Higher-Level Appell Functions, Modular Transformations, and Characters

  title={Higher-Level Appell Functions, Modular Transformations, and Characters},
  author={A. M. Semikhatov and A. Taorimina and I. Yu. Tipunin},
  journal={Communications in Mathematical Physics},
We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the “period.” Generalizing the well-known interpretation of theta functions as sections of line bundles, the function enters the construction of a section of a rank-(ℓ+1) bundle . We evaluate modular transformations of… 
Superconformal Algebras and Mock Theta Functions
It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to
The mock modular data of a family of superalgebras
The modular properties of characters of representations of a family of W-superalgebras extending ĝl (1|1) are considered. Modules fall into two classes, the generic type and the non-generic one.
Vafa–Witten invariants from modular anomaly
Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses
Comments on non-holomorphic modular forms and non-compact superconformal field theories
A bstractWe extend our previous work [1] on the non-compact $ \mathcal{N} = {2} $SCF T2 defined as the supersymmetric SL(2,$ \mathbb{R} $)/U(1)-gauged WZW model. Starting from path-integral
Higher poles and crossing phenomena from twisted genera
A bstractWe demonstrate that Appell-Lerch sums with higher order poles as well as their modular covariant completions arise as partition functions in the cigar conformal field theory with worldsheet
Sheaves on P2 and generalised Appell functions
A closed expression is given for the generating function of (virtual) Poincar\'e polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and
Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center
The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of
On modular properties of the AdS 3 CFT
We study modular properties of the ${\mathrm{AdS}}_{3}$ Wess-Zumino-Novikov-Witten model. Although the Euclidean partition function is modular invariant, the characters on the Euclidean torus diverge
Identities for Generalized Appell Functions and the Blow-up Formula
In this paper, we prove identities for a class of generalized Appell functions which are based on the $${{\rm A}_2}$$A2 root lattice. The identities are reminiscent of periodicity relations for the
Vafa–Witten Theory and Iterated Integrals of Modular Forms
  • J. Manschot
  • Mathematics
    Communications in Mathematical Physics
  • 2019
Vafa–Witten (VW) theory is a topologically twisted version of $${\mathcal{N}=4}$$N=4 supersymmetric Yang–Mills theory. S-duality suggests that the partition function of VW theory with gauge group


Modular invariant representations of infinite-dimensional Lie algebras and superalgebras.
  • V. Kac, M. Wakimoto
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1988
It is shown that the modular invariant representations of the Virasoro algebra Vir are precisely the "minimal series" of Belavin et al.
Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine
Abstract: The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the
Differential equations, duality and modular invariance
We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following
Representation Theory of the Affine Lie Superalgebra at Fractional Level
Abstract:N= 2 noncritical strings are closely related to the Wess-Zumino-Novikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the
Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function
Abstract:We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this
Verma modules, extremal vectors, and singular vectors on the noncritical N=2 string world sheet
We formulate the general construction for singular vectors in Verma modules of the affine sl(2|1) superalgebra. We then construct sl(2|1) representations out of the fields of the non-critical N=2
The Kernel of the Modular Representation and the Galois Action in RCFT
Abstract: It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An
Bits and Pieces in Logarithmic Conformal Field Theory
These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts