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The $$\mathrm {SU}(r)$$ Vafa–Witten partition function, which virtually counts Higgs pairs on a projective surface S, was mathematically defined by Tanaka–Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $$\mu _r$$ -gerbes. In this paper, we instead use Yoshioka’s moduli spaces of twisted sheaves. Using Chern character twisted by rational B-field, we give a new mathematical definition of the $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ Vafa-Witten partition function when r is prime. Our definition uses the period-index theorem of de Jong. S-duality, a concept from physics, predicts that the $$\mathrm {SU}(r)$$ and $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ partition functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all K3 surfaces and prime numbers r.

We report accurate time-resolved measurements of NH3 desorption from Pt(111) and Pt(332) and use these results to determine elementary rate constants for desorption from steps, from (111) terrace sites and for diffusion on (111) terraces. Modeling the extracted rate constants with transition state theory, we find that conventional models for partition functions, which rely on uncoupled degrees of freedom (DOFs), are not able to reproduce the experimental observations. The results can be reproduced using a more sophisticated partition function, which couples DOFs that are most sensitive to NH3 translation parallel to the surface; this approach yields accurate values for the NH3 binding energy to Pt(111) (1.13 ± 0.02 eV) and the diffusion barrier (0.71 ± 0.04 eV). In addition, we determine NH3's binding energy preference for steps over terraces on Pt (0.23 ± 0.03 eV). The ratio of the diffusion barrier to desorption energy is 0.65, in violation of the so-called 12% rule. Using our derived diffusion/desorption rates, we explain why established rate models of the Ostwald process incorrectly predict low selectivity and yields of NO under typical reactor operating conditions. Our results suggest that mean-field kinetics models have limited applicability for modeling the Ostwald process. D.B. and M.S. thank the BENCh graduate school, funded by the DFG (389479699/GRK2455). I.R. gratefully acknowledges the support by Israel Science Foundation, ISF (Grant No. 2187/19), and by the Open University of Israel Research Authority (Grant No. 31044). O.G. acknowledges financial support by the Spanish Ministerio de Ciencia e Innovación (Grant No. PID2019-107396GB-I00/AEI/10.13039/501100011033). T.N.K., G.S., M.S., and J.F. acknowledge support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 833404).

In this work, we calculate the partition functions and thermodynamic quantities of molecular hydrogen isotopologues using the rovibrational energy levels provided by the highly accurate ab initio adiabatic potential energy functions recently determined by Pachucki and Komasa (Pachucki, K.; Komasa, J. J. Chem. Phys.2014, 141, 224103). The partition functions are calculated by including all bound energy levels of the isotopologues, up to their dissociation limits, plus the quasi-bound levels lying below the centrifugal potential barriers. For the homonuclear isotopologues, H2, D2, and T2, we also determine the partition functions and thermodynamic quantities of the normal mixtures using the statistical treatment recently proposed by Colonna et al. (Colonna, G.; D'Angola, A.; Capitelli, M. Int. J. Hydrogen Energy2012, 37, 9656) based on the definition of the partition function of the mixture, which avoids inconsistencies in the values of the thermodynamic quantities depending directly on the internal partition function, in the high-temperature limit.

We show that the determinant of the co-exact $p$-form on spheres and anti-deSitter 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 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., Comment: Enhanced discussion in section 2,appendix added. Revised version as accepted by JHEP

Chan studied the cubic partition function a(n), the number of partitions of a positive integer n where the even parts appear in two colors. Also, he obtained an infinite family of congruences modulo large powers of 3 for a(n), which are analogues to the Ramanujan-type congruences. Let $$\overline{a}_\ell (n)$$ denote the number of $$\ell $$ -regular cubic partitions of n where the first occurrence of each distinct odd part may be overlined. In this paper, we have established many infinite family of congruences modulo powers of 2 and 3 for $$\overline{a}_3(n)$$ and modulo powers of 2 for $$\overline{a}_5(n)$$ . For example, for all $$n \ge 0$$ and $$\alpha , \beta \ge 0$$ , $$\begin{aligned} \overline{a}_3\left(2^{2\alpha +5}\cdot 5^{2\beta +2}n+\dfrac{c_1\cdot 2^{2\alpha +4}\cdot 5^{2\beta +1}-1}{3}\right)\equiv 0 \pmod {27}, \end{aligned}$$ where $$c_1 \in \{11, 17, 23, 29\}$$ .

We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form $Q$. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by $Q$. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories., Comment: 38 pages, 1 figure

In this paper, we prove that if $f(x)=\sum_{k=0}^n{n\choose k}a_kx^k$ is a polynomial with real zeros only, then the sequence $\{a_k\}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k})^2/a_k^2 \leq(a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}^2(1+\sqrt{1-c_k})^2/a_k^2$, where $c_k=a_ka_{k+2}/a_{k+1}^2$. This inequality holds for the coefficients of the Riemann $\xi$-function, the ultraspherical, Laguerre and Hermite polynomials, and the partition function. Moreover, as a corollary, for the partition function $p(n)$, we prove that $p(n)^2-p(n-1)p(n+1)$ is increasing for $n\geq 55$. We also find that for a positive and log-concave sequence $\{a_k\}_{k\geq 0}$, the inequality $a_{k+2}/a_k\leq (a_{k+1}^2-a_ka_{k+2})/(a_k^2-a_{k-1}a_{k+1}) \leq a_{k+1}/a_{k-1}$ is the sufficient condition for both the $2$-log-concavity and the higher order Tur{\'a}n inequalities of $\{a_k\}_{k\geq 0}$. It is easy to verify that if $a_k^2\geq ra_{k+1}a_{k-1}$, where $r\geq 2$, then the sequence $\{a_k\}_{k\geq 0}$ satisfies this inequality.

We prove a distributional limit theorem conjectured in [Journal of Statistical Physics 174, No. 6, 1372-1403 (2019)] for partition functions defining models of directed polymers on diamond hierarchical graphs with disorder variables placed at the graphical edges. The limiting regime involves a joint scaling in which the number of hierarchical layers, $n\in \mathbb{N}$, of the graphs grows as the inverse temperature, $\beta\equiv \beta(n)$, vanishes with a fine-tuned dependence on $n$. The conjecture pertains to the marginally relevant disorder case of the model wherein the branching parameter $b \in \{2,3,\ldots\}$ and the segmenting parameter $s \in \{2,3,\ldots\}$ determining the hierarchical graphs are equal, which coincides with the diamond fractal embedding the graphs having Hausdorff dimension two. Unlike the analogous weak-disorder scaling limit for random polymer models on hierarchical graphs in the disorder relevant $b, Comment: 79 pages, 2 figures. Additional results have been added on the vertex-disorder version of the model (Theorem 3.1) and the rate of distributional convergence (Theorem 7.3). Various corrections and improvements to the presentation have been made in the text based on the suggestions given by three anonymous referees

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

We discuss the optimum communication capacity of bosonic information carriers propagating through a tunnel junction. Based on the multi-contour Keldysh Green function, we evaluate the information generating function, or the Renyi entanglement entropy, of the reduced density matrix subjected to the constraint of local heat quantity in the steady state. The Renyi entanglement entropy of order zero is the partition function, which exponentially depends on the optimum capacity. For the perfect transmission, the self-information and the local heat quantity, i.e., the energy of signals, are perfectly linearly correlated. The water-filling theorem is recovered in the wave-like regime.

We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of arXiv:1709.01928. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO($N$) gauge theories and the pure $G_2$ gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level $9$. At the end we rewrite the O-vertex in a form of a vertex operator., 41+12 pages, minor corrections in v2

The Baldoni--Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function fundamental in representation theory. On the other hand the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a part generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula of flow polytopes., 11 pages, 3 figures. v2: improvements to exposition

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the $N$ site $\mathfrak{sl}_{2}$ $XXX$ spin chain (for infinite dimensional complex spin representations), as well as the $SL_N$ Gaudin system, which reduces, in a limiting case, to that of the $N$-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture., 76 pages, 3 figures, fix typo

We consider string theory on $\text{AdS}_3 \times \text{S}^3 \times \mathbb{T}^4$ in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal $\text{AdS}_3$, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal $\text{AdS}_3$ and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal $\text{AdS}_3$ and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal $\text{AdS}_3$. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately., 49 pages; v2: added references and corrected typos

Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

We study 6d E-string theory with defects on a circle. Our basic strategy is to apply the geometric transition to the supersymmetric gauge theories. First, we calculate the partition functions of the 5d SU(3)$_0$ gauge theory with 10 flavors, which is UV-dual to the 5d Sp(2) gauge theory with 10 flavors, based on two different 5-brane web diagrams, and check that two partition functions agree with each other. Then, by utilizing the geometric transition, we find the surface defect partition function for E-string on $\mathbb{R}^4\times T^2$. We also discuss that our result is consistent with the elliptic genus. Based on the result, we show how the global symmetry is broken by the defects, and discuss that the breaking pattern depends on where/how we insert the defects., Comment: v1: 39 pages, 22 figures; v2: published version

We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius $l$ is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge $c=3l/2G_N$ ($G_N$ is the 3D Newton constant) equals $c=1/2$, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D {\bf 85}, 024032 (2012)]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the $c=1/2$ theory among all those with $c, 54 pages, 12 figures

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991 , Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017 ) relates gauge theoretic invariants of a knot K in the 3-sphere to the representation theory of a quiver $$Q_{K}$$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419–438, 2000), that relates knot invariants to counts of holomorphic curves with boundary on $$L_{K}$$ , the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d $${\mathcal {N}}=2$$ theory $$T[Q_{K}]$$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $$Q_{K}$$ . From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $$L_{K}$$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $$Q_{K}$$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $$Q_{K}$$ and (doubly) refined version of LMOV invariants (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423–449, 2001. arXiv:hep-th/0004196 ; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102 ; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709 ; Fuji et al. in Nucl Phys B 867:506–546, 2013. arXiv:1205.1515 ).

We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large $N$ results are obtained by using Szeg\"o theorem with a Fisher-Hartwig singularity. In the large $N$ (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space., Comment: 22 pages

There is a study on a square lattice that can predict the existence of multiple phase transitions on a complex plane. We extend the study on the different types of ZQ-symmetric model and different lattices in order to provide more evidence to the existence of multiple phase transitions. We focus on the ZQ-symmetric model with the nearest neighbour interaction on the six spin directions between molecular dipole, i.e. Q = 6 on a triangular lattice. Mainly, the model is defined on the triangular lattice graph with the nearest neighbour interaction. By using the transfer matrix approach, the partition functions are computed for increasing lattice sizes. The roots of polynomial partition function are also computed and plotted in the complex Argand plane. The specific heat equation is used for further comparison. The result supports the existence of the multiple phase transitions by the emergence of the multiple line curves in the locus of zeros distribution for specific type of energy level.

We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of ${\mathcal N}=2$ abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $\tau$. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of $\tau$. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. We comment on S-dualities which relate different theories of supersymmetric graphene. We identify a couple of self-dual theories for which the complexified conductivity associated to the U(1) gauge symmetry is $\tau/2$., Comment: 36 pages, 6 figures; v2 refs added

Protein design algorithms that model continuous sidechain flexibility and conformational ensembles better approximate the in vitro and in vivo behavior of proteins. The previous state of the art, iMinDEE-\(A^*\)-\(K^*\), computes provable \(\varepsilon \)-approximations to partition functions of protein states (e.g., bound vs. unbound) by computing provable, admissible pairwise-minimized energy lower bounds on protein conformations and using the \(A^*\) enumeration algorithm to return a gap-free list of lowest-energy conformations. iMinDEE-A\(^*\)-\(K^*\) runs in time sublinear in the number of conformations, but can be trapped in loosely-bounded, low-energy conformational wells containing many conformations with highly similar energies. That is, iMinDEE-\(A^*\)-\(K^*\) is unable to exploit the correlation between protein conformation and energy: similar conformations often have similar energy. We introduce two new concepts that exploit this correlation: Minimization-Aware Enumeration and Recursive \(K^{*}\). We combine these two insights into a novel algorithm, Minimization-Aware Recursive \(K^{*}\) (\({ MARK}^{*}\)), that tightens bounds not on single conformations, but instead on distinct regions of the conformation space. We compare the performance of iMinDEE-\(A^*\)-\(K^*\) vs. \({ MARK}^{*}\) by running the \(BBK^*\) algorithm, which provably returns sequences in order of decreasing \(K^{*}\) score, using either iMinDEE-\(A^*\)-\(K^*\) or \({ MARK}^{*}\) to approximate partition functions. We show on 200 design problems that \({ MARK}^{*}\) not only enumerates and minimizes vastly fewer conformations than the previous state of the art, but also runs up to two orders of magnitude faster. Finally, we show that \({ MARK}^{*}\) not only efficiently approximates the partition function, but also provably approximates the energy landscape. To our knowledge, \({ MARK}^{*}\) is the first algorithm to do so. We use \({ MARK}^{*}\) to analyze the change in energy landscape of the bound and unbound states of the HIV-1 capsid protein C-terminal domain in complex with camelid V\(_{\mathrm{{H}}}\)H, and measure the change in conformational entropy induced by binding. Thus, \({ MARK}^{*}\) both accelerates existing designs and offers new capabilities not possible with previous algorithms.

The use of partition function zeros in the study of phase transition is growing in the last decade mainly due to improved numerical methods as well as novel formulations and analysis. In this paper the impact of different parameters choice for the energy probability distribution (EPD) zeros recently introduced by Costa et al is explored in search for optimal values. Our results indicate that the EPD method is very robust against parameter variations and only small deviations on estimated critical temperatures are found even for large variation of parameters, allowing to obtain accurate results with low computational cost. A proposal to circumvent potential convergence issues of the original algorithm is presented and validated for the case where it occurs.

Thermal rate coefficients for the hydrogen-abstraction reactions of 3-butenal by a hydrogen atom were obtained applying multipath canonical variational theory with small-curvature tunneling (MP-CVT/SCT). Torsional anharmonicity due to the hindered rotors was taken into account by calculating the rovibrational partition function using the extended two-dimensional torsional (E2DT) method. For comparison, rovibrational partition functions were also estimated using the multistructural method with torsional anharmonicity based on a coupled torsional potential (MS-T(C)). By contrast, with (E)-2-butenal reactions, the abstraction reactions of 3-butenal proceed via five reaction channels (R1)–(R5). In a conformational search, 45 distinguishable structures of transition states were found, including enantiomers, which were separated into six conformational reaction channels (CRCs). The individual reactive paths were constructed, the recrossing and semiclassical transmission coefficients estimated, and the multipath rate constants were obtained. High torsional barriers between the wells of CRC2/CRC6 indicate a harmonic behavior. Consequently, a difference between the torsional anharmonicity of 3-butenal and the transition states is responsible for the increase in the thermal rate constants for channel (R2). Analysis of the contributions of each conformer of the transition state shows an important contribution of the high-energy rotamers in the total flux of (R1)–(R5). After fitting the rate constants in a four-parameter equation, the activation energy estimation showed a strong temperature dependence.

In this work we extend the holographic correspondence between flat space limit of chiral topological massive gravity and a specific unitary field theory to flat space limit of chiral general minimal massive gravity and chiral exotic general massive gravity. To check the conjecture, we compare the entropy and partition function of flat limit of BTZ black hole from viewpoint of both sides.

The generating functional of stress tensor correlation functions in two-dimensional conformal field theory is the nonlocal Polyakov action, or equivalently, the Liouville or Alekseev-Shatashvili action. I review its holographic derivation within the AdS$_3$/CFT$_2$ correspondence, both in metric and Chern-Simons formulations. I also provide a detailed comparison with the well-known Hamiltonian reduction of three-dimensional gravity to a flat Liouville theory, and conclude that the two results are unrelated. In particular, the flat Liouville action is still off-shell with respect to bulk equations of motion, and simply vanishes in case the latter are imposed. The present study also suggests an interesting re-interpretation of the computation of black hole spectral statistics recently performed by Cotler and Jensen as that of an explicit averaging of the partition function over the boundary source geometry, thereby providing potential justification for its agreement with the predictions of a random matrix ensemble., 24 pages + appendices, 1 figure. v2: corrected typos + new discussion of the variational principle where only the conformal class of boundary metrics is fixed. v3: matches the published version

We propose a novel codimension-n holography, called cone holography, between a gravitational theory in $(d+1)$-dimensional conical spacetime and a CFT on the $(d+1-n)$-dimensional defects. Similar to wedge holography, the cone holography can be obtained by taking the zero-volume limit of holographic defect CFT. Remarkably, it can be regarded as a holographic dual of the edge modes on the defects. For one class of solutions, we prove that the cone holography is equivalent to AdS/CFT, by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. In general, cone holography and AdS/CFT are different due to the infinite towers of massive Kaluza-Klein modes on the branes. We test cone holography by studying Weyl anomaly, Entanglement/R\'enyi entropy and correlation functions, and find good agreements between the holographic and the CFT results. In particular, the c-theorem is obeyed by cone holography. These are strong supports for our proposal. We discuss two kinds of boundary conditions, the mixed boundary condition and Neumann boundary condition, and find that they both define a consistent theory of cone holography. We also analyze the mass spectrum on the brane and find that the larger the tension is, the more continuous the mass spectrum is. The cone holography can be regarded as a generalization of the wedge holography, and it is closely related to the defect CFT, entanglement/R\'enyi entropy and AdS/BCFT(dCFT). Thus it is expected to have a wide range of applications., Comment: 43 pages, 10 figures, minor revision published in PRD

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(\tau)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \pi i \tau}$ approaches a root of unity, i.e. as $\widetilde \tau \equiv m \tau + n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(\tau)$ has terms of the form $\widetilde \tau^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde \tau \to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index., Comment: v3: minor corrections and clarifications added

We explore thermodynamic contributions to the three-dimensional de Sitter horizon originating from metric and Chern-Simons gauge field fluctuations. In Euclidean signature these are computed by the partition function of gravity coupled to matter semi-classically expanded about the round three-sphere saddle. We investigate a corresponding Lorentzian picture - drawing inspiration from the topological entanglement entropy literature - in the form of an edge-mode theory residing at the de Sitter horizon. We extend the discussion to three-dimensional gravity with positive cosmological constant, viewed (semi-classically) as a complexified Chern-Simons theory. The putative gravitational edge-mode theory is a complexified version of the chiral Wess-Zumino-Witten model associated to the edge-modes of ordinary Chern-Simons theory. We introduce and solve a family of complexified Abelian Chern-Simons theories as a way to elucidate some of the more salient features of the gravitational edge-mode theories. We comment on the relation to the AdS$_4$/CFT$_3$ correspondence., 34 pages + appendices, 5 figures

We investigate gauge anomalies in the context of orbifold conformal field theories. Such anomalies manifest as failures of modular invariance in the constituents of the orbifold partition function. We review how this irregularity is classified by cohomology and how extending the orbifold group can remove it. Working with such extensions requires an understanding of the consistent ways in which extending groups can act on the twisted states of the original symmetry, which leads us to a discrete-torsion like choice that exists in orbifolds with trivially-acting subgroups. We review a general method for constructing such extensions and investigate its application to orbifolds. Through numerous explicit examples we test the conjecture that consistent extensions should be equivalent to (in general multiple copies of) orbifolds by non-anomalous subgroups., 37 pages, 2 appendices, 3 figures

We present a new expression for the partition function of the refined Chern-Simons theory on S3 with an arbitrary gauge group, which is explicitly equal to 1 when the coupling constant is zero. Using this form of the partition function we show that the previously known Krefl-Schwarz representation of the partition function of the refined Chern-Simons theory on S3 can be generalized to all simply laced algebras.For all non-simply laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with the refined topological strings.

The fractal family $\left\{p\left(n, k\right), k \in \mathbb{N}\right\}$, describe a rule to calculate the number of partitions obtained by decomposing $n\in \mathbb{N}$, into exactly $k$ parts. In this paper, we will present a novel method for proving that polynomials $\left\{p\left(n, k\right), k \in \mathbb{N} \right\}$ have fractal form. For each class $k$, up to the $LCM\left(1, 2, 3, \dots, k\right)$, different polynomials of degree $k-1$ are needed to form one quasi-polynomial $p\left(n, k\right)$. All the polynomials (needed for the same class $k$) have all coefficients of the higher degrees ending with the $ \left[\frac{k}{2}\right] $ degree in common. Moreover, we will prove that, for a fixed value of $k$, all the first, second, etc. coefficients of the common part of the fractal family have a general form, showing the vertical connection between the corresponding coefficients of all fractal family $\left\{ p\left(n, k\right), k\in \mathbb{N}\right\} $. Furthermore, for a fixed value of $k$, all the coefficients within the same polynomial have a unique general form, showing the horizontal connection of the coefficients of the polynomial $p\left(n, k\right)$. The partition function is not real a polynomial, but it can be written as a fractal polynomial which can be obtained from the general form of the partition class functions $\left\{p\left(n, k\right)\right\}$. In that case, the partition function for each $n$ uses a different polynomial. We show that all these polynomials can be combined with one single in which each member can be a formula for calculating the total number of partitions of all natural numbers.

We develop several ensemble learning models to predict the partition function of fluids adsorbed in nanoporous materials. The partition functions so obtained allow rapid access to all thermodynamic...

We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin$_\pm$ case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin$_-$ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the $\mathbb{Z}_8$ classification of (1+1)d topological superconductors. We also compute the indicator formula of $\mathbb{Z}_{16}$ valued time-reversal anomaly for (2+1)d pin$_+$ TQFT based on our construction., 26 pages, 6 figures

We briefly present the results of our computation of special Kahler geometry for polynomial deformations of Berglund–Hubsch type Calabi–Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that its partition function computed by supersymmetric localization coincides with exponent of the Kahler potential of the special metric.

We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d N=2 and 5d N=1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO(N) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part., 35+15 pages, 3 figures, v2: references added, minor corrections

The connection matrix of the Ising model on a d -dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.

We study the symmetric six-vertex model on a finite square lattice with partial domain wall boundary conditions. We use the known connection of the model to the off-shell Bethe states of the Heisenberg XXZ spin chain. We obtain various formulas for the partition function, and also discuss the model in the limit of semiinfinite lattice.

In his famous 2007 paper on three-dimensional quantum gravity, Witten defined candidates for the partition functions \( \mathit Z_k(q) = \sum_{n=-k}^{\infty} \omega_k(n)q^n \) of potential extremal conformal field theories (CFTs) with central charges of the form c = 24k. Although such CFTs remain elusive, he proved that these modular functions are well defined. In this note, we point out several explicit representations of these functions. These involve the partition function p(n), Faber polynomials, traces of singular moduli, and Rademacher sums. Furthermore, for each prime p ≤ 11, the p series Zk(q), where k e {1, . . . , p−1}∪{p+1}, possess a Ramanujan congruence. More precisely, for every non-zero integer n we have that .

We study the probability distribution, $P_N(T)$, of the coincidence time $T$, i.e. the total local time of all pairwise coincidences of $N$ independent Brownian walkers. We consider in details two geometries: Brownian motions all starting from $0$, and Brownian bridges. Using a Feynman-Kac representation for the moment generating function of this coincidence time, we map this problem onto some observables in three related models (i) the propagator of the Lieb Liniger model of quantum particles with pairwise delta function interactions (ii) the moments of the partition function of a directed polymer in a random medium (iii) the exponential moments of the solution of the Kardar-Parisi-Zhang equation. Using these mappings, we obtain closed formulae for the probability distribution of the coincidence time, its tails and some of its moments. Its asymptotics at large and small coincidence time are also obtained for arbitrary fixed endpoints. The universal large $T$ tail, $P_N(T) \sim \exp(- 3 T^2/(N^3-N))$ is obtained, and is independent of the geometry. We investigate the large deviations in the limit of a large number of walkers through a Coulomb gas approach. Some of our analytical results are compared with numerical simulations., 33 pages, 6 figures

We consider the Fr\"ohlich model of the Polaron whose path integral formulation leads to the transformed path measure $$ \widehat{\mathbb P}_{\alpha,T}(\mathrm d\omega)= Z_{\alpha,T}^{-1}\,\, \exp\bigg\{\frac{\alpha}{2}\int_{-T}^T\int_{-T}^T\frac{e^{-|t-s|}}{|\omega(t)-\omega(s)|} \, d s \, d t\bigg\}\,\mathbb P(\mathrm d\omega) $$ with respect to $\mathbb P$ which governs the law of the increments of the three dimensional Brownian motion on a finite interval $[-T,T]$, and $ Z_{\alpha,T}$ is the partition function or the normalizing constant and $\alpha>0$ is a constant. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$ g_0=\lim_{\alpha \to\infty}\frac{1}{\alpha^2}\bigg[\lim_{T\to\infty}\frac{\log Z_{\alpha,T}}{2T}\bigg] $$ exists and has a variational formula. In this article we show that for any $\alpha>0$, the infinite-volume limit $\widehat{\mathbb P}_{\alpha}=\lim_{T\to\infty}\widehat{\mathbb P}_{\alpha,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $\alpha>0$ and as $T\to\infty$) for the distribution of $\frac{\omega(T)-\omega(-T)}{\sqrt{2T}}$ both under the finite-volume Polaron measure $\widehat{\mathbb P}_{\alpha,T}$ and its infinite-volume counterpart $\widehat{\mathbb P}_\alpha$, and obtain an expression for the limiting variance., Comment: Theorem 4.5 (in the current version) was earlier stated (as Thm 4.8 in v3 and in the published version) for $\alpha\in (0,\alpha_0) \cup (\alpha_1,\infty)$, but its proof for $\alpha>\alpha_1$ there had a gap. In v4 (v5-v6 contain some simplifications) this gap has been fixed where Theorem 4.5 is shown for all coupling parameter $\alpha>0$. Consequently, the main results hold also for all $\alpha$

We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating function of the number of generalized lattice paths and a new version of Chaundy-Bullard identity for the vector partition function., Comment: 10 pages. The article will appear in JDEA - Journal of Difference Equations and Applications (GDEA 1649396)

The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.

$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the $\hat W$ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for $\hat W$ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between $\beta$- and $q,t$-deformations -- which is not always seen at the accidently simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots., Comment: 11 pages

Topological phase transitions such as the Berezinskii-Kosterlitz-Thouless (BKT) transition are difficult to characterize due to the difficulty in defining an appropriate order parameter or to unravel its critical properties. In this paper, we discuss the application of a newly introduced numerical algorithm that was inspired by the Fisher zeros of the partition function and is based on the partial knowledge of the zeros of the energy probability distribution (EPD zeros). This iterative method has proven to be quite general, furnishing the transition temperature with great precision and a relatively low computational effort. Since it does not need the a priori knowledge of any order parameter it provides an unbiased estimative of the transition temperature being convenient to the study of this kind of phase transition. Therefore, we applied the EPD zeros approach to the 2D XY model, which is well known for showing a BKT transition, in order to demonstrate its effectivity in the study of the BKT transition. Our results are consistent with the real and imaginary parts of the pseudo-transition temperature, T(L), having a different asymptotic behavior, which suggests a way to characterize a BKT like transition.

A diagonally connected two-dimensional lattice is the objective of the research. The model draws interest because each of its spins has 6 connects as in a 3D lattice. On the other hand, the planarity of the model allows us to use a strict polynomial algorithm to find its partition function and other characteristics. The Kasteleyn-Fisher algorithm is employed to carry out a computer simulation which enables us to see how the heat capacity behaves with the increasing lattice dimensionality. Given a finite lattice dimensionality, it is impossible to draw a definite conclusion, yet there is every reason to believe that the heat capacity diverges logarithmically at the critical point.

The mirror curves enable us to study B-model topological strings on non-compact toric Calabi--Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with single brane. In this paper, we discuss two types of geometries; one is the chain of $N$ $\mathbb{P}^1$'s which we call `$N$-chain geometry,' the other is the chain of $N$ $\mathbb{P}^1$'s with a compactification which we call `periodic $N$-chain geometry.' We calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization. Through the computation, we find some difference equations of (elliptic) hypergeometric functions. We also find a relation between the periodic chain and $\infty$-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the large $N$ limit., 28 pages, 10 figures, v2: references added, text corrected, published in JHEP

Polymer chains undergoing adsorption are expected to show universal critical behavior which may be investigated using partition function zeros. The focus of this work is the adsorption transition for a continuum chain, allowing for investigation of a continuous range of the attractive interaction and comparison with recent high-precision lattice model studies. The partition function (Fisher) zeros for a tangent-hard-sphere $N$-mer chain (monomer diameter $\ensuremath{\sigma}$) tethered to a flat wall with an attractive square-well potential (range $\ensuremath{\lambda}\ensuremath{\sigma}$, depth $\ensuremath{\epsilon}$) have been computed for chains up to $N=1280$ with $0.01\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}2.0$. In the complex-Boltzmann-factor plane these zeros are concentrated in an annular region, centered on the origin and open about the real axis. With increasing $N$, the leading zeros, ${w}_{1}(N)$, approach the positive real axis as described by the asymptotic scaling law ${w}_{1}(N)\ensuremath{-}{y}_{c}\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\phi}}$, where ${y}_{c}={e}^{\ensuremath{\epsilon}/{k}_{B}{T}_{c}}$ is the critical point and ${T}_{c}$ is the critical temperature. In this work, we study the polymer adsorption transition by analyzing the trajectory of the leading zeros as they approach ${y}_{c}$ in the complex plane. We use finite-size scaling (including corrections to scaling) to determine the critical point and the scaling exponent $\ensuremath{\phi}$ as well as the approach angle ${\ensuremath{\theta}}_{c}$, between the real axis and the leading-zero trajectory. Variation of the interaction range $\ensuremath{\lambda}$ moves the critical point, such that ${T}_{c}$ decreases with $\ensuremath{\lambda}$, while the results for $\ensuremath{\phi}$ and ${\ensuremath{\theta}}_{c}$ are approximately independent of $\ensuremath{\lambda}$. Our values of $\ensuremath{\phi}=0.479(9)$ and ${\ensuremath{\theta}}_{c}=56.8{(1.4)}^{\ensuremath{\circ}}$ are in agreement with the best lattice model results for polymer adsorption, further demonstrating the universality of these constants across both lattice and continuum models.

A new method is presented for one to obtain the molar heat capacities of diatomic macroscopic gas with a full set of microscopic molecular rovibrational energies. Based on an accurate experimental vibrational energies subset of a diatomic electronic ground state, the full vibrational energies can be obtained by using the variational algebraic method (VAM), the potential energy curves (PECs) will be constructed by the Rydberg-Klein-Rees (RKR) method, the full set of rovibrational energies will be calculated by the LEVEL program, and then the partition functions and the molar heat capacities of macroscopic gas can be calculated with the help of the quantum statistical ensemble theory. Applying the method to the ground state HBr and HCl gases, it is found that the relative errors of the partition functions calculated in the temperature range of 300 ∼ 6000 K are in excellent agreement with those obtained from TIPS database, and the calculated molar heat capacities are closer to the experimental values than those calculated by other methods without considering the energy levels of highly excited quantum states. The present method provides an effective new way for one to obtain the full set of molecular rovibrational energies and the molar heat capacities of macroscopic gas through the microscopic spectral information of a diatomic system.

We apply the theory of harmonic analysis on the fundamental domain of $SL(2,\mathbb{Z})$ to partition functions of two-dimensional conformal field theories. We decompose the partition function of $c$ free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space $\mathbb H/SL(2,\mathbb Z)$, and of target space moduli space $O(c,c;\mathbb Z)\backslash O(c,c;\mathbb R)/O(c)\times O(c)$. This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS$_3$ gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies., 35+24 pages, v2: corrected a mistake in Sec 3