gbg −1 a} = {g ∈ G|(r, s, t − as + rb) = (r, s, t − as + rb)} = G. Moreover, we have g −1 ag = a for g ∈ G and gbg −1 = (g1 , g2 , g3 )(r, s, 0)(−g1 , −g2 , −g3 − g1 g2 ) = (r, s, g2 r − g1 s) by the formula for χ, χi,j (gbg −1 ) = ξprx+sy where x, y ∈ G are fixed for each character χi,j . We can change the notation x and y to u1 to u2 . On the other hand, Γi (0, 0, t) = ξpa3 t where we can define Γi (r, s, t) = ξpvs ξpa3 t where v and a3 are fixed for each character Γi . We recall that, for (x, y, z) ∈ CG (r, s, 0), x and y are the same up to a constant while z is independent than the other coordinates. For the character Γi of Z/p × Z/p we thus consider the second and third coordinates without loss of generality. We use this fact again in the next calculation. As a result, we obtain S((0,0,t),χi,j )((r,s,0),Γi ) = 1/p2 ξp−ru1 −su2 −ta3 . Hence if r = 0 this block coincides with (1, 2) and (2, 1) blocks of the S α matrix. If r = 0 it coincides with the (1, 3) and (3, 1) blocks if we change r, s and t to a1 , a2 and u3 , respectively. The remaining blocks that we need to check are the four blocks at the center. Let us first check S((0,s,0),Γi )((r ,s ,0),Γj ) where r = 0. In this case we see that G(a, b) = {g ∈ G|agbg −1 = gbg −1 a} = {g ∈ G|(r , s + s , g2 r − g1 s ) = (r , s + s , g2 r − g1 s − sr ) = ∅ because sr = 0. Hence  43  Chapter 3. The Fusion Algebra of an Extraspecial p-group S((0,s,0),Γi )((r ,s ,0),Γj ) = 0 corresponds to the zero blocks ((2, 3) and (3, 2) blocks) at the center. We continue with the block corresponding to S((0,s,0),Γi )((0,s ,0),Γj ) . Here we have G(a, b) = {g ∈ G|agbg −1 = gbg −1 a} = {g ∈ G|(0, s + s , −g1 s ) = (0, s + s , −g1 s )} = G, gbg −1 = (0, s , −s g1 ) and g −1 ag = (0, s, sg1 ) where g = (g1 , g2 , g3 ), and recall that we defined Γi and Γj as Γi (r, s, t) = ξpvs ξpa3 t and Γj (r, s, t) = ξpv s ξpb3 t . If a3 s = b3 s then Γi (gbg −1 )∗ Γj (g −1 ag)∗ = ξp−vs ξpa3 s g1 ξp−v s ξp−b3 sg1 = ξp−vs −v s ξpg1 (s a3 −b3 s) = ξp−vs −v s . If a3 s = b3 s which means δ(a3 s − b3 s) = 0, then Γi (gbg −1 )∗ Γj (g −1 ag)∗ = ξp−vs −v s ξpg1 (s a3 −b3 s) . We now consider the S-matrix formula ξp−vs −v s ξpg1 (s a3 −b3 s) .  S((0,s,0),Γi )((0,s ,0),Γj ) = 1/p4 g∈G g (s a3 −b3 s)  If a3 s = b3 s, then ξp1  is summed while g1 is covering all Z/p p2  times. As the sum of all the roots of unity is 0, the entry S((0,s,0),Γi )((0,s ,0),Γj ) must be zero. Otherwise S((0,s,0),Γi )((0,s ,0),Γj ) = 1/pξp−vs −v s . Any non-zero power of ξp is also a pth root unity. Hence we can choose v = ua−1 2 and v = u b−1 2 . If we change the notation s = a2 and s = b2 , we obtain the (3, 3) block of the twisted S matrix of (Z/p)3 . Note that this assignment does not change the order of the entries in the first column or in the first row as they are determined by second factor ξpa3 t of the character Γi . 44  Chapter 3. The Fusion Algebra of an Extraspecial p-group The (2, 2) block is the only block left in the twisted S α of (Z/p)3 . We need to calculate S((r,s,0),Γi )((r ,s ,0),Γj ) . Here G(a, b) = {g ∈ G|agag −1 = gbg −1 a} = {g ∈ G|(r + r , s + s , g2 r − g1 s − r s) = (r + r , s + s , −g1 s − s r) is G if and only if r s = s r. Otherwise G(a, b) is an empty set, a case which is covered by examining the Kronecker delta functionδ(s − r−1 sr ), as these terms are in the abelian group Z/p. We now assume that r s = s r. We also have gbg −1 = (r , s , r g2 − s g1 ) and g −1 ag = (r, s, sg1 − rg1 ) where g = (g1 , g2 , g3 ). By the formulae Γi (r, s, t) = ξpvs ξpa3 t and Γj (r, s, t) = ξpv s ξpb3 t we get the multiplication Γi (gbg −1 )∗ Γj (g −1 ag)∗ = ξp−vs −v s ξpg1 (s a3 −b3 s)+g2 (r a3 −rb3 ) . We note that s a3 − b3 s = 0 if and only if r a3 − rb3 as r s = s r. If these exponents are non-zero, the entry in our S is zero because gi is covering the set G by resolving the power sum of the ξp which is equal to zero. Hence another Kronecker delta function appears in our expression which is δ(r a3 − rb3 ). If δ(r a3 − rb3 ) = 1 then S((r,s,0),Γi )((r ,s ,0),Γj ) = 1/pξp−vs −v s . We now make the conventional choice of v and v as ua−1 2 − a1 a3 /2 and u b2−1 − b1 b3 /2. Hence 2−1 −b b /2)s −(ua−1 1 3 2 −a1 a3 /2)s −(u b  S((r,s,0),Γi )((r ,s ,0),Γj ) = 1/pξp  .  We also change the notation r = a1 , s = a2 , r = b1 and s = b2 to obtain −1 −ua−1 2 b2 −u b2 a2 +(a1 a3 b2 +b1 b3 b2 )/2  ξp  −1 −ua−1 1 b1 −u b1 a1 +(a2 a3 b1 +b2 b3 a1 )/2,  = ξp  where the equality follows from b1 a2 = a1 b2 and b2 a3 − b3 a2 = 0. The choice of v and v is the same as the last part we have calculated because r = a1 = 0 45  Chapter 3. The Fusion Algebra of an Extraspecial p-group in that case. We note that the change in variables does not change the order of the entries in first column or row as they are determined by second factor ξpa3 t of the character Γi . This completes our calculation of the S matrix of H which as a result is the same as the twisted S α matrix of the group (Z/p)3 We can write the change of variables as a map from the primary field ((r, s, 0), Γi ) to ((a1 , a2 , a3 ), χu ) in the twisted fusion algebra where r and s are mapped to a1 and a2 , respectively. The two factors in the character Γi are mapped to a3 and u. We next calculate the T matrix. We again start with the primary field of type ((0, 0, t), χi,j ) for which T((0,0,t),χi,j )((0,0,t),χi,j ) = χi,j (0, 0, t)/χi,j (0, 0, 0) = 1. If the field is ((0, 0, t), χzi ) then T((0,0,t),χzi )((0,0,t),χzi ) = χzi (0, 0, t)/χzi (0, 0, 0) = ξptzi . Remember that we replace (0, 0, t) by (0, 0, tzi−1 ). Hence T((0,0,t),χzi )((0,0,t),χzi ) is ξpt . If the field is of type ((0, s, 0), Γi ) then T((0,s,0),Γi )((0,s,0),Γi ) = Γi (0, s, 0)/Γi (0, 0, 0) = ξpsv . u By our previous change of v and s to ua−1 2 and a2 , respectively, we obtainξp  for this primary field. The last primary field to check is the field of type ((r, s, 0), Γi ) for which T((r,s,0),Γi )((r,s,0),Γi ) = Γi (r, s, 0)/Γi (0, 0, 0) = ξpsv , 46  Chapter 3. The Fusion Algebra of an Extraspecial p-group where we replace s and v by a2 and ua−1 2 − a1 a3 /2, respectively. Therefore u−a1 a2 a3 /2  we obtain the term ξp  . We conclude that the matrix T α of the group  (Z/p)3 for a particular twist α and the T matrix of the extraspecial p-group H of order p3 with exponent p coincide. As a result, we have proved the following theorem. Theorem 3.4.1. The twisted modular data of the group (Z/p)3 are the same as the modular data of the extraspecial p-group of order p3 with exponent p. We also have the following corollary. Corollary 3.4.2. The twisted fusion algebra of (Z/p)3 , which is isomorphic to the twisted orbifold K-theory α Korb (∧[∗/G]) in the sense of [8], is isomorphic to the fusion algebra of the extraspecial p-group of order p3 with exponent p, which is isomorphic to the orbifold K-theory Korb (∧[∗/H]).  47  Chapter 4  Cohomology of Toroidal Orbifolds 4.1  Introduction  Let G be a finite group and ϕ : G → GLn (Z) an integral representation of G. In this way G acts linearly on Rn preserving the integral lattice Zn , thus inducing a G-action on the torus Xϕ = X := Rn /Zn . The quotient X → X/G naturally has the structure of an orbifold as a global quotient, and these kind of orbifolds are usually referred to as toroidal orbifolds. The goal of this chapter is to compute the cohomology groups H ∗ (X/G; Z) for the particular case where G = Z/p for a prime number p. The quotients of the form X/G appear naturally in different contexts. For example, given a topological space Y , the m-th cyclic product of Y is defined to be the quotient CP m (Y ) := Y m /Z/m, where Z/m acts by cyclically permuting the product Y m . In the particular case where the representation ϕ : G → GLn (Z) induces the ZG-module (ZG)n , the associated torus X is (S 1 )p )n , where G acts cyclically on each  48  Chapter 4. Cohomology of Toroidal Orbifolds (S 1 )p and diagonally on the product ((S 1 )p )n . In this case X/G = (((S 1 )p )n )/Z/p ∼ = ((S 1 )n )p /Z/p where now Z/p acts cyclically on the ((S 1 )n )p . Therefore X/G ∼ = CP p ((S 1 )n ). The homology groups of quotient spaces of the form X m /K, where K ⊂ Σm , have long been studied. In particular, in [38] Swan formulated a method for the computation of homology of cyclic products of topological spaces.  4.2  Preliminaries  Let G be a finite group and ϕ : G → GLn (Z) an integral representation of G. Consider X = Xϕ the standard torus with the action of G induced from ϕ. Then there is a fibration sequence X → X ×G EG → BG.  (4.1)  Using the long exact sequence in homotopy groups associated to this fibration, it follows that X ×G EG is an Eilenberg-Maclane space of type K(Γ, 1), where Γ := π1 (X ×G EG). The G-action on X makes L := π1 (X) ∼ = H1 (X; Z) ∼ = Zn into a ZG-module. Moreover, since [0] ∈ Rn /Zn is always a fixed point for the action of G on X, it follows that the fibration (4.1) has a section. The existence of such a section implies that the short exact sequence 1 → π1 (X) → π1 (X ×G EG) → π1 (BG) → 1 49  Chapter 4. Cohomology of Toroidal Orbifolds has a section and therefore Γ ∼ =L  Gis a semi-direct product. For example,  when the representation ϕ is injective, the group Γ is a crystallographic group. The cohomology groups of the groups of the form Γ ∼ = L  G, when  G = Z/p for a prime number p, were computed in [4, Theorem 1.1] where it is shown that a certain special free resolution  : F → Z of Z as a  Z[L]-module admits an action of G compatible with ϕ. This implies that the Lyndon-Hochshild-Serre spectral sequence associated to the short exact sequence 1→L→Γ→G→1 collapses on the E2 -term without extension problems, thus for any k ≥ 0 j  H (Γ; Z) ∼ = k  i  H (G;  (L∗ )),  i+j=k  where L∗ denotes the dual module Hom(L, Z).  One application of this, by [4, Theorem 1.2] is when G = Z/p acts on X via a representation ϕ : G → GLn (Z), then H k (X ×G EG; Z) ∼ =  H i (G; H j (X; Z)). i+j=k  for each k ≥ 0. The strategy we will use to compute the cohomology groups H ∗ (X/G; Z) is as follows. Let F denote the subspace of fixed points under the G-action. In general, F will be a disjoint union of product of circles. Then the long exact sequence in cohomology associated to the pair (X/G, F ) can be used to compute the cohomology groups H ∗ (X/G; Z). To make this 50  Chapter 4. Cohomology of Toroidal Orbifolds work, the relative cohomology groups H ∗ (X/G, F ; Z) need to be determined. For this, we consider the equivariant projection π1 : X × EG → X. On the level of orbit spaces π1 induces a map φ : X ×G EG → X/G. By [17, Proposition V 1.1], φ induces the isomorphism ∗ φ∗ : H ∗ (X/G, F ; Z) → HG (X, F ; Z).  This reduces the problem to one of determining the cohomology groups ∗ (X, F ; Z). These groups will be computed using representation theory HG  and the fact that the Lyndon-Hochshild-Serre spectral sequence associated to the fibration sequence X → X ×G EG → BG collapses on the E2 -term without extension problems. Let L := H 1 (X; Z) ∼ = π1 (X). As explained, L has the structure of a ZG-lattice; this structure determines the cohomology groups of X/G. Let R = Z(p) be the ring of integers localized at the prime p. Then (see [21]) there are only three distinct isomorphism classes of RG-lattices, namely the trivial module R, the augmentation ideal IG and the group ring RG. Moreover, if L is any finitely generated ZG-lattice, then there is a ZG-lattice L ∼ = Zr ⊕ ZGs ⊕ IGt and ZG homomorphism f : L → L such that f is an isomorphism after tensoring with R. In general, a ZG-module of the form Zr ⊕ ZGs ⊕ IGt is called a module of type (r, s, t).  51  Chapter 4. Cohomology of Toroidal Orbifolds A fundamental tool in the computation of the cohomology groups of the toroidal orbifolds is the next lemma. Lemma 4.2.1. Suppose that p is a prime number. Let G = Z/p act on a finite dimensional space X with fixed point set F . If there is an integer N k (X, F ; Z) = 0 for k > N . such that H k (X, F ; Z) = 0 for k > N , then HG  Proof.  This follows by applying [17, Exercise III.9] and [17, Proposition  VII 1.1]. From now on G denotes the group Z/p for a prime number p. Given a representation ϕ : G → GLn (Z), X denotes the G-space Rn /Zn with the G-action induced by ϕ and F denotes the fixed point set under this action. The representation ϕ makes L : H 1 (X; Z) into a ZG-module whose structure completely determines the cohomology groups H ∗ (X/G; Z) as it will be shown in the next section in the case of L = IGm .  4.3  The case L = IGm  In this section the particular case where the ZG-lattice L equals IGm is considered. Let ρm := ⊕m ρ : G → GLN (Z) be the integral representation inducing the ZG-module L, where N = m(p − 1). The fixed point set of such an action is easily identified by the following straightforward lemma. Lemma 4.3.1. If X = X⊕m ρ is the G-space induced by the representation ⊕m ρ then the fixed point set F under this G-action is a discrete set with pm points. Proof.  Consider the short exact sequence of G-modules defining the G-  space X 0→L→L  R → (L  R)/L = X → 0. 52  Chapter 4. Cohomology of Toroidal Orbifolds This short exact sequence induces a long exact sequence on the level of group cohomology 0 → H 0 (G, L) → H 0 (G, L R) → H 0 (G, X) → H 1 (G, L) → H 1 (G, L R) → . . . Note that H 1 (G, L  R) = 0 and H 0 (G, L) = H 0 (G, L  R) = 0, thus  there is an isomorphism F = H 0 (G, X) ∼ = H 1 (G, L) ∼ = (Z/p)m . ∗ (X, F ). These We are interested in computing the cohomology groups HG  can be computed using the Serre spectral sequence for the pair (X, F ) i+j E2i,j = H i (G, H j (X, F )) =⇒ HG (X, F ).  Therefore the structure of H j (X, F ) as a G-module needs to be studied. To this end, we consider the long exact sequence in cohomology associated to the pair (X, F ). Since H j (F ) = 0 when j ≥ 1, it follows that H j (X, F ) ∼ = H j (X) for j ≥ 2 and there is a short exact sequence m −1  0 → Zp  → H 1 (X, F ) → (IG)m → 0.  (4.2)  By the classification theorem for ZG-modules in [21], there are integers aj , bj and cj ≥ 0, ideals A1 , ..., Aaj of ZG rank p−1 and projective indecomposable modules P1 , ..., Pbj such that   aj  H (X, F ) ∼ = j  Ai 1  bj  ⊕    cj  Pj  ⊕ 1  Z . 1  53  Chapter 4. Cohomology of Toroidal Orbifolds In particular,    Zbj +cj   H i (G, H j (X, F )) ∼ = (Z/p)aj     (Z/p)cj  if  i = 0,  if  i is odd and i > 0,  if  i is even and i > 0.  On the other hand, for j ≥ 2 we obtain j  H (X, F ) ∼ = H j (X) ∼ = j  Hence the rank of (  bj 1  Pi ) ⊕ (  cj 1  j  (IG)m .  (IG)m is equal to the rank of H j (X, F ) ∼ =(  aj 1  Ai ) ⊕  Z). This implies the following equation when j ≥ 2: m(p−1) j  = aj (p − 1) + bj p + cj .  (4.3)  For each m, j ≥ 0, we define pm (j) to be the number of all possible sequences of integers l1 , ..., lm such that 0 ≤ lr ≤ p − 1 and l1 + · · · lm = j. This forces pm (j) = 0 for j > N = (p − 1)m, pm (0) = 1, pm (1) = m and by induction it is easy to see that m(p−1)  pm (j) = pm . j=0  When j ≥ 2, it follows from [3, Proposition 1.10] that   (Z/p)pm (j) H i (G, H j (X, F )) ∼ =  0  if  i + j is even and i > 0,  if  i + j is odd and i > 0.  54  Chapter 4. Cohomology of Toroidal Orbifolds This result implies   pm (j) aj =  0  if j is odd and  j ≥ 2,  if j is even and  j ≥ 2,     if j is odd and  j ≥ 2,  and  cj =  0  if j is even and j ≥ 2   p (j) m  We also need to find H 0 (G, H j (X, F )) ∼ = Zbj +cj . When j ≥ 2 by equation (4.3) we obtain  bj =      1 p  m(p−1) j 1 p  − (p − 1)pm (j)  m(p−1) j  − pm (j)  if  j is odd,  if  j is even.  The E2i,j of the this spectral sequence is described below  E2i,j    0       Zbj +cj       (Z/p)pm (j) =   0       (Z/p)a1      (Z/p)c1  if j = 0, if  i = 0 and j > 0,  if  i + j is even and i > 0, j ≥ 2,  if  i + j is odd and i > 0,  j ≥ 2,  if  i is odd and  j = 1,  if  i is even and  j = 1.  Consider the Serre spectral sequence ˜ i,j = H i (G, H j (X)) =⇒ H i+j (X) E 2 G  55  Chapter 4. Cohomology of Toroidal Orbifolds associated to the fibration sequence X → X ×G EG → BG. As it was pointed out before this sequence collapses on the E2 -term. The inclusion ˜ri,j . Notice X → (X, F ) defines a map of spectral sequences fri,j : Eri,j → E that f2i,j is an isomorphism when j ≥ 2. This implies that the only possible nontrivial differentials in the spectral sequence {Eri,j } must land in Eri,1 . On k (X, F ) = 0 for k > N ; this the other hand, by Lemma 4.2.1 it follows that HG  means that there are no permanent cocycles of total degree k with k > N . We notice that all the differentials that end at E22k+1,1 ∼ = (Z/p)a1 start at trivial groups. This implies that if 2k + 1 > N then E22k+1,1 must be the trivial group and thus a1 = 0. On the other hand, when 2 ≤ j ≤ N and i + j even, the only possible nonzero differential starting at E2i,j ∼ = (Z/p)pm (j) is dj : Eji,j → Eji+j,1 . If i + j > N , there are no permanent cocycles in Eji,j and thus dj must be injective. We note that the spectral sequence Eri,j is a spectral sequence of H ∗ (BG)-modules and we can find t ∈ H 2 (BG) such that multiplication by tk is an isomorphism tk : Eji,j → Eji+2k,j . Since dj is a map of H ∗ (BG) modules, we are able to choose k big enough such that i + j + 2k > N ; it follows that dj : Eji,j → Eji+j,1 is injective for all i + j > 0 and its image has rank cj = pm (j) when i = 0 and j is even. Moreover Zbj ⊂ (H j (X, F ))G lies inside the image of the norm map. We next consider the transfer map associated with the trivial subgroup {1} → G. This map preserves the filtrations that induce the Serre spectral sequence and thus it induces a map of the corresponding spectral sequences τ1G : H i ({1}, H j (X, F )) → H i (G, H j (X, F )). 56  Chapter 4. Cohomology of Toroidal Orbifolds Since the image of the transfer map τ1G : H 0 ({1}, H j (X, F )) → H 0 (G, H j (X, F )) consists of elements in the image of the norm map, it follows that all the differentials in the Serre spectral sequence vanish on the elements that are in the image of the norm map; in particular, dj is trivial on the summand Zbj . 2k,1 We conclude that when 2k > N , E22k,1 ∼ = (Z/p)c1 , E3 ∼ = (Z/p)c1 −pm (2) , 2k,1 ∼ c1 −( ... ,EN +1 = (Z/p)  PN  j=2  pm (j))  2k,1 . Also, if 2k + 1 > N , the group EN +1 must  vanish because its elements are not permanent cocycles and there are no 2k,1 i,j nontrivial differentials with target EN +1 as EN +1 = 0 for j ≥ 2. This shows  that c1 = (  N j=2 pm (j))  = pm − 1 − m. By the short exact sequence (4.2),  we obtain b1 = m. 0,1 ∼ ( Therefore E∞ =Z  PN  j=1  pm (j))  m −1  = Zp  k,1 ∼ , E∞ = (Z/p)(  PN  j=k+1  pm (j))  for  0,k ∼ bk +ck for 1 < k ≤ N and the other groups are 0 < k ≤ N even, E∞ =Z  trivial.  By studying the Serre spectral sequence for the pair (X, F ) with coefficients in Z/q with q a prime number different from p and also with rational coefficients, we can conclude that there are no extension problems and the following theorem is obtained. Theorem 4.3.2. Suppose that G = Z/p acts on X ∼ = ((S 1 )p−1 )m via the  57  Chapter 4. Cohomology of Toroidal Orbifolds representation ⊕m ρ. If F denotes the fixed point set under this action, then  k HG (X, F ) ∼ =           0 pm −1  Z    Zγk    P  pm (j))  Zγk ⊕ (Z/p)( m(p−1) j=k  where γk = bk + ck = 1/p[  m(p−1) k  if  k = 0,  if  k = 1,  if  k is even and k > 0,  if  k is odd and k > 0.  + (−1)k (p − 1)pm (k).  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