arXiv subjects

Antun Milas

Publications and source records attributed to Antun Milas.

At least 37 records · Page 2Linked to original sources

arXiv2019

Permutation Orbifolds of the Heisenberg Vertex Algebra $\mathcal{H}(3)$

We study the $S_3$-orbifold of a rank three Heisenberg vertex algebras in terms of generators and relations. By using invariant theory we prove that the orbifold algebra has a minimal strongly generating set of vectors whose conformal weights are $1,2,3,4,5,6^2$ (two generators of degree $6$). The structure of the cyclic $\mathbb{Z}_3$-oribifold is determined by similar methods. We also study modular properties of characters of modules for these vertex algebras.

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arXiv2018

On some vertex algebras related to $V_{-1}(\frak{sl} (n) )$ and their characters

We consider several vertex operator (super)algebras closely related to $V_{-1}(\frak{sl} (n) )$, $n \ge 3$ : (a) the parafermionic subalgebra $K(\frak{sl}(n),-1)$ for which we completely describe its inner structure, (b) the vacuum algebra $Ω(V_{-1}(\frak{sl} (n) ) )$, and (c) an infinite extension $\mathcal U$ of $V_{-1}(\frak{sl} (n) )$ constructed by combining certain irreducible ordinary modules with integral weights. It turns out that $\mathcal U$ is isomorphic to the coset vertex algebra $\frak{psl}(n|n) _1 / \frak{sl}(n)_1$, $n \ge 3$. We show that $V_{-1}(\frak{sl}(n))$ admits precisely $n$ ordinary irreducible modules, up to isomorphism. This leads to the conjecture that ${\mathcal U}$ is {\em quasi-lisse}. We present evidence in support of this conjecture: we prove that the (super)character of $\mathcal U$ is quasi-modular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for $\frak{g}=\frak{sl}(3)$ and outlined for general $n$. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras $\frak{psl}(n|n) _1$, $n \geq 2$. We finish with a theorem pertaining to characters of $\frak{psl}(n|n)_1$ and $\mathcal U$-modules.

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arXiv2018

On characters of $L_{\frak{sl}_n}(-Λ_0)$-modules

We use recent results of Rolen, Zwegers, and the first author to study characters of irreducible (highest weight) modules for the vertex operator algebra $L_{\frak{sl}_\ell}(-Λ_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{\frak{sl}_\ell}(-Λ_0)$-modules. As a consequence we prove that their quantum dimensions are one, as predicted by representation theory. We also establish a full asymptotic expansion of irreducible characters for $\frak{sl}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.

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arXiv2018

Higher depth quantum modular forms, multiple Eichler integrals, and $\frak{sl}_3$ false theta functions

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of "double error" functions as coefficients. In particular, we prove that the false theta of $\frak{sl}_3$, appearing in the character of the vertex algebra $W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.

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arXiv2016

Some applications and constructions of intertwining operators in LCFT

We discuss some applications of fusion rules and intertwining operators in the representation theory of cyclic orbifolds of the triplet vertex operator algebra. We prove that the classification of irreducible modules for the orbifold vertex algebra W(p)^{A_m} follows from a conjectural fusion rules formula for the singlet vertex algebra modules. In the p=2 case, we computed fusion rules for the irreducible singlet vertex algebra modules by using intertwining operators. This result implies the classification of irreducible modules for W(2)^{A_m}, conjectured previously in [4]. The main technical tool is a new deformed realization of the triplet and singlet vertex algebras, which is used to construct certain intertwining operators that can not be detected by using standard free field realizations.

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arXiv2016

Logarithmic Link Invariants of $\overline{U}_q^H(\mathfrak{sl}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projective modules of $\overline{U}_q^H(\mathfrak{sl}_2)$. These relate to the colored Alexander tangle invariants studied in [ADO, M1]. It follows that the regularized asymptotic dimensions of characters of $\mathcal M(r)$ coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of $\mathcal M(r)$-mod in connection to braided tensor categories.

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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.

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arXiv2016

Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters

We prove several asymptotic results for partial and false theta functions arising from Jacobi forms, as the modular variable $τ$ tends to $0$ along the imaginary axis, and the elliptic variable $z$ is unrestricted in the complex plane. We observe that these functions exhibit Stokes' phenomenon - the asymptotic behavior of these functions sharply differs depending on where the elliptic variable $z$ is located within the complex plane. We apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the $(1,p)$-singlet vertex operator algebra coming from conformal field theory. This, in particular, recovers and extends several known results pertaining to regularized quantum dimensions, which served as a main source of motivation.

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arXiv2015

Vertex Algebras $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$ and Constant Term Identities

We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^Γ$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

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arXiv2014

On the Fourier expansion of Bloch-Okounkov

In this paper, we study algebraic and analytic properties of Fourier coefficients, expressed as $q$-series, of the so-called Bloch-Okounkov $n$-point function. We prove several results about these series and explain how they relate to Rogers' false theta function. Then we obtain their full asymptotics, as $τ\rightarrow 0$, and use this result to derive asymptotic properties of the coefficients in the $q$-expansion. At the end, we also introduce and discuss higher rank generalization of Bloch-Okounkov's functions.

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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.

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arXiv2014

Vertex-algebraic structure of principal subspaces of standard A_2^{(2)}-modules, I

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac-Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra A_2^{(2)}, which we expect can be extended to higher rank algebras. As a "test case," we consider the principal subspace of the basic A_2^{(2)}-module and explore its structure in depth.

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arXiv2013

ADE subalgebras of the triplet vertex algebra W(p): A-series

Motivated by \cite{am1}, for every finite subgroup $Γ\subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for $m=2$. We also present a rigorous proof of the fact that the full automorphism group of $\triplet$ is $PSL(2,\mathbb{C})$.

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arXiv2013

ADE subalgebras of the triplet vertex algebra W(p): D_m-series

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules including the twisted Zhu's algebra $A_Ψ (\bar{M(1)})$, which is of independent interest. The category of admissible $Ψ$-twisted $\bar{M(1)}$-modules is expected to be semisimple. We also prove $C_2$-cofiniteness of $\triplet^{D_m}$ for all $m$, and give a conjectural list of irreducible $\triplet^{D_m}$-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.

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arXiv2012

C_2-cofinite W-algebras and their logarithmic representations

We discuss our recent results on the representation theory of $\mathcal{W}$--algebras relevant to Logarithmic Conformal Field Theory. First we explain some general constructions of $\mathcal{W}$-algebras coming from screening operators. Then we review the results on $C_2$--cofiniteness, the structure of Zhu's algebras, and the existence of logarithmic modules for triplet vertex algebras. We propose some conjectures and open problems which put the theory of triplet vertex algebras into a broader context. New realizations of logarithmic modules for $\mathcal{W}$-algebras defined via screenings are also presented.

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arXiv2012

The 3-state Potts model and Rogers-Ramanujan series

We explain the appearance of Rogers-Ramanujan series inside the tensor product of two basic $A_2^{(2)}$-modules, previously discovered by the first author in [F]. The key new ingredients are $(5,6)$ Virasoro minimal models and twisted modules for the Zamolodchikov $\WW_3$-algebra.

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arXiv2012

An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}

By extending the methods used in our earlier work, in this paper, we present an explicit realization of logarithmic $\mathcal{W}_{p,p'$}-modules that have L(0) nilpotent rank three. This was achieved by combining the techniques developed in \cite{AdM-2009} with the theory of local systems of vertex operators \cite{LL}. In addition, we also construct a new type of extension of $\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm several claims in the physics literature regarding the structure of projective covers of certain irreducible representations in the principal block. This approach can be applied to other models defined via a pair screenings.

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arXiv2011

The structure of Zhu's algebras for certain W-algebras

We introduce a new approach that allows us to determine the structure of Zhu's algebra for certain vertex operator (super)algebras which admit horizontal $\mathbb{Z} $-grading. By using this method and an earlier description of Zhu's algebra for the singlet W-algebra, we completely describe the structure of Zhu's algebra for the triplet vertex algebra W(p). As a consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson algebra P(W(p)) have the same dimension. We also completely describe Zhu's algebras for the N=1 triplet vertex operator superalgebra SW(m). Moreover, we obtain similar results for the c=0 triplet vertex algebra W_{2,3} important in logarithmic conformal field theory. Because our approach is "internal" we had to employ several constant term identities for purposes of getting right upper bounds on dimension of Zhu's algebras.

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