W-algebras for Argyres–Douglas theories Thomas Creutzig∗ 7 1 0 Abstract 2 n The Schur-index of the (A1,Xn)-Argyres–Douglas theory is conjecturally a a character of a vertex operatoralgebra. Here such vertex algebrasare found for J 0 theAoddandDeven-typeArgyres–Douglastheories. Thevertexoperatoralgebra 2 corresponding to A2p−3-Argyres-Douglas theory is the logarithmic p-algebra B of[1],while the onecorrespondingto D2p, denotedby p, is realizedasanon- ] W h regular QuantumHamiltonian reduction of Lk(slp+1) at level k = (p2 1)/p. t − − - For all n one observes that the quantum Hamiltonian reduction of the vertex p e operatoralgebraofDnArgyres–Douglastheoryisthevertexoperatoralgebraof h An−3 Argyres–Douglastheory. As corollary,one realizes the singlet and triplet [ algebras (the vertex algebras associated to the best understood logarithmic 1 conformal field theories) as Quantum Hamiltonian reductions as well. Finally, v 6 charactersofcertainmodulesofthesevertexoperatoralgebrasandthemodular 2 properties of their meromorphic continuations are given. 9 5 0 . 1 0 7 1 : v i X r a ∗DepartmentofMathematicalandStatisticalSciences,UniversityofAlberta,Edmonton,Alberta T6G 2G1, Canada. email: [email protected] 1 1 Introduction Vertex operator algebras are a mathematicians attempt to formalize the notion of the symmetry algebra of two dimensional conformal field theory. Recently a close connection between supersymmetric four-dimensional conformal field theories and vertex operator algebras has been observed [2]. In physics terminology a vertex operator algebra is often named a chiral algebra. Two examples are that certain protected states of the four-dimensional = 2 super conformal field theory are N associated to a chiral algebra [3]; and that one associates vertex operator algebras to two-dimensional surfaces with boundary. Gluing of surfaces then corresponds to maps between vertex operator algebras as e.g. cosets or quantum Hamiltonian reductions, see [4]. One instant of this first phenomenon are Argyres-Douglas theories [5] whose Schur-indexisbelievedtobethecharacter ofavertexoperatoralgebra, forliterature on the topic see e.g. [6, 7, 8, 9, 10, 3, 11]. The objective of this note is to find such vertex operator algebras for the missing type (A ,X )-Argyres–Douglas theories. 1 n These are the cases of D and A . The relevant vertex operator algebras are 2n 2n+1 the -algebra of [1] and a certain non-regular Quantum Hamiltonian reduction p p B W of the simple affine vertex operator algebra L (sl ) at level k = (p2 1)/p. k p+1 − − For details on Quantum Hamiltonian reduction see [12, 13]. The main result is the following statement, which combines Theorems 4.1, 3.1 and 5.7. Theorem. Let p in Z . ≥2 1. The Schur-index of (A ,A ) Argyres-Douglas theory and the character of 1 2p−3 the -algebra agree in the following sense p B ch[Bp] = q−cp2[B4p]IA1,A2p−3 q;z−1 (p−1)2 (cid:0) (cid:1) with c [ ] = 2 6 the central charge of the -algebra. p Bp − p Bp 2. The closed formula of the Schur-index of (A ,D ) Argyres-Douglas theory as 1 2p conjectured by [8] and the character of the -algebra agree in the following p W sense q−cp[2W4p]I(A1,D2p)(x,z;q) = ch[Wp](x,z;q). with c [ ] = 4 6p the central charge of the -algebra. p p p W − W It has been understood by Christopher Beem and Leonardo Rastelli that the vertex operator algebras of (A ,A ) Argyres-Douglas theories are generalized 1 2p−3 Bershadsky-Polyakov algebras and this will be mentioned in [9]. A correspondence between seemingly unrelated concepts like vertex operator algebras and four-dimensional super symmetric conformal quantum field theory is potentially benefitial for both areas. I will in a moment list a few patterns con- cerning the vertex operator algebras and . A physics interpretation of these p p W B observations would be interesting. 2 A key present vertex operator algebra problem is the understanding of the non semi-simple representation category of logarithmic vertex operator algebras, see [14] for the current picture. I find it very interesting that categorical data of the vertex operator algebra as for example fusion products [16] and modular data [17] appear in the four-dimensional setting. For this reason, I provide some modular data for modules that are obtained from the vertex operator algebra itself via twisting by spectral flow automorphism, see Theorem 3.6. This modular data is associated to the meromorphic continuation of characters of modules. A priori, the character of a module is a formal power series and it might converge in a suitable domain. Here, it turns out that this is indeed the case and these characters then can be meromorphi- cally continued to certain vector-valued Jacobi forms of indefinite index. In analogy to the modular story of admissble level L (sl ) (see Theorem 21 of [18]) I expect k 2 that the modular properties of these Jacobi forms capture the semi-simplification of the log-modular tensor category of the corresponding vertex operator algebra. 1.1 Quantum Hamiltonian reduction at boundary admissible level Thereareveryinterestingpatternsemergingoftheinterplay of and andother p p B W vertex operator algebras related to Argyres-Douglas theory. Recall that given an affinevertexoperatoralgebraV (g)atlevelk ofsomeLiealgebragandaLiealgebra k homomorphism from sl to g one associates a W-algebra via Quantum Hamiltonian 2 reduction [13]. If the homomorphism is trivial, then this is just the affine vertex operator algebra again. For the principal embedding of sl in g one obtains the 2 principalor regular W-algebra and other non-trivial embeddings lead to W-algebras that are called non-regular. A level k is called boundaryadmissible [19] if it satisfies h∨ k+h∨ = p for h∨ the dual Coxeter number of g and p a positive integer co-prime to h∨. At such levels, the characters of some simple vertex operator algebra modules allow for particularly nice meromorphic extensions to multi-variable meromorphic Jacobi forms. Nice productforms of characters also appear in thecontext of chiral algebras coming from M5 branes [11]. Remark 2.1 tells us that Observation 1. The W-algebras corresponding to (A ,X ) Argyres-Douglas the- 1 n ories are of boundary admissible level; in the cases of A (n 4) and E this 2n+1 7 ≥ observation is at the moment only conjectural. The case of A can be verified in a similar manner as Theorem 5.7 and that odd will be part of a general study of the -algebras [15]. p B An affine vertex operator algebra is said to be conformally embedded in a larger vertex operator algebra W if both vertex operator algebras have the same Virasoro field. Conformal embeddings in larger affine vertex operator algebras and minimal W-algebras have been of recent interest [20, 21, 22, 23]. The next observation is Corollary 5.8. 3 Observation 2. Let p in Z . The simple affine vertex operator algebra L (gl ) ≥2 k 2 for k+2= 1 embeds conformally in . p Wp Note, that 2+ 1 is not an admissible level of sl . − p 2 The vertex operator algebras of type (A ,D ) all contain an affine vertex opera- 1 n c toralgebraofsl assubvertexoperatoralgebra. OnecanthusconsidertheQuantum 2 Hamiltonian reduction of these vertex operator algebras. These must then be ex- tensions of the Virasoro vertex operator algebra. The next observation is Corollary 5.9 and the beginning of Section 2. Observation 3. The vertex operator algebra of type (A ,A ) Argyres-Douglas the- 1 n ory is isomorphic to the Quantum Hamiltonian reduction of the type (A ,D ) 1 n+3 Argyres-Douglas theory. If n is even this is an isomorphism of vertex operator alge- bras and if n is odd this is an isomorphism of Virasoro modules. I conjecture that also in the case n odd this is a vertex operator algebra iso- morphism. The work in progress [24], see also [25], on the (p)-algebra by Drazen R Adamovi´c should be useful here. This observation is very natural from the physics point of view. Firstly equation (5.20) of [8] relates the Schur-index of ype (A ,D )-Argyres-Douglas theory to the 1 2n one of type (A ,A ) in an analogous way as one relates affine vertex operator 1 2n−3 algebra characters to characters of the W-algebra obtained via Quantum Hamilto- nian reduction. Secondly, in [17], Riemann surfaces are asscoiated to the Argyres- Douglas theories andthesurfacescorrespondingto(A ,A )Argyres-Douglas theory 1 n and (A ,D ) Argyres-Douglas theory only differ by an extra regular puncture. 1 n+3 The authors of that article find a natural cap state that closes this puncture and thus maps the (A ,D ) Argyres-Douglas theory to the (A ,A ) Argyres-Douglas 1 n+3 1 n theory. 1.2 Singlet and triplet algebras as Quantum Hamiltonian reduction The triplet algebras W(p), for p in Z are the best understood C -cofinite but ≥2 2 non-rational vertex operator algebras [26, 27, 28]. The singlet algebra M(p) is its U(1)-orbifold [29, 30]. These algebras are closely related to the vertex operator algebras of Argyres-Douglas theories. TheM(p)andW(p)algebras areinfinteorder extensions of theVirasoro algebra Vir at central charge c = 1 6(p 1)2/p. W(p) actually carries an action of p p − − PSL(2,C) [31] and as PSL(2,C) Vir -module p ⊗ ∞ W(p)= ρ L(h ,c ) ∼ 2n+1⊗ 2n+1,1 p n=0 M withρ then-dimensionalirreduciblerepresentationofPSL(2,C)[31]andL(h ,c ) n n,1 p the simple highest-weight module of the Virasoro vertex operator algebra at central 4 charge c at hightest-weight h = ((n2 1)p 2(n 1))/4. The singlet on the p n,1 − − − other hand decomposes as Virasoro module as ∞ M(p) = L(h ,c ). ∼ 2n+1,1 p n=0 M Corollary 5.9 tells us the following. Let k = 1 2, then the two L (sl )-modules p − k 2 inherit each the structure of a simple vertex operator algebra from p W ∞ ∞ := Com( , )= L(k,2m) and := (2m+1)L(k,2m). Cp H Wp ∼ Yp m=0 m=0 M M Here L(k,n) denotes the Weyl-module at level k induced from the n+1 dimension irreducible representation of sl . It is simple, see Section 3.1.1. Proposition 5.10 2 follows: As Virasoro vertex operator algebra-modules, we have H ( ) = , H ( )= (p) and H ( ) = (p). QH Wp ∼ Bp QH Cp ∼ M QH Yp ∼ W Here H is the so-called plus reduction functor of the Quantum Hamiltonian re- QH ductionfromtheaffinevertex operator algebraofsl totheVirasorovertex operator 2 algebra. It follows directly from [32, Theorem 9.1.4]. I strongly belief that this iso- morphism is even a vertex operator algebra-isomorphism. Moreover it is tempting to conjecture as well that contains PSL(2,C) as subgroup of automorphisms and p Y that as PSL(2,C) L (sl )-module k 2 ⊗ ∞ = ρ L(k,2m). (1) Yp ∼ 2m+1 ⊗ m=0 M Remark, that vertex operator algebras with continuous groups of outer automor- phisms are important in Davide Gaiotto’s picture of dualities in = 2 super N conformal gauge theories [33]. These dualities relate to the geometric Langlands correspondence and so it is worth speculating that the resemblance of to chiral p Y Hecke algebras of sl is not a coincidence.1 2 1.3 Conjectural generalizations The vertex operator algebras of this work allow for two conjectural generalizations. Firstly it seems that [20, 21, 22, 23] and this work together are first members of a large family of conformal embeddings of affine vertex operator algebras into bound- ary admissible level W-algebras. We now define a family of W-algebras parameter- ized by two positive integers n,m so let us call this algebra . Consider sl , n,m n+m W thenwecanembedsl gl intheobviousway. ConsidertheQuantumHamiltonian m n ⊕ reduction of V (sl ) for the sl embedded as follows, embed sl principally in the k n+m 2 2 sl sub algebra and map it trivially in gl so that the affine sub vertex operator m n 1I thankTomoyuki Arakawa for pointing out that Y is similar to chiral Heckealgebras of sl . p 2 5 algebraisV (gl ). ThecentralchargeofthisW-algebracanbecomputedusing k+m−1 n equation (2.3) of [13] and one gets k((n+m)2 1) c = − (m 1) km(m+1)+m(m2 m 1)+n(m2 2m 1) . n,m,k k+n+m − − − − − − (cid:0) (cid:1) is defined to be this W-algebra at level k, such that n,m W n+m k+n+m = m+1 so that this is a boundary admissible level provided n+m and m+1 are co-prime. The central charge then becomes n(mn+1+m) which coincides with the central − charge of V (gl ). I thus conjecture k+m−1 n Conjecture 1.1. Let n,m be positive integers, n 2. Then the simple affine vertex ≥ operator algebra L (gl ) embeds conformally in . −nmm++11 n Wn,m This conjecture for m = 1 has been proven in [21, Theorem 5.1], for m = 2 in [20, Theorem 1.1] and we treated the case of n = 2. Generalizations of the singlet and triplet algebras exist as well [34, 35] and have been called narrow W-algebras W0(p) . Here Q denotes the root lattice of a simply Q laced Lie algbera and p is a positive integer greater or equal to two. Consider the regular quantum Hamiltonian reduction of . By this we mean the principal n,m W embedding of sl in sl and the corresponding Quantum Hamiltonian reduction of 2 n . n,m W Conjecture 1.2. Let m+1 = p(n 1). Then the Heisenberg coset of the regular − Quantum Hamiltonian reduction H ( ) of is isomorphic to the narrow reg n,m n,m W W W-algebra of type sl and parameter p: n Com( ,H ( )) = W0(p) . H reg Wn,m ∼ An−1 In this work, this conjecture has been proven in the case of n = 2, but only as isomorphism of Virasoro modules. 1.4 Organization In Section 2 the Schur-indices of Argyres-Douglas theories are quickly reviewed. Section 3 starts with adiscussion of properties of Weyl modules for L (sl )when k 2 k +2 = 1 and p in Z . Next the L (sl ) -module is introduced. Here p ≥2 k 2 ⊗H Xp H denotestherankoneHeisenbergvertexoperatoralgebra. Thereasonforintroducing is that its character coincides with the Schur index of type (A ,D )-Argyres- p 1 2p X Douglastheory. IthendiscussmodularandJacobipropertiesofcharactersoftwisted versions of using Appell-Lerch sums. p X 6 In Section 4 properties of the -algebra are discussed and especially it is ob- p B servedthatits character coincides withtheSchurindexoftype(A ,A )-Argyres- 1 2p−3 Douglas theory. Finally in Section 5 the properties of mentioned in the introduction are p W derived; especially its character coincides with the Schur index of type (A ,D )- 1 2p Argyres-Douglas theory. AcknowledgementsMost ofall, Iamvery gratefultoShu-HengShaoformany discussionsonthistopic. IalsowouldliketothankTomoyukiArakawa, ChrisBeem, LeonardoRastelliandDavideGaiottoforexplanationsontherelationofvertexoper- ator algebras and four-dimensional quantum field theory. I thank Drazen Adamovi´c for his useful comments on the draft and discussion. Finally I very much appreciate discussions with Wenbin Yan and Ke Ye and sharing [17] with me. I am supported by NSERC RES0020460 and have benefitted from the workshop ”exact operator algebras in superconformal field theories” on the topic at Perimeter Institute. 2 Schur-indices of (A ,X ) Argyres-Douglas theories 1 n I collect some data from [8, 6]. First recall the known cases. The Schur index of (A ,A ) Argyres-Douglas theories is identified with the 1 2n vacuum character of the simple and rational (2,2n + 3) Virasoro vertex operator algebra at central charge c = 1 6(2n + 1)2/(4n + 6) [36]. The Schur index of − (A ,D ) Argyres-Douglas theories is identified with the vacuum character of the 1 2n+1 simple affine vertex operator algebra of sl at level k = 4n/(2n + 1), denoted 2 − by L (sl ). Note, that k + 2 = 2/(2n + 1) and hence the quantum Hamiltonian k 2 reduction of the (A ,D ) Argyres-Douglas theory vertex operator algebra is the 1 2n+1 vertex operator algebraofthe(A ,A )Argyres-Douglas theory. TheSchurindex 1 2n−2 of the (A ,E )-Argyres-Douglas theory coincides with the vacuum character of the 1 n simple regular W-algebra of sl at level k+3= 3/7 for E and at level k+3 = 3/8 3 6 for E . In the case of E it seems to correspond to the sub-regular W-algebra of sl 8 7 3 at level k+3 = 3/5 (Shu-Heng Shao has verified this for the leading terms). Remark 2.1. We will see that all W-algebras corresponding to (A ,X ) Argyres- 1 n Douglas theories are of boundary admissible level (in the case of A and E this is odd 7 conjectural). A level k associated to a simply-laced Lie algebra g is called boundary admissible if it satisfies k +h∨ = h∨ for h∨ the dual Coxeter number of g and p p a positive integer co-prime to the dual Coxeter number. The notion of boundary admissible level appeared in [19] and at such a level the meromorphic continuation of characters has a particular nice form. 7 2.1 (A ,A ) Argyres-Douglas theories 1 2p−3 The central charge for this case is (p 1)2 c =2 6 − p − p and equation (5.14) of [8] is easily modified to 2 2 p n+1− 1 p n+1+ 1 q−c2p4I(A1,A2p−3)(z;q) = η(1q)2  q (cid:16) p 2n+21p−(cid:17)1 − q (cid:16) p 2n+21p+(cid:17)1 . (2) nX∈Z 1−zq (cid:16) 2 2p(cid:17) 1−zq (cid:16) 2 2p(cid:17)   Thedomain forz andq is z±1qp−21 < 1. Wewillobserve thatthisformulacoincides | | with the vacuum character of the -algebra of [1]. This algebra is conjecturally a p B subregular W-algebra of sl at level k+p 1 = p−1 and this conjecture is true in p−1 − p the first cases p = 3,4,5. The case p = 5 follows due to the recent identification of Naoki Genra [37] of these subregular W-algebras with Feigin-Semikhatov algebras [38]. 2.2 (A ,D ) Argyres-Douglas theories 1 2p TheSchur-indexofthe(A ,D )Argyres-Douglastheoryisjustthevacuumcharacter 1 2 oftheranktwoβγ-vertexoperatoralgebra 2,whichwewilldiscussasaninstructive S baby example in Section 5.1. The central charges for this series are c = 4 6p. p − A closed formula for the Schur-Index is conjectured in equation (1.5) of [8]. It reads ∞ (x,z;q) = f˜p (q;x)f (q;z) (3) I(A1,D2p) ρm ρm m=0 X with ρ the m-dimensional irreducible representation of sl and m 2 p(m2−1) f˜ρpm(q;x) = ∞q (14 qn)trρm xhq−ph42 n=1 − (cid:18) (cid:19) where our sl convention is that Qh,e,f form a basis with commutation relations 2 [h,e] = 2e, [h,f]= 2f, [e,f] = h. − f (q;z) = P.E. q ch[ρ ](z) ch[ρ ](z) is defined using the plethystic exponential ρm 1−q 3 m h i ∞ F(qk;zk) P.E.[F(q;z)] = exp k ! k=1 X 8 and it can be simplified as follows q f (q;z) = P.E. ch[ρ ](z) ch[ρ ](z) ρm 1 q 3 m (cid:20) − (cid:21) q = P.E. z2+1+z−2 ch[ρ ](z) m 1 q (cid:20) − (cid:21) ∞ (cid:0) (cid:1) = P.E. qn z2+1+z−2 ch[ρ ](z) m (4) " # n=1 X (cid:0) (cid:1) ∞ = ch[ρ ](z) exp qn z2+1+z−2 m n=1 Y (cid:0) (cid:0) (cid:1)(cid:1) ∞ = ch[ρ ](z) 1 z2qn −1(1 qn)−1 1 z−2qn −1. m − − − n=1 Y(cid:0) (cid:1) (cid:0) (cid:1) 3 The L (sl ) -module k 2 p ⊗ H X The aim is to find a vertex operator algebra for each Schur-index. For this, we will first find an L (sl ) -module whose character coincides with the (A ,D ) k 2 p 1 2p ⊗ H X Schur-index. 3.1 The affine vertex operator algebra L (sl ) k 2 I use [39, 18] for conventions on the affine vertex operator algebra L (sl ). k 2 We denote the generators of sl by h ,e ,f ,K and d for n integer. As usual we 2 n n n identify the action of d with L . K is central and the modes satisfy 0 − [h ,e ]= +2e , [h ,h ] = 2mδ K, [e ,e ] = 0, m n n+m m n n+m,0 m n [h ,f ]= 2f , [e ,f ]= h +mδ K, [f ,f ] = 0. m n n+m m n n+m n+m,0 m n − The affine vertex operator algebra L (sl ) is strongly generated by fields k 2 e(z),h(z),f(z) with operator products 2e(w) 2k 2f(w) h(z)e(w) , h(z)h(w) , h(z)f(w) , ∼ (z w) ∼ (z w)2 ∼ −(z w) − − − k h(w) e(z)f(w) ,+ . ∼ (z w)2 (z w) − − The Virasoro field for k+2 = 0 is 6 1 1 L(z) = : h(z)h(z) :+ : e(z)f(z) :+ : f(z)e(z); 2(k+2) 2 (cid:18) (cid:19) and it has central charge 3k 6 c = = 3 k+2 − t 9 with k+2 = t. Let n = span (e ,f ,h n 1) C n−1 n n | ≥ be a nilpotent subalgebra, h = Ch CK Cd 0 ⊕ ⊕ the Cartan subalgebra and b = h n the corresponding Borel subalgebra. Let C λ ⊕ the one-dimensional b-module on which n acts as zero, h acts by multiplication 0 with λ, K by k and L by h where 0 λ λ(λ+2) h = . λ 4(k+2) Denote by V(k,λ) = Indsl2C b λ c and its simple quotient by L(k,λ). We will surpress the level k if its value is clear. Singular vectors are determined by the determinant of the Shapovalov form on the weight spaces. This determinant for the weight space of weight (λ µ,k,h +m) is λ − given by the formula of Kac and Kazhdan [40], which I take from (2.8) of [41]. ∞ det (µ,m) = (λ+1 ℓ)P(2ℓ−µ,m)(λ+1+n(k+2) ℓ)P(2ℓ−µ,m−nℓ) λ − − (5) ℓ,n=1 Y ( λ 1+n(k+2) ℓ)P(−2ℓ−µ,m−nℓ)(n(k+2))P(−µ,m−nℓ) · − − − P(µ,m) is the multiplicity of the weight (µ,m) in the level zero vacuum module V (sl ). A singular vector is a highest-weight vector of the Verma module and it 0 2 appears if one of the factors vanishes as well as the arguments of P appearing in the exponent of the corresponding factor. 3.1.1 The case k+2= 1 p For this work, the relevant levels are those for which k+2 = 1 and p is a positive p integer. In this case, the universal affine vertex operator algebra is simple, i.e. V (sl ) = L (sl ) [42]. For the purpose of this article it remains to understand k 2 k 2 irreducibility of all Weyl modules as suggested in [20, Remark 6.2]. We need to analyze the factors of (5) of V(k,λ) for λ > 0. 1. The first factor vanishes for ℓ = λ + 1 and the arguments of P vanish if µ = 2ℓ = 2(λ + 1) and m = 0. We thus have a singular vector of weight ( λ 2,k,h ). λ − − 2. The second factor vanishes if λ + 1 + n ℓ = 0 and the argument of the p − corresponding P vanish if 2ℓ = µ and m = nℓ. Set n = pr. The condition of beingasingularvector constraintstheconformalweighttosatisfyh = h + λ−µ λ m implying that 4m = µp(µ 2 λ). It follows that 2rµ = 4rℓ = µ(µ 2 λ). − − − − Now µ = 0 implies ℓ = 0, which cannot be since ℓ > 1. We thus have 2r =µ 2 λ and thus − − 0 = 2λ+2+2r 2ℓ = 2λ+2+µ 2 λ µ = λ. − − − − 10