A commutant realization of Odake's algebra - Creutzig, Thomas; Linshaw, Andrew R. (Transformation Groups, 2013)
The bcβγ-system W of rank 3 has an action of the affine vertex algebra V0(sl2), and the commutant vertex algebra C = Com(V0(sl2),W) contains copies of V-3/2(sl2) and Odake's algebra O. Odake's algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V-3/2(sl2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of sl2. © 2013 Springer Science+Business Media New York.
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Affine Laumon Spaces and Iterated W -Algebras - Creutzig, Thomas; Diaconescu, Duiliu Emanuel; Ma, Mingyang (Communications in Mathematical Physics, 2023)
A family of vertex algebras whose universal Verma modules coincide with the cohomology of affine Laumon spaces is found. This result is based on an explicit expression for the generating function of Poincaré polynomials of these spaces. There is a variant of quantum Hamiltonian reduction that realizes vertex algebras which we call iterated W -algebras and our main conjecture is that the vertex algebras associated to the affine Laumon spaces are subalgebras of iterated W -algebras.
CRIS-Team Scopus Importer:2024-06-24

Airy Ideals, Transvections, and W(sp2N) -Algebras - Bouchard, Vincent; Creutzig, Thomas; Joshi, Aniket (Annales Henri Poincaré, 2024)
In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ-adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W-algebra of sp2N at level -N-1/2, following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function.
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A Kazhdan–Lusztig Correspondence for L-32(sl3) - Creutzig, Thomas; Ridout, David; Rupert, Matthew (Communications in Mathematical Physics, 2023)
The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for L-32(sl3) via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of U¯iH(sl3), the unrolled restricted quantum group of sl3 at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of WA20(2)-modules. Here, WA20(2) is an orbifold of the octuplet vertex algebra WA2(2) of Semikhatov, the latter being the natural sl3-analogue of the well known triplet algebra. Moreover, WA20(2) is the parafermionic coset of the affine vertex algebra L-32(sl3). We formulate an explicit conjecture relating the representation theory of WA20(2) and U¯iH(sl3) and work out the resulting structures of the corresponding L-32(sl3)-modules. In particular, we obtain conjectural Loewy diagrams for the latter’s projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde’s formula. Finally, we give analogous results for WA2(2).
CRIS-Team Scopus Importer:2024-06-24

A quasi-Hopf algebra for the triplet vertex operator algebra - Creutzig, Thomas; Gainutdinov, Azat M.; Runkel, Ingo (Communications in Contemporary Mathematics, 2020)
We give a new factorizable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sa(2) at a 2pth root of unity. The representation category of U is conjecturally ribbon equivalent to that of the triplet vertex operator algebra (VOA) (p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet VOA a(p), and our construction is parallel to extending a(p) to (p). We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra a'a to a quasi-Hopf algebra for a'a;2p, which corresponds to passing from the Heisenberg VOA to a lattice extension.
CRIS-Team Scopus Importer:2024-06-24
Keywords: braided tensor categories; logarithmic modules; quantum groups; quasi-Hopf algebras; Vertex operator algebras

Braided Tensor Categories of Admissible Modules for Affine Lie Algebras - Creutzig, Thomas; Huang, Yi Zhi; Yang, Jinwei (Communications in Mathematical Physics, 2018)
Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is sl^ 2, we prove the rigidity and modularity conjectures.
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Braided Tensor Categories Related to Bp Vertex Algebras - Auger, Jean; Creutzig, Thomas; Kanade, Shashank; Rupert, Matthew (Communications in Mathematical Physics, 2020)
The Bp-algebras are a family of vertex operator algebras parameterized by p∈ Z≥ 2. They are important examples of logarithmic CFTs and appear as chiral algebras of type (A1, A2p-3) Argyres–Douglas theories. The first member of this series, the B2-algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra. We study categories related to the Bp vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl2. These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.
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CLASSICAL FREENESS OF ORTHOSYMPLECTIC AFFINE VERTEX SUPERALGEBRAS - Creutzig, Thomas; Linshaw, Andrew R.; Song, Bailin (Proceedings of the American Mathematical Society, 2024)
The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name “classical freeness” by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra Ln(ospm|2r) for all positive integers m, n, r satisfying Formula presented. In particular, it holds for the rational vertex superalgebras Ln(osp1|2r) for all positive integers r, n.
CRIS-Team Scopus Importer:2024-09-20
Keywords: arc space; associated scheme; classical freeness; coset construction; Vertex algebra

Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models - Creutzig, Thomas; Hikida, Yasuaki (Journal of High Energy Physics, 2021)
We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a “first order formulation” of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset (SL(n)k ⊗SL(n)−1)/SL(n)k−1 and sl(n) Toda field theory with generic level k. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset SL(n + 1)k /(SL(n)k ⊗ U(1)), which could be done only for n = 2, 3 in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rapčák dualities of corner vertex operator algebras.
CRIS-Team Scopus Importer:2024-06-24
Keywords: BRST Quantization; Conformal and W Symmetry; Conformal Field Theory; String Duality

Correspondences among CFTs with different W-algebra symmetry - Creutzig, Thomas; Genra, Naoki; Hikida, Yasuaki; Liu, Tianshu (Nuclear Physics B, 2020)
W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra g and an sl(2)-embedding into g. We derive correspondences among correlation functions of theories having different W-algebras as symmetry algebras. These W-algebras are associated to the same g but distinct sl(2)-embeddings. For this purpose, we first explore different free field realizations of W-algebras and then generalize previous works on the path integral derivation of correspondences of correlation functions. For g=sl(3), there is only one non-standard (non-regular) W-algebra known as the Bershadsky-Polyakov algebra. We examine its free field realizations and derive correlator correspondences involving the WZNW theory of sl(3), the Bershadsky-Polyakov algebra and the principal W3-algebra. There are three non-regular W-algebras associated to g=sl(4). We show that the methods developed for g=sl(3) can be applied straightforwardly. We briefly comment on extensions of our techniques to general g.
CRIS-Team Scopus Importer:2024-06-24

Correspondences between WZNW models and CFTs with W-algebra symmetry - Creutzig, Thomas; Hikida, Yasuaki; Rønne, Peter B. (Journal of High Energy Physics, 2016)
Abstract: We study theories with W-algebra symmetries and their relation to WZNW-type models on (super-)groups generalizing the H3+ WZNW to Liouville correspondence. Correlation functions of the WZNW models are expressed in terms of correlators of CFTs with W-algebra symmetry. The symmetries of the theories involved in these correspondences are related by the Drinfeld-Sokolov reduction of Lie algebras to W-algebras. The W-algebras considered in this paper are the Bershadsky-Polyakov algebra for sl(3) and the quasi-superconformal algebra for generic sl(N|M). The quantum W-algebras obtained from affine sl(N) are constructed using embeddings of sl(2) into sl(N), and these can in turn be characterized by partitions of N. The above cases correspond to N+ 2 = 2 + N1 and its supergroup extension. Finally, sl(2N) and the correspondence corresponding to 2N = N2 is also analyzed. These are all W-algebras that are generated by fields of at most dimension two.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal and W Symmetry; Conformal Field Models in String Theory

Correspondences of Categories for Subregular W -Algebras and Principal W -Superalgebras - Creutzig, Thomas; Genra, Naoki; Nakatsuka, Shigenori; Sato, Ryo (Communications in Mathematical Physics, 2022)
Based on the Kazama–Suzuki type coset construction and its inverse coset between the subregular W-algebras for sln and the principal W-superalgebras for sl1|n, we prove weight-wise linear equivalences of their representation categories. Our main results are then improvements of these correspondences incorporating the monoidal structures. Firstly, in the rational case, we obtain the classification of simple modules and their fusion rules via simple current extensions from their Heisenberg cosets. Secondly, beyond the rational case, we use certain kernel VOAs together with relative semi-infinite cohomology functors to get functors from categories of modules for the subregular W-algebras for sln to categories of modules for the principal W-superalgebras for sl1|n and vice versa. We study these functors and in particular prove isomorphisms between the superspaces of logarithmic intertwining operators. As a corollary, we obtain correspondences of representation categories in the monoidal sense beyond the rational case as well.
CRIS-Team Scopus Importer:2024-06-24

Cosets, characters and fusion for admissible-level osp(1|2) minimal models - Creutzig, Thomas; Kanade, Shashank; Liu, Tianshu; Ridout, David (Nuclear Physics B, 2019)
We study the minimal models associated to osp(1|2), otherwise known as the fractional-level Wess–Zumino–Witten models of osp(1|2). Since these minimal models are extensions of the tensor product of certain Virasoro and sl2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp(1|2). In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles.
CRIS-Team Scopus Importer:2024-06-24

COSETS FROM EQUIVARIANT W-ALGEBRAS - Creutzig, Thomas; Nakatsuka, Shigenori (Representation Theory, 2023)
The equivariant W-algebra of a simple Lie algebra g is a BRST reduction of the algebra of chiral differential operators on the Lie group of g. We construct a family of vertex algebras A[g, k, n] as subalgebras of the equivariant W-algebra of g tensored with the integrable affine vertex algebra Ln(ǧ) of the Langlands dual Lie algebra ǧ at level n ∈ Z>0. They are conformal extensions of the tensor product of an affine vertex algebra and the principal W-algebra whose levels satisfy a specific relation. When g is of type ADE, we identify A[g, k, 1] with the affine vertex algebra Vk−1(g) ⊗ L1(g), giving a new and efficient proof of the coset realization of the principal W-algebras of type ADE.
CRIS-Team Scopus Importer:2024-06-24

Cosets of affine vertex algebras inside larger structures - Creutzig, Thomas; Linshaw, Andrew R. (Journal of Algebra, 2019)
Given a finite-dimensional reductive Lie algebra g equipped with a nondegenerate, invariant, symmetric bilinear form B, let Vk(g,B) denote the universal affine vertex algebra associated to g and B at level k. Let Ak be a vertex (super)algebra admitting a homomorphism Vk(g,B)→Ak. Under some technical conditions on Ak, we characterize the coset Com(Vk(g,B),Ak) for generic values of k. We establish the strong finite generation of this coset in full generality in the following cases: Ak=Vk(g′,B′), Ak=Vk−l(g′,B′)⊗F, and Ak=Vk−l(g′,B′)⊗Vl(g″,B″). Here g′ and g″ are finite-dimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B′ and B″ which extend B, l∈C is fixed, and F is a free field algebra admitting a homomorphism Vl(g,B)→F. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple N=2 superconformal algebra with c=[Formula presented] for all positive integers k.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine vertex algebra; Commutant construction; Coset construction; Finite generation; Invariant theory; N=2 superconformal algebra; Orbifold construction; W-algebra

Cosets of Bershadsky–Polyakov algebras and rational W -algebras of type A - Arakawa, Tomoyuki; Creutzig, Thomas; Linshaw, Andrew R. (Selecta Mathematica-New Series, 2017)
The Bershadsky–Polyakov algebra is the W-algebra associated to sl3 with its minimal nilpotent element fθ. For notational convenience we define Wℓ= Wℓ-3/2(sl3, fθ). The simple quotient of Wℓ is denoted by Wℓ, and for ℓ a positive integer, Wℓ is known to be C2-cofinite and rational. We prove that for all positive integers ℓ, Wℓ contains a rank one lattice vertex algebra VL, and that the coset Cℓ= Com (VL, Wℓ) is isomorphic to the principal, rational W(sl2ℓ) -algebra at level (2 ℓ+ 3) / (2 ℓ+ 1) - 2 ℓ. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, C2-cofinite vertex superalgebras from Wℓ.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Coset construction; Vertex algebra; W-algebra

Cosets of the Wk(Sl4, fsubreg)-algebra - Creutzig, Thomas; Linshaw, Andrew R. (Contemporary Mathematics, 2018)
Let Wk(sl4, fsubreg) be the universal W-algebra associated to sl4 with its subregular nilpotent element, and let Wk(sl4, fsubreg) be its simple quotient. There is a Heisenberg subalgebra H, and we denote by Ck the coset Com(H,Wk(sl4, fsubreg)), and by Ck its simple quotient. We show that for k = −4 + (m + 4)/3 where m is an integer greater than 2 and m + 1 is coprime to 3, Ck is isomorphic to a rational, regular W-algebra W (slm, freg). In particular, Wk(sl4, fsubreg) is a simple current extension of the tensor product of W(slm, freg) with a rank one lattice vertex operator algebra, and hence is rational.
CRIS-Team Scopus Importer:2024-06-24

Direct limit completions of vertex tensor categories - Creutzig, Thomas; McRae, Robert; Yang, Jinwei (Communications in Contemporary Mathematics, 2022)
We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of (1|2) and the N = 1 super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine lie superalgebras; Braided tensor categories; Commutative algebra objects; Direct limits; Ind-completions; Super virasoro algebra; Vertex operator algebras; Vertex tensor categories

Duality of subregular W-algebras and principal W-superalgebras - Creutzig, Thomas; Genra, Naoki; Nakatsuka, Shigenori (Advances in Mathematics, 2021)
We prove Feigin-Frenkel type dualities between subregular W-algebras of type A,B and principal W-superalgebras of type sl(1|n),osp(2|2n). The type A case proves a conjecture of Feigin and Semikhatov. Let (g1,g2)=(sln+1,sl(1|n+1)) or (so2n+1,osp(2|2n)) and let r be the lacity of g1. Let k be a complex number and ℓ defined by r(k+h1∨)(ℓ+h2∨)=1 with hi∨ the dual Coxeter numbers of the gi. Our first main result is that the Heisenberg cosets Ck(g1) and Cℓ(g2) of these W-algebras at these dual levels are isomorphic, i.e. Ck(g1)≃Cℓ(g2) for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W-algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W-algebra at level k times the lattice vertex superalgebra VZ is the principal W-superalgebra at the dual level ℓ. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level ℓ times the lattice vertex superalgebra V−1Z is the subregular W-algebra at the dual level k. Again this is proven for the universal W-algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W-superalgebra and its Heisenberg coset at level ℓ are rational and/or C2-cofinite if the same is true for the simple subregular W-algebra at dual level ℓ. This gives many new C2-cofiniteness and rationality results.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine W-algebra; Coset; Duality; Vertex algebra

False theta functions and the Verlinde formula - Creutzig, Thomas; Milas, Antun (Advances in Mathematics, 2014)
We discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of W-algebras and vertex algebras in general. More precisely, motivated by clues from conformal field theory, first, we are able to determine modular-like transformation properties of regularized partial and false theta functions. Then, after suitable identification of regularized partial/false theta functions with the characters of atypical modules for the singlet vertex algebra W(2,2p-1), we formulate a Verlinde-type formula for the fusion rules of irreducible W(2,2p-1)-modules. © 2014 Elsevier Inc.
CRIS-Team Scopus Importer:2024-06-24
Keywords: False and partial theta functions; Logarithmic conformal field theories; Modular forms; Verlinde formula; Vertex operator algebras

Fusion categories for affine vertex algebras at admissible levels - Creutzig, Thomas (Selecta Mathematica-New Series, 2019)
The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.
CRIS-Team Scopus Importer:2024-06-24

FZZ-triality and large N=4 super Liouville theory - Creutzig, Thomas; Hikida, Yasuaki (Nuclear Physics B, 2022)
We examine dualities of two dimensional conformal field theories by applying the methods developed in previous works. We first derive the duality between SL(2|1)k/(SL(2)k⊗U(1)) coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large N=4 super Liouville theory and a coset of the form Y(k1,k2)/SL(2)k1+k2, where Y(k1,k2) consists of two SL(2|1)ki and free bosons or equivalently two U(1) cosets of D(2,1;ki−1) at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden SL(2)k′ in SL(2|1)k or D(2,1;k−1)1. The relation of levels is k′−1=1/(k−1). We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with SL(n|1)k/(SL(n)k⊗U(1)) for arbitrary n>2.
CRIS-Team Scopus Importer:2024-06-24

Generalized parafermions of orthogonal type - Creutzig, Thomas; Kovalchuk, Vladimir; Linshaw, Andrew R. (Journal of Algebra, 2022)
There is an embedding of affine vertex algebras Vk(gln)↪Vk(sln+1), and the coset Ck(n)=Com(Vk(gln),Vk(sln+1)) is a natural generalization of the parafermion algebra of sl2. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter W∞-algebra of type W(2,3,…). In this paper, we consider an analogous structure of orthogonal type, namely Dk(n)=Com(Vk(so2n),Vk(so2n+1))Z2. We realize this algebra as a one-parameter quotient of the two-parameter even spin W∞-algebra of type W(2,4,…), and we classify all coincidences between its simple quotient Dk(n) and the algebras Wℓ(so2m+1) and Wℓ(so2m)Z2. As a corollary, we show that for the admissible levels k=−(2n−2)+[Formula presented](2n+2m−1) for soˆ2n the simple affine algebra Lk(so2n) embeds in Lk(so2n+1), and the coset is strongly rational. As a consequence, the category of ordinary modules of Lk(so2n+1) at such a level is a braided fusion category.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine Lie algebra; Coset construction; Vertex algebra; W-algebra

Gluing vertex algebras - Creutzig, Thomas; Kanade, Shashank; McRae, Robert (Advances in Mathematics, 2022)
We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠Crev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X′⊠X is a commutative algebra, where X′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi⁎. These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Braid-reversed equivalences; Commutative algebra objects; Coset conformal field theory; Kazhdan-Lusztig categories; Tensor categories; Vertex operator algebras

Harmonic analysis and free field realization of the takiff supergroup of GL(1|1) - Babichenko, Andrei; Creutzig, Thomas (Symmetry Integrability and Geometry-Methods and Applications, 2015)
Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1j1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Free field realization; Harmonic analysis; Logarithmic CFT

Hecke operators on vector-valued modular forms - Bouchard, Vincent; Creutzig, Thomas; Joshi, Aniket (Symmetry Integrability and Geometry-Methods and Applications, 2019)
We study Hecke operators on vector-valued modular forms for the Weil representation ρL of a lattice L. We first construct Hecke operators Tr that map vector-valued modular forms of type ρL into vector-valued modular forms of type ρL(r), where L(r) is the lattice L with rescaled bilinear form (·, ·)r = r(·, ·), by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vectorvalued Hecke operators Tr have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators Tr. In the particular case when r = n2 for some positive integer n, we compose Tn 2 with a projection operator to construct new Hecke operators Hn 2 that map vector-valued modular forms of type ρL into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators Hn 2, and compare our operators with the alternative construction of BruinierStein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].
CRIS-Team Scopus Importer:2024-06-24
Keywords: Hecke operators; Vector-valued modular forms; Weil representation

Higgs and Coulomb branches from vertex operator algebras - Costello, Kevin; Creutzig, Thomas; Gaiotto, Davide (Journal of High Energy Physics, 2019)
We formulate a conjectural relation between the category of line defects in topologically twisted 3d N = 4 supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local operators for the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal Field Theory; Supersymmetric Gauge Theory; Topological Field Theories

Higgs phenomenon for higher spin fields on AdS3 - Creutzig, Thomas; Hikida, Yasuaki (Journal of High Energy Physics, 2015)
Abstract: In a previous work, a marginal deformation of 2d coset type model with N=3 N=3 superconformal symmetry was studied, and it was interpreted as a change of boundary conditions for bulk fields in the dual higher spin theory. The deformation breaks generic higher spin gauge symmetry, and the generated mass of a spin 2 field was computed. The deformation might be related to the introduction of finite string tension in a superstring theory. In this paper, we extend the analysis and compute the masses of generic higher spin fields at the leading order of 1/c (c is the CFT central charge) but at the full order of the deformation parameter. We find that the masses are not generated for so(3)R singlet higher spin fields at this order and the spectrum is the Regge-like one for so(3)R triplet higher spin-charged fields.
CRIS-Team Scopus Importer:2024-06-24
Keywords: AdS-CFT Correspondence; Conformal and W Symmetry; Higher Spin Gravity; Higher Spin Symmetry

Higher Airy Structures, W Algebras and Topological Recursion - Borot, Gaëtan; Bouchard, Vincent; Chidambaram, Nitin K.; Creutzig, Thomas; Noshchenko, Dmitry (Memoirs of the American Mathematical Society, 2024)
We define higher quantum Airy structures as generalizations of the Kontsevich–Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of W(g) algebras at self-dual level, with g = glN+1, so2N or eN. We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these W constraints have already appeared in the literature, but we find many new ones. For glN+1 our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard–Eynard topological recursion gives symmetric ωg,ns and is thus well defined. For all such cases, we show that the topological recursion is equivalent to W(gl) constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Airy structures; enumerative geometry; topological recursion; W algebras

Higher rank FZZ-dualities - Creutzig, Thomas; Hikida, Yasuaki (Journal of High Energy Physics, 2021)
We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the sl(2)/u(1) coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from sl(2) Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type sl(N+1)/(sl(N)×u(1)) and investigate the equivalence to a theory with an sl(N+ 1 | N) structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for sl(N) and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal and W Symmetry; Conformal Field Theory; String Duality

Higher rank partial and false theta functions and representation theory - Creutzig, Thomas; Milas, Antun (Advances in Mathematics, 2017)
We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to representation theory of W-algebras. We derive modular transformation properties of regularized higher rank partial and false theta functions as well as Kostant's version of these. Modulo conjectures in representation theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of “narrow” logarithmic W-algebras associated to rescaled root lattices. This paper substantially generalize our previous work [19] pertaining to (1,p)-singlet W-algebras (the sl2 case). Results in this paper are very general and are applicable in a variety of situations.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Partial theta functions; Theta functions; W-algebras

Higher spin AdS3 holography with extended supersymmetry - Creutzig, Thomas; Hikida, Yasuaki; Rønne, Peter B. (Journal of High Energy Physics, 2014)
Abstract: We propose a holographic duality between a higher spin AdS3 gravity with so(p) extended supersymmetry and a large N limit of a 2-dimensional Grassmannian-like model with a specific critical level k = N and a non-diagonal modular invariant. As evidence, we show the match of one-loop partition functions. Moreover, we construct symmetry generators of the coset model for low spins which are dual to gauge fields in the supergravity. Further, we discuss a possible relation to superstring theory by noticing an N = 3 supersymmetry of critical level model at finite k, N. In particular, we examine BPS states and marginal deformations. Inspired by the supergravity side, we also propose and test another large N CFT dual obtained as a Z2 automorphism truncation of a similar coset model, but at a non-critical level.
CRIS-Team Scopus Importer:2024-06-24
Keywords: AdS-CFT Correspondence; Conformal and W Symmetry; Extended Supersymmetry; Higher Spin Gravity

Hilbert schemes of nonreduced divisors in Calabi–Yau threefolds and W-algebras - Chuang, Wu Yen; Creutzig, Thomas; Diaconescu, Duiliu Emanuel; Soibelman, Yan (European Journal of Mathematics, 2021)
A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of the Hilbert scheme of points on a nonreduced plane in three-dimensional affine space. The resulting W-module is then identified to the vacuum module. The construction is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel–Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Hilbert schemes; Quiver representations; W-algebras

INVARIANT SUBALGEBRAS OF THE SMALL N = 4 SUPERCONFORMAL ALGEBRA - Creutzig, Thomas; Linshaw, Andrew R.; Riedler, Wolfgang (Transformation Groups, 2021)
Various aspects of orbifolds and cosets of the small N = 4 superconformal algebra are studied First, we determine minimal strong generators for generic and specific levels As a corollary, we obtain the vertex algebra of global sections of the chiral de Rham complex on any complex Enriques surface We also identify orbifolds of cosets of the small N = 4 superconformal algebra with Com (Vl(sl2); Vl+1(sl2) W-5=2(sl4; frect)) and in addition at special levels with Grassmanian cosets and principal W-algebras of type A at degenerate admissible levels These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.
CRIS-Team Scopus Importer:2024-06-24

Logarithmic conformal field theory, log-modular tensor categories and modular forms - Creutzig, Thomas; Gannon, Terry (Journal of Physics A: Mathematical and Theoretical, 2017)
The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C 2-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde's formula. We illustrate our ideas with the -triplet algebras and the symplectic fermions.
CRIS-Team Scopus Importer:2024-06-24
Keywords: category; logarithmic VOA; modular forms; Verlindes formula

Logarithmic link invariants of U‾qH(sl2) and asymptotic dimensions of singlet vertex algebras - Creutzig, Thomas; Milas, Antun; Rupert, Matt (Journal of Pure and Applied Algebra, 2018)
We study relationships between the restricted unrolled quantum group U‾q H(sl2) at q=eπi/r, and the singlet vertex operator algebra M(r), r≥2. We use deformable families of modules to efficiently compute (1,1)-tangle invariants colored with projective U‾q H(sl2)-modules. These invariants relate to the colored Alexander tangle invariants studied in [6,40]. It follows that the regularized asymptotic dimensions of characters of M(r), studied previously by the first two authors, coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of M(r)-mod in connection to braided tensor categories.
CRIS-Team Scopus Importer:2024-06-24

Logarithmic W-algebras and Argyres-Douglas theories at higher rank - Creutzig, Thomas (Journal of High Energy Physics, 2018)
Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of S-duality for four-dimensional gauge theories. In the case of type A and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur-indices of type IV Argyres-Douglas theories. Further, there is a conformal embedding pattern of these vertex operator algebras that nicely matches the RG-flow of Argyres-Douglas theories as discussed by Buican and Nishinaka.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal and W Symmetry; Conformal Field Theory; Supersymmetric Gauge Theory

Mathieu moonshine and the geometry of k3 surfaces - Creutzig, Thomas; Höhn, Gerald (Communications in Number Theory and Physics, 2014)
We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M24 and the complex elliptic genus of a K3 surface with the symmetries of geometric structures on K3 surfaces. Two main results are that the complex elliptic genus of a K3 surface is a virtual module for the Mathieu group M24 and also for a certain vertex operator superalgebra VG where G is the holonomy group.
CRIS-Team Scopus Importer:2024-06-24

Modular data and Verlinde formulae for fractional level WZW models II - Creutzig, Thomas; Ridout, David (Nuclear Physics B, 2013)
This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl̂(2) at an arbitrary admissible level k. Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce non-negative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck fusion rules are determined explicitly. These rules reproduce the well-known "fusion rules" of Koh and Sorba, negative coefficients included, upon quotienting the Grothendieck fusion ring by a certain ideal. © 2013 Elsevier B.V.
CRIS-Team Scopus Importer:2024-06-24

Modularity of logarithmic parafermion vertex algebras - Auger, Jean; Creutzig, Thomas; Ridout, David (Letters in Mathematical Physics, 2018)
The parafermionic cosets Ck= Com (H, Lk(sl2)) are studied for negative admissible levels k, as are certain infinite-order simple current extensions Bk of Ck. Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to Ck, irreducible Ck- and Bk-modules are obtained from those of Lk(sl2). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible Bk-modules. The irreducible Ck- and Bk-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the Bk are C2-cofinite vertex operator algebras.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal field theory; Coset constructions; Modular transformations; Parafermions; Vertex algebras

N = 4 Superconformal Algebras and Diagonal Cosets - Creutzig, Thomas; Feigin, Boris; Linshaw, Andrew R. (International Mathematics Research Notices, 2022)
Coset constructions of W-algebras have many applications and were recently given for principal W-algebras of A, D, and E types by Arakawa together with the 1st and 3rd authors. In this paper, we give coset constructions of the large and small N = 4 superconformal algebras, which are the minimal W-algebras of d(2, 1; α) and psl(2|2), respectively. From these realizations, one finds a remarkable connection between the large N = 4 algebra and the diagonal coset Ck1,k2 = Com(Vk1+k2 (sl2),Vk1 (sl2) ⊗ Vk2 (sl2)), namely, as two-parameter vertex algebras, Ck1,k2 coincides with the coset of the large N = 4 algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various Walgebras. As a corollary, we give new examples of strongly rational principalW-algebras of type C at degenerate admissible levels.
CRIS-Team Scopus Importer:2024-06-24

Negative index Jacobi forms and quantum modular forms - Bringmann, Kathrin; Creutzig, Thomas; Rolen, Larry (Research in the Mathematical Sciences, 2014)
In this paper, we consider the Fourier coefficients of a special class of meromorphic Jacobi forms of negative index considered by Kac and Wakimoto. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family of rank-crank type partial differential equations generalizing the famous example of Atkin and Garvan. We then describe the modularity properties of these coefficients, showing that they are ‘mixed partial theta functions’, along the way determining a new class of quantum modular partial theta functions which is of independent interest. In particular, we settle the final cases of a question of Kac concerning modularity properties of Fourier coefficients of certain Jacobi forms.
CRIS-Team Scopus Importer:2024-06-24

On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions - Creutzig, Thomas; Milas, Antun; Wood, Simon (International Mathematics Research Notices, 2017)
We study a family of non-C 2 -cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ε. We then apply modular transformation properties of regularisedpartial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ε. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ε, while for sufficiently negative real part of ε its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra can be also detected from vector-valued quantum modular forms formed by distinguished atypical characters.
CRIS-Team Scopus Importer:2024-06-24

On Ribbon Categories for Singlet Vertex Algebras - Creutzig, Thomas; McRae, Robert; Yang, Jinwei (Communications in Mathematical Physics, 2021)
We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra M(p) , p≥ 2. The first category consists of all finite-length M(p) -modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra W(p). We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.
CRIS-Team Scopus Importer:2024-06-24

On the structure of W-algebras in type A - Creutzig, Thomas; Fasquel, Justine; Linshaw, Andrew R.; Nakatsuka, Shigenori (Japanese Journal of Mathematics, 2025)
We formulate and prove examples of a conjecture which describes the W−algebras in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type W−algebras are building blocks of the W−algebras in type A. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular W−algebras. In contrast, they are provided by singlet-type extensions of W−algebras at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different W−algebras are established.
CRIS-Team Scopus Importer:2025-04-01
Keywords: affine W-algebras; collapsing levels; inverse Hamiltonian reductions; reductions by stage; universal W-algebras

Orbifolds and Cosets of Minimal W -Algebras - Arakawa, Tomoyuki; Creutzig, Thomas; Kawasetsu, Kazuya; Linshaw, Andrew R. (Communications in Mathematical Physics, 2017)
Let g be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of sl2 inducing the minimal gradation on g. The corresponding minimal W-algebra Wk(g, e-θ) introduced by Kac and Wakimoto has strong generators in weights 1 , 2 , 3 / 2 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra Vk′(g♮), where g♮⊂ g denotes the centralizer of sl2. Therefore, Wk(g, e-θ) has an action of a connected Lie group G0♮ with Lie algebra g0♮, where g0♮ denotes the even part of g♮. We show that for any reductive subgroup G⊂G0♮, and for any reductive Lie algebra g′⊂ g♮, the orbifold Ok=Wk(g,e-θ)G and the coset Ck= Com (V(g′) , Wk(g, e-θ)) are strongly finitely generated for generic values of k. Here V(g′) denotes the affine vertex algebra associated to g′. We find explicit minimal strong generating sets for Ck when g′= g♮ and g is either sln, sp2n, sl(2 | n) for n≠ 2 , psl(2 | 2) , or osp(1 | 4). Finally, we conjecture some surprising coincidences among families of cosets Ck which are the simple quotients of Ck, and we prove several cases of our conjecture.
CRIS-Team Scopus Importer:2024-06-24

Orbifolds of symplectic fermion algebras - Creutzig, Thomas; Linshaw, Andrew R. (Transactions of the American Mathematical Society, 2017)
We present a systematic study of the orbifolds of the rank n symplectic fermion algebra A(n), which has full automorphism group Sp(2n). First, we show that A(n)Sp(2n) and A(n)GL(n) are W-algebras of type W(2, 4,…, 2n) and W(2, 3,…, 2n + 1), respectively. Using these results, we find minimal strong finite generating sets for A(mn)Sp(2n) and A(mn)GL(n) for all m, n ≥ 1. We compute the characters of the irreducible representations of A(mn)Sp(2n)ï¿½SO(m) and A(mn)GL(n)ï¿½GL(m) appearing inside A(mn), and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for A(n); we show that for any reductive group G of automorphisms, A(n)G is strongly finitely generated.
CRIS-Team Scopus Importer:2024-06-24

Ordinary modules for vertex algebras of osp1|2n - Creutzig, Thomas; Genra, Naoki; Linshaw, Andrew (Journal für die reine und angewandte Mathematik, 2024)
We show that the affine vertex superalgebra Vk(osp1|2n) at generic level k embeds in the equivariant W-algebra of sp2n times 4n free fermions. This has two corollaries: (1) it provides a new proof that, for generic k, the coset Com(Vk(sp2n), Vk(osp1|2n)) is isomorphic to (Wl(sp2n) for l = -(n + 1) + (k + n + 1)=(2k + 2n + 1), and (2) we obtain the decomposition of ordinary Vk (osp1|2n)-modules into Vk(sp2n) ⊗ Wl(sp2n)-modules. Next, if k is an admissible level and is a non-degenerate admissible level for sp2n, we show that the simple algebra Lk(osp1|2n) is an extension of the simple subalgebra Lk(sp2n) ⊗ Wl(sp2n). Using the theory of vertex superalgebra extensions, we prove that the category of ordinary Lk(osp1/2n)-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of Wl(sp2n)-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary Lk(sp2n)-modules are rigid.
CRIS-Team Scopus Importer:2025-02-04

Quasi-lisse extension of affine sl2 à la Feigin–Tipunin - Creutzig, Thomas; Nakatsuka, Shigenori; Sugimoto, Shoma (Advances in Mathematics, 2024)
We study the affine analogue FTp(sl2) of the triplet algebra. We show that FTp(sl2) is quasi-lisse and the associated variety is the nilpotent cone of sl2. We realize FTp(sl2) as the global sections of a sheaf of vertex algebras in the spirit of Feigin–Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama–Suzuki dual superalgebra sWp(sl2|1) of FTp(sl2) and their singlet type subalgebras sMp(sl2|1) and Mp(sl2) and show their correspondence of categories. For p=1, we show the logarithmic Kazhdan–Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to sMp(sl2|1) is the unrolled restricted quantum supergroup u−1H(sl2|1) as suggested by Semikhatov and Tipunin.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Affine vertex algebras; Feigin-Tipunin type extensions; Logarithmic Kazhdan-Lusztig correspondence; Nilpotent cone

Rectangular W algebras and superalgebras and their representations - Creutzig, Thomas; Hikida, Yasuaki (Physical Review D, 2019)
We study W algebras obtained by quantum Hamiltonian reduction of sl(Mn) associated to the sl(2) embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with M×M matrix valued fields. In our previous work, we examined the basic properties of the W algebra and claimed that the algebra can be realized as the symmetry of Grassmannian-like coset even with finite central charge based on a proposal of holography. In this paper, we extend the analysis in the following ways. First, we compute the operator product expansions among low spin generators removing the restriction of n=2. Second, we investigate the degenerate representations in several ways, and see the relations to the coset spectrum and the conical defect geometry of the higher spin gravity. For these analyses, we mainly set M=n=2. Finally, we extend the previous analysis by introducing N=2 supersymmetry.
CRIS-Team Scopus Importer:2024-06-24

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs - Creutzig, Thomas; Hikida, Yasuaki (Journal of High Energy Physics, 2019)
We analyze the asymptotic symmetry of higher spin gravity with M × M matrix valued fields, which is given by rectangular W-algebras with su(M) symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as s = 2, 3,…, n, we evaluate the central charge c of the algebra and the level k of the affine currents with finite c, k. For the simplest case with n = 2, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing c, k from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for n = 2. We interpret the map of parameters by decomposing the algebra in the coset description.
CRIS-Team Scopus Importer:2024-06-24
Keywords: AdS-CFT Correspondence; Conformal and W Symmetry; Higher Spin Gravity

Rectangular W-algebras of types so(M) and sp(2M) and dual coset CFTs - Creutzig, Thomas; Hikida, Yasuaki; Uetoko, Takahiro (Journal of High Energy Physics, 2019)
We examine rectangular W-algebras with so(M) or sp(2M) symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of so(M) or sp(2M) affine subalgebras by applying the Hamiltonian reductions of so or sp type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several strong supports. The analysis can be regarded as a check of extended higher spin holographies including full quantum corrections. We also extend the analysis by introducing N = 1 supersymmetry.
CRIS-Team Scopus Importer:2024-06-24
Keywords: AdS-CFT Correspondence; Conformal and W Symmetry; Higher Spin Gravity

Relaxed and logarithmic modules of sl3^ - Adamović, Dražen; Creutzig, Thomas; Genra, Naoki (Mathematische Annalen, 2024)
In Adamović (Commun Math Phys 366:1025–1067, 2019), the affine vertex algebra Lk(sl2) is realized as a subalgebra of the vertex algebra Virc⊗Π(0), where Virc is a simple Virasoro vertex algebra and Π(0) is a half-lattice vertex algebra. Moreover, all Lk(sl2)-modules (including, modules in the category KLk, relaxed highest weight modules and logarithmic modules) are realized as Virc⊗Π(0)-modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case g=sl3 and present realization of the VOA Lk(g) for k∉Z≥0 as a vertex subalgebra of Wk⊗S⊗Π(0), where Wk is a simple Bershadsky–Polyakov vertex algebra and S is the βγ vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand–Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain g-modules which are not Gelfand–Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of Wk from Adamović et al. (Lett Math Phys 111(2), Paper No. 38, arXiv:2007.00396 [math.QA], 2021) and obtain a realization of logarithmic modules for Wk of nilpotent rank two at most admissible levels. Beyond admissible levels, we get realization of logarithmic modules up to a existence of certain Wk(sl3,fpr)-modules. Using logarithmic modules for the βγ VOA, we are able to construct logarithmic Lk(g)-modules of rank three.
CRIS-Team Scopus Importer:2024-06-24

Representation theory of lk (Osp(1|2)) from vertex tensor categories and jacobi forms - Creutzig, Thomas; Frohlich, Jesse; Kanade, Shashank (Proceedings of the American Mathematical Society, 2018)
The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let Lk (osp(1|2)) be the simple affine vertex operator superalgebra of osp(1|2) at an admissible level k. We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of Lk (sl2) ⊗ Vir(p, (p + p′)/2) where k +3/2 = p/(2p′) and Vir(u, v) denotes the regular Virasoro vertex operator algebra of central charge c = 1− 6(u − v)2/(uv). Especially, for a positive integer k, we get a regular vertex operator superalgebra, and this case is studied further. The interplay of the theory of vertex algebra extensions and modular data of the vertex operator subalgebra allows us to classify all simple local (untwisted) and Ramond twisted Lk (osp(1|2))-modules and to obtain their super fusion rules. The latter are obtained in a second way from Verlinde’s formula for vertex operator superalgebras. Finally, using again the theory of vertex algebra extensions, we find all simple modules and their fusion rules of the parafermionic coset Ck = Com(VL, Lk (osp(1|2))), where VL is the lattice vertex operator algebra of the lattice L = √2kℤ.
CRIS-Team Scopus Importer:2024-06-24

Ribbon tensor structure on the full representation categories of the singlet vertex algebras - Creutzig, Thomas; McRae, Robert; Yang, Jinwei (Advances in Mathematics, 2023)
We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z>1, is equal to the category OM(p) of C1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since OM(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory OM(p)T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories OM(p) and OM(p)T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Braided tensor categories; Ribbon categories; Singlet algebras; Typical modules; Vertex operator algebras

Rigid Tensor Structure on Big Module Categories for Some W-(super)algebras in Type A - Creutzig, Thomas; McRae, Robert; Yang, Jinwei (Communications in Mathematical Physics, 2023)
We establish rigid tensor category structure on finitely-generated weight modules for the subregular W-algebras of sln at levels -n+nn+1 (the Bn+1 -algebras of Creutzig–Ridout–Wood) and at levels -n+n+1n (the finite cyclic orbifolds of the βγ -vertex algebra), as well as for their Feigin–Semikhatov dual principal W-superalgebras of sln|1 . These categories are neither finite nor semisimple, and in the W-algebra case they contain modules with infinite-dimensional conformal weight spaces and no lower bound on conformal weights. We give complete lists of indecomposable projective modules in these tensor categories and fusion rules for simple modules. All these vertex operator (super)algebras are simple current extensions of singlet algebras tensored with a rank-one Heisenberg algebra, so we more generally study simple current extensions in direct limit completions of vertex algebraic tensor categories. Then our results for W-(super)algebras follow from the known ribbon category structure on modules for the singlet algebras. Our results include and generalize those of Allen-Wood on the βγ -vertex algebra, as well as our own on the affine vertex superalgebra of gl1 | 1 . Our results also include the first examples of ribbon category structure on all finitely-generated weight modules for an affine vertex algebra at a non-integral admissible level, namely for affine sl2 at levels -43 and -12 .
CRIS-Team Scopus Importer:2024-06-24

SCHUR–WEYL DUALITY FOR HEISENBERG COSETS - Creutzig, Thomas; Kanade, S.; Linshaw, A. R.; Ridout, D. (Transformation Groups, 2019)
Let V be a simple vertex operator algebra containing a rank n Heisenberg vertex algebra H and let C = Com(H;V) be the coset of H in V. Assuming that the module categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of C are found and every simple C-module is shown to be contained in at least one V-module. A corollary of this is that if V is rational, C 2 -cofinite and CFT-type, and Com(C;V) is a rational lattice vertex operator algebra, then C is likewise rational. These results are illustrated with many examples and the C 1 -cofiniteness of certain interesting classes of modules is established.
CRIS-Team Scopus Importer:2024-06-24

S-duality for the Large N = 4 Superconformal Algebra - Creutzig, Thomas; Gaiotto, Davide; Linshaw, Andrew R. (Communications in Mathematical Physics, 2020)
We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the ST2S duality transformation in SU(2) gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small N= 4 superconformal algebra at central charge - 9 as well as for the generic large N= 4 superconformal algebra at central charge - 6. Moreover we obtain the affine vertex superalgebra of osp(1 | 2) and the N= 1 superconformal algebra times a free fermion as quantum Hamiltonian reductions of the large N= 4 superconformal algebras at c= - 6.
CRIS-Team Scopus Importer:2024-06-24

Self-dual vertex operator superalgebras and superconformal field theory - Creutzig, Thomas; Duncan, John F.R.; Riedler, Wolfgang (Journal of Physics A: Mathematical and Theoretical, 2018)
Recent work has related the equivariant elliptic genera of sigma models with K3 surface target to a vertex operator superalgebra that realizes moonshine for Conway's group. Motivated by this we consider conditions under which a self-dual vertex operator superalgebra may be identified with the bulk Hilbert space of a superconformal field theory. After presenting a classification result for self-dual vertex operator superalgebras with central charge up to 12 we describe several examples of close relationships with bulk superconformal field theories, including those arising from sigma models for tori and K3 surfaces.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conway group; K3 sigma model; Superconformal field theory; Vertex operator superalgebra

Simple current extensions beyond semi-simplicity - Creutzig, Thomas; Kanade, Shashank; Linshaw, Andrew R. (Communications in Contemporary Mathematics, 2020)
Let V be a simple vertex operator algebra (VOA) and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that V ⊕ J is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are C2-cofinite and non-rational are then given and induced modules listed.
CRIS-Team Scopus Importer:2024-06-24
Keywords: simple current extensions; Vertex operator algebras

Tensor categories arising from the Virasoro algebra - Creutzig, Thomas; Jiang, Cuipo; Orosz Hunziker, Florencia; Ridout, David; Yang, Jinwei (Advances in Mathematics, 2021)
We show that there is a braided tensor category structure on the category of C1-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge c=13−6(t+t−1), with t∉Q, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Non C; Universal Virasoro vertex algebras; Vertex tensor categories; Virasoro algebra

Tensor Categories for Vertex Operator Superalgebra Extensions - Creutzig, Thomas; Kanade, Shashank; McRae, Robert (Memoirs of the American Mathematical Society, 2024)
Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V. We employ tensor categories to study untwisted (also called local) A-modules in C, using results of Huang-Kirillov-Lepowsky that show that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V-modules to Amodules is a vertex tensor functor. Two applications are given: First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 12Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively. Second, we analyze parafermionic cosets C = Com(VL,V) where L is a positive definite even lattice and V is regular. If the vertex tensor category of either V-modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.
CRIS-Team Scopus Importer:2024-06-24
Keywords: braided tensor categories; commutative algebra objects; coset vertex operator algebras; Verlinde formula; Vertex operator superalgebras

Tensor categories of affine Lie algebras beyond admissible levels - Creutzig, Thomas; Yang, Jinwei (Mathematische Annalen, 2021)
We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are C1-cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) g at level k and the category KLk(g) of its finite length generalized modules, we discover several families of KLk(g) at non-admissible levels k, having braided tensor category structures. In particular, KLk(g) has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories KLk(g) , including the category KL- 1(sln). Using these results, we construct a rigid tensor category structure on a full subcategory of KL1(sl(n| m)) consisting of objects with semisimple Cartan subalgebra actions.
CRIS-Team Scopus Importer:2024-06-24

Tensor categories of weight modules of sl̂2 at admissible level - Creutzig, Thomas (Journal of the London Mathematical Society-Second Series, 2024)
The category of weight modules (Formula presented.) of the simple affine vertex algebra of (Formula presented.) at an admissible level (Formula presented.) is neither finite nor semisimple and modules are usually not lower-bounded and have infinite-dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with (Formula presented.), the simple principal (Formula presented.) -algebra of (Formula presented.) at level (Formula presented.) (the (Formula presented.) super conformal algebra) where the levels are related via (Formula presented.). Every weight module of (Formula presented.) is lower-bounded and has finite-dimensional conformal weight spaces. The main technical result is that every weight module of (Formula presented.) is (Formula presented.) -cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on (Formula presented.) for any admissible level (Formula presented.). As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that (Formula presented.) is a ribbon category if and only if (Formula presented.) is, in particular, it follows that for admissible levels (Formula presented.) and (Formula presented.) and (Formula presented.), the category (Formula presented.) is a ribbon category.
CRIS-Team Scopus Importer:2025-02-04

Tensor category KLk(sl2n) via minimal affine W-algebras at the non-admissible level k=−[Formula presented] - Adamović, Dražen; Creutzig, Thomas; Perše, Ozren; Vukorepa, Ivana (Journal of Pure and Applied Algebra, 2024)
We prove that the Kazhdan-Lusztig category of slˆm at level k, KLk(slm), is a semi-simple, rigid braided tensor category for all even m≥4, and k=−[Formula presented]. Moreover, all modules in KLk(slm) are simple-currents and they appear in the decomposition of conformal embeddings glm↪slm+1 at level k=−[Formula presented]. For this we inductively identify minimal affine W-algebra Wk−1(slm+2,θ) as simple current extension of Lk(slm)⊗H⊗M, where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c=−2. The proof uses previously obtained results for the tensor categories of singlet algebra. We also classify all irreducible ordinary modules for Wk−1(slm+2,θ). The semi-simple part of the category of Wk−1(slm+2,θ)–modules comes from KLk−1(slm+2), using quantum Hamiltonian reduction, but this W-algebra also contains indecomposable ordinary modules.
CRIS-Team Scopus Importer:2024-06-24

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We show that the Kazhdan-Lusztig category KLk of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1) has vertex algebraic braided tensor supercategory structure and that its full subcategory Okfin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)

The McKay-Thompson series of Mathieu Moonshine modulo two - Creutzig, Thomas; Höhn, Gerald; Miezaki, Tsuyoshi (Ramanujan Journal, 2014)
In this note, we describe the parity of the coefficients of the McKay-Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan, and Harvey stated in connection with umbral moonshine for the case of Mathieu moonshine. © 2014 Springer Science+Business Media New York.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Mock modular forms; Moonshine

The super W1+∞ algebra with integral central charge - Creutzig, Thomas; Linshaw, Andrew R. (Transactions of the American Mathematical Society, 2015)
The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension SD. For each c ∈ C, the vacuum module Mc(SD) of central charge c admits a vertex superalgebra structure,and Mc(SD) ∼= M−c(SD). The irreducible quotient Vc(SD) of the vacuummodule is known as the super W1+∞ algebra. We show that for each integern > 0, Vn(SD) has a minimal strong generating set consisting of 4n fields, andwe identify it with aW-algebra associated to the purely odd simple root systemof gl(n|n). Finally, we realize Vn(SD) as the limit of a family of commutantvertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.
CRIS-Team Scopus Importer:2024-06-24

The Vertex Algebras R(p) and V(p) - Adamović, Dražen; Creutzig, Thomas; Genra, Naoki; Yang, Jinwei (Communications in Mathematical Physics, 2021)
The vertex algebras V(p) and R(p) introduced in Adamović (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra V(p) (respectively, R(p)) is a large extension of the simple affine vertex algebra Lk(sl2) (respectively, Lk(sl2) times a Heisenberg algebra), at level k= - 2 + 1 / p for positive integer p. Particularly, the algebra V(2) is the simple small N= 4 superconformal vertex algebra with c= - 9 , and R(2) is L- 3 / 2(sl3). In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on V(p) and we decompose V(p) as an Lk(sl2) -module and R(p) as an Lk(gl2) -module. The decomposition of V(p) shows that V(p) is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of V(p) is the logarithmic doublet algebra A(p) introduced in Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of R(p) yields the B(p)-algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize V(p) and R(p) from A(p) and B(p) via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category KLk of ordinary Lk(sl2) -modules at level k= - 2 + 1 / p is a rigid vertex tensor category equivalent to a twist of the category Rep (SU(2)). This finally completes rigid braided tensor category structures for Lk(sl2) at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both R(p) and B(p) are certain non-principal W-algebras of type A at boundary admissible levels. The same uniqueness result also shows that R(p) and B(p) are the chiral algebras of Argyres-Douglas theories of type (A1, D2p) and (A1, A2p-3).
CRIS-Team Scopus Importer:2024-06-24

Trialities of orthosymplectic W-algebras - Creutzig, Thomas; Linshaw, Andrew R. (Advances in Mathematics, 2022)
Trialities of W-algebras are isomorphisms between the affine cosets of three different W-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of W-(super)algebras of types B, C, and D. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin W∞-algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigin-Frenkel duality in types B, C, and D, and the coset realization of principal W-algebras of type D due to Arakawa and us. It also provides a new coset realization of principal W-algebras of types B and C. As an application, we prove the rationality of the affine vertex superalgebra Lk(osp1|2n), the minimal W-algebra Wk−1/2(sp2n+2,fmin), and the coset Com(Lk(sp2m),Lk(sp2n)), for all integers k,n,m≥1 with m<n. We also prove the rationality of some families of principal W-superalgebras of osp1|2n and osp2|2n, and subregular W-algebras of so2n+1.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Coset construction; Nonlinear Lie conformal algebra; Vertex algebra; W-algebra

Trialities of W-algebras - Creutzig, Thomas; Linshaw, Andrew R. (Cambridge Journal of Mathematics, 2022)
We prove the conjecture of Gaiotto and Rapčák that the Y-algebras YL,M,N [ψ] with one of the parameters L, M, N zero, are simple one-parameter quotients of the universal two-parameter W1+∞-algebra, and satisfy a symmetry known as triality. These Y-algebras are defined as the cosets of certain non-principal W-algebras and W-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on W-algebras of type A. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal W-algebras due to Arakawa and us, (3) Feigin and Semikha-tov’s conjectured triality between subregular W-algebras, principal W-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular W-algebras due to Arakawa and van Ekeren, and (5) the identification of Heisenberg cosets of subregular W-algebras with principal rational W-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Procházka and Rapčák on the explicit truncation curves realizing the simple Y-algebras as W1+∞-quotients, and on their minimal strong generating types.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Coset construction; Poisson vertex algebra; Vertex algebra; W-algebra

Unitary and non-unitary N = 2 minimal models - Creutzig, Thomas; Liu, Tianshu; Ridout, David; Wood, Simon (Journal of High Energy Physics, 2019)
The unitary N = 2 superconformal minimal models have a long history in string theory and mathematical physics, while their non-unitary (and logarithmic) cousins have recently attracted interest from mathematicians. Here, we give an efficient and uniform analysis of all these models as an application of a type of Schur-Weyl duality, as it pertains to the well-known Kazama-Suzuki coset construction. The results include straight-forward classifications of the irreducible modules, branching rules, (super)characters and (Grothendieck) fusion rules.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal and W Symmetry; Conformal Field Theory; Sigma Models

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry - Afshar, Hamid; Creutzig, Thomas; Grumiller, Daniel; Hikida, Yasuaki; Rønne, Peter B. (Journal of High Energy Physics, 2014)
We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov Wn 2-algebra. This algebra is conjecturally a quantum Hamiltonian reduction corresponding to a non-principal nilpotent element. We conjecture that this algebra admits a unitary real form for even n. Our main result is that this conjecture is consistent with the known part of the operator product algebra, and especially it is true for n = 2 and n = 4. Moreover, we find certain ranges of allowed levels where a positive definite inner product is possible. We also find a unitary conformal field theory for every even n at the special level k + n = (n + 1)/(n - 1). At these points, the Wn 2-algebra is nothing but a compactified free boson. This family of W-algebras admits an 't Hooft limit. Further, in the case of n = 4, we reproduce the algebra from the higher spin gravity point of view. In general, gravity computations allow us to reproduce some leading coefficients of the operator product. © 2014 The Author(s).
CRIS-Team Scopus Importer:2024-06-24
Keywords: AdS-CFT Correspondence; Conformal and W Symmetry; Higher Spin Symmetry

Uprolling unrolled quantum groups - Creutzig, Thomas; Rupert, Matthew (Communications in Contemporary Mathematics, 2022)
We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group ŪqH() of a simple Lie algebra at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the = 2 case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra WQ(r) of Feigin and Tipunin and the BQ(r) algebras.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Algebra objects; Quantum groups; Vertex operator algebras

Urod algebras and Translation of W-algebras - Arakawa, Tomoyuki; Creutzig, Thomas; Feigin, Boris (Forum of Mathematics. Sigma, 2022)
In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with, we have the vertex algebra isomorphism, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of on. The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations for the moduli space of framed torsion free sheaves on of an arbitrary rank.
CRIS-Team Scopus Importer:2024-06-24
Keywords: 2020 Mathematics subject classification 17B65 17B69

Vertex Algebras for S-duality - Creutzig, Thomas; Gaiotto, Davide (Communications in Mathematical Physics, 2020)
We define new deformable families of vertex operator algebras A[g, Ψ , σ] associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The A[g, Ψ , σ] vertex operator algebras are equipped with two g affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a g affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for A[g, Ψ , σ] is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the A[g, Ψ , σ] vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.
CRIS-Team Scopus Importer:2024-06-24

Vertical D4–D2–D0 Bound States on K3 Fibrations and Modularity - Bouchard, Vincent; Creutzig, Thomas; Diaconescu, Duiliu Emanuel; Doran, Charles; Quigley, Callum; Sheshmani, Artan (Communications in Mathematical Physics, 2017)
An explicit formula is derived for the generating function of vertical D4–D2–D0 bound states on smooth K3 fibered Calabi–Yau threefolds, generalizing previous results of Gholampour and Sheshmani. It is also shown that this formula satisfies strong modularity properties, as predicted by string theory. This leads to a new construction of vector valued modular forms which exhibit some of the features of a generalized Hecke transform.
CRIS-Team Scopus Importer:2024-06-24

W-algebras as coset vertex algebras - Arakawa, Tomoyuki; Creutzig, Thomas; Linshaw, Andrew R. (Inventiones Mathematicae, 2019)
We prove the long-standing conjecture on the coset construction of the minimal series principal W-algebras of ADE types in full generality. We do this by first establishing Feigin’s conjecture on the coset realization of the universal principal W-algebras, which are not necessarily simple. As consequences, the unitarity of the “discrete series” of principal W-algebras is established, a second coset realization of rational and unitary W-algebras of type A and D are given and the rationality of Kazama–Suzuki coset vertex superalgebras is derived.
CRIS-Team Scopus Importer:2024-06-24

W-algebras for Argyres–Douglas theories - Creutzig, Thomas (European Journal of Mathematics, 2017)
The Schur index of the (A1, Xn) -Argyres–Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the Aodd and Deven-type Argyres–Douglas theories. The vertex operator algebra corresponding to A2 p - 3-Argyres–Douglas theory is the logarithmic [InlineEquation not available: see fulltext.]-algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014), while the one corresponding to D2 p, denoted by [InlineEquation not available: see fulltext.], is realized as a non-regular quantum Hamiltonian reduction of Lk(slp + 1) at level [InlineEquation not available: see fulltext.]. For all n one observes that the quantum Hamiltonian reduction of the vertex operator algebra of Dn-Argyres–Douglas theory is the vertex operator algebra of An - 3-Argyres–Douglas theory. As a corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Conformal embeddings; Gauge theories; Logarithmic VOA; W-algebras

Weight representations of affine Kac-Moody algebras and small quantum groups - Arakawa, Tomoyuki; Creutzig, Thomas; Kawasetsu, Kazuya (Advances in Mathematics, 2025)
We study the weight modules over affine Kac-Moody algebras from the view point of vertex algebras, and determine the abelian category of weight modules for the simple affine vertex algebra Lk(sl2) at any non-integral admissible level k. In particular, we show that the principal block of the category of weight modules over admissible Lk(sl2) is equivalent to that of the corresponding (unrolled) small quantum group.
CRIS-Team Scopus Importer:2025-06-06
Keywords: Quantum groups; Vertex algebras

Whittaker vectors for W-algebras from topological recursion - Borot, Gaëtan; Bouchard, Vincent; Chidambaram, Nitin K.; Creutzig, Thomas (Selecta Mathematica-New Series, 2024)
We identify Whittaker vectors for Wk(g)-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P2 for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N=2 four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.
CRIS-Team Scopus Importer:2024-06-24
Keywords: Airy structures; Gaiotto vectors; Instanton partition function; Spectral curves; Supersymmetric gauge theories; Topological recursion; W-algebras