The Physics Superselection Principle in Vertex Operator Algebra Theory

Home , Operator algebra, Superselection, Vertex operator algebra

JOURNAL OF ALGEBRA 196, 436᎐457Ž. 1997 ARTICLE NO. JA977126

Haisheng Li*

Department of Mathematical Sciences, Rutgers Uni¨ersity, Camden, New Jersey 08102 View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Georgia Benkart provided by Elsevier - Publisher Connector Received March 4, 1996

We formulate an interpretation of the theory of physics superselection sectors in terms of vertex operator algebra language and prove some initial results. As one of the main results we give a construction of simple currents from a weight-one primary semisimple element. By applying our results to vertex operator algebras associated to affine Lie algebras or to positive-definite even lattices, we construct their simple currents. ᮊ 1997 Academic Press

1. INTRODUCTION

In mathematical physics, there are various approaches to two-dimensional quantum field theory, among which are the ރ*-algebra approach through the superselection principlewx HK and the chiral algebraŽ vertex operator algebra. approachwx BPZ, MSe . This paper studies the application of the theory of superselection sectors to vertex operator algebra theory wxB, FLM2, FHL . Although this paper is motivated by some mathematical physics papers such aswx FRS, MSc in the theory of superselection sectors, the main results in this paper are purely algebraic in the context of vertex operator algebras. In local quantum field theory one considers a Hilbert space H of

physical states which decomposes into inequivalent, irreducible modules Hi Ž.superselection sectors for the observable algebra A, possibly with some multiplicitieswx HK . Among the superselection sectors, there is a distinguished sector H0 which contains the vacuum vector and carries the vacuum representation ␲ 0. In general, A admits infinitely many inequiva-

* E-mail address: [email protected].

436

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. SUPERSELECTION PRINCIPLE IN VOA THEORY 437 lent irreducible modules, so a criterion is needed to rule out the physically irrelevant modules. If U is an element of A with a left inverse U*, then we have an Ž. endomorphism ␺UUof A defined by ␺ a s UaU* for any a g A. Conse- quently, we have a representation ␲␺0 U on H0 of A. Roughly speaking, in the algebraic theory of superselection sectorsŽ see, e.g.,wx HK. , the physics superselection sectors by definition consist of each equivalent class of irreducible representation ␲ which is equivalent to ␲␺0 U for some Ž. Ž.Ž. so-called localized endomorphism ␺Uiof A.IfWs H00,␲␺iis1, 2 are superselection sectors, an intertwiner from W12to W is defined to be Ž. an A-homomorphism, and W s H0021, ␲␺␺ is defined to be the tensor product module of W12with W . Such a tensor product module is in general reducible, but it is assumed to be decomposable into irreducible Ž.Ž. ones. Furthermore, if H00, ␲␺i is1, 2, 3 are three superselection sectors, then an intertwining operator of typeŽ.W3 is defined to be an WW12 Ž. intertwiner or an A-homomorphism ␾ from W s H0, ␲␺␺ 021to W 3and fusion rules are defined accordingly. In mathematics, the notion of vertex operator algebrawx B, FLM2 natu- rally arose from the vertex operator construction of the moonshine module wxFLM1 for the Monster group, the largest sporadic finite simple group. On the other hand, vertex operator algebras are essentially chiral algebras formulated inwx BPZ in two-dimensional conformal field theory. Vertex operator algebras provide a powerful algebraic tool for studying the general structure of conformal field theory. For vertex operator algebra theory, the notions of module, intertwining operator, and fusion rule have been defined inwx FLM2, FHL . Furthermore, the notions of tensor product for modules have been also developed inwx HL0᎐HL3, Hua, Li3 . The purpose of this paper is to apply the physics superselection theory to vertex operator algebra theorywx B, FLM2, FHL . Note that if ␴ is an endomorphism of a vertex operator algebra V, then by definition ␴ preserves both the vacuum and the Virasoro element so that ␴ preserves each homogeneous subspace of V Žseewx FHL, Sect. 2.4. . If V is simple, i.e., V is an irreducible V-module, it follows from the Schur lemma that any nonzero endomorphism is a scalar. Then the twisting of V by ␴ is isomorphic to V. Therefore, it is impossible to obtain all irreducible modules by twisting V unless V is holomorphic, i.e., any irreducible V-module is isomorphic to V. Having known the above fact, we turn to a certain associative algebra. For any vertex operator algebra V, Frenkel and Zhuwx FZ constructed a topological ޚ-graded associative algebra UVŽ., which was called the universal enveloping algebra of V. Roughly speaking, UVŽ.is the associative algebra with identity generated by all anŽ.Žlinear in a ., for a g V, n g ޚ 438 HAISHENG LI with certain defining relations coming from the Jacobi identity and the Virasoro algebra relations. Then there is a natural 1-1 correspondence between the set of equivalence classes of lower truncated ޚ-graded weak V-modules and the set of equivalence classes of continuous1 lower truncated ޚ-graded UVŽ.-modules. It is reasonable to believe that UV Ž. should play the role of the observable algebra in the algebraic quantum field theory. Note that for all known rational vertex operator algebrasŽ the definition is given in Section 2. there are only finitely many inequivalent irreducible modules and all irreducible modules are exactly those which are needed in conformal field theory. For instance, it was provedwx DL, FZ, Li1 that for any positive integer l, the set of equivalence classes of irreducible LlŽ.,0 - modules is exactly the set of equivalence classes of unitary highest weight ˜g-modules of level l. It was also provedwx DMZ, W that if c s 1 y 6Žp y 2 q. rpq, where p, q g Ä42, 3, . . . and p and q are relatively prime, then the set of equivalence classes of irreducible LcŽ., 0 -modules is exactly the set of equivalence classes of lowest weight Virasoro modules in the minimal series given inwx BPZ . Therefore, at least for a rational vertex operator algebra V, each irreducible V-module or sector is physically relevant so that each irreducible V-module should be a superselection sector. Based on this interpretation we conjecture that each irreducible V-module is isomorphic to some twisting of V by an endomorphism of UVŽ.. As mentioned, twisting the adjoint module by an endomorphism of a vertex operator algebra V does not give a new module. Notice that an endomorphism of V is an element of EndރV satisfying certain conditions. Now we consider certain elements ⌬Ž.z g ŽEndރVz .Ä4satisfying certain conditions so that ŽV, Y Ž⌬ Žz ., и , z .. is a V-module.Ž To study the twisting of intertwining operators we more generally consider ⌬Ž.z g UV Ž .Ä4 z.. We prove that ŽŽŽ...V, Y ⌬ z и , z is a weak V-module if and only if the conditionsŽ.Ž. 2.17 ᎐ 2.20 hold. This implies that such a ⌬ Ž.z induces an endomorphism ␺ of UVŽ.defined by

␺ Ž.Ž.YaŽ.,z sY⌬ Ž.za,z for any a g V.1.1Ž. Our first theorem claims that if ⌬Ž.z is invertible, M˜is isomorphic to a tensor product module of M with V˜ in the sense ofwx HL0᎐HL3, Hua, Li3 . This implies that V˜ is a simple currentwx SY1, SY2, FG , i.e., the tensor functor associated to V˜ gives a permutation on the set of equivalence classes of irreducible V-modules. In Section 3, we construct such a ⌬Ž.h, z satisfying the conditions Ž.Ž.2.17 ᎐ 2.20 from a primary weight-one semisimple element h of a vertex operator algebra V. Applying our results to a vertex operator algebra LlŽ., 0 associated to an affine Lie algebra ˜g, we prove that if the funda-

1 It was pointed out by C. Dong that this condition is necessary. SUPERSELECTION PRINCIPLE IN VOA THEORY 439

mentalŽ. dominant integral weight ␭i is cominimalwx FG , then for any Ž.Ž.Ž. complex number l / y⍀ the dual Coxeter number , Ll,l␭i is a weak LlŽ., 0 -module and it is a simple current if l is a positive integer. This result has been proved inwx FG by calculating four-point functionsŽ see also wxF, FGV1, FGV2. . We also apply our results to the vertex operator algebra VL associated to a positive-definite even lattice L to find all the fusion rules. This result has been previously obtained inwx DL by using a different method.

2. THE SUPERSELECTION PRINCIPLE IN TERMS OF VERTEX OPERATOR ALGEBRAS AND MODULES

In this section we formulate an interpretation of the physics superselection principle in terms of vertex operator algebras and modules and prove some initial results. We recall the following definition fromwx FLM2, Sect. 8.10 . A ¨ertex operator algebra is a ޚ-graded vector space V V ; for V , n wt ;2Ž..1 s @Žn.Ž¨ g n.s ¨ ngޚ such that

dim VŽn.- ϱ for n g ޚ,2.2Ž.

VŽn.s0 for n sufficiently small,Ž. 2.3 equipped with a linear map 1 VªŽ.End Vzwx,zy yny1 ¨¬YŽ.¨,zsÝ¨nnz Žwhere ¨ g End V . Ž.2.4 ngޚ and with two distinguished homogeneous vectors 1, ␻ g V, satisfying the following conditions for u, ¨ g V,

un¨ s 0 for n sufficiently large;Ž. 2.5 YŽ.1,zs1;Ž. 2.6 YŽ.¨,z1gVzwx and lim Y Ž.¨ , z 1 s ¨ ;2.7 Ž. zª0

z12z y1 y z01␦ YuŽ.Ž.,zY¨,z2 ž/z 0

z21z y1 y yz02␦ YŽ.Ž.¨,zYu,z1 ž/z y0

z10z y1 y sz20␦ YYuŽ.Ž.,z ¨,z2;2.8Ž. ž/z 2 440 HAISHENG LI

Ž.Ž.␦ Ý n ␦ŽŽ . . the Jacobi identity where z s ng ޚ z and where z120y z rz is to be expanded as a formal power series in the second term in the numerator, z2 , and analogously for the other ␦-function expressions; when each expression inŽ. 2.8 is applied to any element of V, the coefficient of each monomial in the formal variables is a finite sum; 1 3 LmŽ.Ž.Ž.Ž.Ž,Ln s mynLmqnq mym .Ž␦mn,0 rank V . 12 q Ž.2.9 for m, n g ޚ, where

␻ ޚ ␻ yny2 LnŽ.s nq1 for n g , i.e., Y Ž, z .s ÝLnz Ž. Ž2.10 . ngޚ and

rank V g ރ; Ž.2.11

LŽ.0¨sn¨s Žwt ¨¨ . for n g ޚ and ¨ g VŽn.;Ž. 2.12 d YuŽ.,zsYLŽ. Ž.y1u,z.Ž. 2.13 dz This completes the definition. We shall just use V for a vertex operator algebra. A weak V-module is a vector space M together with a linear map Ž. Ž . y 1 Ž. YM и,zfrom V to End Mzww ,zxx such that YM 1, z s ŽŽ . . Ž . Ž . 1, YLMMy1a,zsdrdz Y a, z for any a g V and that a suitably adjusted Jacobi identity holds. We shall just use M for the weak module. A Ž . ŽŽ .. subspace U of M is called a submodule if YaM ,zugUz for any agV,ugU. IfÄ4 0 and M are the only submodules, M is said to be irreducible. Let ŽŽ..Ž.W , Y и, zi1, 2 be two weak V-modules. A V- iWi s homomorphism from W to W is a linear map ␺ such that ␺ ŽŽYa,zu .. 12 W1 YaŽ.Ž.,z␺ufor any a V, u W . Furthermore, if ␺ is a linear sW2 g g 1 isomorphism, ␺ is called a V-isomorphism. A V-module is a weak V-module M which is a ރ-graded vector space Ž. Ms@hg ރ MŽh.Ž, where M h.is the eigenspace of L 0onMwith ރ ϱ eigenvalue h, such that for any h g , dim MŽh.Ž- and M nqh.s 0 for sufficiently large integer n. A lower truncated ޚ-graded weak V-module is a weak V-module M ޚ Ž. Ž. together with a -grading M s @ ng ޚ Mn such that Mns0 for sufficiently small integer n and that

aMmn Ž.:Mm Ž qkyny1. for a g VŽk., m, n, k g ޚ. SUPERSELECTION PRINCIPLE IN VOA THEORY 441

If any lower truncated ޚ-graded weak V-module is a direct sum of irreducible ޚ-graded weak V-modules, we say that V is rational. If any weak V-module W is a direct sum of irreducibleŽ. ordinary V-modules, we say that V is regular. It was proved inwx DLM that vertex operator algebras VL associated to a nondegenerate even lattice L, LlŽ., 0 associated to standard modules of a positive integral level l for an affine Lie algebra ˜g, LcŽ., 0 associated to unitary highest weight Virasoro modules with central charge 0 - c - 1, and Vh, Frenkel, Lepowsky, and Meurman’s moonshine module, are regular. That is, all known rational vertex operator algebras are regular. Ž. Let Wii s1, 2, 3 be three weak V-modules. Then an interesting operator of typeŽ.W3 is defined in FHL to be a linear map IŽ.и, z from W to WW12 wx 1 Ž.Ä4 ŽŽ . . Ž .Ž . Homރ W23, Wzsuch that ILy1u,zsdrdz I u, z for u g W1 and that a suitably adjusted Jacobi identity holds. Denote by IŽ.W3 the space of all intertwining operators of the indi- WW12 cated type. The dimension of this vector space is called the fusion rule of this type. Let IrrŽ.V be the set of equivalence classes of irreducible weak V-modules and for any V-module M, denote the equivalence class of M by EŽ.M. Let V be a vertex operator algebra. Recall Frenkel and Zhu’s construction of the universal enveloping algebra UVŽ.of V as followsw FZ, Sect. 1.3x : First, let A be the free algebra with generators anŽ.Žlinear in a .for ŽŽ.. agV,ngޚ. Define deg an smyny1 for a g VŽ m., n g ޚ. Then ޚ ޚ A becomes a -graded algebra with A s @ ng ޚ An. For any n g , ޚ Ý Ž. ig, we set In, ijs GinAAqjyjn:A. Then иии иии In,2 : In,1 : In,0 : In,y1 : In,y2 : . Ž.Ž Let i, j g ޚ, u, ¨ g Annfor a fixed n. Then for any x g u q I ,il ¨ q In, j., we have

x q In,kn: Ž.u q I ,inl Ž.¨ q I ,jfor k G i, j. Ž.Ž.Ž. Thus u q In, inl ¨ q I ,jxns D x q I ,k, where x runs through each Ž.Ž. element of u q In, inl ¨ q I ,j. Let ␶ be the collection of the empty set and all unions of some u q In, infor u g A , i g ޚ. Then ␶ is closed under finite intersection, so that ␶ defines a topology on An. Thus a sequence Ä4xmnof elements in A converges to zero if and only if for any In,kthere exists a positive integer r such that xmng I ,kfor m G r. Since

AImn,im:Iqn,in, IA,im:I mqn,iym for m, n g ޚ, the multiplication of A is continuous. 442 HAISHENG LI

Let Annbe the completion of A under this topology. Set A s ޚ @ng ޚ An. Then A is a topological -graded associative algebra. Next, we define UVŽ.to be the quotient algebra of A modulo the two-sided ideal generated by the relations 1 ␦ Ž.n s n,y1; Ž.2.14 Ž.LŽ.Ž.y1ansyna Ž n y 1 . ;Ž. 2.15 ϱ im Ž.1am Ž n 1 ibi.Žk 1 . Ýyž/i y y y y y is0 ϱ m i m Ž.1y bm Žk i 1 .Žai n 1 . yyÝ ž/i y yy y y is0 ϱ n 1 yy Ž.Ž.Ž.abnk2i 2.16 sÝž/i iymy1yyyy is0 ޚ Ž. Ž. ޚ for a, b g V, m, n, k g . Then UV s@ng ޚ UV n is a -graded topological associative algebra such that there is a natural 1-1 correspondence between the set of equivalence classes of the lower truncated ޚ-graded continuous UVŽ.-module and the set of equivalence classes of the lower truncated ޚ-graded weak V-module.

PROPOSITION 2.1. Let V be a ¨ertex operator algebra and let ⌬Ž.z g y1 Ž.Endރ Vzww ,zxx such that

1 ⌬Ž.zagVzw ,zy x for any a g V.Ž. 2.17 Suppose that

Ž.V˜˜,YŽ.и,z [Ž.V,YŽ.⌬ Ž.z и,z is a weak V-module. Then the following conditions hold:

⌬Ž.z 1 s 1;Ž. 2.18 d LŽy1, .⌬ Ž.z sy ⌬ Ž.z ;Ž. 2.19 dz

Y Ž.⌬Žz20q za .,z 0⌬ Ž.z 2s⌬ Ž.ŽzYa 2,z 0 . for any a g V.Ž. 2.20

Con¨ersely, if the abo¨e three conditions hold, for any weak V-module ŽŽ..M,YMи,z

Ž.W˜,YMMŽ.и,z[Ž.W,YŽ.⌬Ž.zи,z is a weak V-module. SUPERSELECTION PRINCIPLE IN VOA THEORY 443

Proof. First of all, because ofŽ. 2.17 ,

Y Ž.Ž.⌬Ž.za,zbgVz Ž. for any a, b g V.

We prove the converse first. Suppose thatŽ.Ž. 2.18 ᎐ 2.20 hold. Let ŽW, YW . be a weak V-module. Then for any a, b g V we have

z12y z zy1␦ YŽ.Ž.⌬Ž.za,zY ⌬ Ž.zb,z 0ž/z W11W 22 0

z21z y1yq yz0␦ YWŽ.Ž.⌬Ž.zb22,zYW⌬ Ž.za11,z ž/z 0

z10z y1 y sz2␦ YYWŽ.Ž.⌬Ž.za10,z⌬ Ž.zb 22,z ž/z 2

z10z y1 y sz2␦ YYWŽ.Ž.⌬Ž.Ž.z20qza,z 0⌬zb 2,z 2 ž/z 2

z10z y1 y sz2␦ YWŽ.⌬Ž.ŽzYa202,zb .,z.Ž. 2.21 ž/z 2

This proves the Jacobi identity. Furthermore, the conditionsŽ. 2.18 and Ž.2.19 imply

YWŽ.⌬Ž.z 1, z s 1, d LŽy1, .YWWŽ.⌬ Ž.za,z s Y Ž.⌬Ž.za,z for a g V.Ž. 2.22 dz

Therefore ŽŽ..W˜, YW˜ и, z is a weak V-module. On the other hand, suppose that ŽŽŽ...V, Y ⌬ z и , z is a weak V-module. Since Y˜Ž.1, z s 1, using the skew-symmetry we get

zLŽ 1. e y ⌬Ž.z 1sYŽ.⌬ Ž.z 1, z 1s1.

Thus ⌬Ž.z 1 s 1. Since YL˜˜ŽŽ..Ž.Ž.y1a,zsdrdz Y a, z for a g V we have

Y Ž.⌬Ž.zL Žy1 .a,zsYLŽ.Ž.y1a,z sYLŽ.Ž.Ž.y1⌬za,z qYŽ.⌬Ј Ž.za,z. That is,

YLŽ.Ž.Ž.y1,⌬za,zsyYŽ.⌬ЈŽ.za,z. 444 HAISHENG LI

By a similar argument we obtain w LŽ.Ž.y1,⌬ zxsy⌬ЈŽ.z . Following the argument ofŽ. 2.21 we find that the Jacobi identity for Y˜ holds if and only if

z10z y1 y z22␦ YYŽ.Ž.⌬Ž.Ž.zqza0,z0⌬zb2,z2 ž/z 2

z10z y1 y sz22␦ YŽ.⌬Ž.ŽzYa,zb0 .,z2.Ž. 2.23 ž/z 2

ApplyingŽ. 2.23 to 1, then using the skew-symmetry we obtain

z10zz10z y1 y y1 y z21␦ YŽ.⌬Ž.za,z0⌬ Ž.z2sz22␦ ⌬Ž.ŽzYa,z0 .. ž/zz ž/ 22 Ž.2.24

Then we immediately obtainŽ. 2.20 . This proves that the listed conditions are also necessary and concludes the proof. It follows from Proposition 2.1 that each ⌬Ž.z gives rise to an endomorphism ␺ of UVŽ.such that ␺ ŽŽYa,z ..sY ŽŽ.⌬za,z .for any a g V. The following proposition gives the injective property of ⌬Ž.z as a linear map from V to V m ރŽŽz ... PROPOSITION 2.2. Let V be a simple ¨ertex operator algebra and let y1 ⌬Ž.zg ŽEndރVz .ww ,zxx satisfy the conditions Ž.Ž.2.17 ᎐ 2.20 . Then for a g V,⌬Ž.zas0if and only if a s 0. Proof. Set

ker ⌬Ž.z [ Ä4a g V N ⌬Ž.zas0.

Then it is equivalent to proving that ker ⌬Ž.z s 0. Let a g ker ⌬Ž.z . Then for any u g V, usingŽ. 2.20 we get

⌬Ž.ŽzYu20,za .sYŽ.⌬ Žz 2002qzu .,z⌬ Ž.zas0.

Thus ker ⌬Ž.z is an ideal of V. Since V is simple and ⌬Ž.z 1 s 1, we obtain ker ⌬Ž.z s 0. This concludes the proof.

Notice that for anyŽ. weak V-module W, YW Ž.и, z is an intertwining Ž.W ˜ Ž. operator of typeVW . Then Proposition 2.1 implies that YW и, z is an Ž.W˜ intertwining operator of typeVW˜ . To generalize this for an intertwining operator for three arbitrary modules we shall consider elements ⌬Ž.z g UVŽ.Ä4 z so that ⌬Ž.z can act on any weak V-module. This leads us to the SUPERSELECTION PRINCIPLE IN VOA THEORY 445 following definition:

DEFINITION 2.3. Let V be a vertex operator algebra. Then we define Ž.⌬ Ž. Ý ⌬ r Ž.Ä4 GV to be the set consisting of each z s r g ރ r z g UV z satisfying the following condition for any weak V-module W and any u g W: Ž. There exist finitely many n1,...,nk gރ such that

n n ⌬Ž.zugzWz1 wxqиии qzWzk wx Ž.2.25 and all the conditions in Proposition 2.1 hold. The following propositions are motivated by the theory of superselection sectors:

i PROPOSITION 2.4. Let ⌬Ž.z g GV Ž .,let MŽ. i s 1, 2, 3 be three weak Ž. Ž.M 3 V-modules, and let I и, z be an intertwining operator of type MM12. Then ˜Ž. ŽŽ.. Ž.M˜3 Iи,zsI⌬zи,z is an intertwining operator of type MM12˜.

Proof. The LŽ.y1 -derivative property for I˜Ž.и, z follows from the con- 1 ditionŽ. 2.19 immediately. For any a g V, u g M we have

z12zz21z y1 y y1 y z0␦YaM˜˜3Ž.Ž.,zIu120˜˜,z yz␦ IuŽ.Ž.,zY 2M2 a,z1 ž/z ž/z 00y

z12z y1 y sz0␦ YM3Ž.Ž.⌬Ž.za11,zI⌬ Ž.zu 2,z 2 ž/z 0

z21z y1 y yz02␦ IŽ.Ž.⌬Ž.zu,zY2M2⌬ Ž.za11,z ž/z y0

z10z y1 y sz2␦ IYŽ.M1Ž.⌬Ž.za10,z⌬ Ž.zu 22,z ž/z 2

z10z y1 y sz22␦ IŽ.⌬Ž.zYM1 Ž a,zu02 .,z ž/z 2

z10z y1 y sz2␦ IY˜Ž.M1Ž. a,zu02,z.Ž. 2.26 ž/z 2

Then the proof is complete.

PROPOSITION 2.5. Let ⌬Ž.z g G Ž V . and let ␺ be a V-homomorphism from aŽ. weak V-module W to another V-module M. Then ␺ is also a V-homomorphism from the V-module W˜˜ to the V-module M. 446 HAISHENG LI

Proof. For any a g V, u g W, we have

␺ Ž.YaWW˜Ž.,zus␺Ž.Y Ž.⌬ Ž.za,zu

sYMŽ.⌬Ž.za,z␺ Ž.u

sYaM˜Ž.Ž.,z␺u.Ž. 2.27

Thus ␺ is a V-homomorphism from W˜˜to M. Ž. Ž. Ž . Ž. Ž. Ž . PROPOSITION 2.6. Let ⌬12z , ⌬ z g GV .Then ⌬12z ⌬ z g GV . Proof. By definition we have

⌬12Ž.z ⌬ Ž.z 1 s ⌬ 1 Ž.z 1 s 1,Ž. 2.28

LŽy1, .⌬12 Ž.z⌬ Ž.z

sLŽy1, .⌬12 Ž.z ⌬ Ž.z q⌬ 1 Ž.zL Žy1, .⌬2 Ž.z dd sy ⌬12Ž.z ⌬ Ž.z y⌬ 1 Ž.z ⌬ 2 Ž.z ž/dz ž/dz d sy Ž.⌬12Ž.z ⌬ Ž.z ,Ž. 2.29 dz and

z10y z zy1␦ YŽ.⌬Ž.z⌬ Ž.za,z⌬ Ž.z⌬ Ž.z 21ž/z 12101222 2

z10z y1 y sz21␦ ⌬Ž.zY2Ž.⌬2 Ž.za1,z0⌬2 Ž.z2 ž/z 2

z10z y1 y sz21␦ ⌬Ž.z2⌬2 Ž.ŽzYa20,z . Ž2.30 . ž/z 2 for any a g V. The other conditions follow directly from the definition. Ž. Ž. Ž . Thus ⌬12z ⌬ z g GV . Ž. Ž. It is clear that idV g GV , so that GV is a semigroup. PROPOSITION 2.7. Let ⌬Ž.z g G Ž V . such that ⌬Ž.z has an in¨erse 1 1 ⌬y Ž.zgUV Ž .Ä4 z.Then ⌬yŽ.z g GV Ž .. SUPERSELECTION PRINCIPLE IN VOA THEORY 447

1 1 1 Proof. First, we have ⌬y Ž.z 1 s ⌬yŽ.Ž.z ⌬ z 1 s 1. Since ⌬Ž.z ⌬yŽ.z s 1, we have dd 1 1 0s⌬Ž.z⌬y Ž.zq⌬ Ž.z⌬y Ž.z. Ž 2.31 . ž/dz dz Then dd 11 1 ⌬yŽ.zsy⌬y Ž.z ⌬ Ž.z ⌬y Ž.z dz ž/dz

1 1 syLŽ.y1 ⌬y Ž.z q⌬y Ž.Ž.zLy1 1 sy LŽ.y1,⌬y Ž.z .Ž. 2.32

For any a g V, we have

z10y z zy1␦ ⌬y1Ž.ŽzYa,z . 22ž/z 0 2

z10z y1 y y1 y1 sz22␦ ⌬Ž.zYŽ.⌬ Ž.z1⌬ Ž.za1,z0 ž/z 2

z10z y1 y y1 y1 sz21␦ YŽ.⌬Ž.za,z0⌬ Ž.z2. Ž 2.33 . ž/z 2

1 Thus ⌬yŽ.z g GV Ž .. 0 Define GVl Ž.to be the subset of GVŽ., consisting of each ⌬Ž.z that has a left inverse ⌬*Ž.z in GV Ž .. Denote by GV0Ž.the subgroup of all invertible elements of GVŽ.. Conjecture 2.8. Let V be a vertex operator algebra. Then for any ŽŽ.. Ž.0Ž. irreducible V-module M, YMlи, z , there is a ⌬ z g GVsuch that the V-module ŽV, Y Ž⌬ Žz .и , z .. is isomorphic to ŽM, YM Žи, z ... As mentioned in the Introduction, this conjecture comes from our interpretation of the physics superselection principle in the context of vertex operator algebras. In Section 3 we shall prove that Conjecture 2.8 holds for vertex operator algebras VL associated to positive-definite even lattices L. Inwx HL0᎐HL3, Hua , a theory of a tensor product for the module category of a vertex operator algebra was developed and it involved geometry in a certain natural way. Later, a formal variable construction of tensor products was given inwx Li3 . These two constructions give isomorphic tensor product modules although they appear differently. For our 448 HAISHENG LI purpose, we recall the following definition of a tensor product for modules for a vertex operator algebra V fromwx Li3Ž seewx HL0᎐HL3 for a different version. .

DEFINITION 2.9. Let M 1 and M 2 be two weak V-modules. A tensor product for the ordered pair Ž M 12, M .ŽŽ..is a pair M, F и, z consisting of a Ž. ŽM . weak V-module M and an intertwining operator F и, z of typeMM12 such that the following universal property holds: For any weak V-module W and Ž. ŽW . any intertwining operator I и, z of typeMM12 , there exists a unique V-homomorphism ␺ from M to W such that IŽ.и, z s ␺ ( F Ž.Žи, z . Here ␺ extends canonically to a linear map from MzÄ4to Wz Ä4.. As a direct consequence of Definition 2.9 we haveŽ seewx Li3 for a proof. :

COROLLARY 2.10. IfŽŽ.. M, F и, z is a tensor product for the ordered pair Ž M12,MofweakV. -modules, thenforanyweakV-module M 3, 3 HomV Ž M, M. is linearly isomorphic to the space of intertwining operators of type Ž.M 3 . MM12 Ž. 0Ž. Ž. Let M be a V-module and let ⌬ z g GVlW. Set W s V, Y и, z s YV ŽŽ.⌬zи,z .. In the theory of superselection sectorsŽ see, for example, wxMSc, Sect. 1.ŽŽŽ... , essentially M, YM ⌬ z и , z was defined to be the tensor product module of M with W. Here we have

PROPOSITION 2.11. LetŽŽ.. W, F и, z be a tensor product for a pair 12 0 Ž.M,M of weak V-modules and let ⌬Ž.Ž.ŽŽ..z g GV.Then W˜˜, F и, zisa tensor product of the pairŽ. M 1, M˜2. Proof. From Proposition 2.4 we have an intertwining operator F˜Ž.и, z ŽŽ. . ŽW˜ . Ž. sF⌬zи,zof typeMM12˜ . Let M be any V-module and let I и, z be Ž.M ŽŽ..y1 any intertwining operator of typeMM12˜ . Then I ⌬ z и , z is an inter- Ž.ŽŽ..ŽŽŽ...M˜ ˆ y1 twining operator of typeMM12 , where M, YMMˆи, z s M, Y ⌬ z и , z . By the universal property of ŽŽ..W, F и, z , there is a unique V-homomorphism ␺ from W˜ˆto M such that I ˆŽ.и, z s ␺ ( F ˆ Ž.и, z . By Proposition 2.5, ␺is a V-homomorphism from W to M. Since ⌬Ž.zuonly involves finitely many terms, we have

1 1 IuŽ,z .sIŽ.Ž.⌬ Ž.z⌬y Ž.zu,zs␺(F⌬Ž.z⌬y Ž.zu,zs␺(FuŽ.,z.

It is not difficult to check that ␺ is the unique V-homomorphism from W˜ to Mˆˆˆsuch that IŽ.и, z s ␺ ( F Ž.и, z . Then the proof is complete. 0 COROLLARY 2.12. Let M be a weak V-module and let ⌬Ž.z g GVŽ.. Then the pairŽŽ.. M˜˜, I и, z is a tensor product of M with V˜, where IŽ.и, z is the SUPERSELECTION PRINCIPLE IN VOA THEORY 449

Ž.M Ž . zLŽy1. Ž. intertwining operator of type MV , defined by I u, zaseYaM ,yzu for a g V, u g M. Proof. By Proposition 5.1.6 inwx Li3 , ŽŽ..M, I и, z is a tensor product for the pair Ž.M, V . By Proposition 2.11, ŽŽ..M˜˜, I и, z is a tensor product for Ž.M,V˜ . In general, we have Ž. 0Ž. Conjecture 2.13. Let M be a weak V-module and let ⌬ z g GVl . Then the pair ŽŽ..ŽŽŽ...M˜˜, I и, z [ M, I ⌬ z и , z is a tensor product of Ž.M, V˜, Ž. Ž . zLŽy1. Ž. where I и, z is defined by Iu,zaseYaM ,yzufor a g V, u g M. The following definition is the algebraic counterpart of the physics definition of simple currentsŽ see, for example,wx SY1, SY2, FG. .

DEFINITION 2.14. Let V be a vertex operator algebra. An irreducible weak V-module M is called a simple current if for any irreducible weak V-module W, there exists a tensor product of M and W, which is irreducible. If the associativity of the tensor product is assumed, then one can show that M being a simple current is equivalent to that the tensor functor ‘‘M = ’’ is a permutation acting on the set of equivalence classes of irreducible weak V-modules. By Corollary 2.12 we have:

0 THEOREM 2.15. For any ⌬Ž.z g GVŽ.Ž,V,Y ŽŽ.⌬zи,z .. is a simple current V-module.

Proof. By definition, we only need to prove that ŽŽ..W˜, YW˜ и, z is irreducible for any irreducible weak V-module ŽŽ..W, YW и, z .IfUis a submodule of W˜, then

y1 YaWWŽ.,zusY˜Ž.⌬ Ž.za,zugUzŽ. Ž. for a g V, u g U.

Thus U is also a submodule of W. Since W is irreducible, W˜is irreducible.

In the last section we will apply this result to vertex operator algebras associated to affine Lie algebras later. 0 Next, we define HVŽ.to be the subset of GVl Ž.consisting of each ⌬Ž.z such that ŽŽŽ...V, Y ⌬ z и , z is isomorphic to ŽŽ..V, Y и, z . Ž. 0Ž. Ž. Ž Ž .. PROPOSITION 2.16. Let ⌬ z g GVlH V and let W, YWи, zbe any irreducible weak V-module. ThenŽ W˜, YW˜ Žи, z ..[ ŽW, Y Ž⌬ Žz .и , zis .. V-isomorphic to W. 450 HAISHENG LI

Proof. First, fromwx FHL, Sect. 5.4 we have an intertwining operator Ž. Ž.W Ž . zLŽy1. Ž. Iи,zof typeWV , defined by Iu,zaseYaW ,yzufor any a g V, ugW. By Proposition 2.4, we obtain an intertwining operator I˜Ž.и, z of Ž.W˜ ˜ typeWV˜ . Let ␺ be a V-isomorphism from V onto V. Then we obtain an Ž.˜ Ž. Ž.W˜ intertwining operator I1 и, z [ I и, z ␺ of typeWV . Furthermore, we Ž. Ž.W˜ Ž . obtain an intertwining operator I22и, z of typeVW , defined by Iu,z¨s zLŽy1. Ž . Ž .Ž.ŽŽ.. Ž. eI12¨,yzu. Since drdz I 1, z s IL22y11,zs0, I 1, z is a Ž. constant independent of z. Let ␾ [ I2 1, z . For any a g V, u g W we have the Jacobi identity

z12zz21z y1 y y1 y z01␦ YaŽ.Ž.,zI21,zu2yz02␦ IŽ.Ž.1,zYa2,zu1 ž/z ž/z 00y

z10z y1 y sz220␦ IYaŽ.Ž.,z1,zu2. ž/z 2 Taking Res we obtain z0

YaŽ.Ž.Ž.Ž.,zI121,zu 2yI 21,zYa 2,zu 1 s0. That is, ␾ is a V-homomorphism from W to W˜. Furthermore, since

z12zz21zz10z y1 y y1 yy1 y z002␦ yz␦ sz␦ ž/ž/ž/z zz 002y and

YaŽ.Ž.,zI121,zu 2sI 2 Ž.Ž.1,zYa 2,zu 1 , we obtain

z10zz10z y1 y y1 y z21␦ YaŽ.Ž.,zI21,zu2sz22␦ IYaŽ.Ž.,z01,zu2. ž/zz ž/ 22 Taking Res Res zy1 zy1 we obtain zz0102

Ia2Ž.,zu 2sYa Ž,z202222qzI .Ž.Ž.Ž1,zusI1,zYa,z 20qzu .. Ž. Ž. Since I22и, z / 0, it follows that ␾ s I 1, z / 0. Because both W and W˜ are irreducibleŽ. from the proof of Theorem 2.15 , ␾ is an isomorphism. Then the proof is complete. Ž. Ž. 0Ž. Ž Ž Ž. .. COROLLARY 2.17. Let ⌬12z , ⌬ z gG V such that V, Y ⌬1z и , z is V-isomorphic toŽŽŽ... V, Y ⌬ 2 z и , z . Then for any irreducible weak V-mod- ŽŽ.. ule W, YW и, z , we ha¨e

Ž.W, YWŽ.⌬1Ž.z и , z , Ž.W, YW Ž.⌬ 2Ž.z и , z . SUPERSELECTION PRINCIPLE IN VOA THEORY 451

ŽŽŽ...ŽŽŽ... Proof. Since V, Y ⌬12z и , z , V, Y ⌬ z и , z , we obtain y1 Ž.V,YŽ.⌬12Ž.z⌬ Ž.zи,z,Ž.V,YŽ.и,z. By Proposition 2.16 we have

y1 Ž.W, YWŽ.⌬12Ž.z ⌬ Ž.z и , z , Ž.W, YWŽ.и, z . Then

Ž.W, YWŽ.⌬1Ž.z и , z , Ž.W, YW Ž.⌬ 2Ž.z и , z . B 0 Remark 2.18. Clearly we have an equivalence relation ' on GVlŽ. Ž. Ž. Ž Ž Ž. .. defined by ⌬12z ' ⌬ z if and only if W, YW⌬1z и , z , ŽŽŽ...W,YW ⌬2 zи,zfor any irreducible weak V-module ŽŽ..W, YW и, z .We Ž. Ž. 0Ž. conjecture that Corollary 2.17 holds if ⌬12z , ⌬ z g GVlsuch that ŽŽ ..ŽŽ .. V,Y⌬и12,z ,V,Y⌬и,z. If this is true, then combining with Con- jecture 2.8 we would have a 1-1 correspondence between IrrŽ.V and 0Ž. GVl r'.

3. SIMPLE CURRENT MODULES

In this section, we first give a construction of ⌬Ž.z from a primary weight-one semisimple element of a vertex operator algebra. Then we apply this result to vertex operator algebras associated to a positive- definite even lattice or to affine Lie algebras. Recall the following commutator formula for a vertex operator algebra VŽwxwxFLM2, 8.6.5 ; see also B. , ϱ m a,b Ž.ab Ž.3.1 wxmnsÝž/i imqnyi is0 for a, b g V, m, n g ޚ. PROPOSITION 3.1. Let V be a regular ¨ertex operator algebra and let a g V Ž. ␦ ޚ such that L n a s n,0a for n g q and that a0 acts semisimply on V. Then a0 acts semisimply on any weak V-module. Proof. Since V is regularŽ any weak V-module is a direct sum of irreducible V-modules with finite-dimensional homogeneous subspaces. , it is enough to prove that a0 acts semisimply on each irreducible V-module Ž. Ž. W. From the commutator formula 3.1 we get wL 0,a00xs0, so that a preserves each homogeneous subspace of W. Since W has finite-dimensional homogeneous subspaces, there exists a 0 / u g W such that au0 s r␭for some ␭ g ރ. Since W is irreducible, W as a V-module is generated Ž. by u.If ab00srb for b g V, r g ރ, then wxa , bns ab0nnsrb for ngޚ. Then it follows that a0 acts semisimply on W. Then the proof is complete. 452 HAISHENG LI

Let V be a regular vertex operator algebra and let h g V satisfy the conditions ␦ ␦␥1 ޚ LnhŽ. s n,0h, hhnns ,1 for any n g q,3.2Ž. where ␥ is a fixed integer, and that h0 acts semisimply on V with integral eigenvalues. Then by Proposition 3.1, h0 acts semisimply on any weak V-module. Combining the commutator formula with the conditionŽ. 3.2 we get ␥␦ wxhmn,hsm mqn,0 Ž.3.3 for m, n g ޚ. Ž. From now on, we also freely use hn for hn. For any ␣ g ޑ, set ϱ ␣ hŽ."k ".k EŽ.␣h,zsexp Ý z .3.4Ž. ž/k ks1 Then fromwx LW we have

y␥␣␤ z2 qy y q EŽ.Ž.␣h,zE12␤h,z s1y EŽ␤h,zE21 .Ž.␣h,z. ž/z 1 Ž.3.5

Define

ϱ hkŽ. hŽ0. yk hŽ0. ⌬Ž.h,zsz exp Ý Ž.yz s zEq Žyh,yz .gUV Ž.Ä4 z. ž/yk ks1 Ž.3.6

PROPOSITION 3.2. Let V be a regular ¨ertex operator algebra and let h g V 0 satisfying Ž.3.2 . Then ⌬ Žh, z .g GVŽ.. To prove Proposition 3.2 we first prove the following lemma.

LEMMA 3.3. Let h g V satisfying Ž.3.2 . Then we ha¨e

q q hŽ0. E Ž.Ž.Žh, zYa12,zEyh,z 1 .sYzŽ. 1⌬ Žyh,z 122yza .,z for a g V.3.7Ž.

Proof. For any a g V, using the formula ϱ k hkŽ.,Ya Ž,z . zYhiaky1 Ž.Ž.,z sÝž/i is0 SUPERSELECTION PRINCIPLE IN VOA THEORY 453 we obtain ϱ hkŽ. yk Ýz12,YaŽ.,z k ks1 ϱϱ1 kykkyi sÝÝ zz12 YhiaŽ.Ž. ,z2 kž/i ks1is0 ϱϱ11ϱ ykk kykkyi sÝÝzzYh12Ž.Ž.0a,z2q Ýzz12 YhiaŽ.Ž. ,z2 kkž/i ks1 ks1is1

z2 sylog 1 y YhŽ.Ž.0a,z2 ž/z 1 ϱϱ 1 kqiykyik qÝÝ zzYhia12Ž.Ž. ,z 2 kiž/i ks0is1 q

z2 sylog 1 y YhŽ.Ž.0a,z2 ž/z 1 ϱϱ1 kyiykyik qyÝÝ Ž.1zzYhia12Ž.Ž.,z 2 i ž/k is1ks0 z ϱ 1 2 yi sylog 1 y YhŽ.Ž.0a,z21qÝŽzyzYhia2 .Ž. Ž. ,z 2. ž/zi 1 is1 Ž.3.8

Then

q q E Ž.Ž.Žh, zYa12,zEyh,z 1 .

yhŽ.0 z2 q sY 1y EhŽ.,z12yza,z 2 ž/z ž/1

hŽ0. sYzŽ.1122⌬Ž.yh,zyza,z.B Ž.3.9

Proof of Proposition 3.2. Since whŽ.0,Yu Ž,z .xsYhŽŽ.0u,z .for any u g V, we have

hŽ0. hŽ0. hŽ0. z YuŽ.,zzy sYz Žu,z . for any u g V.Ž. 3.10 Then it follows from the construction of ⌬Ž.h, z and Lemma 3.3 that ⌬Ž.h,zsatisfies Ž. 2.14 . Since

LŽ.Ž.y1,h0 s0, L Ž.Ž.y1,hk sykh Ž k y 1 . for k g ޚ, 454 HAISHENG LI we obtain

ϱ d yk LŽ.Ž.y1,⌬h,z sÝhk Žy1 .Ž.Ž.yz ⌬h,z s ⌬ Ž.h,z. dz ks1

0 It is clear that ⌬Ž.h, z satisfies the other conditions. Thus ⌬Ž.h, z g GVŽ..

At the end of this section, we apply our results to some concrete examples. Let L be a positive-definite even lattice, let P be the dual lattice of L, and let VL be the vertex operator algebra constructed by Borcherdswx B and Frenkel, Lepowsky, and Meurmanw FLM2 x . Then there is a 1-1 correspondence between the set of equivalence classes of irreducible modules for VL and the set of cosets of PrL wxB, FLM2, D . More specifically, VPLis a V -module with the following decomposition into irreducible modules,

V V V ␤иии V ␤,Ž. 3.11 PLLs [ q1[ [ Lqky1 where k s <

1 ⌬Ž.␤ , zhЈs⌬ Ž.Ž.␤,zhЈy11shЈqzy␤ Ž.Ž.hЈ. 3.12

ŽŽ . . Ž .y1 Ž . Then Y ⌬ ␤, zhЈ,z sYhЈ,zqz ␤hЈ. Thus V˜L as a module for the ˜ Ý n ރ Heisenberg algebra H s ng ޚyÄ04 t m H [ is a completely reducible module which is isomorphic to VL, and the set of eigenfunctions of HŽ.0 on V˜L is ␤ q L. Then it follows from Theorem 3.1 ofwx D that ŽŽŽ...⌬␤и VLL,Y,z,zis isomorphic to V q␤.

It follows from Proposition 3.4 that all irreducible VL-modules can be obtained by using some ⌬Ž.␤, z and that ŽVL, Y ŽŽ...⌬ ␤, z и , z is isomor- ŽŽ.. Ž. phic to VL, Y и, z if and only if ␤ g L. It is clear that ⌬ ␤, z is invertible so that each irreducible module is a simple current. It is also clear that ⌬Ž.Ž.␣, z ⌬ ␤, z s ⌬ Ž␣ q ␤, z .for ␣, ␤ g P. Ž. ŽŽ.. Let ␤i g Pis1, 2 . Then Y ⌬ ␤2 , z и , z is a nonzero intertwining operator of type

Ž.VL, Y Ž.⌬Ž.␤12q ␤ , z и , z . ž/Ž.VL,YŽ.Ž.⌬Ž.␤1,zи,zVŽ.L,Y⌬ Ž.␤2,zи,z SUPERSELECTION PRINCIPLE IN VOA THEORY 455

Since each irreducible VL-module is a simple current, all fusion rules are either zero or 1. This result on fusion rules has been obtained in Chapter 12 ofwx DL using a different method. It is clear that Conjectures 2.8 and 2.13 hold for V s VL. Let g be a finite-dimensional simple Lie algebra with a fixed Cartan subalgebra H, let Ä4␣ ,...,␣ be a set of positive roots, and let Äe , f , h 1 niiinN 4 is1,...,n be the Chevalley generators. Let ␪ s Ý aii␣ be the highest is1 positive root and let ⍀ be the dual Coxeter number of g. Let ²:и , и be the ²: Ž . normalized Killing form on g such that ␪, ␪ s 2. Let ␭i i s 1,...,n be g the fundamental weights of and let Pq be the set of dominant integral weights of g. A dominant integral weight ␭ is said to be minimal if it is Ž .␭ minimal in Pq Exercise 13 in Section 13 ofwx Hum . is said to be cominimal wxFG if ␭kis minimal for the dual Lie algebra. Fromw K, Table y1 Aff 1, Chap. 4xw , ␭iiis cominimal if and only if a s 1. Let ˜g s ރ t, t xm g [ރcbe the affine Lie algebra inwx K . For any complex number l and any weight ␭ of g, let LlŽ.,␭ be the irreducible highest weight ˜g-module of level l with lowest weight ␭. It has been well knownŽ cf.w Li1, Theorem 4.3.1xwx ; see also FZ.Ž. that Ll, 0 has a natural vertex operator algebra structure if l / y⍀. Ž. PROPOSITION 3.5. For any positi¨e integer l, Ll,l␭i is a simple current for LŽ. l,0 if ␭i is cominimal. Ž. ² : Proof. Choose h g H such that ␣ jjih s h, h s ␦ ,jfor 1 F j F n. Then we are going to show that ŽŽŽ...V, Y ⌬ h, z и , z is isomorphic to Ž. Ž. Ll⌳ii. Since a s 1, ␪ h s a is 1. By definition we have y1 y1 ⌬Ž.h,zhjjshql␦ i,jiiiiz , ⌬Ž.h,zesze , ⌬ Ž.h, zfsz f, Ž.3.13

y1 ⌬Ž.h,zejjse, ⌬ Ž.h,zfjjsf, ⌬ Ž.h,zf␪␪sz ffor j / i. Ž.3.14 In other words, the corresponding automorphism ␺ of UŽ.˜g or ULl Ž Ž,0 .. satisfies the conditions

␺ Ž.hniinŽ.shn Ž.q␦,0l, ␺Ž.eniiŽ.sen Žq1, .

␺Ž.fniiŽ.sfn Žy1 . ;Ž. 3.15

␺Ž.hnjjŽ.shn Ž., ␺ Ž.en jj Ž.sen Ž.,

␺Ž.fnjjŽ.sfn Ž.for j / i, n g ޚ,Ž. 3.16 and

␺ Ž.fn␪Ž.sfn␪ Žy1 . for n g ޚ.Ž. 3.17 456 HAISHENG LI

Then the vacuum vector 1 in ŽŽŽ...V, Y ⌬ h, z и , z is a highest weight vector of weight l␭i. Thus ŽŽŽ...V, Y ⌬ h, z и , z is isomorphic to LlŽ.,l␭i as a ˜g-module. By Theorem 2.15, LlŽ.,l␭i is a simple current. Remark 3.6. Proposition 3.5 has been proved inwx FG by calculating the four point functions. Remark 3.7. It has been proved inwx F that those are all simple currents except for E8. Remark 3.8. From Propositions 3.2, 2.1, and the proof of Proposition

3.5 we find that LlŽ.,l␭i is always a weak LlŽ., 0 -module for any complex number l / y⍀.

ACKNOWLEDGMENTS

This paper is based on the first part of the preprint ‘‘The theory of physical superselection sectors in terms of vertex operator algebra language,’’ which was circulated in the Spring of 1995. We thank Professors Dong, Lepowsky, and Mason for many useful discussions.

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