arXiv subjects

Thomas Creutzig

Publications and source records attributed to Thomas Creutzig.

At least 73 records · Page 4Linked to original sources

arXiv2021

Direct limit completions of vertex tensor categories

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 $\mathfrak{osp}(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.

math.QA↗

arXiv2021

Tensor Categories arising from the Virasoro Algebra

We show that there is a braided tensor category structure on the category of $C_1$-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 $t \notin \mathbb{Q}$, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.

math.RT↗

arXiv2020

Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra

Various aspects of orbifolds and cosets of the small $\mathcal{N}=4$ 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 $\mathcal{N}=4$ superconformal algebra with $\text{Com}(V^{\ell}(\mathfrak{sl}_2), V^{\ell+1}(\mathfrak{sl}_2) \otimes \mathcal{W}_{-5/2}(\mathfrak{sl}_4, f_{\text{rect}}))$ and in addition at special levels with Grassmanian cosets and principal $\mathcal{W}$-algebras of type $A$ at degenerate admissible levels. These coincidences lead us to a novel level-rank duality involving Grassmannian supercosets.

math.RT↗

arXiv2020

Higher rank FZZ-dualities

We propose new 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 $\mathfrak{sl}(2)/\mathfrak{u}(1)$ coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $\mathfrak{sl}(2)$ 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 $\mathfrak{sl}(N+1)/(\mathfrak{sl}(N) \times \mathfrak{u}(1))$ and propose that the model is dual to a theory with an $\mathfrak{sl}(N+1|N)$ structure. We derive the duality explicitly for $N=2,3$ by applying recent works on the reduction method extended for $\mathfrak{sl}(N)$ 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 $Y_{0, N, N+1}[ψ]$ and $Y_{N, 0, N+1}[ψ^{-1}]$.

hep-th↗

arXiv2020

Urod algebras and Translation of W-algebras

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 $\mathfrak{g}$, we have the vertex algebra isomorphism $H_{DS,f}^0(V\otimes L)\cong H_{DS,f}^0(V)\otimes L$, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of $\widehat{\mathfrak{g}}$ on $V\otimes L$. The proof is based on some new constructionof 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 $\mathbb{CP}^2$ of an arbitrary rank.

math.RT↗

arXiv2020

Correspondences among CFTs with different W-algebra symmetry

W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra $\mathfrak{g}$ and an $\mathfrak{sl}(2)$-embedding into $\mathfrak{g}$. We derive correspondences among correlation functions of theories having different W-algebras as symmetry algebras. These W-algebras are associated to the same $\mathfrak{g}$ but distinct $\mathfrak{sl}(2)$-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 $\mathfrak{g}=\mathfrak{sl}(3)$, 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 $\mathfrak{sl}(3)$, the Bershadsky-Polyakov algebra and the principal $W_3$-algebra. There are three non-regular W-algebras associated to $\mathfrak{g}=\mathfrak{sl}(4)$. We show that the methods developed for $\mathfrak{g}=\mathfrak{sl}(3)$ can be applied straightforwardly. We briefly comment on extensions of our techniques to general $\mathfrak{g}$.

hep-th↗

arXiv2020

Uprolling Unrolled Quantum Groups

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ 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 $\mfg=\mathfrak{sl}_2$ 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 $W_Q(r)$ of Feigin and Tipunin \cite{FT} and the $B_Q(r)$ algebras of \cite{C1}.

math.RT↗

arXiv2020

The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$

The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ (respectively, $L_k(\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. 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 $L_k(\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\mathfrak{gl}_2)$-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 [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of [39]. 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 $KL_{k}$ of ordinary $L_k(\mathfrak{sl}_2)$-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 $L_k(\mathfrak{sl}_2)$ at all 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 $(A_1, D_{2p})$ and $(A_1, A_{2p-3})$.

math.RT↗

arXiv2019

Rectangular W-(super)algebras and their representations

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 $M \times M$ 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 $\mathcal{N}=2$ supersymmetry.

hep-th↗

arXiv2019

$N=4$ superconformal algebras and diagonal cosets

Coset constructions of $\mathcal{W}$-algebras have many applications, and were recently given for principal $\mathcal{W}$-algebras of $A$, $D$, and $E$ types by Arakawa together with the first and third authors. In this paper, we give coset constructions of the large and small $N=4$ superconformal algebras, which are the minimal $\mathcal{W}$-algebras of $\mathfrak{d}(2,1;a)$ and $\mathfrak{psl}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \text{Com}(V^{k_1+k_2}(\mathfrak{sl}_2), V^{k_1}(\mathfrak{sl}_2) \otimes V^{k_2}(\mathfrak{sl}_2))$, namely, as two-parameter vertex algebras, $C^{k_1, k_2}$ 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 $\mathcal{W}$-algebras. As a corollary, we give new examples of strongly rational principal $\mathcal{W}$-algebras of type $C$ at degenerate admissible levels.

math.RT↗

arXiv2019

Rectangular W-algebras of types $so(M)$ and $sp(2M)$ and dual coset CFTs

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 $\mathcal{N}=1$ supersymmetry.

hep-th↗

arXiv2019

Braided Tensor Categories related to $\mathcal{B}_p$ Vertex Algebras

The $\mathcal{B}_p$-algebras are a family of vertex operator algebras parameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples of logarithmic CFTs and appear as chiral algebras of type $(A_1, A_{2p-3})$ Argyres-Douglas theories. The first member of this series, the $\mathcal{B}_2$-algebra, are the well-known symplectic bosons also often called the $βγ$ vertex operator algebra. We study categories related to the $\mathcal{B}_p$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $\mathfrak{sl}_2$. 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 successfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

math.QA↗

arXiv2019

Hecke Operators on Vector-Valued Modular Forms

We study Hecke operators on vector-valued modular forms for the Weil representation $ρ_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $ρ_L$ into vector-valued modular forms of type $ρ_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ 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 $\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer $n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of type $ρ_L$ into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, and compare our operators with the alternative construction of Bruinier-Stein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].

math.NT↗

arXiv2019

W-algebras as coset vertex algebras

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.

math.QA↗

arXiv2019

Unitary and non-unitary $N=2$ minimal models

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 straightforward classifications of the irreducible modules, branching rules, (super)characters and (Grothendieck) fusion rules.

math-ph↗

arXiv2019

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs

We analyze the asymptotic symmetry of higher spin gravity with $M \times M$ 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 $s=2,3,\ldots , n$, 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.

hep-th↗

arXiv2019

Higher rank partial and false theta functions and representation theory

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 Kostant's partial and certain higher rank false theta functions. 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. Results in this paper substantially generalize our previous work [19] pertaining to (1,p)-singlet W-algebras (the sl_2 case).

math.QA↗

arXiv2018

Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1 \vert 2)$ minimal models

We study the minimal models associated to $\mathfrak{osp}(1 \vert 2)$, otherwise known as the fractional-level Wess-Zumino-Witten models of $\mathfrak{osp}(1 \vert 2)$. Since these minimal models are extensions of the tensor product of certain Virasoro and $\mathfrak{sl}_2$ 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 $\mathfrak{osp}(1 \vert 2)$. 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.

hep-th↗