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We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings' worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d N = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions. [ABSTRACT FROM AUTHOR]

We solve the Wheeler-DeWitt equation in the 'cosmological interior' (the past causal diamond of future infinity) of four dimensional dS-Schwarzschild spacetimes. Within minisuperspace there is a basis of solutions labelled by a constant c, conjugate to the mass of the black hole. We propose that these solutions are in correspondence with partition functions of a dual quantum mechanical theory where c plays the role of time. The quantum mechanical theory lives on worldtubes in the 'static patch' of dS-Schwarzschild, and the partition function is obtained by evolving the corresponding Wheeler-DeWitt wavefunction through the cosmological horizon, where a metric component gtt changes sign. We establish that the dual theory admits a symmetry algebra given by a central extension of the Poincaré algebra e (1, 1) and that the entropy of the dS black hole is encoded as an averaging of the dual partition function over the background gtt. [ABSTRACT FROM AUTHOR]

In 2007, Andrews and Paule introduced the broken k -diamond partition function Δ k (n). Many researches on the arithmetic properties for Δ k (n) have been done. In this paper, we prove that D 3 log ⁡ Δ 1 (n − 1) > 0 for n ≥ 5 and D 3 log ⁡ Δ 2 (n − 1) > 0 for n ≥ 7 , where D is the difference operator with respect to n. We also conjecture that for any k ≥ 1 and r ≥ 1 , there exists a positive integer n k (r) such that for n ≥ n k (r) , (− 1) r D r log ⁡ Δ k (n) > 0. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang, and Xie. Furthermore, we obtain that both { Δ 1 (n) } n ≥ 0 and { Δ 2 (n) } n ≥ 0 satisfy the higher order Turán inequalities for n ≥ 6. [ABSTRACT FROM AUTHOR]

We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples. [ABSTRACT FROM AUTHOR]

In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n). [ABSTRACT FROM AUTHOR]

Let W N (β) = E 0 e ∑ n = 1 N β ω (n , S n) - N β 2 / 2 be the partition function of a two-dimensional directed polymer in a random environment, where ω (i , x) , i ∈ N , x ∈ Z 2 are i.i.d. standard normal and { S n } is the path of a random walk. With β = β N = β ^ π / log N and β ^ ∈ (0 , 1) (the subcritical window), log W N (β N) is known to converge in distribution to a Gaussian law of mean - λ 2 / 2 and variance λ 2 , with λ 2 = log (1 / (1 - β ^ 2)) (Caravenna et al. in Ann Appl Probab 27(5):3050–3112, 2017). We study in this paper the moments E [ W N (β N) q ] in the subcritical window, for q = O (log N) . The analysis is based on ruling out triple intersections. [ABSTRACT FROM AUTHOR]

We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the Ω-background with gauge group U(N) and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the Ω-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are also characterised by counting specific holomorphic curves in a Calabi-Yau threefold XN,1 which engineers the LST. Furthermore, using the formulation of the partition function in terms of the Kronecker-Eisenstein series, we argue for new recursive structures which relate higher instanton contributions to products of lower ones. [ABSTRACT FROM AUTHOR]

In 2018, Andrews introduced the partition functions E O(n) and E O(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function E Oe(n) which counts the number of partitions of n which are enumerated by E O(n) together with the partitions enumerated by E O(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for E O(n) and E Oe(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for E O(n) and modulo 4 for E Oe(n). For example, if p ≥ 5 is a prime with Legendre symbol (−3/p ) = -1, then for all integers n ≥ 0 and α ≥ 0, we have E O ( 8 · p 2α+1(pn + j) + 19 · p 2α+2 − 1/3 ) Ⲷ 0 (mod 8); 1 ≤ j ≤ (p − 1) [ABSTRACT FROM AUTHOR]

This paper describes the reconstruction of the topological string partition function for certain local Calabi–Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann–Hilbert problem. The isomonodromic tau-functions associated to these Riemann–Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion of generalised theta series type from which one can extract the topological string partition functions for each chamber. [ABSTRACT FROM AUTHOR]

In the present work, we report an investigation on the importance of using realistic partition functions in the opacity calculations of lanthanide ions whether they are moderately or lowly ionized. In order to do this, atomic data for various moderately charged samarium (Sm V–XI) and lowly charged neodymium (Nd II–IV) ions were calculated by the pseudo-relativistic Hartree–Fock method (HFR) and then, used to compute the expansion opacities for conditions characterizing the ejecta of kilonovae observed as a result of neutron star mergers, with a particular attention given to the partition function computations. [ABSTRACT FROM AUTHOR]

Let b ℓ (n) denote the number of ℓ -regular partitions of n, and b ℓ ′ (n) denote the number of ℓ -regular partitions of n into distinct parts. In this paper we prove some infinite families of new congruences for b ℓ (n) and b ℓ ′ (n) . [ABSTRACT FROM AUTHOR]

We extend a formula for 1-loop black hole determinants by Denef, Hartnoll, and Sachdev (DHS) to spinning fields on any (d + 1)-dimensional static spherically symmetric black hole. By carefully analyzing the regularity condition imposed on the Euclidean eigenfunctions, we reveal an unambiguous bulk-edge split in the 1-loop Euclidean partition function for tensor fields of arbitrary integer spin: the bulk part captures the "renormalized" thermal canonical partition function recently discussed in [1]; the edge part is related to quasinormal modes (QNMs) that fail to analytically continue to a subset of Euclidean modes with enhanced fall-offs near the origin. Since the edge part takes the form of a path integral on Sd−1, this suggests that these are associated with degrees of freedom living on the bifurcation surface in the Lorentzian two-sided black hole geometry. For massive higher spin on static BTZ and massive vector on Nariai black holes, we find that the edge partition function is related to the QNMs with lowest overtone numbers. [ABSTRACT FROM AUTHOR]

If t is a positive integer, then a partition of a non-negative integer n is a t-core if none of the hook numbers of the associated Ferrers-Young diagram is divisible by t. Let a t (n) denote the number of t-core partitions of n. The purpose of this article is to prove some infinite families of congruences modulo 3 for the 5-core partition function a 5 (n) and parity results for the 7-core partition function a 7 (n) by using elementary generating functions and q-series manipulations. We further verify the congruences for the 5-core partition function a 5 (n) using the theory of modular forms. [ABSTRACT FROM AUTHOR]

We study universal properties of the torus partition function of T T ¯ -deformed CFTs under the assumption of modular invariance, for both the original version, referred to as the double-trace version in this paper, and the single-trace version defined as the symmetric product orbifold of double-trace T T ¯ -deformed CFTs. In the double-trace case, we specify sparseness conditions for the light states for which the partition function at low temperatures is dominated by the vacuum when the central charge of the undeformed CFT is large. Using modular invariance, this implies a universal density of high energy states, in analogy with the behavior of holographic CFTs. For the single-trace T T ¯ deformation, we show that modular invariance implies that the torus partition function can be written in terms of the untwisted partition function and its modular images, the latter of which can be obtained from the action of a generalized Hecke operator. The partition function and the energy of twisted states match holographic calculations in previous literature, thus providing further evidence for the conjectured holographic correspondence. In addition, we show that the single-trace partition function is universal when the central charge of the undeformed CFT is large, without needing to assume a sparse density of light states. Instead, the density of light states is shown to always saturate the sparseness condition. [ABSTRACT FROM AUTHOR]

Many methods to accelerate sampling of molecular configurations are based on the idea that temperature can be used to accelerate rare transitions. These methods typically compute equilibrium properties at a target temperature using reweighting or through Monte Carlo exchanges between replicas at higher temperatures. A recent paper [G. M. Rotskoff and E. Vanden-Eijnden, Phys. Rev. Lett. 122, 150602 (2019)] demonstrated that accurate equilibrium densities of states can also be computed through a nonequilibrium "quench" process, where sampling is performed at a higher temperature to encourage rapid mixing and then quenched to lower energy states with dissipative dynamics. Here, we provide an implementation of the quench dynamics in LAMMPS and evaluate a new formulation of nonequilibrium estimators for the computation of partition functions or free energy surfaces (FESs) of molecular systems. We show that the method is exact for a minimal model of N-independent harmonic springs and use these analytical results to develop heuristics for the amount of quenching required to obtain accurate sampling. We then test the quench approach on alanine dipeptide, where we show that it gives an FES that is accurate near the most stable configurations using the quench approach but disagrees with a reference umbrella sampling calculation in high FE regions. We then show that combining quenching with umbrella sampling allows the efficient calculation of the free energy in all regions. Moreover, by using this combined scheme, we obtain the FES across a range of temperatures at no additional cost, making it much more efficient than standard umbrella sampling if this information is required. Finally, we discuss how this approach can be extended to solute tempering and demonstrate that it is highly accurate for the case of solvated alanine dipeptide without any additional modifications. [ABSTRACT FROM AUTHOR]

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I × Σ , with I an interval and dim Σ = 4 l + 2 , in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Σ = 0 is considered as a toy example. We show that one can identify—for certain choices of residual fields—the "physical part" (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton–Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton–Jacobi actions, and quantization, arXiv:2012.13270), without any quantum corrections. This Hamilton–Jacobi action is the action functional of a conformal field theory on Σ . For dim Σ = 2 , this implies a version of the CS-WZW correspondence. For dim Σ = 6 , using a particular polarization on one end of the cylinder, the Chern–Simons partition function is related to Kodaira–Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili. [ABSTRACT FROM AUTHOR]

We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of PSL (2 , R) Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface Σ is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form Σ × S 1 , where Σ can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of n asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over M ¯ g , n , which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton’s constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces Σ . There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization. [ABSTRACT FROM AUTHOR]

An improved Hulburt‐Hirschfelder (IHH) quantum mechanical anharmonic oscillator model which describes the exact analytical potential energy curves (PECs) of diatomic molecules is presented. The IHH analytical potential function that involves a variable parameter, four empirically determined low‐order molecular vibrational and rotational spectral constants and the dissociation energy was applied to the ground electronic states of HF and HCl, yielding a good agreement between the Rydberg‐Klein‐Rees (RKR) potential energy points and the calculations. Correspondingly, with the determined IHH PECs, the partition functions at temperatures 300–6000 Kelvin (K) and rovibrational line intensities are evaluated, and coincide well with those quoted in the HITRAN2020 database compilation. The methodology could be valuable for the detection of molecules in interstellar clouds using infrared spectroscopy, as well as thermochemical reactions. [ABSTRACT FROM AUTHOR]

Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized subtractively by its cumulant or free energy function, or equivalently normalized divisively by its partition function. Both the cumulant and partition functions are strictly convex and smooth functions inducing corresponding pairs of Bregman and Jensen divergences. It is well known that skewed Bhattacharyya distances between the probability densities of an exponential family amount to skewed Jensen divergences induced by the cumulant function between their corresponding natural parameters, and that in limit cases the sided Kullback–Leibler divergences amount to reverse-sided Bregman divergences. In this work, we first show that the α -divergences between non-normalized densities of an exponential family amount to scaled α -skewed Jensen divergences induced by the partition function. We then show how comparative convexity with respect to a pair of quasi-arithmetical means allows both convex functions and their arguments to be deformed, thereby defining dually flat spaces with corresponding divergences when ordinary convexity is preserved. [ABSTRACT FROM AUTHOR]

Utilization of metal tungsten (W) as the structural material or pressure scale requires accurate knowledge of the equation of state (EOS), which is far beyond the available experimental measurements. In the present work, a direct integral approach (DIA) with ultrahigh efficiency was applied to calculate the EOS of W up to 500 GPa and 3000 K with ab initio calculations. Compared with previous static compression experiments up to 150 GPa under room temperature and 35 GPa at high temperatures up to 1673 K, all the deviations of the calculated pressure are within or comparable to the uncertainty of experiments. Predictions for higher-temperature and simultaneously higher-pressure EOS up to 300 GPa and 3000 K differ slightly from the comprehensive analysis by Litasov et al. [J. Appl. Phys. 113, 133505 (2013)] via fitting available experimental data with the empirical equation. These results indicate that the EOS of crystal W obtained from DIA should be convincible, and DIA without any empirical or artificial parameters may find its wide applications for predicting thermodynamic properties of condensed matter in the future. [ABSTRACT FROM AUTHOR]

Andrews [ Generalized Frobenius Partitions , Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers $\ell ,$ all primes $p\geq 5$ and all values $r, 0 such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' memoir, classical q -series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question. [ABSTRACT FROM AUTHOR]

We study the S1 × Σ g topologically twisted index and the squashed sphere partition function of various 3d N ≥ 2 holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative 1/N expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS4 solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS4 BPS black holes. [ABSTRACT FROM AUTHOR]

Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimension-2 singularities analogous to the familiar Euclidean conical singularities. With this understanding, we show that the usual Euclidean-signature black holes (or their complex rotating analogues) define saddle points for the real-time path integrals that compute our partition functions. Furthermore, when the black holes have positive specific heat, we provide evidence that a codimension-2 subcontour of our real Lorentz-signature contour of integration can be deformed so as to show that these black holes saddles contribute with non-zero weight to the semiclassical limit, and that the same is then true of the remaining two integrals. [ABSTRACT FROM AUTHOR]

For rational \alpha, the fractional partition functions p_\alpha (n) are given by the coefficients of the generating function (q; q)_\infty ^\alpha. When \alpha = -1, one obtains the usual partition function. Congruences of the form p(\ell n + c)\equiv 0 \pmod {\ell } for a prime \ell and integer c were studied by Ramanujan. Such congruences exist only for \ell \in \{5, 7, 11\}. Chan and Wang recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as p_\frac {57}{61}(17^2 n - 3) \equiv 0 \pmod {17^2}. [ABSTRACT FROM AUTHOR]

We study the n$n$‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2$\hbar ^2$‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ℏ2$\hbar ^2$‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature. [ABSTRACT FROM AUTHOR]

We compute the sphere and disk partition functions in semiclassical Liouville and analogous quantities in double-scaled matrix integrals. The quantity sphere/disk2 is unambiguous and we find a precise numerical match between the Liouville answer and the matrix integral answer. An application is to show that the sphere partition function in JT gravity is infinite. [ABSTRACT FROM AUTHOR]

We introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal L K , a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in C 3 . [ABSTRACT FROM AUTHOR]

Let Q 8 (n) and Q [ 1 k r k ] (n) count the 8-colored partitions of n into odd parts and the 2k-color partitions of n into odd parts, respectively, where the multiples of r can appear in 2k colors and the others can appear in k colors. In this paper, we find infinite families of congruences modulo powers of 2 for Q 8 (n) , Q [ 1 2 3 2 ] (n) , Q [ 1 4 5 4 ] (n) , and Q [ 1 2 1 1 2 ] (n). [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

We evaluate here the quantum gravity partition function that counts the dimension of the Hilbert space of a simply connected spatial region of a fixed proper volume in the context of Lovelock gravity, generalizing the results for Einstein gravity. It is found that there are sphere saddle metrics for a partition function at a fixed spatial volume in Lovelock theory. Those stationary points take exactly the same forms as in Einstein gravity. The logarithm of Z corresponding to a zero effective cosmological constant indicates that the Bekenstein–Hawking entropy of the boundary area and that corresponding to a positive effective cosmological constant points to the Wald entropy of the boundary area. We also show the existence of zeroth-order phase transitions between different vacua, a phenomenon distinct from Einstein gravity. [ABSTRACT FROM AUTHOR]

In this paper, we provide an algorithm to detect linear congruences of p l k (n) , the number of MacMahon's k -rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the k -rowed plane partition functions. We also show p (n , m) that the number of partitions at most m parts always admits linear congruences. [ABSTRACT FROM AUTHOR]

We establish the relation of partition functions of conformal higher spin fields on Weyl equivalent spaces in d = 4 dimension. We express the partition function of Weyl graviton and conformal higher spin fields as an integral over characters on S1× AdS3, S4, and AdS4. We observe that the partition function of conformal higher spins on hyperbolic cylinders differs from the partition function on S4 by the 'edge' contribution. The logarithmic coefficient obtained from the character integral of the partition function of conformal higher spins on AdS4 is the half of that obtained from the partition function on S4. We evaluate the entanglement entropy and the conformal dimension of the twist operator from the partition function on the hyperbolic cylinder. The conformal dimension of the co-dimension two twist operator enables us to find a linear relation between Hofman-Maldacena variables which we use to show the non-unitarity of the theory. We observe that the spectrum of the quasinormal modes of conformal higher spins obtained from the bulk character contains additional distinct states compared to the spectrum of unitary massless higher spin fields. [ABSTRACT FROM AUTHOR]

We reinterpret the OSV formula for the on-shell action/entropy function of asymptotically flat BPS black holes as a fixed point formula that is formally equivalent to a recent gluing proposal for asymptotically AdS4 black holes. This prompts a conjecture that the complete perturbative answer for the most general gravitational building block of 4d N = 2 supergravity at a single fixed point takes the form of a Nekrasov-like partition function with equivariant parameters related to the higher-derivative expansion of the prepotential. In turn this leads to a simple localization-like proposal for a set of supersymmetric partition functions in (UV completed) 4d N = 2 supergravity theories. The conjecture is shown to be in agreement with a number of available results for different BPS backgrounds with both Minkowski and AdS asymptotics. In particular, it follows that the OSV formula comes from the unrefined limit of the general expression including only the so-called 𝕎 tower of higher derivatives, while the on-shell action of pure (Euclidean) AdS4 with round S3 boundary comes from the NS limit that includes only the 𝕋 tower. Backgrounds preserving less supersymmetry, such as the under-rotating black holes in flat space, the holographic squashed S3, and the static/rotating twisted and non-twisted Kerr-Newman-like black holes in AdS4 lead to a more general refined version of the corresponding gravitational blocks as dictated by the supersymmetric gluing rules. [ABSTRACT FROM AUTHOR]

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators. [ABSTRACT FROM AUTHOR]

We consider finite quantum systems defined by a mixed set of commutation and anti-commutation relations between components of the Hamiltonian operator. These relations are represented by an anti-commutativity graph that contains necessary and sufficient information for computing the full quantum partition function. We derive a second-order differential equation for an extended partition function z[β, β′, J] that describes a transformation from a "parent" partition function z[0, β′, J] (or anti-commutativity graph) to a "child" partition function z[β, 0, J] (or anti-commutativity graph). The procedure can be iterated, and then one forms a pseudo-forest of duality transformations between quantum systems, i.e., a directed graph in which every vertex (or quantum system) has at most one incoming edge (from its parent system). The pseudo-forest has a single tree connected to a constant partition function, many pseudo-trees connected to self-dual systems, and all other pseudo-trees connected to closed cycles of transformations between mutually dual systems. We also show how the differential equation for the extended partition function can be used to study disordered systems. [ABSTRACT FROM AUTHOR]

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results. [ABSTRACT FROM AUTHOR]

In this letter we continue the development of W -representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz τ -functions. We construct W -representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. We further propose the β -deformation of the discussed W -representation for the Hurwitz case for two sets of times as well as for the multi-character case. [ABSTRACT FROM AUTHOR]

Andrews and Newman introduced the mex-function mex A , a (λ) for an integer partition λ of a positive integer n as the smallest positive integer congruent to a modulo A that is not a part of λ . They then defined p A , a (n) to be the number of partitions λ of n satisfying mex A , a (λ) ≡ a (mod 2 A) . They found the generating function for p t , t (n) and p 2 t , t (n) for any positive integer t, and studied their arithmetic properties for some small values of t. In this article, we study the partition function p m t , t (n) for all positive integers m and t. We show that for sufficiently large X, the number of all positive integers n ≤ X such that p m t , t (n) is an even number is at least O (X / 3) for all positive integers m and t. We also prove that for sufficiently large X, the number of all positive integers n ≤ X such that p m p , p (n) is an odd number is at least O (log log X) for all m ≢ 0 (mod 3) and all primes p ≡ 1 (mod 3) . Finally, we establish identities connecting the ordinary partition function to p m t , t (n) . [ABSTRACT FROM AUTHOR]

We construct 3D N = 2 abelian gauge theories on S 2 × S 1 labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in S 3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on S 3, and then our construction provides an explicit correspondence between 3D N = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists. [ABSTRACT FROM AUTHOR]

In this article we study certain degenerations of the mirror curves associated with the Calabi–Yau threefolds X N , M , and the effect of these degenerations on the refined topological string partition function of X N , M . We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreover we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to X N , M - 1 from X N , M . [ABSTRACT FROM AUTHOR]

The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space Sa × AdSb can be mapped to Sa+b provided Sa and AdSb are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on Sa × AdSb and Sa+b. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in d = 6 dimension can be expressed as an integral over 'naive' bulk and 'naive' edge characters. However, the partition function of the conformal vector gauge field on S q 1 × AdS5 contains only the 'naive' bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal p-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the T T ¯ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature. [ABSTRACT FROM AUTHOR]

Accurate atomic and diatomic partition functions are required to determine the level populations for the calculations of radiative properties in thermodynamic equilibrium and nonequilibrium plasmas produced by various atmospheric re‐entries. In this work, a reliable partition functions database was rebuilt for some atomic and diatomic species from 100 K to 50000 K. The atomic partition functions were obtained by a four‐level model, while the diatomic partition functions were predicted based upon a more rigorous approach for the computation of the energy levels. Compared with previous publications, the number of diatomic electronic states considered in our work is as large as possible. Estimates are made for the contributions of each electronic state of the diatomic molecule to the partition function. Moreover, the effect of the number of electronic states on the partition function was also evaluated. Finally, we calculated the specific heat based upon the obtained partition functions. All the results were validated by the available data in recent references and the relative errors were systematically analysed. [ABSTRACT FROM AUTHOR]

We find new bilinear relations for the partition functions of the U(N)k × U(N + M)−k ABJ theory with two parameter mass deformation (m1, m2), which generalize the -Toda-like equation found previously for m1 = m2. By combining the bilinear relations with the Seiberg-like dualities and the duality cascade relations, we can determine the exact values of the partition functions recursively with respect to N. This method is more efficient than the exact calculation by the standard TBA-like approach in the Fermi gas formalism. As an application we study the large N asymptotics of the partition function with the mass parameters in the supercritical regime where the large N expansion obtained for small mass parameters is invalid. [ABSTRACT FROM AUTHOR]

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For a class of radially symmetric potentials with soft edges, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the O (log N) - and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to an additive constant depending on N but not on the background potential. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles. [ABSTRACT FROM AUTHOR]

We show that the determinant of the co-exact p-form on spheres and anti-de Sitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact p-form contains characters of anti-symmetric tensors of rank lower to p all the way to the zero-form. Using this result we evaluate the partition function of p-forms and demonstrate that they obey known properties under Hodge duality. We show that the partition function of conformal forms in even d + 1 dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of p-forms on branched spheres. [ABSTRACT FROM AUTHOR]

Let Q (n , d) be the number of representations of n by all reduced quadratic forms of discriminant d. We give an asymptotic formula for the number of n ≤ x such that 1 2 w (d) Q (k n + l , d) is odd, where k , l are integers and w (d) = 2 if d < − 4 , w (d) = 4 if d = − 4 and w (d) = 6 if d = − 3. As applications, we improve early results on odd values of the number of four-core partitions, the Rogers–Ramanujan's partitions and five-regular partitions. [ABSTRACT FROM AUTHOR]

Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function. [ABSTRACT FROM AUTHOR]

The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 핋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 핋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 핋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly. [ABSTRACT FROM AUTHOR]

Let A ⊂ N + and by P A (n) denotes the number of partitions of an integer n into parts from the set A. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form P A (x) = P B (y) , where A, B are certain finite sets. [ABSTRACT FROM AUTHOR]