Fusion Rules for the Free Bosonic Orbifold Vertex Operator Algebra

Journal of Algebra 229, 333᎐374 Ž2000. doi:10.1006rjabr.2000.8311, available online at http:rrwww.idealibrary.com on

Fusion Rules for the Free Bosonic Orbifold Vertex Operator Algebra Toshiyuki Abe Department of Mathematics, Graduate School of Science, Osaka Uni¨ ersity, Toyonaka, Osaka 560-0043, Japan E-mail: [email protected] Communicated by Geoffrey Mason Received July 27, 1999

Fusion rules among irreducible modules for the free bosonic orbifold vertex operator algebra are completely determined. 䊚 2000 Academic Press

1. INTRODUCTION We determine the fusion rules for the free bosonic orbifold vertex operator algebra M Ž1.q which is the fixed point set of the free bosonic vertex operator algebra M Ž1. under an automorphism ␪ of order 2. In wZx, Zhu introduced an associative algebra AŽ V . Žcalled Zhu’s algebra. associated to a vertex operator algebra V. Zhu’s algebra AŽ V . inherits a part of the vertex operator algebra structure of V which affords much information on V-modules. For example, there exists a one-to-one correspondence between the set of equivalence classes of irreducible ⺞-gradable V-modules and the set of equivalence classes of irreducible AŽ V .-modules. Later, in wFZx, the notion of Zhu’s algebra is generalized to an AŽ V .-bimodule AŽ M . for an ⺞-gradable V-module M, and the fusion rules of rational vertex operator algebras are completely characterized in terms of these bimodules Žsee also wLix.. More precisely, for irreducible V-modules M i Ž i s 1, 2, 3., with nontrivial top level M0i , there exists a natural injection from

I

ž

M3 , M M2 1

/

333 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

334

TOSHIYUKI ABE

which is the space of intertwining operators of type

ž

M3 , M M2 1

/

into the dual space of the contraction Ž M03 .* m AŽV . AŽ M 1 . m AŽV . M02 . Moreover, if V is rational, then this map becomes an isomorphism. By using this isomorphism, fusion rules were calculated for vertex operator algebras associated with finite-dimensional simple Lie algebras wFZx, for the minimal series wWx, etc. Let ᒅ be a d-dimensional complex vector space with a nondegenerate symmetric bilinear form and let ˆ ᒅ be its affinization. Then the Fock space M Ž1. s SŽ ᒅ m ty1 ⺓w ty1 x. is a simple vertex operator algebra with central charge d and has the automorphism ␪ of order 2 lifted from the map ᒅ ª ᒅ, h ¬ yh wFLMx. The fixed point set M Ž1.q of M Ž1. is also a simple vertex operator algebra and the y1-eigenspace M Ž1.y is an irreducible M Ž1.q-module. It is well known that for every ␭ g ᒅ the Fock space M Ž1, ␭. s SŽ ᒅ m ty1 ⺓w ty1 x. m ⺓ e ␭ is an irreducible M Ž1.-module and the set  M Ž1, ␭. N ␭ g ᒅ 4 gives all the inequivalent irreducible M Ž1.modules. Moreover, if ␭ / 0, M Ž1, ␭. is an irreducible M Ž1.q-module, and M Ž1, ␭. and M Ž1, y␭. are isomorphic to each other as the M Ž1.q-module Žsee wDMx.. In addition, the ␪-twisted M-module M Ž1.Ž ␪ . defined as an induced module of the twisted affine Lie algebra ˆ ᒅ wy1x is also an M Ž1.q-module, and the "1-eigenspaces M Ž1.Ž ␪ . " for ␪ give inequivalent irreducible M Ž1.q-modules. It is known that these irreducible modules M Ž1. ", M Ž1, ␭. Ž, M Ž1, y␭.. Ž ␭ / 0., and M Ž1.Ž ␪ . " give all the inequivalent irreducible M Ž1.q-modules Žsee wDN1x and wDN2x.. In this paper, we only consider the rank one, i.e., the d s 1, case and determine the fusion rules among any triples of irreducible M Ž1.q-modules. One of the main results in this paper is the following theorem. THEOREM. Ž1.

Let M, N, and L be irreducible M Ž1.q-modules.

If M s M Ž1.q, then NMLŽ1.q N s ␦ N, L .

Ž2. If M s M Ž1.y, then NMLŽ1.y N is 0 or 1, and NMLŽ1.y N s 1 if and only if the pair Ž N, L. is one of the following pairs: .

Ž M Ž 1. " , M Ž 1. . ,

.

Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . . ,

Ž M Ž 1, ␭ . , M Ž 1, ␮ . . Ž ␭2 s ␮2 . .

FUSION RULES FOR BOSONIC ORBIFOLD

335

Ž3. If M s M Ž1, ␭. Ž ␭ / 0., then NMLŽ1, ␭. N is 0 or 1, and NMLŽ1, ␭ . N s 1 if and only if the pair Ž N, L. is one of the following pairs:

Ž M Ž 1. " , M Ž 1, ␮ . . Ž ␭2 s ␮2 . , Ž M Ž 1, ␮ . , M Ž 1, ␯ . . Ž ␯ 2 s Ž ␭ " ␮ . 2 . , . Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . " . , Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . . . Ž4. If M s M Ž1.Ž ␪ .q, then NMLŽ1.Ž ␪ .q N is 0 or 1, and NMLŽ1.Ž ␪ .q N s 1 if and only if the pair Ž N, L. is one of the following pairs:

Ž M Ž 1. " , M Ž 1. Ž ␪ . " . ,

Ž M Ž 1, ␭. , M Ž 1. Ž ␪ . " . .

Ž5. If M s M Ž1.Ž ␪ .y, then NMLŽ1.Ž ␪ .y N is 0 or 1, and NMLŽ1.Ž ␪ .y N s 1 if and only if the pair Ž N, L. is one of the following pairs: .

Ž M Ž 1. " , M Ž 1. Ž ␪ . . ,

Ž M Ž 1, ␭. , M Ž 1. Ž ␪ . " . .

Let M i Ž i s 1, 2, 3. be irreducible M Ž1.q-modules. Then the classification result of irreducible M Ž1.q-modules in wDN1x and formal characters 3 for irreducible M Ž1.q-modules show that the fusion rule NMM1 M 2 is invariant under any permutation of  1, 2, 34 . In more detail, the explicit forms of the formal characters of irreducible M Ž1.q-modules imply that two irreducible M Ž1.q-modules with the same formal characters are isomorphic to each other, and in particular that every irreducible M Ž1.q-module is isomorphic to its contragredient module. Then the above symmetry-of-fusion rules follow from the fact that NMLN s NNLM s NMNL⬘ holds for modules M, N, L of a vertex operator algebra, where N⬘ and L⬘ are contragredient modules of N and L respectively. Next, we prove that AŽ M 1 . is generated as an AŽ M Ž1.q. -bimodule by at most two elements which are images of singular vectors of M 1 viewed as a module for the Virasoro algebra. This is obtained by using the fact that the M Ž1.q is generated by the Virasoro element and a singular vector of weight 4 Žsee wDGx.. Further, using the 3 Frenkel᎐Zhu injection, we prove that the fusion rule NMM1 M 2 is less than 2. 3 A more detailed study of the contraction Ž M0 .* m AŽV . AŽ M 1 . m AŽV . M02 3 of AŽ M Ž1.q. -modules implies that the fusion rule NMM1 M 2 is in fact less than 1. In wFLMx and wDN1x, the nontrivial intertwining operator of type

ž

M Ž 1, ␭ q ␮ . M Ž 1, ␭ . M Ž 1, ␮ .

/

336

TOSHIYUKI ABE

was constructed for every ␭, ␮ g ⺓. This gives us nontrivial intertwining operators of types M Ž 1.

ž

M Ž 1.

q

and

"

M Ž 1.

ž

"

/ ž ,

M Ž 1, ␭ . M Ž 1.

"

M Ž 1, ␭ .

/

,

M Ž 1, ␭ q ␮ . . M Ž 1, ␭ . M Ž 1, ␮ .

/

The fusion rules of corresponding types are nonzero. In addition, in wFLMx, a twisted vertex operator from M Ž1, ␭. to HomŽ M Ž1.Ž ␪ ., M Ž1.Ž ␪ .. z 4 was obtained for every ␭ g ⺓. This provides the nontrivial intertwining operators of types

ž

M Ž 1. M Ž 1.

"

q

M Ž 1. Ž ␪ .

"

/

and

ž

M Ž 1. Ž ␪ . M Ž 1, ␭ .

␤

M Ž 1. Ž ␪ .

␣

/

for any ␣ , ␤ g  q, y4 . Thus the fusion rules of corresponding types are also nonzero. The study of the contractions also shows that all nonzero fusion rules are derived from these fusion rules by means of the above symmetry-of-fusion rules and the equivalency between M Ž1, ␭. and M Ž1, y␭. for ␭ g ⺓. The organization of this paper is as follows. We recall the definitions of vertex operator algebras, modules, and fusion rules in Section 2.1, and those of Zhu’s algebras and their bimodule in Section 2.2, where we also explain the relation between fusion rules and the bimodules. We review the vertex operator algebra M Ž1.q and its irreducible modules in Section 2.3. In Section 3.1, we describe the irreducible decompositions of the irreducible M Ž1.q-modules as modules for Virasoro algebra and prove that 3 NMM1 M 2 is invariant under any permutation of  1, 2, 34 for irreducible M Ž1.q-modules M i Ž1 F i F 3.. In Section 3.2 we prove some lemmas and a proposition ŽProposition 3.7. which gives a generalization of Zhu’s anti-isomorphism of AŽ V ., and in Section 3.3 we give a set of generators of AŽ M . for an irreducible M Ž1.q-module M and show that all fusion rules among irreducible M Ž1.q-modules are less than 2. In Section 4.1 we explain that the vertex operators and twisted vertex operators constructed in wFLMx give some nonzero intertwining operators among irreducible M Ž1.q-modules, and we state the main theorem. In Section 4.2 we prove the main theorem by studying the structure of the contraction Ž M03 .* ⭈ AŽ M 1 . ⭈ M02 for irreducible M Ž1.q-modules M i Ž1 F i F 3..

337

FUSION RULES FOR BOSONIC ORBIFOLD

2. PRELIMINARIES We recall the definitions of vertex operator algebras, their modules from wFLM, DLM1, and DMZx, and fusion rules from wFHLx in Section 2.1. In Section 2.2, following wZx and wFZx, we review the definition of Zhu’s algebra AŽ V . associated to a vertex operator algebra V and its bimodule AŽ M . for an ⺞-gradable V-module M. In Section 2.3, following wFLMx, we recall the vertex operator algebra M Ž1.q and its irreducible modules. Throughout the paper, ⺞ is the set of nonnegative integers and ⺪ ) 0 is the set of positive integers. For vector space V, the vector space of the formal power series in z is denoted by V z4 s

½Ý

5

¨n z n ¨n g V ,

ng⺓

and we set the subspaces V ww z, zy1 xx and V ŽŽ z .. as V w z, zy1 x s

½Ý

5

¨n z n ¨n g V ,

ng⺪

VŽ Ž z.. s

½

⬁

Ý ¨n z n

5

k g ⺪, ¨ n g V .

nsk

For f Ž z . s Ý ng ⺓ ¨ n z n g V z 4 , ¨ y1 is called the formal residue denoted by Res z f Ž z . s ¨ y1. 2.1. Vertex Operator Algebras, Modules, and Fusion Rules DEFINITION 2.1. A ⺪-graded vector space V s [ ng ⺪ Vn such that dim Vn is finite for all integers n and Vn s 0 for a sufficiently small integer n is called a ¨ ertex operator algebra if V is equipped with a linear map Y : V ª Ž End V . w z, zy1 x ¨ ¬ YŽ¨ , z. s

Ý

¨ n zyny1

Ž ¨ n g End V .

ng⺪

and with two distinguished vectors 1 g V0 and ␻ g V2 such that the conditions Y Ž a, z . b g V Ž Ž z . . and ŽJacobi identity. zy1 0 ␦

ž

z1 y z 2 z0

s zy1 2 ␦

ž

/

Y Ž a, z1 . Y Ž b, z 2 . y zy1 0 ␦

z1 y z 0 z2

/

Y Ž Y Ž a, z 0 . b, z 2 .

ž

z 2 y z1 yz 0

/

Y Ž b, z 2 . Y Ž a, z1 .

338

TOSHIYUKI ABE

hold for a, b g V and m, n g ⺪, where ␦ Ž z . s Ý ng ⺪ z n and all binomial expressions are to be expanded as formal power series in the second variable. Y Ž a, z . 1 g V w z x ,

Y Ž 1, z . s id V ,

Y Ž a, z . 1 N zs0 s a.

and

We set Y Ž ␻ , z . s Ý ng ⺪ LŽ n. zyn y2 , then LŽ n. Ž n g ⺪. forms a Virasoro algebra, LŽ m. , LŽ n. s Ž m y n. LŽ m q n. q

m3 y m 12

␦mq n , 0 c V , Ž 2.1.

for any m, n g ⺪, where c V , which is called the central charge of V, g ⺓. L Ž 0 . a s na

for n g ⺪, a g Vn ,

Y Ž L Ž y1 . a, z . s

d dz

Y Ž a, z . .

Ž 2.2.

The vertex operator algebra is denoted by Ž V, Y, 1, ␻ . or simply by V. An element a g Vn is called a homogeneous element of weight n and we write n s wtŽ a.. An automorphism g of a vertex operator algebra V is a linear automorphism of V such that gY Ž a, z . gy1 s Y Ž g Ž a., z . for all a g V and g Ž ␻ . s ␻ . Set Aut V to be the set of all automorphisms of V and let G be a subgroup of Aut V. Then the fixed point set for G naturally becomes a vertex operator algebra. This vertex operator algebra is called the orbifold of V Žcf. wDVVV, DMx.. Let g be an automorphism of a vertex operator algebra V of order T. Then V is decomposed into the eigenspaces for g: Ty1

Vs

[ V r,

V r s  a g V N g Ž a . s ey2 ␲ i r r Ta4 .

rs0

DEFINITION 2.2. Let V be a vertex operator algebra and let g be an automorphism of order T. A weak g-twisted V-module M is a vector space equipped with a linear map YM : V ª Ž End M .  z 4 , a ¬ YM Ž a, z . s

Ý ng⺡

a nM zyny1

Ž anM g End M . ,

FUSION RULES FOR BOSONIC ORBIFOLD

339

such that the conditions YM Ž a, z . s

Ý

a nM zyny1 ,

YM Ž a, z . ¨ g zyr r T M Ž Ž z . . ,

ngrrTq⺪

YM Ž 1, z . s id M , and Žtwisted Jacobi identity. zy1 0 ␦

ž

z1 y z 2 z0

s zy1 2

ž

/

YM Ž a, z1 . YM Ž b, z 2 . y zy1 0 ␦

z1 y z 0 z2

yrrT

␦

/ ž

z1 y z 0 z2

/

ž

z 2 y z1 yz 0

/

YM Ž b, z 2 . YM Ž a, z1 .

YM Ž Y Ž a, z 0 . b, z 2 .

Ž 2.3.

hold for 0 F r F T y 1, a g V r , b g V, and u g M. The weak g-twisted V-module is denoted by Ž M, YM . or simply by M. In the case g is an identity of V, the weak g-twisted V-module is called a weak V-module. Here and later we denote the component operator a nM Ž a g V, n g ⺡. by a n for simplicity. Let Ž M, YM . be a weak g-twisted V-module and set YM Ž ␻ , z . s Ý ng ⺪ LŽ n. zyn y2 . Then the operators  LŽ n.; Ž n g ⺪., id M 4 also form a Virasoro algebra with central charge c V . Moreover, we also have Ž2.2. Žsee wDLM2x.. DEFINITION 2.3. Let V be a vertex operator algebra and let g be an automorphism of V of order T. A T1 ⺞-gradable g-twisted V-module M is a weak g-twisted V-module which has a T1 ⺞-grading M s [ ng Ž1r T .⺞ Mn such that a m Mn ; MwtŽ a.qnymy1 holds for any homogeneous a g V, n g T1 ⺞, and m g ⺡. In the case g is an identity on V, the T1 ⺞-gradable g-twisted V-module is called an ⺞-gradable V-module. An element u g Mn is called a homogeneous element of degree n and we write n s degŽ ¨ .. DEFINITION 2.4. An ordinary g-twisted V-module M is a weak g-twisted V-module on which LŽ0. acts semisimply, Ms

[ M Ž ␭. ,

␭g⺓

M Ž ␭ . s  u g M N L Ž 0 . u s ␭ u4 ,

such that each eigenspace is finite dimensional and for fixed ␭ g ⺓, M Ž ␭ q nrT . s 0 for a sufficiently small integer n. In the case in which g is an identity of V, the ordinary g-twisted V-module is called an ordinary V-module, or more simply a V-module. An

340

TOSHIYUKI ABE

element u g M Ž ␭. is said to be homogeneous of weight ␭ and we write ␭ s wtŽ u.. The notions of submodules and irreducible modules are defined in the obvious way. Let M be a V-module, then the restricted dual M⬘ s [ ␭g ⺓ M Ž ␭.* is a V-module and the vertex operator YMU Ž a, z . for a g V is defined by ² YMU Ž a, z . u⬘, ¨ : s ² u⬘, YM e z LŽ1. Ž yzy2 .

ž

L Ž0 .

a, zy1 ¨ :

/

for u⬘ g M⬘, ¨ g M. This V-module Ž M⬘, YMU . is called the contragredient module of M. It is known that if M irreducible, then M⬘ is also irreducible Žcf. wFHLx.. DEFINITION 2.5. Let V be a vertex operator algebra and Ž M i, YM i . Ž i s 1, 2, 3. be weak V-modules. An intertwining operator of type

ž

M3 M M2 1

/

is a linear map I : M 1 ª Ž Hom Ž M 2 , M 3 . .  z 4 , ¨ ¬ IŽ ¨ , z. s

Ý

¨ n zyny1

Ž ¨ n g HomŽ M 2 , M 3 . .

ng⺓

such that for a g V, ¨ g M 1 , and u g M 2 the following conditions hold. For fixed n g ⺓, ¨ nq k u s 0 for sufficiently large integer k, zy1 0 ␦

Ž Jacobi identity . y zy1 0 ␦ s zy1 2 ␦

ž ž

z 2 y z1 yz 0 z1 y z 0 z2 d dz

ž

z1 y z 2 z0

/

YM 3 Ž a, z1 . I Ž ¨ , z 2 .

I ¨ , z 2 . YM 2 Ž a, z1 .

/Ž /Ž

I YM 1 Ž a, z 0 . ¨ , z 2 . ,

Ž 2.4.

I Ž ¨ , z . s I Ž L Ž y1 . ¨ , z . .

We denote the vector space composed of the intertwining operators of type

ž

M3 M1 M 2

/

341

FUSION RULES FOR BOSONIC ORBIFOLD

by I

ž

M3 . M M2

/

1

The dimension of this vector space is called a fusion rule of corresponding 3 type and is denoted by NMM1 M 2 . It is well known that fusion rules have the following symmetry Žsee wFHLx and wHLx.. PROPOSITION 2.6.

Let M i Ž i s 1, 2, 3. be V-modules. Then

3

3

NMM1 M 2 s NMM2 M 1

3

2

NMM1 M 2 s NMŽ M1 Ž M.⬘3 .⬘ .

and

2.2. Zhu’s Algebra AŽ V . and the AŽ V .-Bimodule AŽ M . We recall the definition of Zhu’s algebra. Two bilinear products ) and ( on V are defined as follows: For homogeneous a g V and b g V, we define

Ž 1 q z . wt a

Ž .

a) b s Res z

ž ž

z

Ž 1 q z . wt a

/ /

Y Ž a, z . b,

Ž .

a( b s Res z

z2

Y Ž a, z . b,

and extend these to V by linearity. Let O Ž V . be the linear span of a( b Ž a, b g V . and set AŽ V . s VrOŽ V .. Let M be an ⺞-gradable V-module. For M every homogeneous a g V, define oŽ a. s a wtŽ a.y1 and extend to V linearly. The following proposition is due to Zhu Žsee wZx.. PROPOSITION 2.7. Ž1. The bilinear product ) induces AŽ V ., an associati¨ e algebra structure. The ¨ ector 1 q O Ž V . is the identity and ␻ q O Ž V . is in the center of AŽ V .. Ž2. The linear map o: V ª End M0 , ¨ ¬ oŽ ¨ . N M induces an associa0 ti¨ e algebra homomorphism o: AŽ V . ª End M0 . Thus M0 is a left AŽ V .module. Next we recall the AŽ V .-bimodule AŽ M .. Let Ž M, YM . be an ⺞-gradable V-module. Define bilinear maps ): M ª M1 , (: V = M ª M, ): M = V ª M by

Ž 1 q z . wt a

Ž .

a) u s Res z

ž ž

a( u s Res z

z

Ž 1 q z . wt a

Ž .

z2

wtŽ a .

YM Ž a, z . u s

/ /

YM Ž a, z . u s

ž Ý ž

Ý

is0

wt Ž a . is0

wt Ž a . a iy1 u, Ž 2.5. i

/ Ž . /

wt a i

a iy2 u,

342

TOSHIYUKI ABE

and

Ž 1 q z . wt a y1 Ž .

u) a s Res z

ž

z

YM Ž a, z . u s

/

wtŽ a .y1

Ý is0

ž

wt Ž a . y 1 a iy1 u, i

/

Ž 2.6. for homogeneous a g V and u g M respectively, and extend these to linear operations on V. Let O Ž M . be the linear span of a( u for a g V, u g M, and set AŽ M . s MrOŽ M .. We denote the image of u g M in AŽ M . by w u x. Then AŽ M . is an AŽ V .-bimodule, and the left and right actions are given by w ax)w u x s w a) u x and w u) ax s w u) ax respectively for a g V, u g M. DEFINITION 2.8. Let A be an right A-module, an A-bimodule, tensor product of R, B, and L as a contraction of R, B, and L and

associative algebra and R, B, L be a and a left A-module respectively. The an A-module R m A B m A L is called is denoted by R ⭈ B ⭈ L.

⬁ Now let M i s [ ns0 Mni Ž i s 1, 2, 3. be ⺞-gradable V-modules. Suppose that for any i there exists a scalar h i g ⺓ such that LŽ0. acts on Mni as h i q n. Let I be an intertwining operator of type

ž

M3 . M M2 1

/

T h e n for e a ch ¨ g M 1 , w e h a ve I Ž ¨ , z . g z y h 1 y h 2 q h 3 ŽHomŽ M 2 , M 3 ..ww z, zy1 xx Žsee Proposition 1.5.1 in wFZx.. We define oŽ ¨ . s Res z z h1qh 2yh 3qd egŽ ¨ .y1 I Ž ¨ , z . for homogeneous ¨ g M 1 and extend it to M 1 by linearity. Then we have following theorem Žsee Theorem 1.5.2 in wFZx.. ⬁ i Ž . THEOREM 2.9. Let M i s [ ns 0 Mn i s 1, 2, 3 be ⺞-gradable V-modi ules. Suppose that for each M there exists an h i g ⺓ such that LŽ0. acts on ⬁ Ž Mn3 .* be the contragredient Mni as a scalar h i q n. Let Ž M 3 .⬘ s [ ns0 3 module of M . Then the linear map

␲: I

ž

M3 ª Ž Ž M03 . * ⭈ A Ž M 1 . ⭈ M02 . *, M M2

/

1

I ¬ ␲ Ž I . Ž 2.7.

gi¨ en by the property ␲ Ž I .Ž ¨ X3 m w ¨ 1 x m ¨ 2 . s ² ¨ X3 , oŽ ¨ 1 . ¨ 2 : for IgI is well-defined.

ž

M3 , ¨ X3 g Ž M03 . *, ¨ 1 g M 1 , and ¨ 2 g M02 M M2 1

/

FUSION RULES FOR BOSONIC ORBIFOLD

343

If V is rational, that is, all ⺞-gradable V-modules are completely reducible, then the linear map ␲ is an isomorphism. In general, ␲ is not surjective Žsee wLix., but we have the following proposition. ⬁ i Ž . PROPOSITION 2.10. Let M i s [ ns 0 Mn i s 1, 2, 3 be as in Theorem 2 2.9. Suppose that M is irreducible and that M 3 is an irreducible ordinary. Then ␲ is injecti¨ e. Thus we ha¨ e 3

NMM1 M 2 F dim Ž Ž M03 . * ⭈ A Ž M 1 . ⭈ M02 . *. Proof. Let I be an intertwining operator of type

ž

M3 . M M2

/

1

Suppose that ␲ Ž I . s 0. Then we have ² ¨ X3 , I Ž ¨ 1 , z . ¨ 2 : s 0 for any ¨ X3 g Ž M03 .*, ¨ 1 g M 1, and ¨ 2 g M02 . Set W s  u g M 2 N ² ¨ X3 , I Ž ¨ 1 , z . u: s 0 for any ¨ X3 g Ž M03 . *, ¨ 1 g M 1 4 > M02 . Note that for any nonzero u g M02 , we have a n u g W for any homogeneous a g V, n g ⺪. In fact, it is obvious that a n u g W for n G wtŽ a. y 1, and for n - wtŽ a. y 1 we have ² ¨ 3 , a nŽ I Ž ¨ 1 , z . u.: s 0 for any ¨ X3 g Ž M03 .* and ¨ 1 g M 1. Hence we see that ² ¨ X3 , I Ž ¨ 1 , z . a n u: ⬁

s ² ¨ X3 , a n I Ž ¨ 1 , z . u: y

Ý is0

n nyi ² X z ¨ 3 , I Ž a i ¨ 1 , z . u: s 0, i

ž /

from the Jacobi identity Ž2.4.. Thus a n u g W for all n - wtŽ a. y 1. Since M 2 is irreducible, M 2 is spanned by a n u for homogeneous a g V Žsee wDMx., and then W s M 2 . Let w⬘ g Ž M03 .* be a nonzero element. Then for every a g V and n g ⺪, we have ² w⬘, a n Ž I Ž ¨ 1 , z . ¨ 2 . : ⬁

s ² w⬘, I Ž ¨ 1 , z . a n¨ 2 : q

Ý is0

n ny i ² z w⬘, I Ž a i ¨ 1 , z . ¨ 2 : s 0. i

ž /

Hence ² w⬘, YM 3 Ž a, z 0 . I Ž ¨ 1 , z . ¨ 2 : s 0. This implies that ² YMU 3 Ž a, z 0 . w⬘, I Ž ¨ 1 , z . ¨ 2 : s ² w⬘, YM 3 e z 0 LŽy1. Ž yzy2 0 .

ž

L Ž0 .

a, zy1 I Ž ¨ 1 , z . ¨ 2 : s 0. 0

/

344

TOSHIYUKI ABE

Therefore ² aUn w⬘, I Ž ¨ 1 , z . ¨ 2 : s 0 for any a g V and n g ⺪, where YMU 3 Ž a, z . s Ý ng ⺪ aUN zyny1. Since Ž M 3 .⬘ is irreducible, it is spanned by aUn w⬘ for a g V and n g ⺪. Then for every ¨ X3 g Ž M 3 .⬘, ¨ 1 g M 1 , and X ¨ 2 g M 2 , ² ¨ 3 , I Ž ¨ 1 , z . ¨ 2 : s 0, which means I s 0. 2.3. Vertex Operator Algebra M Ž1.q Let ᒅ be a d-dimensional vector space with a nondegenerate symmetric bilinear form ² , :, and let ˆ ᒅ s ᒅ m ⺓w t, ty1 x [ ⺓ K be a Lie algebra with the commutation relation given by w h m t m , h⬘ m t n x s m ␦mq n, 0 ² h, h⬘: K, w K, ˆ ᒅ x s 0. Set ˆ ᒅqs ᒅ m ⺓w t x [ ⺓ K. For ␭ g ᒅ, let ⺓ e ␭ be a one-dimenq sional ˆ ᒅ -module on which ᒅ m t⺓w t x acts trivially, ᒅ acts as ² h, ␭: for h g ᒅ, and K acts as 1. Let M Ž1, ␭. be an ˆ ᒅ-module induced by the ˆᒅq-module ⺓ e ␭ : M Ž 1, ␭ . s U Ž ˆ ᒅ . mUŽ ᒅˆq .⺓ e ␭ , S Ž ᒅ m ty1 ⺓ w ty1 x .

Ž linearly . .

Denote the action of h m t n Ž h g ᒅ, n g ⺪. on M Ž1, ␭. by hŽ n. and set Ž h z . s Ý ng ⺪ hŽ n. zyn y1. For ␭, ␮ g ᒅ, we define a linear map P␭␮ : M Ž1, ␮ . ª M Ž1, ␭ q ␮ . by P␭␮Ž u m e ␮ . s u m e ␭q ␮ for u g SŽ ᒅ m ty1 ⺓w ty1 x.. Then a ¨ ertex operator associated with e ␭ is defined by I␭␮ Ž e ␭ , z . s exp

ž

␭Ž yn .

Ý ng⺪ )0

n

/ ž

z n exp y

␭Ž n .

Ý ng⺪ ) 0

n

zyn P␭␮ z ² ␭ , ␮ : .

/

Ž 2.8. The vertex operator associated with h1Žyn1 . h 2 Žyn 2 . ⭈⭈⭈ h k Žyn k . e ␭ g M Ž1, ␭. Ž n i g ⺪ ) 0 , h i g ᒅ . is defined by I␭␮ Ž h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . e ␭ , z . Ž n y1 . Ž n y1 . Ž n y1 . ␭ s( ( ⭸ 1 h1 Ž z . ⭸ 2 h 2 Ž z . ⭈⭈⭈ ⭸ k h k Ž z . I␭␮ Ž e , z . ( ( , Ž 2.9.

where ⭸ Ž n. s n!1 Ž dzd . n for n g ⺞ and the normal ordering ( (⭈( ( is an operation which reorders so that hŽ n. Ž n - 0., P␭ are placed to the left of hŽ n. Ž n g ⺞.. We extend I␭␮ to M Ž1, ␭. by linearity. Now let  ␣ 1 , ␣ 2 , . . . , ␣ d 4 Ž d s dim ᒅ . be an orthonormal basis of ᒅ and set 1 s 1 m e 0 , ␻ s Ž1r2.Ý dis1 ␣ i Žy1. 2 1 g M Ž1, 0.. Then Ž M Ž1, 0., I00 , 1, ␻ . is a simple vertex operator algebra with central charge d and Ž M Ž1, ␭., I0 ␭ . N ␭ g ᒅ 4 gives all irreducible M Ž1, 0.-modules Žsee wFLMx.. The vertex operator algebra M Ž1, 0. is called the free bosonic ¨ ertex operator algebra and is denoted by M Ž1..

345

FUSION RULES FOR BOSONIC ORBIFOLD

Let ␪ be an automorphism of M Ž1. defined by

␪ Ž h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . 1 . k

s Ž y1 . h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . 1 for h i g ᒅ, n i g ⺪ ) 0 . We denote the orbifold of M Ž1. for ␪ by M Ž1.q and the y1-eigenspace of M Ž1. by M Ž1.y. Then the following proposition is proved in wDN1x. PROPOSITION 2.11. The ¨ ertex operator algebra M Ž1.q is simple, and M Ž1.y, M Ž1, ␭. Ž ␭ / 0. are irreducible M Ž1.q-modules. Moreo¨ er, M Ž1, ␭. is isomorphic to M Ž1, y␭. as an M Ž1.q-module. Next we consider the ␪-twisted M Ž1.-module. Let ˆ ᒅ wy1x s ᒅ m t 1r2 ⺓w t, ty1 x [ ⺓ K be a Lie algebra with commutation relation w h m t m , h⬘ m t n x s m ␦mq n, 0 ² h, h⬘: K, w K, ˆ ᒅ wy1xx s 0 for h, h⬘ g ᒅ, m, n g 1r2 q ⺪. Set ˆ ᒅ wy1xqs ᒅ m t 1r2 ⺓w t x [ ⺓ K. Then ⺓ is viewed as an ˆᒅ wy1xq-module on which ᒅ m t 1r2 ⺓w t x acts trivially and K acts as 1. Set M Ž1.Ž ␪ . to be the induced ˆ ᒅ wy1x-module: y1 r2 q ⺓ , SŽ ᒅ m t M Ž 1. Ž ␪ . s U Ž ˆ ᒅ w y1 x . mUŽ ᒅwy1x ⺓ w ty1r2 x . ˆ .

Ž linearly . .

Denote the action of h m t n Ž h g ᒅ, n g 1r2 q ⺪. on M Ž1.Ž ␪ . by hŽ n. and set hŽ z . s Ý ng 1r2q⺪ hŽ n. zyn y1. For ␭ g ᒅ, a twisted ¨ ertex operator associated with e ␭ g M Ž1, ␭. is defined as I␭␪ Ž e ␭ , z . s zy² ␭ , ␭: r2 exp

ž

= exp y

ž

␭Ž yn .

Ý ng1r2q⺞

␭Ž n .

Ý ng1r2q⺞

n

n zyn .

/

zn

/ Ž 2.10.

For h1Žyn1 . h 2 Žyn 2 . ⭈⭈⭈ h k Žyn k . m e ␭ g M Ž1, ␭. Ž h i g ᒅ, n g ⺪ ) 0 ., set W␭␪ Ž h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . e ␭ , z . Ž n y1 . Ž n y1 . Ž n y1 . ␭ s( ( ⭸ 1 h1 Ž z . ⭸ 2 h 2 Ž z . ⭈⭈⭈ ⭸ k h k Ž z . I␭ Ž e , z . ( (

and extend to a linear operator on M Ž1, ␭., where the normal ordering ( ( Ž . Ž . Ž . ⭈( is an operation which sifts h n n g ⺪ to the right and h n ( )0 Ž n g ⺪ - 0 . to the left. Let c m n g ⺡ Ž m, n g ⺞. be constants defined by

346

TOSHIYUKI ABE

the formal power series expansion

Ý

c m n x y s ylog m

n

m, nG0

ž

Ž1 q x.

1r2

q Ž1 q y.

1r2

2

/

.

and define an operator ⌬ z on M Ž1, ␭. by d

⌬z s

Ý Ý c m n ␣ i Ž m . ␣ i Ž n . zym yn . m, nG0 is0

Then the twisted vertex operator associated with u g M Ž1, ␭. is defined by I␭␪ Ž u, z . s W␭␪ Ž e ⌬ z u, z . .

Ž 2.11.

Now define the action of ␪ on M Ž1.Ž ␪ . by

␪ Ž h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . 1 . k

s Ž y1 . h1 Ž yn1 . h 2 Ž yn 2 . ⭈⭈⭈ h k Ž yn k . 1

Ž 2.12.

for h i g ᒅ, n i g 1r2 q ⺞, and denote the "1-eigenspace of M Ž1.Ž ␪ . by M Ž1.Ž ␪ . " respectively. Then we have following proposition Žsee Theorem 2.5 of wDN1x.. PROPOSITION 2.12. Ž1. Ž M Ž1.Ž ␪ ., I0␪ . is an irreducible ␪-twisted M Ž1.module. Ž2. Ž M Ž1.Ž ␪ . ", I0␪ . are irreducible M Ž1.q-modules. From now on, we consider the case dim ᒅ s 1 and set ␣ 1 s h. For ␭ g ⺓, we denote the irreducible M Ž1.-module M Ž1, ␭ h. by M Ž1, ␭. and e ␭ h by e ␭. Set J s hŽy1. 4 1 y 2 hŽy3. hŽy1.1 q Ž3r2. h Žy2. 2 1 g M Ž1.q which is the lowest weight vector of weight 4 for a Virasoro algebra. Then the vertex operator algebra M Ž1.q is generated by ␻ and J Žsee wDGx., and the Zhu algebra AŽ M Ž1.q. is generated by w ␻ x and w J x. More precisely, ; there is an isomorphism ⺓w x, y xrI ª AŽ M Ž1.q ., x q I ¬ w ␻ x, y q I ¬ w J x, where I is an ideal generated by two polynomials Ž y y 4 x 2 q x .Ž70 y q 908 x 2 y 515 x q 27. and Ž y y 4 x 2 q x .Ž x y 1.Ž x y 1r16.Ž x y 9r16. Žsee wDN1, Theorem 4.4x.. In wDN1, Theorem 4.5x, all equivalence classes of irreducible ⺞-gradable M Ž1.q-modules are classified as follows: THEOREM 2.13. The set  M Ž1. ", M Ž1.Ž ␪ . ", M Ž1, ␭. Ž, M Ž1, y␭..; Ž ␭ / 0.4 gi¨ es all inequi¨ alent irreducible M Ž1.q-modules. Let M be an irreducible M Ž1.q-module. Then the homogeneous space of the lowest weight Žwritten by a M g ⺓. is one-dimensional, and M has

347

FUSION RULES FOR BOSONIC ORBIFOLD

TABLE I The Basis of Top Levels and the Eigenvalues of w ␻ x and w J x M

M Ž1.q

M Ž1.y

M Ž1, ␭. Ž ␭ / 0.

M Ž1.Ž ␪ .q

M Ž1.Ž ␪ .y

¨M

1 0 0

hŽy1.1 1 y6

e␭ ␭ r2 ␭4 y ␭2r2

1 1r16 3r128

hŽy1r2.1 9r16 y45r128

aM bM

2

naturally an ⺞-gradation defined by Mn s M Ž a M q n. for any n g ⺞, where M Ž ␭. is the homogeneous space of weight ␭ of M. One of the basis Žwritten by ¨ M . of M0 is given in Table I, and we denote the dual basis of X Ž M0 .* corresponding to ¨ M by ¨ M . Since M0 is one-dimensional, w J x also acts on this space as a scalar denoted by bM g ⺓. 3. A SPANNING SET OF AŽ M . This section is divided into three subsections. In the first subsection, we describe the irreducible decompositions of irreducible M Ž1.q-modules as modules for the Virasoro algebra. In the second subsection we prove some lemmas. In last subsection we give a spanning set of AŽ M . for the irreducible M Ž1.q-module M. 3.1. Irreducible Decompositions of Irreducible M Ž1.q-Modules Let W be a module for the Virasoro algebra with central charge c g ⺓ such that LŽ0. acts on W semisimply and each eigenspace is finite dimensional. Then the formal character of W is defined by ch W s tr W q LŽ0.yc r24 s qyc r24

Ý Ž dim W Ž ␭. . q ␭ g ⺪  z 4 ,

␭g⺓

where W Ž ␭. is the eigenspace of weight ␭ for LŽ0.. Let LŽ1, ␭. be the irreducible lowest weight module for Virasoro algebra with central charge 1 and lowest weight ␭. Then the formal character of LŽ1, ␭. is given by

¡

ch LŽ1 , ␭.

q␭

if ␭ /

␩Ž q. s~ 1

¢␩ Ž q . Ž q n r4 y qŽ nq2. r4 . 2

2

if ␭ s

n2 4 n2 4

for any n g ⺪ for some n g ⺞,

Ž 3.1. where ␩ Ž q . s q 1r24 Ł⬁ns 1Ž1 y q n . is the Dedekind ␩-function Žsee wKRx..

348

TOSHIYUKI ABE

It is well known that the irreducible M Ž1.-module M Ž1, ␭. Ž ␭ g ⺓. is a completely reducible module for the Virasoro algebra, and decomposes into a direct sum of irreducible modules for the Virasoro algebra as follows Žsee wWYx, wKRx.:

¡

␭2

ž / ~ L 1,

M Ž 1, ␭ . s

2

¢

[⬁ps 0

ž

L 1,

␭2

if

Ž n q 2 p.

2

2

/

4

␭2

if

2

n2

/

4 n2

s

4

for any n g ⺪ for some n g ⺞.

Ž 3.2. Furthermore, in wDGx it is shown that the irreducible M Ž1.q-modules M Ž1. " are also completely reducible as modules for the Virasoro algebra, and the irreducible decompositions are given by q

M Ž 1. s

⬁

y

[ L Ž1, 4 p 2 . ,

M Ž 1. s

ps0

⬁

[ L Ž 1, Ž 2 p q 1. 2 . . Ž 3.3.

ps0

We show that the irreducible M Ž1.q-modules M Ž1.Ž ␪ . " are also completely reducible as modules for the Virasoro algebra. To do this, we consider the character of M Ž1.Ž ␪ .. LEMMA 3.1. We ha¨ e ch M Ž1.Ž ␪ . s

⬁

Ł

ks1

q 1r16y1r24

Ž1 y q

ky

1 2

.

s

⬁

1

Ý q Ž2 pq1. ␩ Ž q . ps0

2

r16

.

Ž 3.4.

Proof. The first equality is clear by the fact that the weight space of weight n Ž n g 1r16 q Ž1r2.⺪. has a basis

 h Ž ym1 . ⭈⭈⭈ h Ž ym k . 1 N k g ⺞, m i g 12 q ⺞, m1 G m 2 G ⭈⭈⭈ G m k ) 0, m1 q ⭈⭈⭈ qm k s n4 . To prove the second equality, we have to show that ⬁

Ž1 y q k .

Ł ky 1r2 . ks1 Ž 1 y q

s

⬁

Ý

q pŽ pq1.r4 .

Ž 3.5.

ps0

From Jacobi’s triple product Ł⬁ns 1Ž1 y q 2 n .Ž1 q q 2 ny1 z .Ž1 q q 2 ny1 zy1 . s 2 Ý ng ⺪ q n z n, we see ⬁

Ł

ns1

Ž 1 y q 4 n .Ž 1 q q 2 n . s

⬁

Ý q n qn ns0

2

FUSION RULES FOR BOSONIC ORBIFOLD

349

by substituting z s q. Replacing q with q 1r4 , we have ⬁

⬁

Ł Ž 1 y q n . Ž 1 q q n r2 . s

ns1

Ý q Ž n qn.r4 . 2

ns0

On the other hand, direct calculations show ⬁

Ž1 y q k .

Ł ky 1r2 . ks1 Ž 1 y q

s

⬁

Ł Ž 1 y q k .Ž 1 q q k r2 . .

ks1

Thus the equality Ž3.5. holds. We define an hermitian form ŽN. : M Ž1.Ž ␪ . = M Ž1.Ž ␪ . ª ⺓ by

Ž h Ž ym k . s

½

pk

p

⭈⭈⭈ h Ž ym1 . 1 1 N h Ž yn l .

k Ł is1 Ž mi . 0

pi

ql

q

⭈⭈⭈ h Ž yn1 . 1 1 .

if k s l and m i s n i , pi s qi for all 1 F i F k otherwise,

pi !

for k, l g ⺞, m i , n j g 1r2 q ⺞, m k G ⭈⭈⭈ G m1 , n l G ⭈⭈⭈ G n1 , and pi , q j g ⺞ Ž1 F i F k, 1 F j F l .. Then we can check that the hermitian form Ž N . is a nondegenerate positive-definite contravariant form on M Ž1.Ž ␪ .. Thus M Ž1.Ž ␪ . is a unitary representation of the Virasoro algebra and is completely reducible as a module for the Virasoro algebra. Let M Ž 1. Ž ␪ . s

[ L Ž 1, h . [m

h

m h g ⺞,

,

hg⺓

be the irreducible decomposition of M Ž1.Ž ␪ . as a Virasoro algebra module, where

L Ž 1, h .

[m h

!

m #

"

h

s L Ž 1, h . [ ⭈⭈⭈ [ L Ž 1, h . .

Since weights which appear in M Ž1.Ž ␪ . are in 1r16 q Ž1r2.⺪, we may assume that M Ž 1. Ž ␪ . s

[

hg

1 16

q ⺪ 1 2

L Ž 1, h .

[m h

,

m h g ⺞.

In addition, we can show that 1r16 q nr2 / m2r4 for any n, m g ⺪. By the formal characters Ž3.1., we find that the formal character of M Ž1.Ž ␪ .

350

TOSHIYUKI ABE

has the form ch M Ž1.Ž ␪ . s

1

␩Ž q.

mh q h ,

Ý hg

1 16

m h g ⺞.

q ⺪ 1 2

Comparing with Ž3.4., we have

mh

¡ ¢0

s~1

if h s

Ž 2 p q 1. 16

2

for some p g ⺞

otherwise.

Consequently we have irreducible decomposition of M Ž1.Ž ␪ ., M Ž 1. Ž ␪ . s

⬁

[L

ps0

ž

1,

Ž 2 p q 1.

2

/

16

.

PROPOSITION 3.2. M Ž1.q-modules M Ž1.Ž ␪ . and M Ž1.Ž ␪ . " are completely reducible as modules for the Virasoro algebra, and the irreducible decompositions are gi¨ en by M Ž 1. Ž ␪ . s

⬁

[L

ps0 q

M Ž 1. Ž ␪ . s

⬁

[

ps0 y

M Ž 1. Ž ␪ . s

⬁

[

ps0

žž žž

L 1,

L 1,

ž

1,

Ž 8 p q 1.

Ž 2 p q 1. 16 2

[ L 1,

16

Ž 8 p q 3. 16

2

2

/ ž / ž

[ L 1,

/

,

Ž 3.6.

Ž 8 p q 7.

2

16

Ž 8 p q 5. 16

2

// //

, Ž 3.7.

. Ž 3.8.

Proof. We have already proved the irreducible decomposition Ž3.6.. By the definitions of M Ž1.Ž ␪ . ", M Ž1.Ž ␪ .q Ž M Ž1.Ž ␪ .y. is spanned by all homogeneous vectors whose weights are in 1r16 q Ž1r2.⺪ Žresp. 9r16 q Ž1r2.⺪.. Since each weight which appears in LŽ1, Ž2 p q 1. 2r16. is in Ž2 p q 1. 2r16 q ⺪, we see that if Ž2 p q 1. 2r16 is in 1r16 q Ž1r2.⺪ Žresp. 9r16 q Ž1r2.⺪., then LŽ1, Ž2 p q 1. 2r16. is contained in M Ž1.Ž ␪ .q Žresp. M Ž1.Ž ␪ .y. . On the one hand, we have

Ž 2 p q 1. 16

2

s

½

1r16 Ž mod ⺪ . 9r16 Ž mod ⺪ .

if p s 0, 3 Ž mod 4 . if p s 1, 2 Ž mod 4 . .

351

FUSION RULES FOR BOSONIC ORBIFOLD

Thus we see that ⬁

[ ž L Ž 1, Ž 8 p q 1. 2r16 . [ L Ž 1, Ž 8 p q 7. 2r16 . / ; M Ž 1. Ž ␪ .

q

, Ž 3.9.

ps0 ⬁

[ ž L Ž 1, Ž 8 p q 3. 2r16 . [ L Ž 1, Ž 8 p q 5. 2r16 . / ; M Ž 1. Ž ␪ .

y

. Ž 3.10.

ps0

Since M Ž1.Ž ␪ . s M Ž1.Ž ␪ .q[ M Ž1.Ž ␪ .y, the irreducible decomposition Ž3.6. shows that the inclusions of Ž3.9. and Ž3.10. are equal. From irreducible decompositions Ž3.2., Ž3.3., Ž3.7., and Ž3.8., we can see that formal characters of irreducible M Ž1.q-modules M Ž1. ", M Ž1.Ž ␪ . ", and M Ž1, ␭. Ž ␭ / 0. are distinct. So together with Theorem 2.13, we have the following lemma. LEMMA 3.3. Let M, N be irreducible M Ž1.q-modules such that the formal character of M coincides with that of N. Then M is isomorphic to N. In particular, if M is an irreducible M Ž1.q-module, then the character of its contragredient module M⬘ is the same as that of M. Thus Lemma 3.3 shows that M and its contragredient module M⬘ are isomorphic as M Ž1.q-modules. Together with Proposition 2.6, we thus have the following proposition. PROPOSITION 3.4. Let M i Ž i s 1, 2, 3. be irreducible M Ž1.q-modules. 3 Then the fusion rule NMM1 M 2 is in¨ ariant under the permutation of  1, 2, 34 . 3.2. Some Useful Lemmas LEMMA 3.5. Let Ž M, YM . be an ⺞-gradable V-module. Then the following hold for any a g V homogeneous, u g M, and homogeneous ¨ ector ¨ g M of weight wtŽ ¨ .: Ž1.

For all m, n g ⺞ such that n G m,

Ž 1 q z . wt a qm Ž .

Res z

z 2qn

YM Ž a, z . u g O Ž M . .

Ž2. For any n g ⺪ ) 0 , w LŽyn. ¨ x s Žy1. ny1 w ␻ ) ¨ y n¨ ) ␻ y wtŽ ¨ . ¨ x. Proof. Part 1 is shown in Lemma 1.5.3 in wFZx. For Part 2, we can show that L Ž yn . ¨ s Ž y1 .

ny 1

Ž L Ž y1. y Ž n y 1. Ž L Ž y2. q L Ž y1. . . ¨

352

TOSHIYUKI ABE

for all u g M and n g ⺪ ) 0 by induction on n and Part 1. Hence L Ž yn . ¨ s Ž y1 .

ny 1

Ž L Ž y2. q 2 L Ž y1. q L Ž 0. y n Ž L Ž y2. qL Ž y1 . . y L Ž 0 . . ¨

s Ž y1 .

ny 1

␻ ) ¨ y n¨ ) ␻ y wt Ž ¨ . ¨ .

LEMMA 3.6. Let M be a V-module and let U be a subspace of M such that a)U ; U and U) a ; U hold for any a g V. If a g V homogeneous of positi¨ e weight and u g U satisfy that a n u g U q O Ž M . for any 1 F n F wtŽ a. y 1, then a n u g U q O Ž M . for all n F wtŽ a. y 1. In particular, if u g M is homogeneous, then LŽyn. u g U q O Ž M . for any n g ⺞. Proof. We show that ayn u g U q O Ž M . ,

Ž 3.11.

for any n g ⺞ by induction on n. From the assumption of this lemma, we have a0 u s a) u y u) a y

wtŽ a .y1

Ý is1

ž

wt Ž a . y 1 ai u g U q O Ž M . . i

/

So Ž3.11. holds for n s 0. In the case n s 1, we have ay1 u s u) a y

wtŽ a .y1

Ý is1

ž

wt Ž a . y 1 a iy1 u. i

/

Since a n u g U q O Ž M . for every 0 F n F wtŽ a. y 1, we see that Ž3.11. holds for n y 1. Assume that l g ⺪ ) 0 and Ž3.11. holds for any 0 F n F l. Then by Lemma 3.5 Ž1., ayly1 u q

wtŽ a .y1

Ý is1

ž

wt Ž a . a iyly1 u g O Ž M . . i

/

By induction hypothesis, we have ayn y1 u g U q O Ž M .. Let M be a V-module. We define a linear endomorphism ␾ M : M ª M by

␾ M Ž u . s e LŽ1. e␲ i LŽ0. u for u g M. We remark that the operator ␾ V induces an anti-automor-

353

FUSION RULES FOR BOSONIC ORBIFOLD

phism on AŽ V . given in wZx Žsee also wDLM1x.. The proof is similar to that of wDLM1x. PROPOSITION 3.7. The linear map ␾ M : M ª M satisfies the properties

␾ M Ž a) u . s ␾ M Ž u . ) ␾ V Ž a . , ␾ M Ž u) a . s ␾ V Ž a . ) ␾ M Ž u . , ␾ M Ž a( u . s y␾ V Ž a . ( ␾ M Ž u . , for any a g V and u g M. Proof. One can find in wFHLx the conjugation formulas Ž0. z1LŽ0. YM Ž a, z 0 . zyL s YM Ž z1LŽ0. a, z1 z 0 . , 1

e z 1 LŽ1. YM Ž a, z 0 . eyz 1 LŽ1. s YM e z 1Ž1yz 1 z 0 . LŽ1. Ž 1 y z1 z 0 .

ž

y2 L Ž0 .

a,

z0 1 y z1 z 0

/

for every a g V. So we have

Ž 1 q z 0 . wt a qn Ž .

Res z 0

z 0m

e LŽ1. e␲ i LŽ0. YM Ž a, z 0 . u

Ž 1 q z 0 . wt a qn Ž .

s Res z 0

z 0m

YM

= e Ž1qz 0 . LŽ1. Ž 1 q z 0 .

ž

y2wt Ž a . ␲ i LŽ0.

e

a,

yz 0 1 q z0

/

␾M Ž u .

for every m, n g ⺪, a g V, and u g M. Here we replace yz 0rŽ1 q z 0 . with w and apply the formula for a change of variables Žsee wZx., Res w g Ž w . s Res z g Ž f Ž z . .

ž

d dz

/

f Ž z. ,

where g Ž w . g M ŽŽ w .. and f Ž z . g ⺓ww z xx. Then we have

Ž 1 q z 0 . wt a qn Ž .

Res z 0

z 0m

e LŽ1. e␲ i LŽ0. YM Ž a, z 0 . u

Ž 1 q w . wt a ynqmy2 Ž .

s Ž y1 . =

ž

mq 1

⬁

Ý ks0

Res w

Ž1 q w. k!

wm

YM

yk

L Ž 1 . e␲ i LŽ0. a, w ␾ M Ž u . k

/

354

TOSHIYUKI ABE

s Ž y1 .

⬁

mq 1

Ž 1 q w . wt a ynqmy2yk Ž .

Res w

Ý

wm

ks0

=

ž

L Ž 1.

YM

k

e␲ i LŽ0. a, w ␾ M Ž u . .

/

k!

Hence if we take m s 1 and n s 0, then

␾ M Ž a) u . s

⬁

Ý

Ž 1 q w . wt a y1yk Ž .

Res w

w

ks0

s

⬁

Ý ␾M Ž u . ) ks0

ž

L Ž 1. k!

YM

ž

L Ž 1.

k

k!

e␲ i LŽ0. a, w ␾ M Ž u .

/

k

e␲ i LŽ0. a

/

s ␾M Ž u . ) ␾ V Ž a. . Similarly, if we take m s 1 and n s y1, then ␾ M Ž u) a. s ␾ V Ž a.) ␾ M Ž u., and if m s 2 and k s 0, then ␾ M Ž a( u. s y␾ V Ž a.( ␾ M Ž u.. Thus ␾ M induces a linear map on AŽ M . Žalso denoted by ␾ M . such that ␾ M Žw a) u x. s ␾ M Žw u x.) ␾ V Žw ax., ␾ M Žw u) ax. s ␾ V Žw ax.) ␾ M Žw u x. hold for all a g V and u g M. Let M be a weak V-module. An element u g M is called a lowest weight ¨ ector of weight wtŽ u. g ⺓ if LŽ n. u s wtŽ u. ␦n, 0 u for any n g ⺞. If a g V is a lowest weight vector, we have the commutation relation L Ž m . , a n s Ž Ž wt Ž a . y 1 . Ž m q 1 . y n . a mqn ,

for m, n g ⺪.

Ž 3.12. LEMMA 3.8. Let M be a V-module, and let a g V and ¨ g M be lowest weight ¨ ectors. Then the subset of M spanned by ¨ ectors a n1 a n 2 ⭈⭈⭈ a n k¨ , k g ⺪ ) 0 and n i g ⺪ Ž i s 1, 2, . . . , k ., is in¨ ariant under the action of LŽ n. for n g ⺞. Proof. Set U s Span a n1 a n 2 ⭈⭈⭈ a n k¨ N k g ⺪ ) 0 , n i g ⺪4 . To show Ž L n.U ; U for all n g ⺞, we prove that LŽ n. a n1 a n 2 ⭈⭈⭈ a n k¨ g U for any n i g ⺪. By the commutation relation Ž3.12., we have k

L Ž n . a n1 a n 2 ⭈⭈⭈ a n k¨ s

Ý an

1

⭈⭈⭈ L Ž n . , a n i ⭈⭈⭈ a n k¨ q a n1 a n 2 ⭈⭈⭈ a n k L Ž n . ¨

is1 k

s

Ý Ž Ž wt Ž a. y 1. Ž n q 1. y n i . an

1

⭈⭈⭈ a n iq n ⭈⭈⭈ a n k¨

is1

q a n1 a n 2 ⭈⭈⭈ a n k L Ž n . ¨ .

Ž 3.13.

355

FUSION RULES FOR BOSONIC ORBIFOLD

Since LŽ n. ¨ s wtŽ ¨ . ␦n, 0¨ for n g ⺞, the right-hand side of Ž3.13. is in U. LEMMA 3.9. Let M, a, and ¨ be as in Lemma 3.8. Then the subspace of M spanned by the ¨ ectors L Ž ym1 . L Ž ym 2 . ⭈⭈⭈ L Ž ym k . a n1 a n 2 ⭈⭈⭈ a n l¨ , k, l g ⺞, m i g ⺪ ) 0 , n j g ⺪,

Ž 3.14.

is in¨ ariant under the actions of LŽ n. Ž n g ⺪. and a m Ž m g ⺪.. Proof. Set U to be the subspace of M spanned by vectors Ž3.14.. By the ŽPBW. Theorem for the Virasoro algebra and Poincare᎐Birkhoff᎐Witt ´ Lemma 3.8, we see that U is invariant under the action of LŽ n. for n g ⺪. Using induction on k, we prove that a m L Ž ym1 . L Ž ym 2 . ⭈⭈⭈ L Ž ym k . a n1 a n 2 ⭈⭈⭈ a n l¨ g U,

Ž 3.15.

for any k, l g ⺞, m, m i g ⺪ ) 0 , and n j g ⺪. The case k s 0 is clear. Assume that Ž3.15. holds for k s p g ⺞. Then if we put ¨ ⬘ s LŽym 2 . ⭈⭈⭈ LŽym pq 1 . a n1 a n 2 ⭈⭈⭈ a n l¨ , we have a m L Ž ym1 . ¨ ⬘ s y L Ž ym1 . , a m ¨ ⬘ q L Ž ym1 . a m¨ ⬘ s Ž Ž wt Ž a . y 1 . Ž m1 q 1 . q m . a nym 1¨ ⬘ q L Ž ym1 . a m¨ ⬘. By the induction hypothesis and the previous paragraph, a ny m 1¨ ⬘, a m¨ ⬘, and LŽym1 . a m¨ ⬘ belong to U. Hence Ž3.15. holds for k s p q 1. 3.3. A Spanning Set of AŽ M . Let M be an irreducible M Ž1.q-module. Then from Lemma 3.9, the subspace of M spanned by vectors L Ž ym1 . L Ž ym 2 . ⭈⭈⭈ L Ž ym k . Jn1 Jn 2 ⭈⭈⭈ Jn l¨ M

Ž 3.16.

for k, l g ⺞, m i g ⺪ ) 0 , and n j g ⺪ is invariant under the actions of LŽ m. and Jn for all m, n g ⺪, where ¨ M is the lowest weight vector of M given in Table I. Since M Ž1.q is generated by ␻ and J, this subspace is invariant under the action of M Ž1.q. Thus this is an M Ž1.q-submodule of M. By the irreducibility of M, we have the following lemma. LEMMA 3.10. Ž3.16..

An irreducible M Ž1.q-module M is spanned by ¨ ectors

One of main results of this subsection is the following proposition.

356

TOSHIYUKI ABE

PROPOSITION 3.11. Let M be an irreducible M Ž1.q-module. Suppose that there exists a lowest weight ¨ ector u g M such that for any n g ⺪ ) 0 , Jn¨ M and Jn u belong to the submodule for the Virasoro algebra of M generated by ¨ M and u. Then M is spanned by the set

 a) ¨ M ) b, a) u) b N a, b g M Ž 1. q 4 q O Ž M . .

Ž 3.17.

To prove the proposition, we need some lemmas. The proof of the proposition is given after Lemma 3.14. LEMMA 3.12. Let M be an M Ž1.q-module. Then for any m, n g ⺪, the commutator w Jm , Jn x is a linear combination of L Ž m1 . L Ž m 2 . ⭈⭈⭈ L Ž m p . Jm pq 1 and L Ž n1 . L Ž n 2 . ⭈⭈⭈ L Ž n q . , for m, n g ⺪. Proof. This is proved by an argument similar to that for Lemma 3.1 of wDN1x. LEMMA 3.13. Let u g M be a lowest weight ¨ ector and set UM , u to be the subspace of M spanned by the set Ž3.17.. Fix a positi¨ e integer l. Suppose that for any 1 F k F l and n1 , . . . , n k g ⺪, Jn1 ⭈⭈⭈ Jn k¨ M and Jn1 ⭈⭈⭈ Jn k u lie in UM , u . Then for any 1 F k F l, 1 F j F k, s g ⺪ ) 0 , and p1 , . . . , ps g ⺪, we ha¨ e Jn1 ⭈⭈⭈ Jn j L Ž p1 . ⭈⭈⭈ L Ž ps . Jn jq 1 ⭈⭈⭈ Jn k¨ M , Jn1 ⭈⭈⭈ Jn j L Ž p1 . ⭈⭈⭈ L Ž ps . Jn jq 1 ⭈⭈⭈ Jn k u g UM , u . Proof. Let ¨ be either ¨ M or u. We have to show that Jn1 ⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ g UM , u ,

Ž 3.18.

for any s g ⺪ ) 0 and n1 , . . . , n k , p1 , . . . , ps g ⺪. By the PBW Theorem for the Virasoro algebra, we may assume that p1 F p 2 F ⭈⭈⭈ F ps . If p1 G 0, then ps G 0. Then we view Ž3.18. as follows: Jn1 ⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ k

s

Ý Jn

1

⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž psy1 . Jn j ⭈⭈⭈ L Ž ps . , Jn i ⭈⭈⭈ Jn k¨

isj

q Jn1 ⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž psy1 . Jn j ⭈⭈⭈ Jn k L Ž ps . ¨ k

s

Ý Ž 3 Ž ps q 1 . y n i . Jn

1

⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž psy1 . Jn j

isj

⭈⭈⭈ J p sqn i ⭈⭈⭈ Jn k¨ q wt Ž ¨ . ␦ p s , 0 Jn1 ⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž psy1 . ⭈⭈⭈ Jn k¨ .

357

FUSION RULES FOR BOSONIC ORBIFOLD

If p1 - 0, then we view it as follows: Jn1 ⭈⭈⭈ Jn jy 1 L Ž p1 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ jy1

sy

Ý Jn

1

⭈⭈⭈ L Ž p1 . , Jn i ⭈⭈⭈ Jn jy 1 L Ž p 2 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨

is1

q L Ž p1 . Jn1 ⭈⭈⭈ Jn jy 1 L Ž p 2 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ . jy1

sy

Ý Ž 3 Ž p 1 q 1 . y n i . Jn

1

⭈⭈⭈ J p 1qn i ⭈⭈⭈ Jn jy 1 L Ž p 2 .

is1

⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ q L Ž p1 . Jn1 ⭈⭈⭈ Jn jy 1 L Ž p 2 . ⭈⭈⭈ L Ž ps . Jn j ⭈⭈⭈ Jn k¨ . Hence by using induction on s, we see that Ž3.18. follows from Lemma 3.6. LEMMA 3.14. Let M and u be as in Proposition 3.11. Then for any positi¨ e integer l and n i g ⺪ Ž i s 1, . . . , l ., we ha¨ e Jn1 ⭈⭈⭈ Jn l¨ M , Jn1 ⭈⭈⭈ Jn l u g UM , u , where UM , u is the subset of M defined in Lemma 3.13. Proof. Let ¨ be either ¨ M or u. By using induction on l, we prove Jn1 ⭈⭈⭈ Jn l¨ g UM , u ,

Ž 3.19.

for any n i g ⺪. In the case l s 1, the assumptions of this lemma imply that for any positive integer n, Jn¨ is a linear combination of LŽymX1 . ⭈⭈⭈ LŽymXp . ¨ M and LŽynX1 . ⭈⭈⭈ LŽynXq . u for mXi , nXj g ⺪ ) 0 . Hence by Lemma 3.5Ž2., Jn u g UM , u holds for any n g ⺪ ) 0 . Thus Ž3.19. for l s 1 follows from Lemma 3.6. Assume that p g ⺪ ) 0 and that Ž3.19. holds for any l F p. Then we have to show Ž3.19. for l s p q 1. By the induction hypothesis, Jn 2 ⭈⭈⭈ Jn pq 1¨ g UM , u . Hence by Lemma 3.6, it is sufficient to prove that Ž3.19. for l s p q 1 holds for any n1 g ⺪ ) 0 . Then we have pq1

Jn1 ⭈⭈⭈ Jn pq 1¨ s

Ý

Jn 2 ⭈⭈⭈ Jn1 , Jn i ⭈⭈⭈ Jn pq 1¨ q Jn 2 ⭈⭈⭈ Jn pq 1 Jn1¨ . Ž 3.20.

is2

By Lemma 3.12, w Jn1, Jn i x is expressed by a linear combination of LŽ mX1 . ⭈⭈⭈ LŽ mXq . JmXqq 1 and LŽ nX1 . ⭈⭈⭈ LŽ nXr . for mXi , nXj g ⺪, and Jn1¨ can be expressed by a linear combination of LŽymY1 . ⭈⭈⭈ LŽymYs . ¨ M and LŽynY1 . ⭈⭈⭈ LŽynYt . u for mYi , nYj g ⺪ ) 0 by assumption of this lemma. Hence by the induction hypothesis and Lemma 3.13, the right-hand side of the equality Ž3.20. lies in UM , u .

358

TOSHIYUKI ABE

Now we prove Proposition 3.11. Proof of Proposition 3.11. Let UM , u be the same set as in Lemma 3.13. Then we show M s UM , u . By Lemma 3.10, it is enough to show that L Ž ym1 . L Ž ym 2 . ⭈⭈⭈ L Ž ym k . Jn1 Jn 2 ⭈⭈⭈ Jn l¨ M g UM , u

Ž 3.21.

for any m i g ⺪ ) 0 and n j g ⺪. But Lemma 3.14 tells us that Jn1 ⭈⭈⭈ Jn l¨ M g UM , u , and since the vectors of this form are homogeneous, Ž3.21. follows from Lemma 3.6. We consider the case M s M Ž1. ", M Ž1, ␭. Ž ␭ / 0, 1r2., and M Ž1.Ž ␪ .q. Since degŽ Jn¨ M . s 3 y n F 2 if n G 1, by the irreducible decomposition Ž3.2., Ž3.3., and Ž3.7., Jn¨ M Ž n G 1. lies in the submodule for the Virasoro algebra of M generated by ¨ M . Thus if we take u s ¨ M in Proposition 3.11, we have M s Span a) ¨ M ) b N a, b g M Ž1.q4 q O Ž M ., then AŽ M . is generated by w ¨ M x as an AŽ M Ž1.q. -bimodule. In the case M s M Ž1, ␭. Ž ␭2 s 1r2., by Ž3.2. we have the irreducible decomposition M s LŽ1, 1r4. [ LŽ1, 9r4. [ LŽ1, 25r4. [ ⭈⭈⭈ . Now let u be the lowest weight vector of LŽ1, 9r4., then degŽ Jn¨ M . s 3 y n F 2 and degŽ Jn u. s 5 y n F 4 for any positive integer n. Hence Jn¨ M and Jn u Ž n G 1. lie in LŽ1, 1r4. [ LŽ1, 9r4.. Thus Proposition 3.11 implies that M Ž1, ␭. s Span a) ¨ M ) b, a) u) b N a, b g M Ž1.q4 q O Ž M .. In this case, AŽ M Ž1, ␭.. is generated by w ¨ M x and w u x as an AŽ M Ž1.q. -bimodule. In the remaining case M s M Ž1.Ž ␪ .y, by Ž3.8., we have the irreducible decomposition M s LŽ1, 9r16. [ LŽ1, 25r16. [ LŽ1, 121r16. [ ⭈⭈⭈ . Let u⬘ be the lowest weight vector of LŽ1, 25r16., then degŽ Jn¨ M . s 3 y n F 2 and degŽ Jn u⬘. s 4 y n F 3 for any positive integer n. Hence Jn¨ M and Jn u⬘ Ž n G 1. are in LŽ1, 9r16. [ LŽ1, 25r16.. Thus by Proposition 3.11, M Ž1.Ž ␪ .y s Span a) ¨ M ) b, a) u⬘) b N a, b g M Ž1.q4 q O Ž M ., and AŽ M Ž1.Ž ␪ .y. is generated by w ¨ M x and w u⬘x as an AŽ M Ž1.q. -bimodule. Summarizing, we have the following proposition. PROPOSITION 3.15. following hold.

Let M be an irreducible M Ž1.q-module. Then the

Ž1. If M s M Ž1. ", M Ž1, ␭. Ž ␭2 / 0, 1r2., or M Ž1.Ž ␪ .q, then AŽ M . is generated by w ¨ M x as an AŽ M Ž1.q. -bimodule. Ž2. If M s M Ž1, ␭. Ž ␭2 s 1r2., then AŽ M . is generated by w ¨ M x and w u x as an AŽ M Ž1.q. -bimodule, where u is the lowest weight ¨ ector of weight 9r4. Ž3. If M s M Ž1.Ž ␪ .y, then AŽ M . is generated by w ¨ M x and w u⬘x as an Ž A M Ž1.q. -bimodule, where u⬘ is the lowest weight ¨ ector of weight 25r16. Remark 3.16. If M is the vertex operator algebra M Ž1.q itself, this result is clear because one has a)1 s a s 1) a for all a g M Ž1.q.

FUSION RULES FOR BOSONIC ORBIFOLD

359

As a corollary of Proposition 3.15, we have: COROLLARY 3.17. Ž1. Ž2.

Let M, N, L be irreducible M Ž1.q-modules.

If M s M Ž1. ", M Ž1, ␭. Ž ␭2 / 0, 1r2., or M Ž1.Ž ␪ .q, then NMLN F 1. If M s M Ž1, ␭. Ž ␭2 s 1r2. or M Ž1.Ž ␪ .y, then NMLN F 2.

Proof. If AŽ M . is generated by n vectors w ¨ 1 xw ¨ 2 x, . . . , w ¨ n x as an AŽ M Ž1.q. -bimodule, then the contraction Ž L0 .* ⭈ AŽ M . ⭈ N0 is spanned by n vectors ¨ LX m w ¨ 1 x m ¨ N , . . . , ¨ XL m w ¨ n x m ¨ N . Hence we have dim ⺓ Ž L0 .* ⭈ AŽ M . ⭈ N0 F n. On the other hand, by Proposition 2.10, NMLN F dim ⺓ Ž Ž L0 . * ⭈ A Ž M . ⭈ N0 . * s dim ⺓ Ž L0 . * ⭈ A Ž M . ⭈ N0 F n. Thus the corollary follows from Proposition 3.15.

4. FUSION RULES This section consists of two subsections. In the first subsection we prove some propositions which give triples of irreducible M Ž1.q-modules whose fusion rules are nonzero and state the main theorem. In the second subsection we prove the main theorem. 4.1. Main Theorem Recall the linear map I␭␮ : M Ž1, ␭. ª z ␭␮ HomŽ M Ž1, ␮ ., M Ž1, ␭ q ..ww ␮ z, zy1 xx defined in Ž2.8. and Ž2.9. for ␭, ␮ g ⺓. By the arguments of Section 8.6 of wFLMx, we see that I␭␮ is an intertwining operator of type

ž

M Ž 1, ␭ q ␮ . . M Ž 1, ␭ . M Ž 1, ␮ .

/

Since e ␮ g M Ž1, ␮ . 0 , we have I␭␮ Ž e ␭ , z . e ␮ s z ␭␮ exp

ž

␭ h Ž yn .

Ý nG0

n

z n e ␭q ␮ ,

/

and its coefficient of z ␭␯ is e ␭q ␮ which is nonzero. Thus this intertwining Ž1, ␭ q ␮ . operator I␭␮ is nonzero. This implies that the fusion rule NMMŽ1, ␭ . M Ž1, ␮ . is nonzero for any ␭, ␮ g ⺓. By Proposition 2.11, there exists an isomorphism f from M Ž1, ␮ . onto M Ž1, y␮ . of M Ž1.q-modules. We define a linear map Y␭ ( f : M Ž 1, ␭ . ª zy␭ ␮ Ž Hom Ž M Ž 1, ␮ . , M Ž 1, ␭ y ␮ . . . w z, zy1 x , u ¬ Y␭ ( f Ž u, z . s Y␭ Ž u, z . ( f .

360

TOSHIYUKI ABE

Then we can easily see that Y␭ ( f gives a nonzero intertwining operator of type M Ž 1, ␭ y ␮ . . M Ž 1, ␭ . M Ž 1, ␮ .

ž

/

Thus the following proposition holds. PROPOSITION 4.1. if ␯ 2 s Ž ␭ " ␮ . 2 .

Ž1, ␯ . For ␭, ␮ , ␯ g ⺓, the fusion rule NMMŽ1, ␭ . M Ž1, ␮ . is nonzero

Ž1, ␯ . 2 2 In particular, if ␭ s 0 we have that NMMŽ1. M Ž1, ␮ . is nonzero if ␮ s ␯ . In fact, we have the following proposition.

PROPOSITION 4.2.

For ␮ , ␯ g ⺓ such that ␮2 s ␯ 2 , the fusion rules Ž1 , ␯ . " NMMŽ1. M Ž1 , ␮ .

are nonzero. Furthermore, the fusion rules "

.

Ž1. M Ž1. q NMMŽ1. M Ž1. " and NM Ž1.y M Ž1. "

are nonzero. Proof. Since Ž M Ž1, ␮ ., I0 ␮ . is an irreducible M Ž1.-module, I0 ␮ gives nonzero intertwining operators of types

ž

M Ž 1, ␮ . M Ž 1.

"

/

M Ž 1, ␮ .

by Proposition 11.9 of wDLx. Because M Ž1, y␮ . is isomorphic to M Ž1, ␮ . as an M Ž1.q-module, the first statement holds. If ␮ s 0, then Y s I00 gives nonzero intertwining operators of types

ž

M Ž 1. M Ž 1.

"

M Ž 1.

/

.

Since ␪ is an automorphism of M Ž1., we have ␪ Y Ž a, z .␪ s Y Ž ␪ Ž a., z . for every a g M Ž1.. This implies that Y Ž a, z . M Ž1. "; M Ž1. "ŽŽ z .. for every a g M Ž1.q and Y Ž a, z . M Ž1. "; M Ž1. .ŽŽ z .. for every a g M Ž1.y. Thus Y gives nonzero intertwining operators of types

ž

M Ž 1. M Ž 1.

q

"

M Ž 1.

"

/

and

ž

M Ž 1. M Ž 1.

y

.

M Ž 1.

"

/

.

361

FUSION RULES FOR BOSONIC ORBIFOLD

Next we recall the linear map I␭␪ : M Ž1, ␭. ª ŽEnd M Ž1.Ž ␪ .. z 4 defined in Ž2.10. and Ž2.11.. The arguments of Chapter 9 of wFLMx show that I␭␪ is an intertwining operator of type M Ž 1. Ž ␪ . . M Ž 1, ␭ . M Ž 1 . Ž ␪ .

ž

/

Let p " be projections from M Ž1.Ž ␪ . onto M Ž1.Ž ␪ . " and let ␫ " be inclusions from M Ž1.Ž ␪ . " into M Ž1.Ž ␪ . respectively. Define a linear map p␤ ( I␭␪ ( ␫␣ by ␣

p␤ ( I␭␪ ( ␫␣ : M Ž 1, ␭ . ª Hom Ž M Ž 1 . Ž ␪ . , M Ž 1 . Ž ␪ .

ž

␤

. /  z4 ,

u ¬ p␤ ( I␭␪ ( ␫␣ Ž u, z . s p␤ Ž I␭␪ Ž u, z . ␫␣ . , where ␣ , ␤ are q or y. Then we have the following lemma. LEMMA 4.3. For any ␣ , ␤ g  q, y4 , p␤ ( I␭␪ ( ␫␣ is a nonzero intertwining operator of type

ž

M Ž 1. Ž ␪ . M Ž 1, ␭ .

␤

M Ž 1. Ž ␪ .

␪

/

if ␭ / 0.

Proof. It is clear that p␤ ( I␭␪ ( ␫␣ is an intertwining operator of type

ž

M Ž 1. Ž ␪ . M Ž 1, ␭ .

␤

M Ž 1. Ž ␪ .

␣

/

.

We next show that p␤ ( I␭␪ ( ␫␣ is nonzero if ␭ / 0. By direct calculation, 2 we can see that the coefficient of zy␭ r2 in I␭␪ Ž e ␭, z .1 is 1 and that of 2 2 zy ␭ r2q1r2 is ␭ hŽy1r2.1, and the coefficient of zy ␭ r2y1r2 in 2 I␭␪ Ž e ␭, z . hŽy1r2.1 is y␭1 and that of zy␭ r2q1 is yŽ2r3. ␭2 hŽy3r2.1 q 2 ␭2 Ž1 y 3r2 ␭2 . hŽy1r2. 31. If ␭ / 0, all of these coefficients are nonzero. This proves that p␤ ( I␭␪ ( ␫␣ Ž e ␭, z . is nonzero for all ␣ , ␤ g  q, y4 if ␭ is nonzero. For any nonzero ␭ g ⺓ and ␣ , ␤ g  q, y4 ,

PROPOSITION 4.4. Ž1.

␤

␪. NMMŽ1Ž1.Ž , ␭ . M Ž1.Ž ␪ . ␣

is nonzero. Ž2.

The fusion rules "

Ž1.Ž ␪ . q NMMŽ1. M Ž1.Ž ␪ . "

are nonzero.

and

.

Ž1.Ž ␪ . y NMMŽ1. M Ž1.Ž ␪ . "

362

TOSHIYUKI ABE

Proof. Part 1 follows from Lemma 4.3. By the definition of the action of ␪ on M Ž1.Ž ␪ . Ž2.12., we can show that ␪ I0␪ Ž a, z .␪ s I0␪ Ž ␪ Ž a., z . holds for every a g M Ž1.. This implies that "

"

"

.

I0␪ Ž a, z . M Ž 1 . Ž ␪ . ; M Ž 1 . Ž ␪ . I0␪ Ž a, z . M Ž 1 . Ž ␪ . ; M Ž 1 . Ž ␪ .

q

Ž Ž z. .

if a g M Ž 1 . ,

Ž Ž z. .

if a g M Ž 1 . .

y

Because I0␪ Ž1, z . s id M Ž1.Ž ␪ . and I0␪ Ž hŽy1.1, z . s hŽ z ., we see that p " ( I0␪ ( ␫ " Ž p " ( I0␪ ( ␫ . . give nonzero intertwining operators of type

ž

M Ž 1. Ž ␪ . M Ž 1.

q

"

M Ž 1. Ž ␪ .

"

/ ž ž resp.

M Ž 1. Ž ␪ . M Ž 1.

y

.

M Ž 1. Ž ␪ .

"

//

.

This shows Part 2. The series of Propositions 4.1, 4.2, and 4.4 together with Proposition 3.4 give us triples Ž M, N, L. of irreducible M Ž1.q-modules for which the fusion rule NMLN is nonzero. In fact, we have following theorem. THEOREM 4.5 ŽMain Theorem.. modules.

Let M, N, and L be irreducible M Ž1.q-

Ž1. If M s M Ž1.q, then NMLŽ1.q N s ␦ N, L . Ž2. If M s M Ž1.y, then NMLŽ1.y N is 0 or 1, and NMLŽ1.y N s 1 if and only if the pair Ž N, L. is one of the following pairs: .

Ž M Ž 1. " , M Ž 1. . ,

.

Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . . ,

Ž M Ž 1, ␭ . , M Ž 1, ␮ . . Ž ␭2 s ␮2 . . Ž3. If M s M Ž1, ␭. Ž ␭ / 0., then NMLŽ1, ␭. N is 0 or 1, and NMLŽ1, ␭ . N s 1 if and only if the pair Ž N, L. is one of the following pairs:

Ž M Ž 1. " , M Ž 1, ␮ . . Ž ␭2 s ␮2 . , Ž M Ž 1, ␮ . , M Ž 1, ␯ . . Ž ␯ 2 s Ž ␭ " ␮ . 2 . , . Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . " . , Ž M Ž 1. Ž ␪ . " , M Ž 1. Ž ␪ . . . Ž4. If M s M Ž1.Ž ␪ .q, then NMLŽ1.Ž ␪ .q N is 0 or 1, and NMLŽ1.Ž ␪ .q N s 1 if and only if the pair Ž N, L. is one of the following pairs:

Ž M Ž 1. " , M Ž 1. Ž ␪ . " . ,

Ž M Ž 1, ␭. , M Ž 1. Ž ␪ . " . .

Ž5. If M s M Ž1.Ž ␪ .y, then NMLŽ1.Ž ␪ .y N is 0 or 1, and NMLŽ1.Ž ␪ .y N s 1 if and only if the pair Ž N, L. is one of the following pairs: .

Ž M Ž 1. " , M Ž 1. Ž ␪ . . ,

Ž M Ž 1, ␭. , M Ž 1. Ž ␪ . " . .

FUSION RULES FOR BOSONIC ORBIFOLD

363

4.2. Proof of the Main Theorem Recall the direct sum decomposition M Ž1.qs LŽ1, 0. [ LŽ1, 4. [ ⭈⭈⭈ Žsee Ž3.3... The lowest weight vector of LŽ1, 4. is J. Hence hŽy3. hŽy1.1 is a linear combination of LŽy4.1, LŽy2. 2 1 and J Žnote that LŽy1.1 s 0.. In fact, hŽy3. hŽy1.l s Žy1r9.Ž J y 3 LŽy4.1 y 4 LŽy2. 2 1.. By Lemma 3.5Ž2., we have the equality in AŽ M Ž1.q. , J y 4␻ * 2 y 17␻ q 9h Ž y3 . h Ž y1 . 1 s 0, n ! #

Ž 4.1.

"

where we use the notation w ax* n s w a* n x s w a) a) ⭈⭈⭈ ) a x for a g M Ž1.q and n g ⺞. Now we prove the main theorem. We divide the proof into five steps, Step 1᎐Step 5, where Ž1. ᎐ Ž5. are proved respectively. Proof of Theorem 4.5. By Theorem 2.13 and Table I, we see that two irreducible M Ž1.q-modules N and L are isomorphic to each other if and only if a N s a L and bN s bL . Step 1. If M s M Ž1.q, then ¨ M s 1 Žsee Table I., and Ž L0 .* ⭈ AŽ M . ⭈ N0 s ⺓ ¨ XL m w1x m ¨ N by Proposition 3.15Ž1.. If the fusion rule NMLN is nonzero, then ¨ XL m w1x m ¨ N is also nonzero. On the one hand, we have the equalities a L¨ LX m w 1 x m ¨ N s ¨ XL m w ␻ x m ¨ N s a N ¨ XL m w 1 x m ¨ N , bL¨ LX m w 1 x m ¨ N s ¨ LX m w J x m ¨ N s bN ¨ LX m w 1 x m ¨ N . This implies that if the fusion rule NMLN is nonzero, then a N s a L and bN s bL , hence M and N are equivalent. Then Ž1. follows from Proposition 4.2 and Proposition 4.4. Step 2. Let M s M Ž1.y. Then we have the irreducible decomposition Ž3.3.. Hence hŽy3. hŽy1.1) ¨ M is in LŽ1, 1. and it is a linear combination of LŽym1 . ⭈⭈⭈ LŽym k . ¨ M Ž m i g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4.. In fact, h Ž y3 . h Ž y1 . 1) ¨ M 2

s 3 ¨ M q 12 L Ž y1 . ¨ M q 12 L Ž y1 . ¨ M y 8 L Ž y3 . ¨ M q 16 L Ž y2 . L Ž y1 . ¨ M y

1 2

L Ž y4 . ¨ M

1 3 2 q L Ž y3 . L Ž y1 . ¨ M q L Ž y2 . L Ž y1 . ¨ M . 4 2

Ž 4.2.

364

TOSHIYUKI ABE

On the other hand, we have ¨ L m L Ž ym1 . ⭈⭈⭈ L Ž ym k . ¨ M m ¨ N s F Ž a L , a N . ¨ L m w ¨ M x m ¨ N X

X

Ž 4.3. for any m i g ⺪ ) 0 by Lemma 2.6, where F g ⺓w x, y x is given by k

Fs

Ł Ž y1.

m iy1

is1

k

ž

x y mi y y

Ý

/

m j y wt Ž ¨ M . .

jsiq1

Ž 4.4.

Hence by Ž4.1. and Ž4.2. ᎐ Ž4.4., we have X

¨L m

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M s f Ž aL , aN ,

bL ¨ LX

.

m ¨N

m w ¨ M x m ¨ N s 0,

where f g ⺓w x, y, z x is given by f s z y 4 x2 q x q

9 4

Ž x y y . Ž 6 x 2 y 18 xy y 12 y 2 y 21 x y 23 y q 11 . .

Proposition 3.7 and Ž4.1. show that X

¨ L m ␾ M Ž ¨ M . ) ␾ M Ž1.q Ž J y 4␻ U 2 y 17␻ q 9h Ž y3 . h Ž y1 . 1 . m ¨ N

s ¨ XL m ␾ M Ž

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M . m ¨ N

s yf Ž a N , a L , bN . ¨ XL m w ¨ M x m ¨ N s 0, since ␾ M Ž1.q Žw ␻ x. s w ␻ x, ␾ M Ž1.q Žw J x. s w J x, and ␾ M Žw ¨ M x. s yw ¨ M x. Moreover, the Verma module of the Virasoro algebra with central charge 1 and lowest weight 1 has a singular vector 2 LŽy3. w y 4 LŽy2. LŽy1. w q LŽy1. 3 w of weight 4, where w is the cyclic vector of the Verma module. The image of this singular vector in M Ž1.y is zero, then by Ž4.3. and Ž4.4. we have g Ž a N , a L . ¨ LX m w ¨ M x m ¨ N s 0, where g Ž x, y . s Ž x y y .Ž x 2 y 2 xy q y 2 y 2 x y 2 y q 1.r2. Now suppose that the fusion rule NMLN is nonzero. Then ¨ XL m w ¨ M x m ¨ N is nonzero by Proposition 3.15Ž1.. Hence f Ž a L , a N , bL ., f Ž a N , a L , bN ., and g Ž a N , a L . are necessarily zero. By Ž1. and Proposition 3.4, we assume that N, L g  M Ž1.y, M Ž1, ␭. Ž ␭ / 0., M Ž1.Ž ␪ . "4 . In the cases N s L s M Ž1.y, M Ž1.Ž ␪ .q and M Ž1.Ž ␪ .y, and the cases L s M Ž1.y and N s M Ž1.Ž ␪ . ", we see that f Ž a L , a N , bL . are nonzero. Therefore the fusion rules of corresponding types are zero. In the case L s M Ž1.y and N s M Ž1, ␭., we see that either f Ž1, ␭2r2, y6. or g Ž1, ␭2r2. is nonzero. Then the fusion rule of corresponding type is zero.

365

FUSION RULES FOR BOSONIC ORBIFOLD

In the case L s M Ž1, ␭. and N s M Ž1, ␮ ., assume that the fusion rule is nonzero. Then if we put s s ␭2 and ␮2 s t, we have s t 2 s t s 2 t 9 2 f , ,s y qf , ,t y s Ž s y t . Ž 3s q 3t y 2 . s 0, 2 2 2 2 2 2 16

ž

/ ž

ž

f

s

t

,

2 2 s

, s2 y

/

s

qf

/ ž

2

t

,

s

2 2

, t2 y

t 2

/

q 81 g

s

t

ž / ,

2 2

9

Ž s y t . Ž s q t y 1 . s 0. 2 Hence we have s s t, that is, ␭2 s ␮2 . Thus NMMMŽ1,Ž1,␮ ␭. . s ␦␭ 2␮ 2 holds. In the case L s M Ž1, ␭. and N s M Ž1.Ž ␪ .q, if we put s s ␭2 , then we have s 1 s 27 f , , s2 y s Ž 8 s y 1 . Ž 32 s 2 y 236 s q 205. , 2 16 2 4096 f

ž ž

/

1

,

s

,

3

16 2 128

/

s

27 4096

Ž 8 s y 9 . Ž 384 s 2 y 1160 s q 131. .

But there is no common zero of these two polynomials, and the corresponding fusion rule is zero. If L s M Ž1, ␭. and N s M Ž1.Ž ␪ .y, then we have s 9 s y9 f , , s2 y s Ž 8 s y 9 . Ž 96 s 2 y 996 s q 119. , 2 16 2 4096

ž

f

ž

/

9

,

s

16 2

,

y45 128

/

s

9 8192

Ž 8 s y 1 . Ž 384 s 2 y 2504 s q 2211. .

Thus there is no common zero of these two polynomials, and the corresponding fusion rule is zero in this case too. Now Ž2. follows from Proposition 4.2 and 4.4. Step 3. Let M s M Ž1, ␭. Ž ␭ / 0.. We prove Ž3. by dividing the problem into the following four cases: Ži. ␭2 / 1r2, 2, 9r2, Žii. ␭2 s 2, Žiii. ␭2 s 9r2, and Živ. ␭2 s 1r2. Ži. First we assume that ␭2 / 1r2, 2, 9r2. Then we can see that Ž . h y3 hŽy1.1) ¨ M belongs to LŽ1, ␭2r2. by the direct sum decomposition Ž3.2.. Thus hŽy3. hŽy1.1) ¨ M can be expressed by a linear combination of LŽym1 . ⭈⭈⭈ LŽym k . ¨ M Ž m i g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4.. Then by using Ž4.3. and Ž4.4., we have the following equalities in the contraction LU0 ⭈ AŽ M . ⭈ N0 , X

¨L m

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M s f Ž aL , aN ,

bL ¨ LX

.

m w ¨ M x m ¨ N s 0,

m ¨N

Ž 4.5.

366

TOSHIYUKI ABE X

¨ L m ␾ M Ž ¨ M . ) ␾ M Ž1.q Ž J y 4␻ * 2 y 17␻ q 9h Ž y3 . h Ž y1 . 1 . m ¨ N

s ¨ XL m ␾ M Ž

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M . m ¨ N

s e ␭ ␲ i r2 f Ž a N , a L , bN . ¨ XL m w ¨ M x m ¨ N s 0, 2

Ž 4.6.

where f s f Ž x, y, z . g ⺓w x, y, z x is given by f s z y 4 x2 q x q

9 Ž ␭4 y 4 ␭2 Ž x q y . q 4 Ž x y y .

2

.

32 ␭2 Ž ␭2 y 2 . Ž 2 ␭2 y 9 . Ž 2 ␭2 y 1 . 3

= y3 Ž ␭2 y 2 . q 4 Ž 8 ␭4 y 29␭2 q 6 . x

ž

2

q4 Ž 7␭2 q 6 . y y 4 Ž 16 ␭2 q 3 . Ž x y y . .

/

We consider the case N s M Ž1.Ž ␪ .q and L s M Ž1, ␮ . Ž ␮ / 0.. If we assume that the fusion rule NMLN is nonzero, then ¨ LX m w ¨ M x m ¨ N is nonzero by Proposition 3.15Ž1.. Hence we have f Ž a L , a N , bM . s f Ž a N , a L , bN . s 0. This shows the equations p Ž t , u . s Ž 8u y 1 . Ž 8u y 9 . Ž Ž 1024t q 192 . u 2 y Ž 2048t 2 q 512 t q 624 . u q 1024t 3 y 256t 2 q 816t q 75 . s 0, 2

q Ž t , u . s Ž Ž 8t q 8u y 1 . y 256tu . Ž Ž 1024t q 192 . u 2 y Ž 1024t 2 y 3456t q 816 . u q 1192 t 2 q 864t q 675 . s 0, respectively. If t g  1r2, 2, 9r24 , then there is no common solution of pŽ t, u. s q Ž t, u. s 0. Hence we may assume that t f  1r2, 2, 9r24 and discuss it by interchanging ␭ and ␮. Thus we have pŽ u, t . s q Ž u, t . s 0. If t s 1r8, 9r8, then pŽ t, u. and q Ž t, u. have no common solution. Therefore t / 1r8, 9r8. Similarly, u / 1r8, 9r8. If we put pŽ t, u. s Ž u y 1r8.Ž u y 9r8. r Ž t, u., then r Ž t, u. s r Ž u, t . s 0 hold. Since r Ž t, t . and q Ž t, t . have no common solution, we may also assume that t / u. From r Ž t, u. y r Ž u, t . s q Ž t, u. y q Ž u, t . s q Ž t, u. q q Ž u, t . s 0, we have following three equations: 32 ␣ 2 y 14␣ q 45 y 128 ␤ s 0,

Ž 128 ␤ q 3 . Ž 64␣ 2 y 16 ␣ q 1 y 256 ␤ . s 0,

Ž 64␣

2

Ž 4.7.

y 16 ␣ q 1 y 256 ␤ .Ž 64␣ y 280 ␣ q 225 q 1024␤ . s 0, 2

where we put ␣ s t q u and ␤ s tu. If 64␣ 2 y 16 ␣ q 1 y 256 ␤ / 0, then ␤ s y3r128. But in this case, Ž4.7. and 64␣ 2 y 280 ␣ q 225 q

367

FUSION RULES FOR BOSONIC ORBIFOLD

1024␤ s 0 don’t have a common solution. So we have 64␣ 2 y 16 ␣ q 1 y 256 ␤ s 0 and 32 ␣ 2 y 14␣ q 45 y 128 ␤ s 0. This implies ␣ s 89r12 and ␤ s 30625r2304. But the solutions Ž t, u. g ⺓ 2 of the equation x 2 y ␣ x q ␤ s 0 don’t satisfy pŽ t, u. s 0. Therefore we see that the corresponding fusion rule is zero. In the case N s M Ž1.Ž ␪ .y and L s M Ž1, ␮ . Ž ␮ / 0. we can show that the fusion rule NMLN is zero by the same method as that in the preceding case. We next prove that NMŽ1,Ž1,␯ .␭. M Ž1, ␮ . s ␦␯ 2 , Ž ␭ " ␮ . 2 if ␭2 / 1r2, 2, 9r2. By Proposition 4.1 and Corollary 3.17, it is enough to prove that if Ž1, ␯ . 2 Ž .2 Ž . Ž . NMMŽ1, ␭ . M Ž1, ␮ . / 0 then ␯ s ␭ " ␮ . Let N s M 1, ␮ and L s M 1, ␯ , L and assume that NM N is nonzero. By Proposition 3.15, we see that X ¨ L m w ¨ M x m ¨ N / 0. Then the equalities Ž4.5. and Ž4.6. follow two equations

Ž s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu . p Ž s, t , u . s Ž s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu . p Ž s, u, t . s 0,

Ž 4.8.

where s s ␭2 , t s ␯ 2 , u s ␮ 2 , and p g ⺓w x, y, z x is given by p s Ž y3 x q 16 xy q z q 32 yz y z 2 . = Ž y2 y 12 x q 58 xy q 16 xy 2 y 12 y q 3 y 2 y 6 yz y 12 z q 3 z 2 . . Suppose that s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu / 0, then we have pŽ s, t, u. s pŽ s, u, t . s 0. In addition, we assume that ␭2 / 8. Then the circle relations hŽy3. hŽy1.1( ¨ M and hŽy2. 2 1( ¨ M belong to LŽ1, ␭2r2. and can be expressed by linear combinations of LŽym1 . ⭈⭈⭈ LŽym k . ¨ M Ž m i g ⺪ ) 0 , m i q ⭈⭈⭈ qm k F 5.. By Ž4.3. and Ž4.4., we have the equalities ¨ L m h Ž y3 . h Ž y1 . 1( ¨ M m ¨ N s g 1 Ž a L , a N . ¨ L m w ¨ M x m ¨ N s 0, X

X

Ž 4.9. ¨ L m h Ž y2 . 1( ¨ M m ¨ N s g 2 Ž a L , a N . ¨ L m w ¨ M x m ¨ N s 0, X

2

X

Ž 4.10.

where g 1Ž a L , a N . and g 2 Ž a L , a N . become g 1 Ž a L , a N . s Ž s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu . Ž t y u . q Ž s, t , u . , g 2 Ž a L , a N . s Ž s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu . r Ž s, t , u . ,

368

TOSHIYUKI ABE

and q, r g ⺓w x, y, z x are given by q s y12 q 24 x y 5 x 2 q 12 y y 16 xy q 4 x 2 y q 3 y 2 y 4 xy 2 q 12 z y 16 xz q 4 x 2 z q 6 yz q 8 xyz y 3 z 2 y 4 yz 2 , r s 192 y 245 x q 108 x 2 y 18 x 3 q x 4 y 240 y y 28 xy q 8 x 2 y q x 3 y q 96 y 2 q 86 xy 2 y 21 x 2 y 2 y 12 y 3 q 19 xy 3 y 144 z q 460 xz y 152 x 2 z q 11 x 3 z y 96 yz y 100 xyz q 14 x 2 yz q 36 y 2 z y 152 xy 2 z q 14 xz 2 q 7x 2 z 2 q 57xyz 2 y 36 y 2 z 2 q 12 z 3 y 19 xz 3 . Since s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu is nonzero, we have Ž t y u. q Ž s, t, u. s r Ž s, t, u. s 0. Similarly, we have Ž u y t . q Ž s, u, t . s r Ž s, u, t . s 0 by interchanging ␮2 and ␯ 2 . Assume that t s u. Then r Ž s, t, t . s 0 follows s s 6Ž1 y 2 t ., and pŽ6Ž1 y 2 t ., t, t . s 0 implies that t f  1r2, 2, 9r2, 84 . Hence we can interchange s and t, and we have r Ž t, 6Ž1 y 2 t ., t . s 0 and r Ž t, t, 6Ž1 y 2 t .. s 0. But the common solutions of these equations don’t satisfy pŽ6Ž1 y 2 t ., t, t . s 0. Hence we see that t / u, which shows q Ž s, t, u. s q Ž s, u, t . s 0. Next assume that t s n2r2 Ž n s 1, 2, 3, 4.. Then pŽ s, n2r2, u. s 0 follows ss

16u 2 y Ž 16 n2 q 1 . u 8 n2 y 3

or y

12 u 2 y 12 Ž n2 q 4 . u q 3n4 y 24n2 y 8 4 Ž 4 n2 q 29n2 y 12 .

.

But in any case, there is no common solution of equations pŽ s, u, n2r2. s 0 and q Ž s, u, n2r2. s 0. Thus t f  1r2, 2, 9r2, 84 . Similarly, we have u f  1r2, 2, 9r2, 84 . Therefore if we put x 1 s s, x 2 s t and x 3 s u, p Ž x i1 , x i 2 , x i 3 . s q Ž x i1 , x i 2 , x i 3 . s r Ž x i1 , x i 2 , x i 3 . s 0 and x i / x j if i / j

Ž 4.11.

hold for every permutation  i1 , i 2 , i 34 of  1, 2, 34 and i, j s 1, 2, 3. In particular, we have Ž t y u.Ž q Ž s, t, u. y q Ž t, s, u.. y Ž t y s .Ž q Ž u, t, s . y q Ž t, u, s .. s 0 and this follows s q t q u q 5 s 0 since s, t, u are distinct from each other. On the other hand, if we put r Ž s, t, u. y r Ž t, s, u. s Ž s y t . ␣ Ž s, t, u., then we have ␣ Ž s, t, u. s ␣ Ž u, t, s . s 0. If we next put ␣ Ž s, t, u. y ␣ Ž u, t, s . s Ž s y u. ␤ Ž s, t, u., we have ␤ Ž s, t, u. s ␤ Ž s, u, t . s 0. Now we see that ␤ Ž s, t, u. y ␤ Ž s, u, t . s y16Ž t y u.Ž3s q 3t q 3u y 10. s 0 and this follows 3s q 3t q 3u y 10 s 0. It is inconsistent with s q t q u q 5 s 0. Thus we see that there is no solution that satisfies Ž4.11.. Hence s 2 q t 2 q u 2 y 2 st y 2 su y 2 tu s 0. By substituting ␭2 for s, ␯ 2 for t, and ␮ 2 for u, we have ␯ 2 s Ž ␭ " ␮ . 2 if ␭ / 1r2, 2, 9r2, 8. In the case ␭2 s 8, we may assume that N and L are any of M Ž1, ␮ . Ž ␮2 s 1r2, 2, 9r2, 8., that is to say, that ␮2 and ␯ 2 are any of 1r2, 2, 9r2,

369

FUSION RULES FOR BOSONIC ORBIFOLD

and 8. But then pŽ8, ␮ 2 , ␯ 2 . is nonzero. Hence we have ␯ 2 s Ž ␮ " 2'2 . 2 by Ž4.8.. Consequently we see that Ž3. holds if ␭2 / 1r2, 2, 9r2. Žii. We next consider the case ␭2 s 2. Then we have the irreducible decomposition for the Virasoro algebra M Ž1, ␭. s LŽ1, 1. [ LŽ1, 4. [ LŽ1, 9. [ ⭈⭈⭈ Žsee Ž3.2... Put u s '2 hŽy3. ¨ M y 3hŽy2. hŽy1. ¨ M q '2 Žy1. 3 ¨ M which is the lowest weight vector of LŽ1, 4.. Since hŽy3. hŽy1.1) ¨ M g LŽ1, 1. [ LŽ1, 4., it is a linear combination of LŽy1. u, u and LŽym1 . ⭈⭈⭈ LŽym k . ¨ M Ž m i g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4.. Then by formulas Ž4.3. and Ž4.4., we have the equation X

¨L m

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M

m ¨N

s f 1 Ž a L , a N . ¨ LX m w u x m ¨ N q f 2 Ž a L , a N , bL . ¨ LX m w ¨ M x m ¨ N s 0, where f 1 s f Ž x, y . g ⺓w x, y x and f 2 s f Ž x, y, z . g ⺓w x, y, z x are given by f1 s f2 s z q q

95 8

135 4

99

xy

8

xy 2 y

yy

27 2

3 2

173 8

q

21 8

x2 y

Ž x y y. ,

9 4

xy q

207 8

y2 q

47 4

x 3 y 27x 2 y

y3.

Since ␾ M Žw u x. s w u x and ␾ M Žw ¨ M x. s yw ¨ M x, we have yf 1 Ž a N , a M . ¨ XN m w u x m ¨ N q f 2 Ž a N , a L , bN . ¨ LX m w ¨ M x m ¨ N s 0. We may assume that N and L are any of M Ž1, ␮ . Ž ␮2 s 1r2, 2, 9r2., M Ž1.Ž ␪ . ". But then we can see that the determinant of the matrix

ž

f 1Ž aL , aN .

f 2 Ž aL , aN , aL .

yf 1 Ž a N , a L .

f 2 Ž aN , aL , aN .

/

is nonzero except for the cases

Ž N, L . s Ž M Ž 1, ␮ . , M Ž 1, ␯ . . ;

Ž ␮2 , ␯ 2 . s Ž 1r2, 9r2. , Ž 9r2, 1r2. , Ž 1r2, 1r2. , Ž M Ž 1. Ž ␪ .

␣

, M Ž 1. Ž ␪ .

␤

.;

␣ , ␤ g  q, y 4 . Note that 9r2 s Ž'2 q 1r '2 . 2 and 1r2 s Ž'2 y 1r '2 . 2 s Ž'2 y

370

TOSHIYUKI ABE

3r '2 . 2 . Then we have NMMŽ1Ž1, ,␭␯..M Ž1 , ␮ . s ␦␯ 2 , Ž ␭ " ␮ . 2 ,

"

␪. NMMŽ1Ž1.Ž , ␭ . M Ž1 , ␮ . s 0

for every ␮ 2 , ␯ 2 s 1r2, 2, 9r2 by Corollary 3.17 and Proposition 4.1. Together with the results of Ži. and Proposition 4.4, we see that Ž3. holds for ␭2 s 2. Žiii. We consider the case ␭2 s 9r2. In this case, the submodule for the Virasoro algebra of M s M Ž1, ␭. generated by ¨ M is isomorphic to the irreducible module LŽ1, 9r4.. The Verma module for the Virasoro algebra with central charge 1 and lowest weight 9r4 has a singular vector 18 LŽy4. w y 14 LŽy3. LŽy1. w y 9LŽy2. 2 w q 10 LŽy2. LŽy1. 2 w y LŽy1. 4 w, where w is the cyclic vector of the Verma module. Since the image of the singular vector in M Ž1, ␭. is zero, by using Ž4.3. and Ž4.4. we have the equality f Ž a L , a N . ¨ XL m w ¨ M x m ¨ N s 0, where f s f Ž x, y . g ⺓w x, y x is given by f s Ž 81 y 72 Ž x q y . q 16 Ž x y y .

2

.Ž 1 y 8 Ž x q y . q 16 Ž x y y . 2 . .

By the results of Step 1, Step 2, and Ži., Žii. of Step 3, we may assume that N and L are any of M Ž1, ␮ . Ž ␮2 s 1r2, 9r2., M Ž1.Ž ␪ . ". If a L , a N g  1r4, 9r4, 1r16, 9r164 , then f Ž a L , a N . is nonzero except for the pairs Ž a L , a N . s Ž1r16, 1r16., Ž1r16, 9r16., Ž9r16, 1r16., Ž9r16, 9r16.. By Corollary 3.17 and Proposition 4.1, we see that for ␭2 s 9r2, ␮2 , ␯ 2 s 1r2, Ž1, ␯ . M Ž1.Ž ␪ . " 9r2, NMMŽ1, ␭ . M Ž1, ␮ . s NM Ž1, ␭ . M Ž1, ␮ . s 0. Then Proposition 4.4 implies that Ž3. holds for ␭2 s 9r2. Živ. We prove that Ž3. holds for ␭2 s 1r2. In this case we have the irreducible decomposition M Ž1, ␭. s LŽ1, 1r4. [ LŽ1, 9r4. [ LŽ1, 25r4. [ ⭈⭈⭈ Žsee Ž3.2... Let u s '2 hŽy2. ¨ M y 2 hŽy1. 2 ¨ M which is the lowest weight vector of LŽ1, 9r4.. Then hŽy3. hŽy1.1) ¨ M belongs to LŽ1, 1r4. [ LŽ1, 9r4. and is a linear combination of LŽym1 . ⭈⭈⭈ LŽym k . ¨ M and LŽyn1 . ⭈⭈⭈ LŽyn l . u Ž m i , n j g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4, n1 q ⭈⭈⭈ qn l F 3.. By Ž4.3. and Ž4.4., we have the equality X

¨L m

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M

m ¨N

s f 1 Ž a L , a N . ¨ LX m w u x m ¨ N q f 2 Ž a L , a N , bL . ¨ LX m w ¨ M x m ¨ N s 0,

Ž 4.12.

371

FUSION RULES FOR BOSONIC ORBIFOLD

where f 1 s f Ž x, y . g ⺓w x, y x and f 2 s f Ž x, y, z . g ⺓w x, y, z x are given by 27

f1 s

128

f2 s z y

q 3 2

13 16

xy

19 16

yq

11 16

2 Ž x y y. ,

2

q 12 Ž x q y . y 24 Ž x y y . .

Since ␾ M Žw u x. s e␲ i r4 w u x and ␾ M Žw ¨ M x. s e␲ i r4 w ¨ M x, we have also f 1 Ž a N , a L . ¨ LX m w u x m ¨ N q f 2 Ž a N , a L , bN . ¨ LX m w ¨ M x m ¨ N s 0. In addition, we have LŽy1. 2 ¨ M q LŽy2. ¨ M s 0, since the Verma module of central charge 1 and highest weight 1r4 has nontrivial singular vector of weight 9r4. Therefore we have the equality X

Ž L Ž y1. 2 ¨ M q L Ž y2. ¨ M .

¨L m

m ¨N

s g Ž a L , a N . ¨ XL m w ¨ M x m ¨ N s 0,

Ž 4.13.

where g s g Ž x, y . s y1r16 q Ž x q y .r2 y Ž x y y . 2 . By Step 1, Step 2 and Ži. ᎐ Žiii. of Step 3, we may assume that N and L are any of M Ž1, ␭. Ž ␭2 s 1r2., M Ž1.Ž ␪ . ". In the case N s L s M Ž1, ␭. Ž ␭2 s 1r2., since both f 1Ž a L , a N . and Ž g a L , a N . are nonzero, ¨ XL m w u x m ¨ N s ¨ XL m w ¨ M x m ¨ N s 0 by Ž4.12. and Ž4.13.. Hence we have dim LU0 ⭈ AŽ M . ⭈ N0 s 0 by Proposition 3.15. Thus Proposition 2.10 shows that the fusion rule NMLN is zero. In the case L s M Ž1, ␮ . Ž ␮2 s 1r2. and N s M Ž1.Ž ␪ . ", the determinant of the matrix

ž

f 1Ž aL , aN .

f 2 Ž aL , aN , aL .

f 1Ž aN , aL .

f 2 Ž aN , aL , aN .

/

is nonzero. Hence by Proposition 2.10, the fusion rule NMLN is zero in this case too. In the case that N and L are either M Ž1.Ž ␪ .q or M Ž1.Ž ␪ .y, ¨ XL m w u x m X ¨ N and ¨ L m w ¨ M x m ¨ N are linearly dependent, so the dimension of U L0 m AŽ M . m N0 is less than one by Proposition 3.15Ž3.. Now Proposition 4.4 Ž1. shows that the fusion rule NMLN is 1 in this case. Consequently, Ž3. holds for ␭2 s 1r2. Thus we see that Ž3. holds for all ␭ g ⺓ y  04 . Step 4. In the case M s M Ž1.Ž ␪ .q, we have M Ž1.Ž ␪ .qs LŽ1, 1r16. [ LŽ1, 49r16. [ LŽ1, 81r16. [ ⭈⭈⭈ Žsee Ž3.7... We put u s 9hŽy5r2. hŽy1r2.1 y 5hŽy3r2. 2 ¨ M y 10 hŽy3r2. hŽy1r2. 3 q 4 hŽy1r2. 6 1 which is the lowest weight vector of LŽ1, 49r16.. Since hŽy3. hŽy1.1) ¨ M is in LŽ1, 1r16. [ LŽ1, 49r16., it can be expressed as a linear combination of LŽy1. u, u and LŽym1 . ⭈⭈⭈ LŽym k . ¨ M Ž m i g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4..

372

TOSHIYUKI ABE

So we have equality X

¨L m

Ž J y 4␻ *2 y 17␻ q 9h Ž y3. h Ž y1. 1 . ) ¨ M

m ¨N

s f Ž a L , a N . ¨ LX m w u x m ¨ N q g Ž a L , a N , bL . ¨ XL m w ¨ M x m ¨ N s 0, where f s f Ž x, y . g ⺓w x, y x and g s g Ž x, y . g ⺓w x, y, z x are given by fs g s 5z y q

32 7

135 1792

1

y

56

1 2

q

xq

73 28

8 7

Ž x y y. ,

yy

2

Ž x y y . Ž 5 x q 12 y . y

Since ␾ M Žw u x. s ye

␲ i r16 w

82 7

x2 q

256 7

212 7

xy y

180 7

y2

4 Ž x y y. .

u x and ␾ M Žw ¨ M x. s e␲ i r16 w ¨ M x, we have equality

yf Ž a L , a N . ¨ XL m w u x m ¨ N q g Ž a L , a N , bL . ¨ LX m w ¨ M x m ¨ N s 0. By the results of Step 1᎐Step 3 and Proposition 3.4, we assume that N and L are either M Ž1.Ž ␪ .q or M Ž1.Ž ␪ .y. But then the determinant of the matrix

ž

f Ž aL , aN .

g Ž aL , aN , aL .

yf Ž a N , a L .

g Ž aN , aL , aN .

/

is nonzero. Hence we have ¨ LX m w u x m ¨ N s ¨ XL m w ¨ M x m ¨ N s 0. This proves that ␤

Ž1.Ž ␪ . NMMŽ1.Ž ␪ .q M Ž1.Ž ␪ . ␣ s 0

for any ␣ , ␤ g  q, y4 . Thus we see that Ž4. holds. Step 5. By the results of Step 1᎐Step 4; it is enough to show that NMLN s 0 for M s N s L s M Ž1.Ž ␪ .q. Since we have the direct product decomposition M Ž1.Ž ␪ .ys LŽ1, 9r16. [ LŽ1, 25r16. [ LŽ1, 121r16. [ ⭈⭈⭈ Žsee 3.8., if we put u s yŽ1r2. hŽy3r2.1 q hŽy1r2. 3 1 which is the lowest weight vector of LŽ1, 25r16., then hŽy3. hŽy1.1) ¨ M can be expressed as a linear combination of LŽym1 . ⭈⭈⭈ LŽym k . ¨ M and LŽyn1 . ⭈⭈⭈ LŽyn l . u Ž m i , n j g ⺪ ) 0 , m1 q ⭈⭈⭈ qm k F 4, n1 q ⭈⭈⭈ qn l F 3.. Calculating the vector ¨ LX m wŽ J y 4␻ * 2 y 17␻ q 9hŽy3. hŽy1.1.) ¨ M x m ¨ N by means of Ž4.3. and Ž4.4. we have the linear equalities 75 224

¨L m w ux m ¨N y X

135 256

¨ L m w ¨ M x m ¨ N s 0. X

FUSION RULES FOR BOSONIC ORBIFOLD

373

Since ␾ M Žw u x. s ye 9␲ i r16 w u x and ␾ M Žw ¨ M x. s e 9␲ i r16 w ¨ M x, by Proposition 3.7, we have y

75 224

¨L m w ux m ¨N y X

135 256

¨ L m w ¨ M x m ¨ N s 0. X

This follows that ¨ LX m w u x m ¨ N s ¨ XL m w ¨ M x m ¨ N s 0. So we see that the fusion rule NMLN s 0 if M s N s L s M Ž1.Ž ␪ .q by Proposition 3.15 Ž3.. The proof of the theorem is complete.

ACKNOWLEDGMENTS I thank Professor Kiyokazu Nagatomo for useful discussions and suggestions. I also thank Doctor Yoshiyuki Koga and Akihiko Ogawa for their many helpful opinions.

REFERENCES wDGx

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