Higher genus characters of orbifolds

Physics Letters B 282 ( 1992 ) 349-351 North-Holland

PHYSICS LETTERS B

Higher genus characters of orbifolds P. Bantay Institute for Theoretical Physics, EOtvOs University, H-1088 Budapest, Hungary

Received 24 February 1992

Quantum group techniques are used in the study of the space of higher genus characters of holomorphic orbifold models.

Holomorphic orbifolds [ 1,2] are an interesting class of CFTs for which a quantum group description of the fusion rules and modular properties is available [3,4]. The aim of this paper is to show how the quantum group may be used to study the space of higher genus characters of these models. To a finite group G and a 3-cocycle ~ Z 3 ( G ) is associated the quantum group d ( G , ¢), which describes the modular properties and fusion rules of the corresponding orbifold model [3 ]. Let Y denote a compact Riemann surface of genus g, and dggthe space of genus g characters. The space ~ of genus 1 characters is naturally identified with the space Cg(G, ~) of characters of the quantum group ~¢ (G, q~). So for g = 1 the structure of ~g is well known, and the problem is to find a similar characterization of the space ~g for arbitrary g. What do we know about d(g? First of all, the knowledge of the fusion rules and modular properties allows us to compute its dimension. It is given by [ 1 ] dim ~ = ~ i

7t,(Y-)=(al, b, .... ,ag, bg: ,=l f i [a~, b~]= 1 ) ,

d, J

7gl ( ~ ) = / X l - , y I .... , X g , yg:

(1)

where the sum is over the irreducible representations of d (G, ~) (the index 0 labeling the trivial one ), dr is the dimension of the ith irreducible representation, and S is the matrix representing the modular transformation S (which interchanges the homology cycles) in the basis of irreducible characters. Moreover, we know that the space ~g should afford a uni-

(2)

where the generators at, b~ ( i = 1, ..., g) correspond to the curves of a canonical homology basis, and [a~, b~] =aT~bT~a~b~ is the commutator of a~ and bi. It turns out that a different presentation of nl (X) is more convenient for our purposes:

ISiol 2~t-g)

( [ G [ ~ 2(g-l>

=~\

tary representation of the mapping class group Fg of E. Finally, we know ~g in the case where the cocycle q~characterizing the model is trivial, i.e. can be taken to be identically 1. In this case ~g may be identified with the space of complex valued functions over the moduli space J//(Z, G) of G-bundles on Z. ~# (Z, G) is a finite set, which consists of the equivalence classes of homomorphisms from G to n~(Z), where two homomorphisms are equivalent iff their images are conjugate in n~ (Z) [ 1,2]. The standard presentation ofn~ (Z) is [ 5 ]

i=lfiXi=fix~i) ' i = 1

(3)

where xY=-y- lxy is the conjugate o f x by y. The generators a~, b~ may be expressed in terms of the generators x~, y~ through the conversion formula ai = x zi,

bi = y zi ,

where zi = I-[ xj . j> i

(4)

The inverse transformation is similar. Thus a homomorphism from G to nl (Z) is deter-

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

349

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PHYSICS LETTERSB

mined by a 2g-tuple xi, yi ( i = 1..... g) of elements of G which satisfy 1-If=l x~ = 1-[f= l x~i. Two such 2g-tupies correspond to equivalent homomorphisms if they are related by simultaneous conjugation of their elements. It follows from what have been said above that, when the cocycle ~ is trivial, the space ~g may be identified with the space of functions ~u(~ . . . xy,g ) of the 2g variables x~, y~, which satisfy

(

x~ ...Xg = 0 YL Y g /

ifl-[ xs¢l--[ x~'~ i=l

i=1

(5a) '

xl ... g =~/ . Y~ ... Y~ Yl ... Yg

(5b)

When the cocycle ~ is non-trivial, the space should still consist of functions ~ satisfying (5a). What changes is the form of (5b), namely we should allow for the appearance of a cocycle factor OOzon the RHS of (5b):

... y~J

4;:

. .xg. . . .

Xg

5

.

The problem of characterizing the space ~g has now been reduced to that of finding an explicit expression for the factor o~z appearing in (5c) in terms of the cocycle 0 which characterizes the model. The result is

;:) gl~i' ~/z(x~, Uj>~xj) ~ qz(y~, xf') i=1

qz(Xi, Yi)

(6) '

where the quantity rlz(X, y) is given by [ 3]

¢(x, z, y:) rb(x, y) = O ( x , y , z ) ~ ( z , x : , y : ) "

(7)

In summary, the space ~g consists of those complex valued functions that satisfy (5a) and (5c), where the factor o9~ is given by (6). Before presenting the line of thought that leads to (6), let us see the evidence supporting its validity. First, in case ~ is trivial it leads to the correct answer as explained above. For g = 1 it reduces to the known 350

28 May 1992

answer, because in that case conditions (5a) and (5c) characterize the characters of d (G, 0). Moreover, the dimension of the space of functions that satisfy (5a) and (5c) is equal to the dimension of ~g as given by ( 1 ). Finally, one can show that this space affords a unitary representation of the mapping class group Fg. All these facts together support strongly that the above identification of ~g is correct. To explain why the last requirement is non-trivial, we recall that the mapping class group Fg acts on the fundamental group 7q (Z) by (outer) automorphisms [ 5 ]. This induces in a natural way an action of Fg on the space of functions that satisfy (5a), by acting on their arguments. When 0 is non-trivial, extra cocycle factors should be included, similarly to the case for g = l [3 ]. But it is not obvious at all that the subspace characterized by (5c) should be invariant under this action. That this is indeed the case may be traced back to the 3-cocycle property of 0. Let us now see the reasoning that leads to (6). As is well known, a quantum group is an associative algebra A together with an imbedding A: A - , A ® A (the comultiplication) which satisfies certain additional requirements (like quasi-triangularity and quasicoassociativity). The comultiplication A allows us to construct a sequence ig of imbeddings of A into A ®g, the gth tensor power of A. i~ :A-~A is the identity mapping, and we define ig recursively:

i~= (1®1®..J®3)0/~_,.

(8)

The image ig(A) is then a subalgebra of A ®g isomorphic to A itself. Consider now the commutant C (ig(A) ) of ig(A ), that is the subalgebra consisting of those elements of A ®gwhich commute with all elements of ig (A). When g = 1, C(il ( A ) ) is just the center of A, and it is well known that it can be identified with the space ~ of genus 1 characters. Our claim is that, at least for the holomorphic orbifold models considered in this paper, there is a one-to-one correspondence between C(ig(A) ) and ~g. This is the argument that leads to the expression (6) for (oz, and, as we argued previously, there is strong supporting evidence in favor of its validity. The above approach offers a bonus, namely that C(ig(A) ) is an associative algebra, that is ~ comes endowed naturally with a convolution product. This

Volume 282, number 3,4

PHYSICS LETTERS B

extra structure might prove useful in deeper studies of the structure of ~g. In summary, we have identified the space ~g of genus g characters holomorphic orbifold models from solely q u a n t u m group information. Whether the above ideas could be generalized to other classes of CFTs or to non-compact R i e m a n n surfaces is not known to us. If such a generalization exists, it would to a better u n d e r s t a n d i n g of the relationship between CFTs and q u a n t u m groups.

28 May 1992

References

[ 1] R. Dijkgraaf, C. Vafa, E. Verlindeand H. Verlinde,Commun. Math. Phys, 123 (1989) 485. [2] R. Dijkgraaf and E. Witten, Commun. Math. Phys. 129 (1990) 393. [3] P. Bantay, Phys. Lett. B 245 (1990) 477; Lett. Math. Phys. 22(1991) 187. [4] R. Dijkgraaf, V. Pasquier and P. Roche, preprint PUPT-1169 (1990). [5 ] J. Stillwell,Classicaltopologyand combinatorialgroup theory (Springer, Berlin, 1980).

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