On classification of rational vertex operator algebras of central charge 1 Chongying Dong University of California at Santa Cruz (based on joint works with Cuipo Jiang) May, 2014 Nashville
Outline Background Recent progress New result Further study
Background A vertex operator algebra V is called rational ([Zhu], J. AMS.,1996], [Dong-Li-Mason], Math. Ann. 1998) if all the admissible modules of V are completely reducible. A vertex operator algebra V is called C 2 -cofinite ([Zhu], J. AMS, 1996), if V/C 2 (V) is finite-dimensional.
Background Theorem ([Zhu], J. AMS, 1996, [Dong-Li-Mason], Math. Ann, 1998, [Anderson-Moore], Comm. Math. Phys., 1988) Let V be a simple rational vertex operator algebra with central charge c. Then (1) The Zhu algebra A(V) is semisimple. (2) V has finitely many irreducible admissible modules M i = where Mλ i i 0. (3) dimmn+λ i i <. Mn+λ i i,i = 0,1,,p, n=0 (4) λ i,c Q,i = 0,1,,p.
Background Let V be a simple rational and C 2 -cofinite vertex operator algebra and M i = n=0 Mn+λ i i, i = 0,1,,p be its all irreducible modules. Let λ min be the minimum of λ i s. The effective central charge c is defined to be c = c 24λ min.
Background Remark Even for rational vertex operator algebras, c and c can be quite different. For example, by choosing suitable p,q,m,n Z +, one can easily construct infinitely many rational vertex operator algebras L(c p,q,0) m L(1/2,0) n. such that c > c.
Background Theorem ([Zhu], J. AMS. 1996, [Dong-Lin-Ng, arxiv:1209.1168], ) Let V be rational and C 2 -cofinite. (1) For homogeneous u V of weight k with respect to the operator L[0], the p + 1-tuple Z(u,q) = (Z M 0(u,q),...,Z M p(u,q)) is a vector-valued modular form of weight k, where Z M j(u,q) = tr M jo(u)q L(0) c/24 = q c/24+λ j tr M j o(u)q n. n+λ n 0 j (2) If λ i > 0 if i 0 the each Z M j(u,q) is a modular form of weight k for a congruence subgroup which depends on V only. (3) p i=0 Z i(q) 2 is SL 2 (Z)-invariant where Z i (q) = Z M i(1,q).
Background There are two different directions in classification of rational vertex operator algebras currently. One direction is to classify simple holomorphic vertex operator algebras ( see [Schellekens, Comm. Math. Phys., 1993], [Dong-Griess-Hohn, Comm. Math. Phys., 1998], [Dong-Mason, IMRN, 2004], [Dong-Griess-Lam, Amer. J. Math., 2007], [Lam-Yamauchi, Comm. Math. Phys., 2008], [Lam-Shimakura-1, Proc. London Math. Soc., 2012], [Lam-Shimakura-2, arxiv: 1209.4677], [Lam, Comm. Math. Phys., 2011], [Sagaki-Shimakura, arxiv: 1302. 4826], etc.) The other direction is to classify rational vertex operator algebras with small central charge.
Background Theorem ([Dong-Zhang, J. Alg. 2008]) Let V be a simple, rational and C 2 -cofinite vertex operator algebra with central charge 0 < c = c < 1, then V is an extension of L(c p,p+1,0). Remark 1. Classification of rational vertex operator algebra with c = c < 1 is the classification of unitary vertex operator algebras with c = c < 1 [Dong-Lin, J. Algebra 2014] The classification of rational vertex operator algebra with c = c < 1 is equivalent to the classification of extensions of L(c p,p+1,0). [Lam, Ching Hung-Lam-Ngau Yamauchi, Int. Math. Res. Not. 2003]
Backgorund Remark Extensions of L(c p,p+1,0) have been classified in the theory of conformal nets in [Kawahigashi-Longo, Ann. Math., 2004]. At the character level, the A-D-E classification has been accomplished in [Cappelli-Itzykson-Zuber, Comm. Math. Phys. 1987].
Main results Next, we consider the case that c = c = 1. Remark In the theory of conformal nets, classification of conformal nets of central charge 1 with a spectrum condition has been estabilished in [Xu, Pacific J. Math. 2005].
Main results In the character level, we have the following result. Theorem (Kiritsis, Phys. Lett., 1989) Let V be a rational CFT type vertex operator algebra such that c = 1, each irreducible character Z i (q) is a modular function on a congruence subgroup and Z i (q)z i (q) is modular invariant, then the q-character of V is equal to the character of one of the following vertex operator algebras V L,V L + and VG Zα, where L is any positive definite even lattice of rank 1, Zα is the root lattice of type A 1 and G is a finite subgroup of SO(3) isomorphic to A 4,S 4 or A 5.
Conjecture V L,V L + and VG Zα should give a complete list of simple and rational vertex operator algebras with c = c = 1.
Main results We have the following result [Dong-Griess, J. Algebra 1998], [Dong-Griess-Ryba, J. Algebra 1999] Theorem Let V {V + Zβ,VA 4 Zα,VS 4 Zα,VA 5 Zα (β,β) 2Z +,(α,α) = 2} and G a finite automorphism of V then V G L {V + Zβ,VA 4 Zα,VS 4 Zα,VA 5 Zα (β,β) 2Z +,(α,α) = 2}.
Main results Remark From the orbifold theory conjecture, V G is rational if V is rational and G is a finite automorphism group of V. So from the orbifold theory point of view, is complete. {V + Zβ,VA 4 Zα,VS 4 Zα,VA 5 Zα (β,β) 2Z +,(α,α) = 2}
Main results Rationality of V + L. Abe, Math. Z. 2005. Dong-Jiang-Lin, Proc. London Math. Soc. 2012.
Main results Theorem (Dong-Mason, IMRN, 2005) Let V be a strongly CFT type rational vertex operator algebra. Then (1) The Lie algebra V 1 is reductive, and the rank l of V 1 satisfies that l c. (2) The following are equivalent: (i) c = l = c. (ii) There is a positive-definite even lattice L such that V is isomorphic to V L.
Main results By the above result, to classify simple rational vertex operator algebras with c = c = 1, we may assume that V 1 = 0.
Main results The following result is obtained in [Zhang-Dong, Comm. Math. Phys. 2008]. [Dong-Jiang, Comm. Math. Phys. 2010]. Theorem Assume that V is a rational, C 2 -cofinite CFT type vertex operator algebra satisfying the following conditions: (a) c = c = 1, (b) dimv 2 2, (c) V is a sum of highest weight modules for the Virasoro algebra. Then V is isomorphic to V + Zα with (α,α) = 4.
Main results A complete characterization of V Zα + is achieved in [Dong-Jiang, J. Reine Angew. Math. 2014], [Dong-Jiang, Adv. Math., 2013] Theorem Assume that V is a simple, rational and C 2 -cofinite CFT type vertex operator algebra with c = c = 1. If V satisfies the following conditions: (a) V 1 = 0 and dimv 2 = 1, (b) dimv 4 3, (c) V is a sum of highest weight modules for the Virasoro algebra. Then V is isomorphic V Zα + with (α,α) = 2n for n 3.
Main results The main ideas and techniques are the following fusion rules: Theorem ([Milas, J. Alg. 2002], =Dong-Jiang, Comm. Math. Phys. 2010], [Rehern-Tuneke, Lett. Math. Phys. 2000] We have ( dimi L(1,0) dimi L(1,0) ( where n,m Z +. L(1,k 2 ) L(1,m 2 )L(1,n 2 ) L(1,k 2 ) L(1,m 2 )L(1,n 2 ) ) ) = 1, k Z +, n m k n + m, = 0, k Z +, k < n m or k > n+m,
Main results For n Z + such that n p 2 for all p Z +, we have ( ) L(1, n) dimi L(1,0) L(1,m 2 = 1, )L(1,n) dimi L(1,0) ( for k Z + such that k n. L(1, k) L(1,m 2 )L(1,n) ) = 0,
Main results To finish the classification of rational CFT type vertex operator algebras with c = c = 1 one has to show that if dimv 4 < 3 then V is isomorphic to VZβ G with (β,β) = 2, where G is one of the three groups {A 4,S 4,A 5 }.
Main results Theorem (Dong-Jiang, J. Alg., 2012) V A 4 L 2 is C 2 -cofinite and rational.
Main results Theorem (Dong-Jiang, J. Alg., 2012) There are exactly 21 irreducible modules of V A 4 L 2. We give them by the following tables 1-4. (V + Zβ )0 (V + Zβ )1 (V + Zβ )2 V Zβ V Zβ+ 1 8 β V Zβ+ 3 8 β 1 9 ω 0 4 4 1 16 16 (V Zβ+ 1 4 β )0 (V Zβ+ 1 4 β )1 (V Zβ+ 1 4 β )2 ω 1 4 9 4 9 4
Main results Theorem (Dong-Jiang, J. Alg., 2012) W 1,T 1,1 W 1,T 1,2 W 1,T 1,3 W 2,T 1,1 W 2,T 1,2 W 2,T 1,3 ω 1 36 25 36 49 36 W 1,T 2,1 W 1,T 2,2 W 1,T 2,3 W 2,T 2,1 W 2,T 2,2 W 2,T 2,3 ω 1 36 25 36 49 36 1 9 1 9 4 9 4 9 16 9 16 9
Main results Fusion rules for V A 4 L 2 2013] is given in [Dong-Jiang-Jiang-Jiao-Yu,
New result Theorem (Dong-Jiang, 2013) Let V be a simple, rational and C 2 -cofinite CFT type vertex operator algebra with c = c = 1. If V satisfies the following condition: (1) V is a sum of highest weight modules for the Virasoro algebra; (2) The lowest weight of non-identity primary vectors is 9, then V is isomorphic to V A 4 L 2.
New result In the characterization of V L +, Dong-Lin-Ng s result on congruence subgroup property for rational vertex operator algebra was not available, so we did not use Kiritsis s result on the irreducible character of rational vertex operator algebras with c = 1. In the characterization of V A 4 L 2 we use both results to assume the characters of the abstract VOA satisfying conditions (1)-(2) and V A 4 L 2 are the same.
Further Study Remark Since V S 4 L 2 is the subvoa of fixed points under an automorphism of V A 4 L 2 of order 2, by a recent result of Miyamoto, V S 4 L 2 is C 2 -cofinite and rational.
Further Study Construct all the irreducible modules of V S 4 L 2 and V A 5 L 2 ; C 2 -cofiniteness and Rationality of V A 5 L 2 ; Characterize V S 4 L 2 and V A 5 L 2. The study of V A 5 L 2 group. may be very difficult, as A 5 is a simple
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