Urod Algebras and Translation of W-Algebras

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Let us describe our results in more details. 1.1. Main Theorem. Let g be a simple Lie algebra, g = g((t)) CK be the affine ⊕ Kac-Moody Lie algebra associated with g defined by the commutation relation b [xf,yg] = [x, y]fg + (x, y)Rest=0(gdf)K, [K, g] = 0, where ( ) is the normalized inner product of g (it is 1/2h∨ times the | k Killing form of g). For k C, let V (g)= U(g) C C, where C is regarded ∈ ⊗U(g[[t]]⊗ K) as ab one-dimensional representation of g[[t]] CK on which g[[t]] acts trivially and ⊗ K acts via multiplication by k. V k(g) is naturallyb a vertex algebra, and is called the universal affine vertex algebra associated with g at level k. Any (graded) quotient V of V k(g) inherits the vertex algebra structure from k k V (g). Let Lk(g) be the unique simple (graded) quotient of V (g). The vertex algebra Lk(g) is integrable as an g-module if and only if k Z>0. • ∈ For a nilpotent element f of g, let HDS,f (M) be the BRST cohomology of the quantized Drinfeld-Sokolov reductionb associated with (g,f) with coefficients in a g-module M ([FF90a, KRW03]). The W -algebra associated with (g,f) at level k is by definition the vertex algebra b Wk 0 k (g,f)= HDS,f (V (g)). For any smooth g-module M of level k, Hi (M), i Z, is a module over Wk(g,f). DS,f ∈ More generally, for any vertex algebra V equipped with a vertex algebra homomor- k i phism V (g) bV and a V -module M, HDS,f (M), i Z, is a module over the → 0 ∈ vertex algebra HDS,f (V ). Theorem 1. Let V be a quotient of the universal affine vertex algebra V k(g) and ℓ Z> . We have a vertex algebra isomorphism ∈ 0 (2) H0 (V L (g)) = H0 (V ) L (g) DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of g on V L (g). More generally, let V be a vertex algebra equipped ⊗ ℓ with a vertex algebra homomorphism V k(g) V . Then we have an isomorphism → b • • (3) H (V L (g)) = H (V ) L (g), DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ of graded vertex algebras, and for any V -module M, Lℓ(g)-module N, there is an isomorphism H• (M N) = H• (M) N. DS,f ⊗ ∼ DS,f ⊗ as modules over H• (V L) = H• (V ) L. DS,f ⊗ ∼ DS,f ⊗ In the case that g = sl2 and L = L1(sl2), the isomorphism (3) was established in [BFL16]. Our argument only requires certain properties of integrable representations and especially also works for superalgebras, see section 5. We note that the existence of the isomorphism (3) as vector spaces is not difficult to see. However, it is not a priori clear at all why there should exist an isomorphism of vertex algebras. We also note that (3) is not compatible with the standard conformal gradings of both sides. To remedy this, we need to change the conformal UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 3 vector of Lℓ(g) on the right-hand-side to a new conformal vector ωUrod, which we call the Urod conformal vector (see Section 6). The proof of Theorem 1 is based on some new construction of automorphisms of vertex algebras, which may be of independent interest, see Section 2 for the details.

1.2. Translation for W -algebras. By applying Theorem 1 to V = V k(g), k C, ∈ we obtain the vertex algebra isomorpshim H0 (V k(g) L (g)) = Wk(g,f) L (g). DS,f ⊗ ℓ ∼ ⊗ ℓ Consequently, the natural vertex algebra homomorphism V k+ℓ(g) V k(g) L (g) → ⊗ ℓ induces a vertex algebra homomorpshim Wk+ℓ(g,f) H0 (V k(g) L (g)) ∼ Wk(g,f) L (g) → DS ⊗ ℓ −→ ⊗ ℓ Therefore, for any Wk(g,f)-module M and any integrable representation L of g of level ℓ, M L is has the structure of an Wk+ℓ(g,f)-module. As a consequence, we ⊗ obtain the translation by L, that is, the exact functor (1) as we wished. b Recall that the Zhu algebra of Wk(g,f) is isomorphic to the finite W -algebra [Pre02] U(g,f) associated with (g,f) ([Ara07, DSK06]). Also, the Zhu algebra of Lℓ(g) is isomorphic to the quotient Uℓ(g) of U(g) by the two-sided ideal generated by ℓ+1 eθ ([FZ92]), and so is that of Lℓ(g) with the Urod conformal structure ([Ara15b]). Therefore, by taking the Zhu algebras we obtain from (1) an algebra homomorphism (4) U(g,f) U(g,f) U (g). → ⊗ ℓ Since any finite-dimensional g-module is an Uℓ(g)-module for a sufficiently large ℓ, (4) gives M E a structure of U(g,f)-module for any U(g,f)-module M and a ⊗ finite-dimensional g-module E. We expect that this U(g,f)-module structure of M E does not depend on the choice of a sufficiently large ℓ, and coincides with ⊗ the one obtained by Goodwin [Goo11].

1.3. Higher rank Urod algebras. We denote also by Wk(g) the W -algebra k W (g,fprin) associated with a principal nilpotent element fprin of g. Let g be ∨ ∨ simply-laced and suppose that k + h 1 Q6 , where h is the dual Coxeter − 6∈ 0 number of g. By the coset construction [ACL19] of the principal W -algebra Wk(g), we have a conformal vertex algebra embedding ,(V k(g) Wℓ(g) ֒ V k−1(g) L (g (5) ⊗ → ⊗ 1 where ℓ is the number deﬁned by the formula 1 1 (6) + =1. k + h∨ ℓ + h∨ By taking the Drinfeld-Sokolov reduction with respect to the level k action of g, (5) gives rise to the full vertex algebra embedding b ,(Wk(g,f) Wℓ(g) ֒ H (V k−1(g) L (g)) = Wk−1(g,f) (g (7) ⊗ → DS,f ⊗ 1 ∼ ⊗U where the last isomorphism follows from Theorem 1 and (g)= (g,f) is the vertex U U algebra L1(g) equipped with the Urod conformal vector ωUrod. We call the vertex 4 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN operator algebra (g) the Urod algebra. In the case that g = sl and f = f , U 2 prin (g) is exactly the Urod algebra introduced in [BFL16]. U Let L be a level one integrable representation of g, which is naturally a module over the Urod algebra (g). By (41), for any Wk(g,f)-module M, the tensor U product M L has the structure of a Wk(g,f) Wbℓ(g)-module. We are able to ⊗ ⊗ describe the decomposition of M L as Wk(g,f) Wℓ(g)-modules for various M ⊗ ⊗ and L (Theorems 8.1, 8.2, 8.4, 8.7 and Corollaries 8.4, 8.8). In particular, Corollary 8.4 states that when M is a generic Verma module of Wk−1(g), then M L decomposes into a direct sum of tensor products of Verma k ⊗ ℓ modules of W (g) and W (g). In the case that g = sl2 with an appropriate choice of L, this provides the decomposition that was used in [BFL16] to give a representation theoretic interpretation of the Nakajima-Yoshioka blowup equations for the moduli space of rank two framed torsion free sheaves on CP2. In a forthcoming paper we show how the decomposition for g = sln stated in Corollary 8.4 can be used to give a representation theoretic interpretation of Nakajima-Yoshioka blowup equations for the moduli space of framed rank n sheaves on CP2 via the AGT conjecture established by Schiﬀmann and Vasserot [SV13].

1.4. Higher rank Urod algebras and VOA[M4]. The Urod algebra is proposed to be important for general smooth 4-manifolds. This appeared in the recent work [FG18] of Gukov and the third named author. For a compact simply-laced Lie group G, one can conjecturally [FG18] associate a vertex operator algebra VOA[M] = VOA[M, G] to every smooth 4-manifold M and an cateogry of modules for VOA[M] to every boundary component of M. The vertex algebra VOA[M] should then act on the cohomology of the moduli space of G-instantons on M. Moreover, the invariant VOA[M] should have the following property: Glueing 4-manifolds along a common boundary amounts to extending the tensor product of the two associated vertex algebras along the categories of modules, see [CKM19] for the theory of these vertex algebra extensions. When G = SU(2) we have [FG18] that VOA[M #CP2]= (sl ) VOA[M ], 4 U 2 ⊗ 4 2 2 where M4#CP is the connected sum of M4 and CP . One expects the same type of formula for any simply laced G with (sl ) replaced by the corresponding higher U 2 rank Urod algebra.

1.5. Higher rank Urod algebras and Vertex algebras for S-duality. The present work was originally motivated by a conjecture that appeared in the context of vertex algebras for S-duality [CG17]. The problem lives inside four-dimensional supersymmetric GL-twisted gauge theories and vertex algebras appear on the intersection of three-dimensional topolog- ical boundary conditions. Such vertex algebras are typically constructed out of W -algebras and aﬃne vertex algebras associated to the Lie algebra g of the gauge group G, and the coupling constant Ψ relates to the level shifted by the dual Cox- eter number h∨. Diﬀerent boundary conditions can be concanated to yield other UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 5 boundary conditions and corresponding vertex algebras are related via vertex algebra extensions. Most of [CG17] is dealing with simply-laced g and the discussion at the bottom of page 22 of [CG17] is concerned with the concanation of boundary conditions called B1,0,B−1,1 and B0,1. The main expectation is that the resulting corner vertex algebra coincides with the corner vertex algebra between the boundary conditions B1,0 and B0,1 dressed by extra decoupled degrees of freedom corresponding to L1(g). In vertex algebra language this expectation is precisely the statement of Theorem 8.1 with µ =0= ν. There are further conjectures around vertex algebras and S-duality. These are mainly the construction of junction or corner vertex algebras which are typically large extensions of products of two vertex algebras associated to g. Theorem 9.1, which gives a lattice type construction of vertex operator algebras of CFT type using W -algebras in place of Heisenberg algebras, proves them in a series of cases.

1.6. Rigidity of vertex tensor categories. One of the most difficult problems of the subject of vertex algebras is the understanding of tensor categories of modules of a given vertex algebra. The theory of tensor categories of modules of vertex algebras has been developed in the series of papers [HL94, HL95]. In particular, Yi-Zhi Huang has shown the existence of vertex tensor categories for lisse vertex algebras without the rationality assumption [Hua09]. The most challenging technical problem for vertex tensor categories is to prove the rigidity of objects. This is crucial as rigidity gives the categories substantial structure and many useful theorems only hold for rigid tensor categories. While this problem was settled by Huang for rational, lisse vertex algebras [Hua08], it is wide open for non-rational lisse vertex algebras. In fact, it is expected that the tensor categories of modules should exist for much more general vertex algebras. A strong evidence was given in [CHY18] that showed the category of ordinary modules over an admissible affine vertex algebra associated with a simply laced Lie algebra has the structure of a vertex tensor category. We conjecture that the category of ordinary modules over a quasi-lisse vertex algebra [AK18] has the structure of a vertex tensor category (Conjecture 1). Note that an admissible affine vertex algebra is quasi-lisse. Assuming this conjecture, and using an idea of [Cre19], we use the decomposition stated in Theorem 8.1 to prove that certain categories of quasi-lisse W -algebras at admissible level are fusion (Theorem 10.4). Since the conjecture is valid for lisse W - algebras, this gives a strong evidence for the rationality conjecture [KW08, Ara15a] of lisse W -algebras at admissible levels, which has been settled only in some special cases [Ara15b, AvE]. It seems that the translation functor is a good tool to prove rationality of vertex operator algebras in suitable cases. We will explain in forthcoming work how to employ the translation functor in order to get new rational W-algebras at non- admissible levels associated to Deligne’s exceptional series [ACK]. 6 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

Acknowledgements T. A. is partially supported by by by JSPS KAKENHI Grant Numbers 17H01086, 17K18724, 19K21828. T.C. appreciates discussions with Sergei Gukov and Emanuel Diaconescu. T.C. is supported by NSERC #RES0048511.

2. A construction of automorphisms of vertex algebras Let V be a vertex algebra. As usual, we set a = Res zna(z) for a V , (n) z=0 ∈ where a(z) is the quantum ﬁeld corresponding to a. We have m (8) [a ,b ]= (a b) (m) (n) j (j) (m+n−j) > Xj 0 for a,b V , m,n Z. In particular, [a ,b ] = (a b) . ∈ ∈ (0) (n) (0) (n) Let A be an element of V such that its zero mode A(0) acts semisimply on V so that V = V [λ], where V [λ] = v V A v = λv . We assume that λ∈C { ∈ | (0) } A V [0]. Set V [> λ]= V [µ] V [> λ]= V [µ]. L ∈ µ∈λ+R>0 ⊃ µ>λ Suppose that there existsL another element Aˆ L V [> 0] such that Aˆ A ˆ ∈ ˆ ≡ (mod V [> 0]) and A(0) acts locally ﬁnitely on V . Then, A(0) acts semisimply on V , and for v V [λ], there exist a unique eigenvectorv ˜ of Aˆ eigenvalue λ such ∈ (0) thatv ˜ v (mod V [> λ]). By extending this correspondence linearly, we obtain a ≡ linear map

(9) V V, v v.˜ → 7→ Lemma 2.1. The map (9) is an automorphism of V .

Proof. Clearly, (9) is a linear isomorphism. We wish to show that (9) is a homomorphism of vertex algebras. It is clear that 0 = 0 . Let v V [λ], w V [µ], ˜ | i | i ∈ ∈ n Z. By(8),v ˜(n)w˜ is an eigenvector of A(0) of eigenvalue λ+µ andv ˜(n)w˜ v(n)w ∈ f ≡ (mod V [> λ + µ]). Therefore,v ˜(n)w˜ = v^(n)w. Similarly, T v˜ is an eigenvector of A˜ of eigenvalue λ such that T v˜ T v (mod V [> λ]). Hence T v˜ = T v. This (0) ≡ completes the proof. f We also have the following. ˆ ˜ Lemma 2.2. The action of A(0) on V coincides with that of A(0). ˆ ˆ ˜ Proof. For an eigenvector v V [> λ] of A(0) of eigenvalue λ, (A(0) A(0))v is also ˆ ∈ ˆ −˜ an eigenvector of A(0) of eigenvalue λ. On the other hand, (A(0) A(0))v belongs ˆ − to V [> λ]. Since all eigenvalues of A(0) on V [> λ] are greater than λ, the vector (Aˆ A˜ )v must be zero. This completes the proof. (0) − (0) ˆ Let M be a V -module on which both A(0) and A(0) act semisimply. Set M[λ]= m M A m = λm , M[> λ] = M[µ]. We can deﬁne a linear isomor- { ∈ | (0) } µ>λ phism P (10) M ∼ M, m m,˜ → 7→ UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 7 that sends m M[λ] to a unique eigenvectorm ˜ of Aˆ of eigenvalue λ such that ∈ (0) m˜ m (mod M[> λ]). The following assertion can be shown in the same manner ≡ as Lemma 2.1.

Lemma 2.3. We have a^m =a ˜ m˜ for a V , m M, n Z, that is, (10) is (n) (n) ∈ ∈ ∈ an isomorphism of V -modules.

3. Preliminaries on Drinfeld-Sokolov reduction Let g be a simple Lie algebra over C, and let V k(g) be the universal affine vertex algebra associated with g at level k as in the introduction. A V k(g)-module is the same as a smooth module M of level k over the affine Kac-Moody algebra g. Here n −n−1 a g-module M is called smooth if x(z)= n∈Z(xt )z is a (quantum) field on M for all x g, that is, (xtn)m = 0 for a sufficiently large n for any m Mb. ∈ P ∈ bLet f be a nilpotent element of g. Let

(11) g = gj j∈ 1 Z M2 be a good grading ([KRW03]) of g for f, that is, f g , ad f : g g is ∈ −1 j → j−1 injective for j > 1/2 and surjective for j 6 1/2. Denote by x0 the semisimple element of g that defines the grading, i.e., (12) g = x g [x , x]= jx . j { ∈ | 0 } We write deg x = d if x g . ∈ d Let e,h,f be an sl -triple associated with f in g. Then the grading defined { } 2 by x0 =1/2h is good, and is called a Dynkin grading. Fix a Cartan subalgebra h of g that is contained in the Lie subalgebra g0. Let ∆ be the set of roots of g, g = h g the root space decomposition. Set ⊕ α∈∆ α ∆j = α ∆ gα gj , so that ∆ = j∈ 1 Z ∆j . Put ∆>0 = j>0 ∆j . Let { ∈ | ⊂ } L 2 I = 1, 2,..., rk g , and let x a I ∆ be a basis of g such that x g , { } { a | ∈ ⊔ F} F α ∈ α α ∆, and x h, i I. Denote by cd the corresponding structure constant. ∈ i ∈ ∈ ab Set g>1 = j>1 gj , g>0 = j>1/2 gj, and let χ : g>1 C be the character → −1 defined by χ(x) = (f x). We extend χ to the characterχ ˆ of g>1[t,t ] by setting n L | L χˆ(xt )= δn,−1χ(x). Define −1 F = U(g [t,t ]) −1 C , χ >0 ⊗U(g>0[t]+g>1[t,t ]) χˆ −1 where Cχˆ is the one-dimensional representation of g>0[t]+ g>1[t,t ] on which −1 g>1[t,t ] acts by the characterχ ˆ and g>0[t] acts trivially. Since it is a smooth −1 g>0[t,t ]-module, the space Fχ is a module over the vertex subalgebra V (g>0) k ⊂ V (g) generated by xα(z) with α ∆>0. For α ∆>0, let Φα(z) denote the image −1 ∈ ∈ of xα(z) in (End Fχ)[[z,z ]]. Then

Φ (z)= χ(x ) for α ∆> α α ∈ 1 and χ([x , x ]) Φ (z)Φ (w) α β . α β ∼ z w − 8 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

There is a unique vertex algebra structure on F such that 0 =1 1 is the vacuum χ | i ⊗ vector and Y ((Φ ) 0 ,z)=Φ (z) α (−1)| i α for α ∆ . (Note that (Φ ) 0 = χ(x ) 0 for α ∆> .) In other words, ∈ >0 α (−1)| i α | i ∈ 1 Fχ has the structure of the βγ-system associated with the symplectic vector space g1/2 with the symplectic form (13) g g C, (x, y) χ([x, y]). 1/2 × 1/2 → 7→ ∞/2+• Next, let (g>0) be the vertex superalgebra generated by odd fields ψα(z), ψ∗ (z), α ∆ , with the OPEs α ∈ V>0 δ ψ (z)ψ∗ (z) α,β , ψ (z)ψ (z) ψ∗ (z)ψ∗(z) 0 α β ∼ z w α β ∼ α β ∼ − Let ∞/2+•(g )= ∞/2+n(g ) be the Z-gradation defined by deg 0 = 0, >0 n∈Z >0 | i deg(ψ ) = 1, deg(ψ∗ ) = 1. Vα (n) − L α (nV) For a smooth g-module M, set ∞/2+• (14) C(M) := M F (g ). b ⊗ χ⊗ >0 i i ^ ∞/2+i k Then C(M)= C (M), C (M)= M Fχ (g>0). Note that C(V (g)) is i∈Z ⊗ ⊗ naturally a vertexL superalgebra and C(M) is aV module over the vertex superalgebra C(V k(g)) for any smooth g-module M. Define 1 Q(z)= x (z)ψ∗ (z)+ Φ (z)ψ∗ (z) cγ ψ∗ (z)ψ∗(z)ψ (z), α α b α α − 2 α,β α β γ α∈X∆>0 α∈X∆>0 α,β,γX∈∆>0 where we have omitted the tensor product symbol. Then Q(z)Q(w) 0 and we 2 k ∼ have Q(0) =0 on any C(V (g))-module. The cohomology • • HDS,f (M) := H (C(M),Q(0)) is called the BRST cohomologyof the Drinfeld-Sokolov reduction associated with f with coefficients in M ([FF90a, KRW03], see also [Ara05]). By definition [Fe˘ı84], • −1 ∞ +• −1 H (M) is the semi-infinite g [t,t ]-cohomology H 2 (g [t,t ],M F ) DS,f >0 >0 ⊗ χ with coefficients in the diagonal g [t,t−1]-module M F . >0 ⊗ χ The vertex algebra Wk 0 k (g,f) := HDS,f (V (g)) is called the W -algebra associated with (g,f) at level k, which is conformal provided that k = h∨. The vertex algebra structure of Wk(g,f) does not depend on the 6 − choice of a good grading ([AKM15]), however, its conformal structure does. The central charge of Wk(g,f) is given by

1 ρ ∨ 2 (15) dim g dim g1/2 12 √k + h x0 , − 2 − |√k + h∨ − | where ρ is the half sum of positive roots of g. k Let Wk(g,f) be the unique simple graded quotient of W (g,f). UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 9

Let KL be the full subcategory of g-modules consisting of objects on which g acts semisimply and tg[t] acts locally nilpotently, and let KLk be the full subcategory of KL consisting of modules of levelbk. Let Q be the root lattice of g, Qˇ the coroot lattice, P the weight lattice, Pˇ the coweight lattice, P+ the set of dominant integral weights, Pˇ+ the set of dominant integral coweights of g. For λ P , set ∈ + k V := U(g) C E KL , λ ⊗U(g[t]⊕ K) λ ∈ k where Eλ is the irreducible ﬁnite-dimensional g-module of highest weight λ lifted to b a g[t]-module by letting g[t]t act trivially and K by multiplication with the scalar k. k k k Note that V (g)= V0 as a g-module. We denote by Lλ the unique simple quotient k k of Vλ. More generally, for any weight λ of g we denote by Lλ the irreducible highest weight representation of g withb highest weight λ and level k.

Theorem 3.1 ([FG10, Ara15a]). We have Hi (M)=0 for i =0 and M KL . b DS,f 6 ∈ k In particular, the functor KL Wk(g,f) -Mod,M H0 (M), k → 7→ DS,f is exact.

Note that for M KL ,N KL , we have M N KL . Therefore, ∈ k ∈ ℓ ⊗ ∈ k+ℓ Hi (M N)=0 for i = 0. In particular, Hi (M L)=0 for i = 0 if M KL DS,f ⊗ 6 DS,f ⊗ 6 ∈ and L is an integrable representation of g.

4. Proof ofb Theorem 1

Let k C, ℓ Z> , and set ∈ ∈ 0 ∞/2+• C := C(V k(g) L (g)) = V k(g) L (g) F (g ). ⊗ ℓ ⊗ ℓ ⊗ χ⊗ >0 For t C, deﬁne the element Q C by ^ ∈ t ∈ 2α(x0) ∗ (16) Qt(z)= (π1(xα)(z)+ t π2(xα)(z))ψα(z) α∈X∆>0 1 + Φ (z)ψ∗ (z) cγ ψ∗ (z)ψ∗(z)ψ (z), α α − 2 α,β α β γ α∈X∆>0 α,β,γX∈∆>0 where π (x )(z) (resp. π (x )(z)) denotes the action of x (z), a I ∆, on V k(g) 1 a 2 a a ∈ ⊔ (resp. on L (g)). Then Q (z)Q (w) 0, and therefore, (Q )2 = 0. It follows that ℓ t t ∼ t (0) (C, (Qt)(0)) is a diﬀerential graded vertex algebra, and the corresponding cohomol- • ogy H (C, (Qt)(0)) is naturally a vertex algebra. Clearly, (17) Hi(C, (Q ) ) = Hi (V k(g)) L (g)= δ Wk(g,f) L (g), t=0 (0) ∼ DS,f ⊗ ℓ i,0 ⊗ ℓ (18) Hi(C, (Q ) ) = Hi (V k(g) L (g)) = δ H0 (V k(g) L (g)), t=1 (0) ∼ DS,f ⊗ ℓ i,0 DS,f ⊗ ℓ see Theorem 3.1.

By [Ara05, 3.7], the diﬀerential (Qt)(0) decomposes as (19) (Q ) = dst + dχ, (dst)2 = (dχ)2 = dst, dχ =0, t (0) t t { t } 10 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN where

st 2α(x0) ∗ dt = (π1(xα)(n) + t π2(xα)(n))ψα,(−n) Z α∈X∆>0 nX∈ ∗ (20) + Φα,(n)ψα,(−n) n<0 α∈X∆1/2 X 1 γ ∗ ∗ cα,β ψα,(m)ψβ,(n)ψγ,(−m−n), − 2 Z α,β,γX∈∆>0 m,nX∈ χ ∗ ∗ (21) d = Φα,(n)ψα,(−n) + χ(xα)ψα,(0). > α∈X∆1/2 nX0 αX∈∆1 Deﬁne the Hamiltonian H on C by

V k(g) H = H + (ωL (g))(1) + (ωF )(1) + (ω ∞/2+• )(1) π1(x0)(0) π2(x0)(0), standard ℓ χ V (g>0) − − k where HV (g) is the standard Hamiltonian of V k(g) that gives π (x), x g, standard 1 ∈ conformal weight one, ωLℓ(g) is the Sugawara conformal vector of Lℓ(g), 1 ω (z)= : ∂ Φα(z)Φ (z):, Fχ 2 z α α∈X∆1/2 ∗ ∗ ω ∞/2+• (z)= j : ψ (z)∂ψα(z):+ (1 j) : (∂ψ (z))ψα(z): . V (g>0) α − α j>0 j>0 X αX∈∆j X αX∈∆j α α Here, Φ is a dual basis to Φα , that is, Φ (z)Φβ(w) δαβ/(z w). Then st{ } χ { } ∼ − [H, dt ] = 0 = [H, dt ], and thus, H deﬁnes a Hamiltonian on the vertex algebra • H (C, (Qt)(0)). Following [FBZ04, KRW03], deﬁne

a γ ∗ J (z)= π1(xa)(z)+ ca,β : ψγ (z)ψβ(z): β,γX∈∆+ x a for a I ∆. We also denote by J the linear combination of J , a I ∆6 , ∈ ⊔ ∈ ⊔ 0 corresponding to x g60 := j60 gj . Let C60 be the vertex subalgebra of C x ∈ ∗ generated by J (z) (x g6 ), π (x)(z) (x g), ψ (z) (α ∆ ), Φ (z) (α ∈ 0 L2 ∈ α ∈ >0 α ∈ ∆1/2), and let C>0 be the vertex subalgebra of C generated by ψα(z) and

α α(h) ((Qt)(0)ψα)(z)= J (z)+ t π2(xα)(z)+Φα(z) (α ∆ ). ∈ >0 As in [FBZ04, KRW03], we ﬁnd that both C60 and C>0 are subcomplexes of (C, (Q ) ) and that C = C6 C as complexes. Moreover, we have t (0) ∼ 0⊗ >0

i C for i =0 H (C>0, (Qt)(0))= (0 for i =0, 6 and therefore,

• • (22) H (C, (Qt)(0)) ∼= H (C60, (Qt)(0)) UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 11 as vertex algebras. Since the cohomological gradation takes only non-negative val- 0 0 ues on C60, it follows that H (C, (Qt)(0))= H (C60, (Qt)(0)) is a vertex subalgebra of C60. Note that the vertex algebra C60 does not depend on the parameter t C. Also, st χ ∈ C60 is preserved by the action of both dt and d . Let C6 = C6 C , so that C6 = C6 . 0,∆ 0 ∩ ∆ 0 ∆ 0,∆

Lemma 4.1. For each ∆, C60,∆ is a ﬁnite-dimensionalL subcomplex of C60.

x ∗ Proof. The generators J (z) (x g6 ), ψ (z) (α ∆ ), and Φ (z) (α ∆ ) ∈ 0 α ∈ >0 α ∈ 1/2 have positive conformal weights with respect to the Hamiltonian H. Therefore, it is suﬃcient to show that the vertex subalgebra L (g) that is generated by π (x)(z), x ℓ 2 ∈ g, has ﬁnite-dimensional weight spaces L (g) := L (g) C6 and the conformal ℓ ∆ ℓ ∩ 0,∆ weights of Lℓ(g) is bounded from the above. On the other hand, the action of

H on Lℓ(g) is the same as the twisted action of (ωLℓ(g))(1) corresponding to Li’s delta operator associated with x . Since L (g) is rational and lisse, the conformal − 0 ℓ weights of this twisted representation are bounded from above and the weight spaces are ﬁnite-dimensional. This completes the proof.

st χ Since both dt abd d preserve C60,∆, we can consider the spectral seqeunce • χ st E H (C6 ) such that d = d and d = d , which is converging since each r ⇒ 0 0 1 t C60,∆ is ﬁnite-dimensional. As in [FBZ04, KW04, KW05], we ﬁnd that

•,q q χ k♮ f (23) E = H (C6 , d )= δ V (g ) L (g), 1 0 q,0 ⊗ ℓ k♮ f x f where V (g ) is the vertex subalgebra of C6 generated by J (z), x g . Thus, 0 ∈ the spectral sequence collapses at E1 = E∞, and we get the vertex algebra isomorphism

q k♮ f (24) gr H (C 6 ) = δ V (g ) L (g). t, 0 ∼ q,0 ⊗ ℓ q Here gr H (Ct,60) is the associated graded vertex algebra with respect to the ﬁltra- ♮ tion that deﬁnes the spectral sequence. By (24), for each v V k (gf ) L (g) there ∈ ⊗ ℓ exists a cocycle

vˆ = v0 + v1 + v2 + ... (a ﬁnite sum) such that

v = v, dstv = dχv . 0 i − i+1 Set

(25) A := π (x ) L (g), 2 0 ∈ ℓ where we recall that x0 is defined by (12). Then C60 = 1 Z C60[λ], C60[λ] = λ∈ 2 ˆ c C60 A(0)c = λc , and A C60[0]. Consider the correspondingL cocycle A that { ∈ | } ∈ st χ has cohomolological degree zero. Since d A C6 [> 0] and d C6 [λ] C6 [λ], t ⊂ 0 0 ⊂ 0 we may assume that Aˆ A (mod C6 [> 0]). Moreover, we may assume that Aˆ ≡ 0 12 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN is homogenous with respect to the Hamiltonian H, so that Aˆ preserves each finite- dimensional subcomplex C60,∆. In particular, Aˆ is locally finite on C60. Therefore, we can apply the construction of Section 2 to obtain the automorphism

∼ (26) ϕ : C6 C6 , c c˜ t 0 → 0 7→ of vertex algebras. Note that A˜ A (mod C[> 0]) and the operator A acts semisimply on the ≡ (0) whole complex C, so that C = λ∈ 1 Z C[λ], C[λ] = c C A(0)c = λc . Since 2 { ∈ | } ˜ ϕt is an automorphism of the vertexL algebra, A ϕt(Lℓ(g)) generates a Heisen- ˜ ∈ berg vertex subalgebra. In particular, A(0) acts semisimply on C. Therefore, by replacing Aˆ by A˜ (see Lemma 2.2), we can extend the automorphism (26) to the automorphism (27) C ∼ C c c,˜ → 7→ which is also denote by ϕt.

Proposition 4.2. We have ϕt(Qt=0)(0) = (Qt)(0) on C. In particular, ϕt deﬁnes an isomorphism

(C, (Qt=0)(0)) ∼= (C, (Qt)(0)) of diﬀerential graded vertex algebras. ˜ Proof. We ﬁrst show that ϕt(Qt=0)(0) = (Qt)(0) on C60. Since ϕt(Qt=0)(0)A = ˜ ϕt(Qt=0)(0)ϕt(A) = ϕt((Qt=0)(0)A) = 0, we have [ϕt(Qt=0)(0), A(0)] = 0. Also, [(Q ) , A˜ ] = [(Q ) , Aˆ ]=0on C6 by Lemma 2.2. Let c C6 [> λ] be an t (0) (0) t (0) (0) 0 ∈ 0 eigenvector of A˜ of eigenvalue λ. Then the vector (ϕ (Q ) (Q ) )c is also (0) t t=0 (0) − t (0) an eigenvector of A˜ of eigenvalue λ. On the other hand, note that ϕ (Q ) (0) t t=0 ≡ Q Q (mod C6 [> λ]). Hence, (ϕ (Q ) (Q ) )c C6 [> λ]. Since t=0 ≡ t=1 0 t t=0 (0) − t (0) ∈ 0 all the eigenvalues of A˜ on C6 [> λ] are greater than λ, we get that (ϕ (Q ) (0) 0 t t=0 (0)− (Qt)(0))c = 0. Hence, ϕt(Qt=0)(0) = (Qt)(0) on C60. ˜ ˜ Next, since (Qt)(0)A = 0, we have [(Qt)(0), A(0)] = 0 on the whole space C. Therefore we can repeat the same argument for an eigenvector c C[> λ] of A˜ ∈ (0) of eigenvalue λ to obtain that (ϕ (Q ) (Q ) )c = 0 for all c C. t t=0 (0) − t (0) ∈ The last assertion follows since ϕt preserves the cohomological gradation as Aˆ has cohomological degree zero.

By Proposition 4.2, ϕ := ϕt=1 deﬁnes an isomorphism

(28) (C, (Qt=0)(0)) ∼= (C, (Qt=1)(0)) of diﬀerential graded vertex algebras. We have shown the following assertion.

• Theorem 4.3. The automorphism ϕ induces an isomorphism H (C, (Qt=0)(0)) ∼= • H (C, (Qt=1)(0)). In particular, Wk(g,f) L (g) = H0 (V k(g) L (g)) ⊗ ℓ ∼ DS,f ⊗ ℓ as vertex algebras. UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 13

Proof of Theorem 1. Let V be a vertex algebra equipped with a vertex algebra homomorphism V k(g) V . Since L (g) is rational, both A and A˜ acts semisim- → ℓ (0) (0) ply on C(V L (g)). Hence we can apply Lemma 2.1 to extend ϕ to the isomorphism ⊗ ℓ C(V L (g), (Q ) ) = C(V L (g), (Q ) ) of diﬀerential graded vertex alge- ⊗ ℓ t=0 (0) ∼ ⊗ ℓ t=t (0) bras, which gives the the isomorphism H• (V L (g)) = H• (V ) L (g) of DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ vertex algebras. Similarly, if M is a V -module and N is an Lℓ(g)-module, it follows from Lemma 2.3 that we have an isomorphism H• (M N) = H• (M) N as DS,f ⊗ ∼ DS,f ⊗ modules over H• (V L (g)) = H• (V ) L (g). DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ

Example 4.4. Let g = sl = spanC e,h,f and ℓ = 1, and consider the Drinfeld- 2 { } Sokolov reduction associated with f. Then C = V k(g) L (g) ∞/2+•(g ), and ⊗ 1 ⊗ >0 ∞/2+•(g ) is generated by odd ﬁelds ψ(z)= ψ (z), ψ∗(z)= ψ∗ (z). We have >0 α Vα 2 ∗ ∗ V Qt(z) = (e1(z)+ t e2(z))ψ (z)+ ψ (z), where we have set xi = πi(x) for x g, i = 1, 2. The vertex subalgebra C60 is f1 h1 ∈ ∗ generated by J (z) = f1(z), J (z) = h1(z) + 2 : ψ(z)ψ (z) :, e2(z), h2(z), f2(z), ∗ and ψ (z). We have A = h2/2, and

2 h1 Aˆ(z)= h2(z)+ t J (z)e2(z). We ﬁnd that the isomorphism ∞/2+• ∞/2+• ϕ : V k(g) L (g) (g ) ∼ V k(g) L (g) (g ) t ⊗ 1 ⊗ >0 → ⊗ 1 ⊗ >0 is given by ^ ^ e (z) e (z)(1 t2e (z)), 1 7→ 1 − 2 h (z) h (z) t2k∂ e (z), 1 7→ 1 − z 2 f (z) f (z)(1 + t2e (z)), 1 7→ 1 2 e (z) e (z), 2 7→ 2 h (z) h (z)+ t2J h1 (z)e (z), 2 7→ 2 2 t2 t4 f (z) f (z) J h1 (z)h (z)+ ∂ J h1 (z) :e (z)J h1 (z)2 :, 2 7→ 2 − 2 2 z − 4 2 ψ(z) ψ(z)(1 t2e (z)), 7→ − 2 ψ∗(z) ψ∗(z)(1 + t2e (z)). 7→ 2 Note that we have (1 t2e (z))(1 + t2e (z)) = 1 on L (g). − 2 2 1 5. Remarks on superalgebras We restricted our attention to vertex algebras and here we remark that the results also hold for vertex superalgebras, i.e. Remark 5.1. All results of section 2 also hold if we allow V to be a vertex superalgebra, such that A and Aˆ are even elements of V . Remark 5.2. In the proofs of section 4 we only used the following properties of Lℓ(g): 14 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

(1) ﬁnite dimensionality and boundedness from above of conformal weight

spaces with respect to the twisted action of (ωLℓ(g))(1) corresponding to Li’s delta operator associated with x ; − 0 (2) semisimple action of A , where A := π (x ) L (g). 0 2 0 ∈ ℓ Thus the results are also true by replacing Lℓ(g) by any vertex algebra that carries an action of V ℓ(g) and satisﬁes these two properties. We can also allow g to be a Lie superalgebra and f an even nilpotent element, such that Lℓ(g) satisﬁes above two properties.

Here are examples of simple aﬃne vertex superalgebras that satisfy the two properties of the Remark.

Firstly if we take g = osp1|2 and ℓ a positive integer ℓ, then Lℓ(g) is rational [CFK18]. A similar statement is expected to be true only for g = osp1|2n. Secondly, it is well-known [KW01] that L1(sln|m) is a conformal extension of L (sl ) L (sl ) π with π a rank one Heisenberg vertex algebra. Thus if 1 n ⊗ −1 m ⊗ we choose the sl -triple e,h,f for the reduction to be a subalgebra of the sl 2 { } n subalgebra of sln|m,

e,h,f sl sl , { }⊂ n ⊂ n|m then the two properties of the above remark are satisfied. Moreover for a positive ⊗ℓ integer ℓ one has the embedding of Lℓ(sln) in L1(sln) . It follows that Lℓ(sln) ⊗ℓ embeds into the diagonal affine vertex subalgebra of sln|m of level ℓ of L1(sln|m) and hence especially Lℓ(sln) is a subalgebra of the simple quotient Lℓ(sln|m). As a consequence the two properties of the remark hold for Lℓ(sln|m) if we choose the sl -triple e,h,f for the reduction to be a subalgebra of the sl subalgebra of 2 { } n sln|m. In general, the notion of integrable highest-weight modules of affine Lie super- algberas is due to Kac and Wakimoto and they are necessarily integrable modules with respect to a certain affine vertex subalgebra, call it Lℓ(a), [KW01, Thm. 6.1 and eqn. 8.2], so that if e,h,f a, then our conditions are also satisfied. { }⊂

6. Urod conformal vector In this section we assume that k is not critical, and discuss how the conformal structure match under the isomorphism

ϕ : Wk(g,f) L (g) = H0 (V k(g) L (g)) ⊗ ℓ ∼ DS,f ⊗ ℓ of vertex algebras. On the right-hand-side the conformal vector of of H0 (V k(g) L (g)) is given DS,f ⊗ ℓ by

ω = ω k + ωL + ω + ω ∞/2+• + T π (x )+ T π (x ), total V (g) ℓ(g) Fχ V (g>0) 1 0 2 0 UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 15

k where ωV k(g) is the Sugawara conformal vector of V (g). It has the central charge k dim g ℓ dim g (k + ℓ) dim g (29) + k + h∨ ℓ + h∨ − k + ℓ + h∨

1 ρ ∨ 2 + dim g0 dim g1/2 12 √k + ℓ + h x0 . − 2 − |√k + ℓ + h∨ − | Clearly, ϕ−1(ω ) is a conformal vector of Wk(g,f) L (g). Set total ⊗ ℓ −1 total (30) ω = ϕ (ω ) ωWk , Urod − (g,f) Wk where ωWk(g,f) is the conformal vector of (g,f). Since it commutes with ωWk(g,f), ωUrod deﬁnes a conformal vector of Lℓ(g) ([LL04, 3.11]), which is called the Urod conformal vector of Lℓ(g). It depends on the choice of x0 and k, and has the central charge k dim g ℓ dim g (k + ℓ) dim g (31) + k + h∨ ℓ + h∨ − k + ℓ + h∨

ρ ∨ 2 ρ ∨ 2 +12 √k + h x0 √k + ℓ + h x0 . |√k + h∨ − | − |√k + ℓ + h∨ − | In the case that g is simply-laced and f = fprin, the central charge (31) becomes ℓ(ℓh + h2 1) dim g (32) − , − ℓ + h where h is the Coxeter number and we have used the strange formula ρ 2/2h∨ = | | dim g/24, and so it does not depend on the parameter k. k By deﬁnition, the conformal vertex algebra W (g,f) Lℓ(g) with the conformal 0 k ⊗ vector ωWk + ω is isomorphic to H (V (g) L (g)) as conformal vertex (g,f) Urod DS,f ⊗ ℓ algebras.

Lemma 6.1. The Hamiltonian of Lℓ(g) deﬁned by ωUrod coincides with

H := (ωL ) (x ) . Urod ℓ(g) (1) − 0 (0) Proof. For a homogenous element x g, π (x) C has the conformal weight ∈ 2 ∈ 1 deg x. Hence,x ˜ has the the conformal weight 1 deg x as well. − − Example 6.2. (Continued from Example 4.4.) The Urod conformal vector of L1(g) is given by 2 ω = ωL + Th/2 (k + 1)T e/2, Urod 1(g) − which agrees with [BFL16].

7. Compatibility with twists In this section we show that the various twisted Drinfeld-Sokolov reduction func- tors commute with tensoring integrable representations as well. −1 Forµ ˇ Pˇ, we deﬁne a characterχ ˆ of g> [t,t ] by the formula ∈ µˇ 1 (33)χ ˆ (x f(t)) = χ(x ) Res f(t)thµ,αˇ idt, f(t) C[t,t−1]. µˇ α α · t=0 ∈ Deﬁne −1 F = U(g [t,t ]) −1 C , χ,µˇ >0 ⊗U(g>0[t]+g>1[t,t ]) χˆµˇ 16 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

−1 where Cχˆµˇ is the one-dimensional representation of g>0[t]+ g>1[t,t ] on which −1 g>1[t,t ] acts by the characterχ ˆµˇ and g>0[t] acts trivially. Then Fχ,µˇ is naturally µˇ −1 a V (g>0)-module and we denote by Φα(z) the image of xα(z) in (End Fχ,µˇ))[z,z ]. We have Φµˇ (z)= zhµ,αˇ iχ(x ) for α ∆ , α α ∈ >0 whµ,αˇ +βiχ([x , x ]) Φµˇ (z)Φµˇ(w) α β α β ∼ z w. − For a smooth g-module M of level k, we define • ∞/2+• −1 (34) H (M)= H (g>0[t,t ],M Fχ,µˇ), b DS,f,µˇ ⊗ where g [t,t−1] acts diagonally on M F . By definition, H• (M) is the co- >0 ⊗ χ,µˇ DS,f,µˇ homology of the complex (C (M), (Q ) ), where C (M)= M F ∞/2+•(g ) µˇ µˇ (0) µˇ ⊗ χ,µˇ⊗ >0 and (Q ) is the zero-mode of the field µˇ (0) V 1 Q (z)= x (z)ψ∗ (z)+ Φµˇ (z)ψ∗ (z) cγ ψ∗ (z)ψ∗ (z)ψ (z), µˇ α α α α − 2 α,β α β γ α∈X∆>0 α∈X∆>0 α,β,γX∈∆>0 on Cµˇ(M). We define the structure of a Wk(g,f)-module on Hi (M),i Z, as follows. DS,f,µˇ ∈ Let ∆(J {µˇ},z) be Li’s delta operator corresponding to the field 1 (35) J {µˇ}(z) :=µ ˇ(z)+ α, µˇ :ψ (z)ψ∗ (z): α, µˇ :Φ (z)Φα(z): h i α α −2 h i α α∈X∆>0 α∈X∆1/2 in C(V k(g)), where Φα(z) is a linear sum of Φβ(z) corresponding to the vector xα dual to x with respect to the symplectic form (13), that is, (f [x , xβ]) = δ . α | α α,β We have α, µˇ α, µˇ J {µˇ}(z)x (w) h ix (w), J {µˇ}(z)Φ (w) h i Φ (w), α ∼ z w α α ∼ z w α − − α, µˇ α, µˇ J {µˇ}(z)ψ (w) h i ψ (w), J {µˇ}(w)ψ∗ (z) h i ψ∗ (w), α ∼ z w α α ∼−z w α − − see [KRW03]. For any smooth g-module M, we can twist the action of C(V k(g)) on C(M) by the correspondence b (36) σ : Y (A, z) Y (∆(J {µˇ},z)A, z). µˇ 7→ C(M) for any A C(V κ(g)). Since we have ∈ −hα,µˇi −hα,µˇi σµˇ(xα(z)) = z xα(z), σµˇ(Φα(z)) = z Φα(z), −hα,µˇi ∗ hα,µˇi ∗ σµˇ(ψα(z)) = z ψα(z), σµˇ(ψα(z)) = z ψα(z), it follows that the resulting complex (C(M), σµˇ (Q(0))) is naturally identified with (Cµˇ(M), (Qµˇ)(0)). Therefore, (Cµˇ(M), (Qµˇ)(0)) has the structure of a differential k graded module over the differential vertex algebra (C(V (g)),Q(0))), and hence, i Wk 0 each cohomology HDS,f,µˇ(M),i Z, is a module over (g,f)= HDS,f (M). 0 ∈ The functor HDS,f,µˇ(?) was introduced in [AF19] in the case that f is a principal nilpotent element. UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 17

Let V be vertex algebra equipped with a vertex algebra homomorphism V k(g) i → V , and let M be a V -module. Then the same construction as above gives HDS,f,µˇ(M) 0 a structure of a module over the vertex algebra HDS,f (V ). Theorem 7.1. Let V be a quotient of the universal affine vertex algebra V k(g), ℓ Z> , and let µˇ Pˇ. For any V -module M, L (g)-module N, and i Z, there ∈ 0 ∈ ℓ ∈ is an isomorphism Hi (M N) = Hi (M) σ∗ N DS,f,µˇ ⊗ ∼ DS,f,µˇ ⊗ µˇ as modules over H0 (V L (g)) = H0 (V ) L (g), where σ∗ N is the twist of DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ µˇ N on which A L (g) acts as σ (A(z)). ∈ ℓ µˇ Proof. Since L (g) is rational, both A and A˜ acts semisimply on C (M N) ℓ (0) (0) µ ⊗ with respect to the twisted action of C(V k(g) L (g)) described above, where A is ⊗ ℓ defined in (25). Therefore the assertion follows immediately from Lemma 2.3. More generally, let w = (y, µˇ) be an element of the extented affine Weyl group W˜ = W ⋉ P ∨, where W is the Weyl group of g, and lety ˜ be a Tits lifting of y to and automorphism of g, so that y˜(x )= c x for some c C∗. Then α α y(α) α ∈ σ : x tn y˜(x )tn−hα,µˇi w α 7→ α defines a Tits lifting of w to an automorphism of g. Set −1 −1 Fχ,w = U(˜w(g>0[t,t ])) U(w ˜(g [t]+g [t,t ])) Cχˆµˇ , ⊗ >0 b >1 −1 where Cχˆµˇ is the one-dimensional representation ofw ˜(g>0[t]+ g>1[t,t ]) on which −1 n n −1 w˜(g> [t,t ]) acts by the characterχ ˆ : x t (x t w˜(ft )) andw ˜(g [t]) acts 1 w α 7→ α | >0 trivially. Then (37) Hi (M)= H∞/2+i(y(g )[t,t−1],M F ), DS,f,w >0 ⊗ χ,w is equipped with a Wk(g,f)-module structure. The proof of the following assertion is the same as that of Theorem 7.1. Theorem 7.2. Let V be a quotient of the universal affine vertex algebra V k(g), ℓ Z> , and let w W˜ . For any V -module M, L (g)-module N, and i Z, there ∈ 0 ∈ ℓ ∈ is an isomorphism Hi (M N) = Hi (M) σ∗ N DS,f,w ⊗ ∼ DS,f,w ⊗ w as modules over H0 (V L (g)) = H0 (V ) L (g). DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ In the above σ∗ N is the twist of N on which A L (g) acts as σ (A(z)), which w ∈ ℓ w is again an integrable representation of g of level ℓ. Suppose that the grading the (11) is even, that is, gj = 0 unless j Z. ∨ ∈ Then x0 P . For a smooth g-moduleb M of level k,“ ”-Drinfeld-Sokolov re- ∈• − duction HDS,f,−(M) ([FKW92, Ara11]) is nothing but the twisted reduction for w = (w , x ), where w is the longestb element of W : 0 − 0 0 • • (38) HDS,f,−(M)= HDS,f,(w0,−x0)(M) 18 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

Corollary 7.3. Let V be a quotient of the universal aﬃne vertex algebra V k(g), ℓ Z> . Suppose that the grading the (11) is even. For any V -module M, L (g)- ∈ 0 ℓ module N, and i Z, there is an isomorphism ∈ Hi (M N) = Hi (M) σ∗ N DS,f,− ⊗ ∼ DS,f,− ⊗ (w0,−x0) as modules over H0 (V L (g)) = H0 (V ) L (g). DS,f ⊗ ℓ ∼ DS,f ⊗ ℓ 8. Higher rank Urod algebras

k In the case that f is a principal nilpotent element fprin, we denote by W (g) k the principal W -algebra W (g,fprin), and by Wk(g) the unique simple quotient of Wk(g). We have the Feigin-Frenkel duality ([FF91, AFO18], see also [ACL19]) which states that Wk Wkˇ L (39) (g) ∼= ( g), where Lg is the Langlands dual Lie algebra, kˇ is deﬁned by the formula r∨(k + h∨)(kˇ + Lh∨) = 1. Here, r∨ is the lacety of g and Lh∨ is the dual Coxeter number of g. ∨ Suppose that g is simply laced and k + h 1 Q6 . By [ACL19], we have − 6∈ 0 Wℓ(g) = (V k−1(g) L (g))g[t], ∼ ⊗ 1 that is, Wℓ(g) is isomorphic to the commutant of V k(g) in V k−1(g) L (g), where ⊗ 1 V k(g) is considered as a vertex subalgebra of V k−1(g) L (g) by the diagonal em- ⊗ 1 bedding and ℓ is deﬁned by (6). In other words, we have a conformal vertex algebra embedding .(V k(g) Wℓ(g) ֒ V k−1(g) L (g (40) ⊗ → ⊗ 1 By Main Theorem 1, (40) induces the conformal vertex algebra homomorphism

ϕ−1 (41) Wk(g,f) Wℓ(g) H0 (V k−1(g) L (g)) ∼ Wk−1(g,f) (g,f). ⊗ → DS,f ⊗ 1 → ⊗U which is again embedding by Theorem 3.1. Here, (g,f) is the vertex algebra U L1(g) equipped with the Urod conformal vector ωUrod, which we call the Urod algebra associated with (g,f). We set (g)= (g,f ). U U prin 8.1. Generic decompositions. In this subsection we assume that k is irrational. For (λ, µˇ) P Pˇ , deﬁne ∈ + × + T k = H0 (Vk ) Wk(g) -Mod . λ,µˇ DS,fprin,µˇ λ ∈ Wk k It was shown in [AF19] that the (g)-modules Tλ,µˇ are simple, and the isomorphism

k kˇ (42) Tλ,µˇ ∼= Tµ,λˇ holds under the Feigin-Frenkel duality (39). UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 19

Let ℓ be non-negative integer. Then Lℓ λ P ℓ gives the complete set of { λ | ∈ +} isomorphism classes of simple Lℓ(g)-modules, where P ℓ = λ P λ, θ∨ 6 ℓ P + { ∈ + | h i }⊂ + is the set of integrable dominant weight of g of level ℓ. In particular, L1 λ P 1 { λ | ∈ +} gives the complete set of isomorphism classes of simple (g,f)-modules. U Let (43) Vk = H0 (Vk ) Wk(g,f) -Mod . µ,f DS,f µ ∈ Theorem 8.1. Let g be simply-laced and let f be any nilpotent element of g. For λ, µ P and ν P 1 , we have ∈ + ∈ + Vk−1 (g,f) = Vk T ℓ µ,f ⊗U ∼ λ,f ⊗ λ,µ λ∈P+ λ−µM−ν∈Q as Wk(g,f) Wℓ(g)-modules (see (41)). ⊗ In the case f = fprin we have the following more general statement. Theorem 8.2. Let g is simply-laced. For λ,µ,µ′ P and ν P 1 , we have ∈ + ∈ + k−1 k ℓ (44) T ′ (g)= T ′ T µ,µ ⊗U λ,µ ⊗ λ,µ λ∈P+ λ−µ−Mµ′−ν∈Q as Wk(g) Wℓ(g)-modules. ⊗ Note that Theorem 8.2 is compatible with (42) since (k ˇ 1) = ℓ 1. − − Proof of Theorem 8.1 and Theorem 8.2. Let λ, µ P and ν P 1 , By [ACL19, ∈ + ∈ + Main Theorem 3], we have (45) Vk−1 L1 = Vk T ℓ µ ⊗ ν λ⊗ λ,µ λ∈P+ λ−µM−ν∈Q k+1 Wℓ k−1 1 as V (g) (g)-modules. Applying Main Theorem 1 to Vµ Lν we obtain ⊗ 1 ⊗ ∗ 1 ′ L ′ Theorem 8.1. Next, under the identiﬁcation P/Q ∼= P+, we have σµ ν+µ +Q ∼= 1 Lν+Q. Hence we obtain Theorem 8.2 by applying Theorem 7.1. Let πk be the Heisenberg vertex subalgebra of V k(g) generated by h(z), h h, k k ∈ and let πλ be the irreducible highest weight representation of π with highest weight λ. There is a vertex algebra embedding ∨ Wk(g) ֒ πk+h → ∨ k+h Wk called the Miura map ([FF90b], see also [Ara17]), and hence each πλ is a (g)- module. Theorem 8.3. Let g is simply-laced, µ h∗ be generic, ν P 1 . We have the ∈ ∈ + isomorphism k−1+h∨ 1 k+h∨ ℓ+h∨ π L = π π ∨ µ ⊗ ν ∼ λ ⊗ λ−(ℓ+h )µ λ∈h∗ λ−Mµ−ν∈Q 20 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN as Wk(g) Wℓ(g)-modules. ⊗ ∨ ∗ Wk k+h For a generic λ h , the (g)-module πλ is irreducible and isomorphic ∈ k to a Verma module M (χλ) with highest weight ([FF90a, Fre92, Ara07]). Here χ : Zhu(Wk(g)) C is described in [ACL19, (27)], where Zhu(Wk(g)) is the Zhu λ → algebra of Wk(g). Hence the following assertion immediately follows from Theorem 8.3.

Corollary 8.4. Let g is simply-laced, µ h∗ be generic, ν P 1 . We have the ∈ ∈ + isomorphism

k−1 1 k ℓ M (χ ) L = M (χ ) M (χ ∨ ) µ ⊗ ν ∼ λ ⊗ λ−(ℓ+h )µ λ∈h∗ λ−Mµ−ν∈Q as Wk(g) Wℓ(g)-modules. ⊗ k Let Wλ be the Wakimoto module [FF90b] of g at level k with highest weight λ ∗ k k k ∈ ֒ (h . For λ = 0, W := W0 is a vertex algebra, and we have an embedding V (g ∨ → Wk of vertex algebras ([FF90b, Fre05]). We haveb Hi (Wk ) δ πk+h , and DS,fprin λ = i,0 λ ∼ ∨ the Miura map is by deﬁnition [FF90b] the map Wk(g) H0 (Wk) = πk+h → DS,fprin ∼ .induced by the embedding V k(g) ֒ Wk → Proof of Theorem 8.3. Since ν is generic, Wk−1 L1 is a direct sum of irreducible µ ⊗ ν Verma modules as a diagonal g-module, or equivalently, a direct sum of irreducible k k−1 1 k λ Wakimoto modules W . So we can write W L = ∗ W m , where λ µ ⊗ ν λ∈h λ⊗ µ,ν mλ be the multiplicity of Wbk in Wk−1 L1 . Note that mλ is a Wℓ(g)-module µ,ν λ µ ⊗ ν Lµ,ν by (40). We have

λ k+1+h∨ ∞/2+0 −1 k−1 1 m = Hom k+1+h∨ (π ,H (n[t,t ], W L ) µ,ν ∼ π λ µ ⊗ ν ℓ+h∨ πµ−(ℓ+h∨)λ if λ µ ν Q, ∼= − − ∈ (0 otherwise, as Wℓ(g)-modules by [ACL19, Proposition 8.3]. The assertion follows by applying Main Theorem 1 to

k−1 1 k (46) W L = W π ∨ . µ ⊗ ν λ⊗ µ−(ℓ+h ) λ∈h∗ λ−Mµ−ν∈Q

8.2. Decomposition at admissible levels. Let k be an admissible number for g, that is, Lk(g) is admissible ([KW89]) as a g-module. In the case that g is simply- laced, this condition is equivalent to b ∨ p b ∨ (47) k + h = , p,q Z> , (p, q)=1, p > h . q ∈ 1

For an admissible number k, a simple module over Lk(g) need not be ordinary, that is, in KL, unless k is a non-negative integer. The classiﬁcation of simple highest weight representations of Lk(g) was given in [Ara16a]. For our purpose, we UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 21 need only the ordinary representations of Lk(g). By [Ara16a], the complete set of isomorphism classes of ordinary simple Lk(g)-modules is given by

k k L λ AdmZ , { λ | ∈ } k k where AdmZ = λ P L is admissible . We have { ∈ + | λ } ∨ k p−h (48) AdmZ = P+ , if g is simply-laced and k is of the form (47). k Let k be an admissible number and λ AdmZ. By [Ara15a], the associated k ∈ variety XLk [Ara12] of L is the closure of some nilpotent orbit Ok which depends λ λ only on the denominator q of k. More explicitly, we have

2q (49) XLk = x g (ad x) =0 λ { ∈ | } in the case that g is simply-laced. By [Ara15a], we have

0 k (50) HDS,f (Lλ) =0 f XLk = Ok. 6 ⇐⇒ ∈ λ

An admissible number k is called non-degenerate if XLk equals to the nilpotent λ k k cone of g for some λ AdmZ, or equivalently, for all AdmZ. In the case that g is N ∈ simply-laced, this happens if and only if the denominator q of k is equal or greater ∨ than h . In this is the case, the simple principal W -algebra Wk(g)= Wk(g,fprin) is rational and lisse ([Ara15a, Ara15b]), and the complete set of the isomorphism classes of Wk(g)-module is given by

k k kˇ (51) L [λ, µˇ] (AdmZ AdmZ)/W˜ , { [λ,µˇ] | ∈ × +} where

k 0 k L L ∨ (52) [λ,µˇ] := HDS,fprin ( λ−(k+h )ˇµ) and W˜ + is the subgroup of the extended aﬃne Weyl group of g consisting of elements k kˇ of length zero that acts on the set AdmZ AdmZ diagonally. We have × Lk Lkˇ (53) [λ,µˇ] ∼= [ˇµ,λ] under the Feigin-Frenkel duality. The following assertion is new for nonzeroµ ˇ.

k Theorem 8.5. Let k be a non-degenerate admissible number, and let λ AdmZ, kˇ ∈ µˇ AdmZ. We have ∈ Lk for i =0, Hi (Lk ) = [λ,µˇ] DS,fprin,µˇ λ ∼ (0 for i =0 6 as Wk(g)-modules.

Proof. The caseµ ˇ = 0 has been proved in [Ara04, Ara07]. In particular,

(54) H0 (L (g)) = W (g). DS,fprin k ∼ k 22 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

i k By [AF19, Theorem 2.1], we have H (V ) = 0 for all i = 0, λ P+. It DS,fprin,µˇ λ 6 ∈ follows that Hi6=0 (M) = 0, i = 0, for any object M in KL that admits a Weyl DS,fprin,µˇ 6 ﬂag. This implies that (55) Hi (M)=0 for i> 0,M KL, DS,fprin,µˇ ∈ see the proof of [Ara04, Theorem 8.3]. k By [Ara14], the admissible representation Lλ admits a two-sided resolution (56) ...C−1 C0 C1 ... → → → i k of the form C = Ww◦λ, where W (λ + kΛ0) is the integral Weyl group w∈W (λ+kΛ ) c 0 ∞ ℓ /L2(w)=i ∞/2 c of λ + kΛ0 and ℓ (w) is the semi-inﬁnite length of w. By [AF19, Lemma 3.2], we have k π ∨ for i =0, H0 (Wk ) = λ−(k+h )ˇµ DS,fprin,µˇ λ ∼ (0 for i =0 6 as Wk(g)-modules. It follows that Hi (Lk ) is the i-th cohomology of the DS,fprin,µˇ λ 0 complex obtained by applying the functor HDS,fprin,µˇ(?) to the resolution (56). In i Lk Wk g particular, HDS,fprin,µˇ( λ) is a subquotient of the ( )-module

0 i k H (C ) = π ∨ . DS,fprin,µˇ ∼ w◦λ−(k+h )ˇµ w∈W (λ+kΛ ) c 0 ∞ ℓ /M2(w)=i On the other hand, since Lk is a L (g)-module, each Hi (Lk ) is a module λ k DS,fprin,µˇ λ over the simple W -algebra W (g) = H0 (L (g)). Since W (g) is rational, k DS,fprin k k i Lk HDS,fprin,µˇ( λ) is a direct sum of simple modules of the form (51). However, such k a module appears in the local composition factor of πw◦λ−(k+h∨)ˇµ if and only if w W ([Ara07]), where W W (kΛ ) is the Weyl group of g. As ∈ ⊂ 0 ℓ∞/2(w) > 0 for w W and the equality holds if and only if w = 1, ∈ c i Lk i Lk HDS,fprin,µˇ( λ) must vanish for i< 0. Together with (55), we get HDS,fprin,µˇ( λ)= k W 0 for i = 0. Finally, since L[λ,µˇ] is the unique simple k(g)-module that appears 6 k in the local composition factor of πλ−(k+h∨)ˇµ and it appears with multiplicity one i Lk Lk ([Ara07]), HDS,fprin,µˇ( λ) is either zero or isomorphic to [λ,µˇ]. The assertion • L g follows since the Euler character of HDS,fprin ( k( )) is equal to the character of k L[λ,µˇ].

k For an admissible number k and λ AdmZ, deﬁne ∈ (57) Lk = H0 (Lk ) Wk(g,f) -Mod . λ,f DS,f λ ∈ ˇ Note that Lk = Lk = Lk . λ,fprin [λ,0] ∼ [0,λ] Let g be simply-laced. Observe that if k 1 is an admissible number, then so is − k, and ℓ is a non-degenerate admissible number. Moreover, we have ∨ ∨ k−1 ℓˇ p−h k ℓ p+q−h AdmZ = AdmZ = P+ , AdmZ = AdmZ = P+ UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 23 if k h∨ +1= p/q with p > h∨, q > 1, (p, q) = 1. − Theorem 8.6. Let g be simply-laced, and let k 1 be admissible. Suppose that k−1 1 − f XL = XL . For µ Adm , ν P , we have ∈ k(g) k−1(g) ∈ Z ∈ + Lk−1 L1 Lk Lℓ µ,f ν ∼= λ,f [λ,µ] ⊗ k ⊗ λ∈AdmZ λ−Mµ−ν∈Q as Wk(g,f) Wℓ(g)-modules. ⊗

In the case f = fprin we have the following more general statement.

Theorem 8.7. Let g be simply-laced, and let k 1 be non-degenerate admissible. k−1 ′ k−ˇ 1 kˇ 1− For µ Adm , µ Adm = AdmZ, ν P , we have ∈ Z ∈ Z ∈ + k−1 1 k ℓ L ′ L L ′ L [µ,µ ] ν ∼= [λ,µ ] [λ,µ] ⊗ k ⊗ λ∈AdmZ λ−µ−Mµ′−ν∈Q as W (g) W (g)-modules. k ⊗ ℓ Corollary 8.8. Let g be simply-laced. Let k + h∨ = (2h∨ + 1)/h∨, so that ℓ + h∨ = (2h∨ + 1)/(h∨ + 1). (1) There is an embedding of vertex algebras ,(W (g) W (g) ֒ L (g k ⊗ ℓ → 1 and Wk(g) and Wℓ(g) form a dual pair in L1(g). (2) For ν P 1 we have ∈ + L1 Lk Lℓ ν ∼= [λ,0] [λ,0] k ⊗ λ∈AdmMZ ∩(ν+Q) as W (g) W (g)-modules. k ⊗ ℓ W k−1 Proof. The assertion follows from Theorem 8.7 noting that k−1(g) = L[0,0] = C if k + h∨ 1 = (h∨ + 1)/h∨ or h∨/(h∨ + 1). − Proof of Theorem 8.6 and Theorem 8.7. By [ACL19, Main Theorem 3], we have Lk−1 L1 Lk Lℓ µ ν ∼= λ [λ,µ] ⊗ k ⊗ λ∈AdmZ λ−Mµ−ν∈Q as L (g) W (g)-modules. Hence the assertion follows from Main Theorem 1 and k ⊗ ℓ Theorem 7.1.

9. Application to the extension problem of vertex algebras In this section we apply Theorem 8.2 to prove the existence of the extensions of vertex algebras that are expected by four-dimensional supersymmetric gauge theories ([CG17, CGL18]). ∗ ∗ For λ P let λ = w (λ), so that E = E ∗ . ∈ + − 0 λ ∼ λ 24 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

Theorem 9.1. Let g be simply-laced. and let k, k′ be irrational complex numbers satisfying 1 1 + = n. k + h∨ k′ + h∨ for n Z> . Then ∈ 1 n k k′ A [g] := T T ∗ λ,0⊗ λ ,0 λ∈MP+∩Q can be equipped with a structure of simple vertex operator algebra of central charge ψ2 2rk g +4h dim g nh dim g 1+ − nψ 1 − where ψ = k + h∨ and h is the Coxeter number (which equals to h∨). The vertex operator algebra An[g] is of CFT type if n > 2.

Proof. We shall prove the assertion on induction on n using the fact that

k′ k′+1 ℓ k′+1 ℓˇ (58) T ′ (g) = T ′ T = T ′ T , µ,µ ⊗U ∼ λ,µ ⊗ λ,µ ∼ λ,µ ⊗ µ,λ λ∈P+ λ∈P+ λ−µM−µ′∈Q λ−µM−µ′∈Q with ℓˇ satisfying the relation 1 1 = +1 ℓˇ+ h∨ k′ + h∨ which follows from Theorem 8.2 and (42). k Let us show the assertion for n = 1. Let IG be the chiral universal centralizer on G at level k ([Ara18]), which was introduced earlier in [FS06] for the g = sl2 case k as the modified regular representation of the Virasoro algebra. By definition, IG is obtained by taking the principal Drinfeld-Sokolov reduction with respect to two commuting actions of g on the algebra of the chiral differential operators [MSV99, BD04] ch on G at level k. The Ik is a conformal vertex algebra of central charge DG,k G 2rk g + 48(ρ ρ∨) = 2rkb g +4h∨ dim g, equipped with a conformal vertex algebra | ∗ homomorphism Wk(g) Wk (g) Ik , where k∗ the dual level of k defined by the ⊗ → G equation 1 1 + =0. k + h∨ k∗ + h∨ k As explained in [Ara18], IG is a strict chiral quantization ([Ara18]) of the universal centralizer g∗ (G ), where is the Kostant-Slodowy slice. Since g∗ (G ) S× ×S Sk S× ×S is a smooth symplectic variety, IG is simple ([AM]). For an irrational k, we have the decomposition k k k∗ I = T T ∗ G ∼ λ,0⊗ λ ,0 λ∈P+ ∗ M as Wk(g) Wk (g)-modules, which follows from the decomposition [AG02, Zhu08] ⊗ of ch as g g-modules. Hence, by (58), DG,k × k k k∗ k k∗+1 ℓˇ I (g) T T ∗ (g) T T T ∗ , G b b ∼= λ,0 λ ,0 ∼= λ,0 µ,0 λ ,µ ⊗U ⊗ ⊗U ∈ ∈ ⊗ ⊗ λ∈P+ λ P+, µ P+ M µ−Mλ∗∈Q UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 25 and ℓˇ satisfies the relation 1 1 + =1. k + h∨ ℓˇ+ h∨ It follows that 1 k∗+1 k k ℓˇ A [g] := Com(W (g), I (g)) = T T ∗ . G⊗U ∼ λ,0⊗ λ ,0 λ∈MP+∩Q Moreover, since Ik (g) is simple A1[g] is simple as well by [CGN, Proposition G⊗U 5.4]. Assuming that the statement is true for n Z>1, we find similarly that ′ ∈ An+1[g] := Com(Wk +1(g), An[g] (g)) ⊗U has the required decomposition. For the central charge computation it is useful to introduce ψ ψ := k + h∨ and φ := , n nψ 1 − so that 1 1 + = n. ψ φn By (32) and the fact that the central charge of Wk+1(g) is ψ2 (1 h(h + 1)(k + h)2/(k + h + 1))rk g = rk g dim gh − − ψ +1 (here we used that (h +1)rk g = dim g) we have h2 + h 1 ψ2 c n+1 = c n − dim g rk g + h ) dim g A [g] A [g] − h +1 − (nψ 1)((n + 1)ψ 1) − − = dim gh + hφ φ dim g, − n n+1 0 k where cV is the central charge of V and we have put A [g]= IG. Note that φ φ = φ φ = ψ(nφ (n + 1)φ ) n − n+1 n n+1 n − n+1 and so by induction for n we have

c n = 2rk g +4h dim g nh dim g(1 + ψφ ). A [g] − n k k′ The conformal dimension of T T ∗ is λ,0⊗ λ ,0 n n ( λ + ρ 2 ρ 2) 2(λ ρ)= λ 2 + (n 2)(λ ρ), 2 | | − | | − | 2 | | − | which is an integer for λ Q. If n > 2, this is clearly non-negative and is equal to ∈ zero if and only if λ = 0. Whence the last assertion. Remark 9.2. More generally, it is expected [CG17, CGL18] that if k and k′ are irrational numbers related by 1 1 + = n Z, k + h∨ k′ + h∨ ∈ then k k′ Vλ,f Vλ∗,f ′ + ⊗ λ∈MQ∩P 26 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN can be given the structure of a simple vertex operator algebra for any nilpotent elements f, f ′.

10. Fusion categories of modules over quasi-lisse W-algebras For a vertex operator algebra V , let ord be the full subcategory of the category CV of finitely generated V -modules consisting of modules M on which L0 acts locally finitely, the L0-eivenvalues of M are bounded from below, and all the generalized L -eigenspaces are finite-dimensional. A simple object in ord is called an ordinary 0 CV representation of V . Recall that a finitely strongly generated vertex algebra V is called quasi-lisse [AK18] if the associate variety XV has finitely many symplectic leaves. By [AK18, Theorem 4.1], if V is quasi-lisse then ord has only finitely many simple objects. CV Conjecture 1. Let V a finitely strongly generated, self-dual, quasi-lisse vertex operator algebra of CFT type. Then the category ord has the structure of a vertex CV tensor category in the sense of [HL94].

In the case that V is lisse, that is, XV is zero-dimensional, Conjecture 1 has been proved in [Hua09]. Conjecture 1 is true if one can show that every object in ord is C -cofinite and if grading-restricted generalized Verma modules for V are of CV 1 finite length [CY20]. If V is a vertex algebra that has an affine vertex subalgebra at admissible level, then another possibility of proving Conjecture 1 is the following: First show that the affine vertex subalgebra is simple, secondly prove that a suitable category of modules of the coset by the affine subalgebra in V has vertex tensor category structure. Then use the theory of vertex algebra extensions [CKM17] to deduce vertex tensor category structure on ord. This is a promising direction as CV for example many simple quotients of cosets of -algebras of type A are rational W and lisse by [CL20, Cor. 6.13] and similar results for -algebras of type B, C and W D are work in progress. In the case that Conjecture 1 is true, we denote by M ⊠V N the tensor product of V -modules M and N in ord, or simply by M ⊠ N if no confusion should occur. CV Now let k be an admissible number for g. As explained in Subsection 8.2,

XL (g) = Ok k b for some nilpotent orbit Ok, and hence, Lk(g) is quasi-lisse. By the conjecture of Adamovi´cand Milas [AM95] that was proved in [Ara16a], the category Cord Lk(g) k k is semisimple and Lλ λ AdmZ gives a complete set of isomorphism classes in ord { | ∈ } L . Note that in the case k is non-negative integer, this is a well-known fact C k(g) [FZ92], and if this is the case Lk(g) is rational and lisse, and hence Conjecture 1 holds by [HL95]. In the case k is admissible but not an integer, Lk(g) is not rational nor lisse anymore. Nevertheless, Conjecture 1 has been proved for V = Lk(g) in [CHY18] provided that g is simply-laced. Moreover, it was shown in [Cre19] that ord L is a fusion category, i,e., any object is rigid, and we have an isomorphism C k(g) ∨ ord ord p−h k (59) K[CL ∨ g ] = K[CL g ], [L ] [Lλ], p−h ( ) ∼ k( ) λ 7→ UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 27 of Grothendieck rings of our fusion categories, where p is the numerator of k + h∨. Let f be a nilpotent element of g. By [Ara15a],

XH0 (L (g)) = XL (g) f , DS,f k k ∩ S where is the Slodowy slice at f in g. Therefore, in the case that k is admissible, Sf XH0 (L (g)) = Ok f is a nilpotent Slodowy slice, that is, the intersection of the DS,f k ∩ S Slodowy slice with a nilpotent orbit closure, provided that H0 (L (g)) = 0, or DS,f k 6 equivalently, O = , that is, f O . In particular, H0 (L (g)) is quasi-lisse k ∩Sf 6 ∅ ∈ k DS,f k if f O , and so is the simple W -algebra W (g,f). If f O , then O is a ∈ k k ∈ k k ∩ Sf point by the transversality of the Slodowy slices, and therefore, Wk(g,f) is lisse. The good grading (11) is called even of gj = 0 unless j Z. If this is the case, k ∈ W (g,f) is Z>0-graded and thus of CFT type. The following assertion was stated in the case that f O in [AvE], but the ∈ k same proof applies.

Theorem 10.1. Let k be admissible and f Ok. Suppose that f admits a good k k ∈ 0 k even graing. Then, for λ AdmZ, Lλ,f = HDS,f (Lλ) is simple. In particular, 0 W ∈ HDS,f (Lk(g)) = k(g,f).

Lemma 10.2. Let f be an even nilpotent element. Then Wk(g,f) is self-dual and of CFT type with respect to the Dynkin grading.

Proof. Since f is an even nilpotent element, the Dynkin grading is even, and so Wk(g,f) is of CFT type. The self-duality follows from the formula in [AvE, Propo- sition 6.1].

Remark 10.3. If the grading is not Dynkin, Wk(f,g) need not be self-dual, see [AvE, Proposition 6.3].

Theorem 10.4. Let g be simply-laced, k admissible, and let f be an even nilpo- W tent element in Ok. Suppose Conjecture 1 is true for k+1(g,f) and also that Wk+1(g,f) is self-dual. Then the functor ord ord 0 L W ,M H (M), C k(g) →C k(g,f) 7→ DS,f k k is a unital braided tensor functor. In particular, the modules L , λ AdmZ, are λ,f ∈ rigid.

Remark 10.5. Note that by Theorem 8.6 we have that W (g,f) L (g) is an ex- k ⊗ 1 tension of W (g,f) W (g), where W (g) is rational and lisse if k is admissible. k+1 ⊗ ℓ ℓ Especially every ordinary module of W (g,f) L (g) is an object in the category k ⊗ 1 of ordinary modules of W (g,f) W (g) and so if the latter has vertex tensor k+1 ⊗ ℓ category structure, then so does the former by [CKM17]. In other words, if Con- jecture 1 is true for Wk+1(g,f) and if k is admissible then Conjecture 1 is also true for Wk(g,f). Moreover, [CKM19, Thm. 5.12] applies to our setting if Wk(g,f) is Z-graded. Thus, if the category of ordinary modules of W (g,f) W (g) is a fusion category, k+1 ⊗ ℓ then so is the one of W (g,f) L (g). If W (g,f) is Z-graded by conformal weight k ⊗ 1 k 28 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN it thus also follows that Wk(g,f) is rational provided that Wk+1(g,f) is rational and lisse.

Note that if f O then Conjecture 1 holds since W (g,f) is lisse. ∈ k k In the case that f = f , f O if and only if O = , the nilpotent cone of g. prin ∈ k k N An admissible number k such that Ok = is called non-degenerate. If this is the k N case W (g) is rational, and the complete fusion rule Wk(g,f) has been determined previously in [FKW92, AvE19, Cre19]. In the case that O is a subregular nilpotent orbit and f O , then W (g,f) is k ∈ k k rational [AvE], and the fusion rules of Wk(g,f) has been determined in [AvE]. The following assertion, which follows immediately from Theorem 10.4, is new except for type A ([AvE]) and the above cases since the conjectural rationality [KW08, Ara15a] of W (g,f) with f O is open otherwise. k ∈ k

Corollary 10.6. Let g be simply-laced, k admissible, and suppose that Ok is an k k even nilpotent orbit, f O . Then L is rigid for all λ AdmZ. ∈ k λ,f ∈ Remark 10.7. Let KL denote the fusion category consisting of objects H0 (M), Wk(g,f) DS,f ord C M L . By Theorem 10.4, ∈C k(g) ord KL 0 L W ,M H (M), C k(g) →C k(g,f) 7→ DS,f is a quotient functor between fusion categories. It gives an equivalence if and only k k if the modules L , λ AdmZ, are distinct. λ,f ∈ The rest of this section is devoted to the proof of Theorem 10.4. Suppose Conjecture 1 holds for V and let be a monoidal full subcategory C of ord. Recall that M,N are said to centralize each other [M¨u03] if the CV ∈ C monodromy of M and N is trivial, i.e., is equal to the identity on M ⊠V N, where the monodromy is the double braiding

b b M ⊠ N M,N N ⊠ M N,M M ⊠ N. V −−−→ V −−−→ V Suppose that V is a vertex operator subalgebra of another quasi-lisse vertex operator algebra W and that W as a V -module is an object of ord. We assume that CV Conjecture 1 holds for W as well. Let be a monoidal full subcategory of such D C that W and any object in centralize each other. Then D (M) := W ⊠ M F V can be equipped with a structure of a module for the vertex operator algebra W , giving rise to the induction functor [CKM17]

Cord,M (M), D → W 7→ F k kˇ Theorem 10.8 ([Cre19]). Let g be simply-laced, and let λ AdmZ, µ AdmZ. ∈ ∈ Lk ⊠ Lk Lk Lk Lkˇ We have [λ,0] [0,µ] ∼= [λ,µ]. Moreover, [λ,0] and [0,µ] centralize each other if k λ AdmZ Q. ∈ ∩ UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 29

Let f be admissible, and let f be an even nilpotent element in O . Then f k ∈ Ok+1 as well since Ok depends only on the denominator of k. Let ℓ be the number deﬁned by 1 1 + =1, k + h∨ +1 ℓ + h∨ that is, k + h∨ +1 ℓ + h∨ = . k + h∨ Then ℓ is a non-degenerate admissible number. Note that

∨ ∨ ℓ k+1 p+q−h ℓˇ k p−h (60) AdmZ = AdmZ = P+ , AdmZ = AdmZ = P+ , where we have put k + h∨ = p/q. Consider the Wℓ(g)-modules

ℓ ℓˇ 0 ℓˇ ℓ L = L = H (L ), µ AdmZ. [0,µ] ∼ [µ,0] DS,fprin µ ∈ ℓ Since the stabilizer of 0 AdmZ of the W˜ +-action is trivial, the simple Wℓ(g)- ℓ ∈ℓ modules L[0,µ], µ AdmZ, are distinct. Therefore, by Theorem 10.4 (that is ∈ ℓ ℓ proved for f = f in [Cre19]), the modules L , µ AdmZ, form a fusion full prin [0,µ] ∈ subcategory of ord that is equivalent to a category that can be called a simple Wℓ(g) C ord ord current twist of L = L (see [Cre19, Thm.7.1] for the details). This simple ℓˇ(g) ∼ k(g) C ord C ord ord current twist of L is the fusion subcategory of L ⊠ L whose simple C k(g) C k(g) C 1(g) k 1 ord,tw objects are the L L . Call this category L . µ,f ⊗ −µ+Q C k(g) Now set W := W (g,f) L (g)= Lk L1 , V := W (g,f) W (g)= Lk+1 L . k ⊗ 1 0,f ⊗ ν k+1 ⊗ ℓ 0,f ⊗ [0,0] By Theorem 8.6, Lk+1 Lℓ (61) W ∼= λ,f [λ,0]. k+1 ⊗ λ∈AdmMZ ∩Q Hence, V is a vertex subalgebra of W . Moreover, each direct summand of W is an ordinary V -module and the sum is ﬁnite, and so W is an object of ord. CV Clearly, the V -modules

k+1 ℓ ℓ ℓˇ L L = W (g,f) L , µ AdmZ, 0,f ⊗ [0,µ] k+1 ⊗ [0,µ] ∈ ord ord form a monoidal full subcategory of that is equivalent to L . We denote CV C k(g) by this fusion category. D By Theorem 10.8 and (61), we ﬁnd that W centralizes any object of . Hence, D we have the induction functor : ord,M W ⊠ M. F D→CW 7→ V ord Theorem 10.9. The induction functor : W is a fully faithful tensor k+1 ℓ k F1 D→C functor that sends L0,f L[0,µ] to Lµ,f L−µ+Q, where µ + Q denotes the class 1 ⊗ ⊗ − in P/Q ∼= P+. 30 TOMOYUKI ARAKAWA, THOMAS CREUTZIG, AND BORIS FEIGIN

Proof. Thanks to [Cre19, Theorem 3.5], it is suﬃcient to show that (Lk+1 Lℓ ) F 0,f ⊗ [0,µ] k 1 ℓˇ is a simple W -module that is isomorphic to L L for all µ AdmZ. As µ,f ⊗ −µ+Q ∈ V -modules we have k+1 ℓ k+1 ℓ k+1 ℓ (L0,f L[0,µ])= (Lλ,f L[λ,0]) ⊠V (L0,f L[0,µ]) F ⊗ k+1 ⊗ ⊗ λ∈AdmMZ ∩Q Lk+1 Lℓ Lk L1 ∼= λ,f [λ,µ] ∼= µ,f −µ+Q k+1 ⊗ ⊗ λ∈AdmMZ ∩Q by Theorem 8.6 and Theorem 10.8. We claim this is indeed an isomorphism of W - modules. To see this, it is suﬃcient to show that there is a non-trivial W -module homomorphism (Lk+1 Lℓ ) Lk L1 since Lk L1 is simple. By F 0,f ⊗ [0,µ] → µ,f ⊗ −µ+Q µ,f ⊗ −µ+Q the Frobenius reciprocity [KO02, CKM17], we have Hom ( (Lk+1 Lℓ ), Lk L1 ) W -Mod F 0,f ⊗ [0,µ] µ,f ⊗ −µ+Q = Hom (Lk+1 Lℓ , Lk L1 ) ∼ V -Mod 0,f ⊗ [0,µ] µ,f ⊗ −µ+Q Lk+1 Lℓ Lk+1 Lℓ ∼= HomV -Mod( 0,f [0,µ], λ,f [λ,µ]). k+1 ⊗ ⊗ λ∈AdmMZ ∩Q It follows that there is a non-trivial homomorphism corresponding to the identity map Lk+1 Lℓ Lk+1 Lℓ . 0,f ⊗ [0,µ] → 0,f ⊗ [0,µ] Proof of Theorem 10.4. By Theorem 10.9, the correspondence k 1 ℓ k+1 ℓ k 1 k ℓˇ L L L L L L L , µ AdmZ = AdmZ, µ⊗ −µ+Q 7→ [0,µ] 7→ 0,f ⊗ [0,µ] 7→ µ,f ⊗ −µ+Q ∈ gives a tensor functor ord,tw ord ord L CW g L g = CW (g,f) ⊠ L g , C k(g) → k( ,f)⊗ 1( ) k C 1( ) where the last ⊠ denotes the Deligne product. This functor is fully faithful if k ord and only if all the L are non-isomorphic. Since L is semi-simple and its µ,f C 1(g) simple objects are all invertible, i.e., simple currents, this corresponds extends to a surjective tensor functor ord ord ord L ⊠ L CW g ⊠ L , C k(g) C 1(g) → k( ,f) C 1(g) which restricts to a surjective tensor functor ord k k 0 k L CW (g,f), L L = H (L ) C k(g) → k µ → µ,f DS,f µ k that again is fully faithful if and only if all the Lµ,f are non-isomorphic. Let us conclude with a remark on the general case.

Remark 10.10. It is a well-known result of Kazhdan-Lusztig that ordinary modules of aﬃne vertex algebras at generic level have vertex tensor category [KL1, KL2, KL3, KL4] and it is reasonable to expect that a similar result might hold for W- algebras as well. However this is only proven for the Virasoro algebra [CJORY20] and the N = 1 super Virasoro algebra [CMY20]. At generic levels one has to deal with inﬁnite order extensions of vertex algebras and so one needs to consider completions of vertex tensor categories. The theory UROD ALGEBRAS AND TRANSLATION OF W-ALGEBRAS 31 of vertex algebra extension also works for such completions [CMY20]. This means that if one can prove the existence of vertex tensor category structure on categories of ordinary modules of W-algebras at generic levels, then one can also derive results for generic levels that are similar to the statements of this section.

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN Email address: [email protected]

University of Alberta Department of Mathematical and Statistical Sciences Edmon- ton, AB T6G 2G1, Canada Email address: [email protected]

National Research University Higher School of Economics, 101000, Myasnitskaya ul. 20, Moscow, Russia, and Landau Institute for Theoretical Physics, 142432, pr. Akademika Semenova 1a, Chernogolovka, Russia Email address: [email protected]