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We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres–Douglas theories on S1 × M3 with a non-trivial holonomy of a discrete global symmetry along the S1. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern–Simons and Rozansky–Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the and Chern–Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d theory. We test our proposal by applying localization techniques to the conjectural UV Lagrangian descriptions of the (A1, A2), (A1, A3) and (A1, D3) theories. [ABSTRACT FROM AUTHOR]

To understand the distribution of the Yang–Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2D Yang–Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature β, can be computed in terms of the thermodynamics Bethe Ansatz based on its exact S-matrix. We extract the Yang–Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit while in the free theories it remains close to 1 even for small values of β, jumping to 0 only at . [ABSTRACT FROM AUTHOR]

We review the holomorphic block factorisation of partition functions of supersymmetric theories on compact manifolds in various dimensions. We then show how to interpret 3d and 5d partition functions as correlation functions with underlying symmetry given by a deformation of the Virasoro algebra. [ABSTRACT FROM AUTHOR]

We present an elementary review of some aspects of Chern–Simons theory with complex gauge group . We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d–3d correspondence. The 3d–3d correspondence relates partition functions (and other aspects) of complex Chern–Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory . Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d–3d correspondence. [ABSTRACT FROM AUTHOR]

The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi–Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local toric Calabi–Yau threefold, the present work shows how the Borel–Padé–Écalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure. [ABSTRACT FROM AUTHOR]

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an matrix of mosaic tiles which are T0 through T10 depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. denotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of for . Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics. [ABSTRACT FROM AUTHOR]