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In this paper, we discuss the simple current orbifold of a rational Narain CFT (Narain RCFT). This is a method of constructing other rational CFTs from a given rational CFT, by 'orbifolding' the global symmetry formed by a particular primary fields (called the simple current). Our main result is that a Narain RCFT satisfying certain conditions can be described in the form of a simple current orbifold of another Narain RCFT, and we have shown how the discrete torsion in taking that orbifold is obtained. Additionally, the partition function can be considered a simple current orbifold with discrete torsion, which is determined by the lattice and the B-field. We establish that the partition function can be expressed as a polynomial, with the variables substituted by certain q -series. In a specific scenario, this polynomial corresponds to the weight enumerator polynomial of an error-correcting code. Using this correspondence to the code theory, we can relate the B-field, the discrete torsion, and the B-form to each other. [ABSTRACT FROM AUTHOR]

In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model, a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is the extension of a recently developed method for the heat kernel of the QRM that uses the Trotter–Kato product formula instead of path integrals or stochastic methods. In addition to the heat kernel formula, we give applications including the explicit formula for the partition function and the Weyl law for the distribution of the eigenvalues, obtained from the corresponding spectral zeta function. [ABSTRACT FROM AUTHOR]

We give a new proof of the equivalence between the dessin partition function and the partition function of the Laguerre unitary ensemble (LUE), originally found by Ambjørn and Chekhov. We also introduce a correction factor for the dessin/LUE partition function, and show that the corrected dessin/LUE partition function is a Dubrovin–Zhang type tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists' terminology, we establish dualities among dessin counting, generalized Penner model, and P 1 -topological sigma model. [ABSTRACT FROM AUTHOR]

We consider a statistical mechanics of rotating ideal gas consisting of classical non-relativistic spinning particles. The microscopic structure elements of the system are massive point particles with a nonzero proper angular momentum. The norm of proper angular momentum is determined by spin. The direction of proper angular momentum changes continuously. Applying the Gibbs canonical formalism for the rotating system, we construct the one-particle distribution function, generalising the usual Maxwellâ€"Boltzmann distribution, and the partition function of the system. The model demonstrates a set of chiral effects caused by interaction of spin and macroscopic rotation, including the change of entropy, heat capacity, chemical potential and angular momentum. [ABSTRACT FROM AUTHOR]

The Fisher zeros of the Baxterâ€"Wu model are examined for the first time and for two series of finite-sized systems, with ‘spherical’ boundary conditions, their location is found to be extremely simple. They lie on the unit circle in the complex sinh[2 βJ 3] plane. This is the same location as the Fisher zeros of the square lattice Ising model with nearest neighbour interactions and Brascampâ€"Kunz boundary conditions. The Baxterâ€"Wu model is an Ising model with three-site interactions, J 3, on the triangle lattice. From the leading Fisher zeros, using finite-size scaling, accurate estimates of the critical exponent 1/ ν are obtained and emphasis is placed on using different variables such as exp[âˆ'2 βJ 3], exp[âˆ'4 βJ 3], and sinh[2 βJ 3] to enhance the accuracy of estimates. Furthermore, using the imaginary parts of the leading zeros versus the real part of the leading zeros, yields different results. This is similar to results of Janke and Kenna for the nearest neighbour, Ising model on the square lattice and extends this behaviour to a multisite interaction system in a different universality class than the pair-interaction cases. [ABSTRACT FROM AUTHOR]

Starting with a gentle approach to the Aldayâ€"Gaiottoâ€"Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School ‘YRISW 2020’ to appear in a special issue of J. Phys. A. Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD (quantum chromodynamics) to non-Lagrangian theories) obtained by twisted compactification of 6d N = (2 , 0) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seibergâ€"Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of a 4d N = 2 S U (2) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyresâ€"Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilsonâ€"’t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries. [ABSTRACT FROM AUTHOR]

The asymptotic partition function for quantized abelian Higgs vortices at high temperature T is found to leading and subleading order, and from this the equation of state of the vortex gas is derived, including the first quantum correction. It is assumed that the Hamiltonian is proportional to the Laplaceâ€"Beltrami operator on the moduli space of static N -vortex solutions. The partition function is calculated using the total volume and total scalar curvature of the moduli space. [ABSTRACT FROM AUTHOR]

Motivated by Gutzwiller’s semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB ‘skeleton’ for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no ‘Lyapunov horizon’; there is only an enumeration of lattice states that contribute to the theory’s partition sum, each a global spatiotemporal solution of system’s deterministic Eulerâ€"Lagrange equations. The reformulation aligns ‘chaos theory’ with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d -dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already one-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, ‘time’-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions. [ABSTRACT FROM AUTHOR]

In this note we present a formula for the equivariant index of the cohomological complex obtained from localization of N = 2 SYM on simply-connected compact four-manifolds with a T   2-action. Knowledge of said index is essential to compute the perturbative part of the partition function for the theory. In the topologically twisted case, the complex is elliptic and its index can be computed in a standard way using the Atiyahâ€"Bott localization formula. Recently, a framework for more general types of twisting, so-called cohomological twisting, was introduced for which the complex turns out to be only transversally elliptic. While the index of such a complex has been computed for some cases where the manifold can be lifted to a Sasakian S 1-fibration in five dimensions, a general four-dimensional treatment was still lacking. We provide a formal, purely four-dimensional treatment of the cohomological complex, showing that the Laplacian part can be globally split off while the remaining part can be trivialized uniquely in the group-direction. This ultimately produces a simple formula for the index applicable for any compact simply-connected four-manifold. Finally, the index formula is applied to examples on S 4, C P 2 and F 1 . For the latter, we use the result to compute the perturbative partition function. [ABSTRACT FROM AUTHOR]

We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological field theory. If one chooses a special conformal boundary condition which we call ‘cloaking boundary condition’, then for each value of the radius the fusion category of topological line defects of the CFT is contained in the lattice model. The fact that the full topological symmetry of the initial CFT is realised exactly is a key feature of our lattice models. We provide an explicit recursive procedure to evaluate the interaction vertex on arbitrary states. As an example, we study the lattice model obtained from the Ising CFT on a torus with one hole, decomposed into two lattice cells. We numerically compare the truncated lattice model to the CFT expression obtained from expanding the boundary state in terms of the hole radius and we find good agreement at intermediate values of the radius. [ABSTRACT FROM AUTHOR]

In previous work, two of the authors determined, completely and rigorously, a solution space S N for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schrammâ€"Löwner evolution (SLE Îş ). The system comprises 2 N null-state equations and three conformal ward identities that govern CFT correlation functions of 2 N one-leg boundary operators or SLE Îş partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE Îş partition functions,’ for the probability that the growing curves of a multiple-SLE Îş process join in a particular connectivity. In a previous article, we rigorously define certain elements of S N , which we call ‘connectivity weights,’ argue that they are in fact pure SLE Îş partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of Îş ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in S N for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in S N for all N ∈ Z + + 1. We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix, we investigate logarithmic singularities of some of these expressions, appearing for certain values of Îş predicted by logarithmic CFT. [ABSTRACT FROM AUTHOR]

A large family of nonsingular rational solutions of the Kadomtsevâ€"Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-defined wave patterns in the xy -plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of PainlevĂ© II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of N objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer N. [ABSTRACT FROM AUTHOR]

We show that the fermionization of the Monster CFT with respect to is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the orbifold, i.e. it enjoys the Kramers–Wannier duality. The Kramers–Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay–Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay–Thompson series is invariant under the genus-zero congruence subgroup 16D0 of. [ABSTRACT FROM AUTHOR]

We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box of given size). The partition function of an inhomogeneous model is given in terms of a determinant. For the homogeneous model, it can be given in terms of a Hankel determinant. We also show that in the homogeneous case the partition function is a τ-function of the sixth Painlevé equation with respect to the rapidity variable of the weights. [ABSTRACT FROM AUTHOR]

A free boson on a lattice is the simplest field theory one can think of. Its partition function can be easily computed in momentum space. However, this straightforward solution hides its integrability properties. Here, we use the methods of exactly solvable models, that are currently applied to spin systems, to a massless and massive free boson on a 2D lattice. The Boltzmann weights of the model are shown to satisfy the Yang–Baxter equation with a uniformization given by trigonometric functions in the massless case, and Jacobi elliptic functions in the massive case. We diagonalize the row-to-row transfer matrix, derive the conserved quantities, and implement the quantum inverse scattering method. Finally, we construct two factorized scattering S matrix models for continuous degrees of freedom using trigonometric and elliptic functions. These results place the free boson model in 2D in the same position as the rest of the models that are exactly solvable ŕ la Yang–Baxter, offering possible applications in quantum computation. [ABSTRACT FROM AUTHOR]

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra. [ABSTRACT FROM AUTHOR]

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]

The Langmann–Szabo–Zarembo (LSZ) matrix model is a complex matrix model with a quartic interaction and two external matrices. The model appears in the study of a scalar field theory on the non-commutative plane. We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes–Wigert matrix model, which is a q-ensemble describing U(N) Chern–Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices, the LSZ matrix model either describes probabilities of large fluctuations in the Chern–Simons partition function, in the unknot invariant or in the two-unknot invariant. We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model. [ABSTRACT FROM AUTHOR]

We study matrix integration over the classical Lie groups U(N), Sp(2N), SO(2N) and SO(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern–Simons theory on S3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern–Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom. [ABSTRACT FROM AUTHOR]

We construct an ϵ-deformation of W algebras, corresponding to the additive version of quiver algebras which feature prominently in the 5D version of the BPS/CFT correspondence and refined topological strings on toric Calabi–Yau's. This new type of algebras fill in the missing intermediate level between q-deformed and ordinary W algebras. We show that ϵ-deformed W algebras are spectral duals of conventional W algebras, in particular the ϵ-deformed conformal blocks manifestly reproduce instanton partition functions of 4D quiver gauge theories in the full Ω-background and give dual integral representations of ordinary W conformal blocks. [ABSTRACT FROM AUTHOR]

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of integrable structure in a modified melting crystalmodel of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of ‘shift symmetries’. They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can also be characterized by a factorization problem of Z×Z matrices. The associated Lax operators turn out to be quotients of first-order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz– Ladik (equivalently, the relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions. [ABSTRACT FROM AUTHOR]

We review recent progress in the construction of black holes in three dimensional higher spin gravity theories. Starting from spin-3 gravity and working our way toward the theory of an infinite tower of higher spins coupled to matter, we show how to harness higher spin gauge invariance to consistently generalize familiar notions of black holes. We review the construction of black holes with conserved higher spin charges and the computation of their partition functions to leading asymptotic order. In view of the antide Sitter/conformal field theory (CFT) correspondence as applied to certain vector-like conformal field theories with extended conformal symmetry, we successfully compare to CFT calculations in a generalized Cardy regime. A brief recollection of pertinent aspects of ordinary gravity is also given. [ABSTRACT FROM AUTHOR]

We evaluate the exact one-loop partition function for fundamental strings whose world surface ends on a cusp at the boundary of AdS4 and has a 'jump' in ℂP³. This allows us to extract the stringy prediction for the ABJM-generalized cusp ABJM anomalous dimension (⌈cuspABJM (ϕ, θ) up to NLO in sigma-model perturbation cusp theory. With a similar analysis, we present the exact partition functions for folded closed string solutions moving in the AdS3 parts of AdS4 x ℂP³ and AdS3 x S³ x S³ x S¹ backgrounds. Results are obtained applying to the string solutions relevant for the AdS4/CFT3 and AdS3/CFT2 correspondence the tools previously developed for their AdS5 x S55 counterparts. [ABSTRACT FROM AUTHOR]

Earlier this year Chan extended the low-density series for the hard-squares partition function κ(z) to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity zd which lies on the negative fugacity axis. We find that the series has a confluent singularity of order at least 2 at zd with exponents θ = 0.833 33(2) and θ'= 1.6676(3). We thus confirm that the exponent è has the exact value ⅚ as observed by Dhar. [ABSTRACT FROM AUTHOR]

We analyze the exact partition function of the anisotropic Ising model on finite M × N rectangular lattices under four different boundary conditions (periodic-periodic (pp), periodic-antiperiodic (pa), antiperiodic-periodic (ap) and antiperiodic-antiperiodic (aa)) obtained by Kaufman (1949 Phys. Rev. 76 1232), Wu and Hu (2002 J. Phys. A: Math. Gen. 35 5189) and Kastening (2002 Phys. Rev. E 66 057103)).We express the partition functions in terms of the partition functions Zα,ß (J, k) with (α, ß) = (0, 0), (1/2, 0), (0, 1/2) and (1/2, 1/2), J is an interaction coupling and k is an anisotropy parameter. Based on such expressions, we then extend the algorithm of Ivashkevich et al (2002 J. Phys. A: Math. Gen. 35 5543) to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. Our result is f = fbulk + Σp∞=0 fp (ρ, k) S-p-1, where f is the free energy of the system, fbulk is the free energy of the bulk, S = MN is the area of the lattice and ρ = M/N is the aspect ratio. All coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ρeff = ρ/ sinh 2Jc and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation ρeff → 1/ρeff. [ABSTRACT FROM AUTHOR]

We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models. [ABSTRACT FROM AUTHOR]

We describe coherent states and associated generalized Grassmann variables for a system of m independent q-boson modes. A resolution of unity in terms of generalized Berezin integrals leads to generalized Grassmann symbolic calculus. Formulae for operator traces are given and the thermodynamic partition function for a system of q-boson oscillators is discussed. [ABSTRACT FROM AUTHOR]

We consider some applications (old and new) of the Patterson-Selberg zeta function to 3D gravity with a negative cosmological constant.We also consider 2D black hole vacua with a parabolic generator of their holonomy. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

This review paper contains a concise introduction to highest weight representations of infinite-dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera of superconformal quantum mechanics and superstring models. The common link of all these concepts and of the many examples considered in this paper is to be found in a very important feature of the theory of infinite-dimensional Lie algebras: the modular properties of the characters (generating functions) of certain representations. The characters of the highest weight modules represent the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. We discuss the role of the unimodular (and modular) groups and the (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated q-series. For mathematicians, elliptic genera are commonly associated with new mathematical invariants for spaces, while for physicists elliptic genera are one-loop string partition function. (Therefore, they are applicable, for instance, to topological Casimir effect calculations.) We show that elliptic genera can be conveniently transformed into product expressions, which can then inherit the homology properties of appropriate polygraded Lie algebras. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of StuartDowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

It is shown that the free energy associated to a finite-dimensional Airy structure is an analytic function at each finite order of the -expansion. Its terms are interpreted as objects living on the zero locus of the classical hamiltonians. The geometry of this variety is studied. The structure of singularities of the free energy is described. To this end topological recursion equations are expressed in a form particularly suitable for semiclassical analysis. It involves a differential operator which is a deformation of the exterior derivative. Its local properties are derived. The developed formalism is applied in several examples. Global properties of the obtained partition functions are investigated. In the case of a divergent -series, a simple resummation is performed. [ABSTRACT FROM AUTHOR]

We compute the instanton partition functions of SCFTs in class . We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of K D(p-4) branes. Starting with D5/D1 setups probing a orbifold singularity we obtain the K instanton partition functions of 6d theories on in the presence of orbifold defects on T2 via computing the 2d superconformal index of the worldvolume theory on K D1 branes wrapping the T2. We then reduce our results to the 5d and to the 4d instanton partition functions. For k  =  1 we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of quivers in class can be obtained from the class mother theory partition functions with gauge factors via imposing the ‘orbifold condition’ with and , on the Coulomb moduli and the mass parameters. [ABSTRACT FROM AUTHOR]

It is shown that the free energy associated to a finite-dimensional Airy structure is an analytic function at each finite order of the -expansion. Its terms are interpreted as objects living on the zero locus of the classical hamiltonians. The geometry of this variety is studied. The structure of singularities of the free energy is described. To this end topological recursion equations are expressed in a form particularly suitable for semiclassical analysis. It involves a differential operator which is a deformation of the exterior derivative. Its local properties are derived. The developed formalism is applied in several examples. Global properties of the obtained partition functions are investigated. In the case of a divergent -series, a simple resummation is performed. [ABSTRACT FROM AUTHOR]

We extend the definition of the refined topological vertex to an n-coloured refined topological vertex that depends on n free bosons, and compute the 5D strip partition function made of N pairs of vertices and conjugate vertices. Using geometric engineering and the AGT correspondence, the 4D limit of this strip partition function is identified with a (normalized) matrix element of a (primary state) vertex operator that intertwines two (arbitrary descendant) states in a (generically non-rational) 2D conformal field theory with parafermion primary states. [ABSTRACT FROM AUTHOR]

We consider an empirical Bayes method for Boltzmann machines and propose an algorithm for it. The empirical Bayes method allows for estimation of the values of the hyperparameters of the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function in this study. However, the maximization is computationally hard because the empirical Bayes likelihood function involves intractable integrations of the partition function. The proposed algorithm avoids this computational problem by using the replica method and the Plefka expansion. Our method is quite simple and fast because it does not require any iterative procedures and gives reasonable estimates at a certain condition. However, our method introduces a bias to the estimate, which exhibits an unnatural behavior with respect to the size of the dataset. This peculiar behavior is supposed to be due to the approximate treatment by the Plefka expansion. A possible extension to overcome this behavior is also discussed. [ABSTRACT FROM AUTHOR]

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of an inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating function of plane partitions with the fixed sums of their diagonals. The obtained results are another example of the connection of integrable models with enumerative combinatorics. [ABSTRACT FROM AUTHOR]

We discuss some physical consequences of the resurgent structure of Painlevé equations and their related conformal block expansions. The resurgent structure of Painlevé equations is particularly transparent when expressed in terms of physical conformal block expansions of the associated tau functions. Resurgence produces an intricate network of inter-relations; some between expansions around different critical points, others between expansions around different instanton sectors of the expansions about the same critical point, and others between different non-perturbative sectors of associated spectral problems, via the Bethe-gauge and Painlevé-gauge correspondences. Resurgence relations exist both for convergent and divergent expansions, and can be interpreted in terms of the physics of phase transitions. These general features are illustrated with three physical examples: correlators of the 2d Ising model, the partition function of the Gross–Witten–Wadia matrix model, and the full counting statistics of one dimensional fermions, associated with Painlevé VI, Painlevé III and Painlevé V, respectively. [ABSTRACT FROM AUTHOR]

We continue to study the matrix model of the Nf   =  2 case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross–Witten–Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres–Douglas type theory from IR. We elaborate upon the method of orthogonal polynomials and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential. [ABSTRACT FROM AUTHOR]

We consider a finite one-dimensional totally asymmetric simple exclusion process with four types of particles, , in contact with reservoirs. Particles of species 0 can neither enter nor exit the lattice, and those of species are constrained to lie at the first and last site. Particles of species 1 enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram. [ABSTRACT FROM AUTHOR]

Distributions of zeros of the grand canonical and canonical partition functions in the complex fugacity and in the complex interaction strength plane are examined numerically for a model of rooted self-interacting self-avoiding walks on a hierarchical graph. It is shown that the pattern of zeros of the grand canonical partition function in the complex fugacity plane has a circular-like form, with the exception of zeros lying in the vicinity of the critical point. Exact values of polymer size critical exponents, in the swollen and dense phase, as well as at the point of collapse transition, are obtained by studying asymptotic behavior of zeros lying closest to the positive real axis in the fugacity plane. Distribution of zeros of the canonical partition function in the complex interaction strength plane is found to be more complicated, and an accurate estimate of crossover exponent is obtained from study of asymptotic behavior of zeros lying closest to the positive real axis in this plane. It is also shown that the next-to-leading term in the asymptotic law for canonical partition function in the dense phase has a stretched-exponential rather than the power-law form. [ABSTRACT FROM AUTHOR]

We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer p  and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter s, while the polymer conformations are modelled by weighted Hamiltonian walks (HWs). Applying an exact recurrence equations method, we have found that partition function ZN for closed HWs consisting of N steps scales as , where constants and depend on both p  and s. We have calculated numerically the stiffness dependence of the polymer persistence length, as well as various thermodynamic quantities (such as free and internal energy, specific heat and entropy) for a large set of members of the MR family. Analysis of these quantities has shown that semi-flexible compact polymers on MR lattices can exist only in the liquid-like (disordered) phase, whereas the crystal (ordered) phase has not appeared. Finally, behavior of the examined system at zero temperature has been discussed. [ABSTRACT FROM AUTHOR]

We derive a one-parameter deformation of the refined topological vertex that, when used to compute non-periodic web diagrams, reproduces the six-dimensional topological string partition functions that are computed using the refined vertex and periodic web diagrams. [ABSTRACT FROM AUTHOR]

This work concerns both the semiclassical and zero temperature asymptotics of quantum weighted double Hurwitz numbers. The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading semiclassical term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit is shown to reproduce the simple single and double Hurwitz numbers studied by Okounkov and Pandharipande (2000 Math. Res. Lett. 7 447–53, 2000 Lett. Math. Phys. 53 59–74). The KP-Toda τ-function that serves as generating function for the quantum Hurwitz numbers is shown to have the τ-function of Okounkov and Pandharipande (2000 Math. Res. Lett. 7 447–53, 2000 Lett. Math. Phys. 53 59–74) as its leading term in the classical limit, and, with suitable scaling, the same holds for the partition function, the weights and expectations of Hurwitz numbers. We also compute the zero temperature limit of the partition function and quantum weighted Hurwitz numbers. The KP or Toda τ-function serving as generating function for the quantum Hurwitz numbers are shown to give the one for Belyi curves in the zero temperature limit and, with suitable scaling, the same holds true for the partition function, the weights and the expectations of Hurwitz numbers. [ABSTRACT FROM AUTHOR]

The zeros of the size-n partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as . Here we use weighted Dyck paths as a simple model of two-dimensional polymer adsorption, and study the behaviour of the partition function zeros, particularly in the thermodynamic limit. The exact solvability of the model allows for a precise calculation of the locus of the zeros and the way in which an edge-singularity on the positive real axis is formed. We also show that in the limit the zeros converge on a limaçon in the complex plane. [ABSTRACT FROM AUTHOR]

We consider 4d supersymmetric (special) unitary Γ quiver gauge theories on compact manifolds which are fibrations over . We show that their partition functions are correlators of vertex operators and screening charges of the modular double version of elliptic algebras. We also consider a generating function of BPS surface defects supported on and show that it can be identified with a particular coherent state in the Fock module over the elliptic Heisenberg algebra. [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]

We review the calculation and properties of the supersymmetric index for four-dimensional theories, illustrating its physical significance in several examples. [ABSTRACT FROM AUTHOR]

We review some exact results for the matrix models appearing in the localization of Chern–Simons-matter theories, focusing on the structure of non-perturbative effects and on the M-theory expansion of ABJM theory. We also summarize some of the results obtained for other Chern–Simons-matter theories, as well as recent applications to topological strings. [ABSTRACT FROM AUTHOR]