arXiv subjects

Thomas Creutzig

Publications and source records attributed to Thomas Creutzig.

At least 109 records · Page 6Linked to original sources

arXiv2016

On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

We study a family of non-C2-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 trans- formation properties of regularised partial 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.

math.QA↗

arXiv2016

Logarithmic conformal field theory, log-modular tensor categories and modular forms

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters on the one hand and its interpretation of modules as objects in a modular tensor category on the other one. 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 that their modular pillar are trace functions with insertions corresponding to intertwiners of the projective cover of the vacuum, and that the categorical pillar are finite tensor categories $\mathcal C$ which are ribbon and whose double is isomorphic to the Deligne product $\mathcal C\otimes \mathcal C^{opp}$. Overarching these pillars is then a logarithmic variant of Verlinde's formula. Numerical data realizing this are the modular $S$-matrix and modified traces of open Hopf links. The representation categories of $C_2$-cofinite and logarithmic conformal field theories that are fairly well understood are those of the $\mathcal W_p$-triplet algebras and the symplectic fermions. We illustrate the ideas in these examples and especially make the relation between logarithmic Hopf links and modular transformations explicit.

math.QA↗

arXiv2015

Simple current extensions beyond semi-simplicity

Let V be a simple 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 $V\oplus J$ 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 $C_2$-cofinite and non-rational are then given and induced modules listed.

math.QA↗

arXiv2015

Higgs phenomenon for higher spin fields on AdS_3

In a previous work, a marginal deformation of 2d coset type model with 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.

hep-th↗

arXiv2015

Correspondences between WZNW models and CFTs with W-algebra symmetry

We study theories with W-algebra symmetries and their relation to WZNW models on (super-)groups. 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 \underline{N+2} = \underline{2} + N \underline{1} and its supergroup extension. Finally, sl(2N) and the correspondence corresponding to \underline{2N} = N \underline{2} is also analyzed.

hep-th↗

arXiv2015

Negative index Jacobi forms and quantum modular forms

In this paper, we consider the Fourier coefficients of a special class of meromorphic Jaocbi forms of negative index. 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 PDEs 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.

math.NT↗

arXiv2015

Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)

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(1|1), 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.

hep-th↗

arXiv2015

Higher spin AdS_3 holography and superstring theory

It has been believed for a long time that the tensionless limit of superstring theory can be described by a higher spin gauge theory. Recently, a concrete realization of this idea was proposed via 3d Aharony-Bergman-Jafferis (ABJ) theory with the help of holographic duality. In this note we review our work on finding a similar relation involving 2d coset type models. We start by proposing and examining holographic dualities between 3d higher spin gauge theories with matrix valued fields and the large N limit of 2d coset type models. After that we discuss possible relations to superstring theory with emphasis on the role of the matrix form of the higher spin fields and the extended supersymmetry.

hep-th↗

arXiv2015

Orbifolds of symplectic fermion algebras

We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\mathcal{A}(mn)^{GL(n)\times GL(m)}$ appearing inside $\mathcal{A}(mn)$, and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for $\mathcal{A}(n)$; we show that for any reductive group $G$ of automorphisms, $\mathcal{A}(n)^G$ is strongly finitely generated.

math.RT↗

arXiv2014

False Theta Functions and the Verlinde formula

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 and prove a Verlinde-type formula for the fusion rules of irreducible W(2,2p-1)-modules.

math.QA↗

arXiv2014

Higher spin AdS_3 holography with extended supersymmetry

We propose a holographic duality between a higher spin AdS_3 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 Z_2 automorphism truncation of a similar coset model, but at a non-critical level.

hep-th↗

arXiv2014

Mathieu Moonshine and the Geometry of K3 Surfaces

We compare the moonshine observation of Eguchi, Ooguri and Tachikawa relating the Mathieu group M_24 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 M_24 and also for a certain vertex operator superalgebra V^G where G is the holonomy group.

math.QA↗

arXiv2014

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov W^(2)_n-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 W^(2)_n-algebra is nothing but a compactified free boson. This family of W-algebras admits an 't Hooft limit that is similar to the original minimal model '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.

hep-th↗

arXiv2014

Coset Constructions of Logarithmic (1,p)-Models

One of the best understood families of logarithmic conformal field theories is that consisting of the (1,p) models (p = 2, 3, ...) of central charge c_{1,p} = 1 - 6 (p-1)^2 / p. This family includes the theories corresponding to the singlet algebras M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized through a coset construction. The W^(2)_n algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras B_p are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p <= 5, the algebra B_p is a homomorphic image of W^(2)_{p-1} at level -(p-1)^2 / p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p>5 as well. The triplet algebra W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.

math.QA↗

arXiv2013

The McKay-Thompson series of Mathieu Moonshine modulo two

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.

math.NT↗

arXiv2013

Extended higher spin holography and Grassmannian models

We propose holographic dualities between higher spin gravity theories extended with Chan-Paton factor on AdS_3 and a large N limit of two dimensional Grassmannian models with and without supersymmetry. These proposals are natural extensions of the duality without Chan-Paton factor, and the extensions are motivated by a higher dimensional version of the duality, which implies a possible relation to superstring theory via ABJ theory. As evidence for the proposals, we show that the free limit of the Grassmannian models have the higher spin symmetry expected from the dual gravity theory. Furthermore, we construct currents in the 't Hooft limit of the supersymmetric Grassmannian model and compare them with the currents from the bulk theory. One-loop partition function of the supergravity theory is reproduced by the 't Hooft limit of the Grassmannian model after decoupling a gauge sector.

hep-th↗

arXiv2013

Modular Data and Verlinde Formulae for Fractional Level WZW Models II

This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl(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.

hep-th↗

arXiv2013

Logarithmic Conformal Field Theory: Beyond an Introduction

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.

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