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We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

We look for the origins of the single equation, which is a peculiar combination of W-constrains, which provides the non-abelian W-representation for generalized Kontsevich model (GKM), i.e. is enough to fix the partition function unambiguously. Namely we compare it with the scalar projection of the matrix Ward identity. It turns out that, though similar, the two equations do not coincide, moreover, the latter one is non-polynomial in time-variables. This discrepancy disappears for the cubic model if partition function is reduced to depend on odd times (belong to KdV sub-hierarchy of KP), but in general such reduction is not enough. We consider the failure of such direct interpretation of the "single equation" as a new kind of anomaly, which should be explained and eliminated in the future analysis of GKM., to Andrei Mironov on occasion of his 60's birthday; 12 pages

The sliding electrical contact interface will generate a huge total heat source including Joule heat and frictional heat when an armature slides along a pair of rails driven by Lorentz forces. The total heat is partitioned at the contact interface flowing into armature and rail respectively. The heat partition at the rail-armature interface and the resulting temperature rise have important effects on the melt wear characteristics of the contact material. A computational method was proposed for determining heat partition at the rail-armature interface based on least squares regression. Temperatures equations of rail and armature were developed in terms of an unknown heat partition function. Solution equations of heat partition function were derived by matching the average temperature of rail and armature at the contact surfaces. The heat partition function was assumed to be a polynomial form and an estimate for the heat partition function was obtained by minimizing the error term in the least squares sense. Then, effects of polynomial order on the heat partition function are analyzed. Furthermore, the heat partition solution from the least squares regression and previous analytical method are compared. Variation characteristics of heat partition coefficient were also analyzed. In the end, the variation mechanism and influencing factors of heat partition coefficient were discussed.

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial., Comment: 42 pages, 4 figures

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637

The partition function of the exactly solvable four-vertex model on a square grid in the presence of a linearly growing external field is calculated. Namely, we study a system when the external field is applied to one of the columns of the grid. The model is considered both for the fixed and periodic boundary conditions.

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$., Comment: 54 pages, Mathematica notebook attached; v2 typos fixed

We show that the fermionic matrix model can be realized by $W$-representation. We construct the Virasoro constraints with higher algebraic structures, where the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is a $\tau$-function of the KP hierarchy. We construct the fermionic Aristotelian tensor model and give its $W$-representation. Moreover, we analyze the fermionic red tensor model and present the $W$-representation and character expansion of the partition function., Comment: 20 pages. Revised version accepted for publication in Nuclear Physics B

In Alexandrov's work [4,5] it has been shown that the extended partition function exp⁡(Fo,ext+Fc) introduced by Buryak in [14,15] is a tau-function of the KP hierarchy. In this work, we compute the affine coordinates of this tau-function on the Sato Grassmannian, and rewrite the Virasoro constraints as recursions for the affine coordinates in the fermionic picture. As applications we derive some formulas for the extended partition function and the connected n-point functions using methods developed by Zhou in [48] based on the boson-fermion correspondence.

We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere., 20 pages + appendices

We discuss two-dimensional quantum gravity coupled to conformal matter and fixed area in a semiclassical large and negative matter central charge limit. In this setup the gravity theory -- otherwise highly fluctuating -- admits a round two-sphere saddle. We discuss the two-sphere partition function up to two-loop order from the path integral perspective. This amounts to studying Feynman diagrams incorporating the fixed area constraint on the round two-sphere. In particular we find that all ultraviolet divergences cancel to this order. We compare our results with the two-sphere partition function obtained from the DOZZ formula., Comment: 16 pages + appendices. arXiv admin note: text overlap with arXiv:2106.01665

We study the large $N$ 't Hooft expansion of the chiral partition function of 2d $U(N)$ Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation., Comment: 30 pages, 7 figures, v2: typos corrected, published version

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $\alpha$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace., Comment: 28 pages

Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential $X^{n+1}$. We propose to use the Hall-Littlewood polynomials at the parameter equal to the $n$-th root of unity as a generalization of the $Q$ Schur functions from $n=2$ to arbitrary $n>2$. They are associated with $n$-strict Young diagrams and are independent of time-variables $p_{kn}$ with numbers divisible by $n$. These are exactly the properties possessed by the generalized Kontsevich model (GKM), thus its partition function can be expanded in such functions $Q^{(n)}$. However, the coefficients of this expansion remain to be properly identified. At this moment, we have not found any "superintegrability" property $\,\sim character$, which expressed these coefficients through the values of $Q$ at delta-loci in the $n=2$ case. This is not a big surprise, because for $n>2$ our suggested $Q$ functions are not looking associated with characters., Comment: 13 pages

We construct a map between a class of codes over F4 and a family of non-rational Narain CFTs. This construction is complementary to a recently introduced relation between quantum stabilizer codes and a class of rational Narain theories. From the modular bootstrap point of view we formulate a polynomial ansatz for the partition function which reduces modular invariance to a handful of algebraic easy-to-solve constraints. For certain small values of central charge our construction yields optimal theories, i.e. those with the largest value of the spectral gap.

We revisit the calculation of the thermal free energy for string theory in three-dimensional anti-de Sitter spacetime with Neveu-Schwarz-Neveu-Schwarz flux. The path integral calculation is exploited to confirm the off-shell Hilbert space and we find that the Casimir of the discrete representations of the isometry group takes values in a half-open interval. We extend the free energy calculation to the case of superstrings, calculate the boundary toroidal twisted partition function in the Ramond-Ramond sector, and prove lower bounds on the boundary conformal dimension from the bulk perspective. We classify Ramond-Ramond ground states and construct their second quantized partition function. The partition function exhibits intriguing modular properties., Comment: 58 pages, 4 figures

In this paper, we study 5d $\mathcal{N}=1$ $Sp(N)$ gauge theory with $N_f ( \leq 2N + 3 )$ flavors based on 5-brane web diagram with $O5$-plane. On the one hand, we discuss Seiberg-Witten curve based on the dual graph of the 5-brane web with $O5$-plane. On the other hand, we compute the Nekrasov partition function based on the topological vertex formalism with $O5$-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten curve and its boundary conditions as well as its relation to the prepotential in terms of the cycle integrals. They coincide with those directly obtained from the dual graph of the 5-brane web with $O5$-plane. This agreement gives further evidence for mirror symmetry which relates Nekrasov partition function with Seiberg-Witten curve in the case with orientifold plane and shed light on the non-toric Calabi-Yau 3-folds including D-type singularities., Comment: v1:60 pages, 22 figures, v2: 67 pages, 26 figures, a new subsection and an appendix added

We investigate the disk partition function for the open string. This is a subtle problem because of the presence of a residual gauge group $\mathrm{PSL}(2,\mathbb{R})$ on the worldsheet even after fixing the conformal gauge. It naively has infinite volume and leads to a vanishing answer. We use different methods that all demonstrate that $\mathrm{PSL}(2,\mathbb{R})$ effectively behaves like a group with finite negative volume in the path integral, which leads to a simple prescription for the computation of the disk partition function. We apply our findings to give a simple rederivation of the D-brane tensions., Comment: 32 pages

$W$-representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models when the relevant operators are of a kind of $W$-operators: for the Hermitian matrix model with the Virasoro constraints, it is a $W_3$-like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is the appearance of an ordered P-exponential for the set of non-commuting operators of different gradings., Comment: 15 pages

We construct the one-dimensional topological sector of $\mathcal N = 6$ ABJ(M) theory and study its relation with the mass-deformed partition function on $S^3$. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at twoloop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge $c_T$ of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs., 38 pages, 2 figures, 3 tables

We explore the conjectured duality between a class of large $N$ matrix integrals, known as multicritical matrix integrals (MMI), and the series $(2m-1,2)$ of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an $S^2$ topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an $S^2$ and a $T^2$ topology using BRST cohomology considerations. Matrix integrals support this finiteness., 42 pages + appendices, comments welcome

We extend the Fermi gas approach to a class of ABJM-like necklace quiver theories without parity invariance. The resulting partition function on $S^3$ retains the form of an Airy function, but now includes a phase that scales as $Nk$ in the large-$N$ limit where $k$ is an overall Chern-Simons level. We demonstrate the presence of this phase both analytically and numerically in the case of a three node quiver., Comment: 20 pages, 2 figures

It is a well-known result by Hanany and Witten that, when two five-branes move across each other, D3-branes stretching between them are generated. Later the same brane configurations played a crucial role in understanding the worldvolume theory of multiple M2-branes. Recently the partition function of multiple M2-branes was transformed to the Fredholm determinant for quantum algebraic curves, where the characteristic 3/2 power law of degrees of freedom is reproduced and the determinant enjoys a large symmetry given by exceptional Weyl groups. The large exceptional Weyl group reproduces the Hanany-Witten brane transitions and, besides, contains brane transitions unknown previously. Aiming at understanding the new brane transitions better, we generalize our previous study on the D5 quantum curve to the E7 case, which requires delicate handling of degeneracies. By combining the results of these two cases, we propose a "local" rule for the brane transitions., 33 pages, 8 eps figures; clarifications added

Using Arakelov geometry, we compute the partition function of the noncompact free boson at genus two. We begin by compiling a list of modular invariants which appear in the Arakelov theory of Riemann surfaces. Using these quantities, we express the genus two partition function as a product of modular forms, as in the well-known genus one case. We check that our result has the expected obstruction to holomorphic factorization and behavior under degeneration., Comment: 21 pages; added subleading order degeneration calculations

We analyze quantum fluctuations around black hole solutions to the Jackiw-Teitelboim model. We use harmonic analysis on Euclidean AdS$_2$ to show that the logarithmic corrections to the partition function are determined entirely by quadratic holomorphic differentials, even when conformal symmetry is broken and harmonic modes are no longer true zero modes. Our quantum-corrected partition function agrees precisely with the SYK result. We argue that our effective quantum field theory methods and results generalize to other theories of two-dimensional dilaton gravity., 47 pages, 2 figures

For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary theory is very similar to a scalar field theory over p-adic numbers., Comment: arguement improved, version published in PLB

We investigate an N=4UNk×UN+M−k $$ \mathcal{N} = 4\;\mathrm{U}{(N)}_k\times \mathrm{U}{\left(N+M\right)}_{-k} $$ Chern-Simons theory coupled to one bifundamental hypermultiplet by employing its partition function, which is given by 2N + M dimensional integration via localization. Surprisingly, by performing the integration explicitly we find that the partition function completely factorizes into that of the pure Chern-Simons theory for two gauge groups and an analogous contribution for the bifundamental hypermultiplet. Using the factorized form of the partition function we argue the level/rank duality, which is also expected from the Hanany-Witten transition in the type IIB brane realization. We also present the all order ’t Hooft expansion of the partition function and comment on the connection to the higher-spin theory.

Geometry optimization and estimation of H/D reduced partition function ratios (RPFRs) of Pt19–H, Pt19–H+, Pt19–H2, and Pt19–H2+ models of hydrogen species adsorbed on surfaces of metallic platinum particles, were carried out based on the density functional theory. Two types of optimized structures were obtained for Pt19–H and Pt19–H+, and three for Pt19–H2 and Pt19–H2+. Stabilization energy consideration suggested that the most probable structure for Pt19–H and Pt19–H+ is the one in which H/H+ is bonded to a Pt atom, and that for Pt19–H2 and Pt19–H2+ is the one in which two H/H+s are bonded to two adjacent Pt atoms with no direct bonding between the two H/H+s. The value of RPFR obtained for those structures ranged from 3.9303 to 4.4014 at 25 °C. Interaction between two adjacent H–Pt bonds seems to slightly enhance the mutual RPFR values.

We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean $AdS_2$ background using the matrix model description recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that the partition function of JT gravity is written as the expectation value of a macroscopic loop operator in the old matrix model of 2d gravity in the background where infinitely many couplings are turned on in a specific way. Based on this expression we develop a very efficient method of computing the partition function in the genus expansion as well as in the low temperature expansion by making use of the Korteweg-de Vries constraints obeyed by the partition function. We have computed both these expansions up to very high orders using this method. It turns out that we can take a low temperature limit with the ratio of the temperature and the genus counting parameter held fixed. We find the first few orders of the expansion of the free energy in a closed form in this scaling limit. We also study numerically the behavior of the eigenvalue density and the Baker-Akhiezer function using the results in the scaling limit., Comment: 44 pages, 6 figures, data of genus and low temperature expansions attached, v2: typos corrected, published version

We consider the Laguerre partition function, and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed from our formulae. As a second result we identify the Laguerre partition function with only positive couplings and a special value of the parameter $\alpha=-1/2$ with the modified GUE partition function, which has recently been introduced as a generating function of Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals., Comment: 41 pages. v5: the value of the normalization constant $C_N$ in Theorem 1.5 has been corrected. No other result in the paper is affected by this change

We derive the partition function of an $\mathcal N=2$ chiral multiplet on topologically twisted $H^2\times S^1$. The chiral multiplet is coupled to a background vector multiplet encoding a real mass deformation. We consider an $ H^2\times S^1$ metric containing two parameters: one is the $S^1$ radius, while the other gives a fugacity $q$ for the angular momentum on $H^2$. The computation is carried out by means of supersymmetric localization, which provides a finite answer written in terms of $q$-Pochammer symbols and multiple Zeta functions. Especially, the partition function of normalizable fields reproduces three-dimensional holomorphic blocks., 14 pages, published version, added clarifications, corrected title

A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity., 30 pages, 5 figures

We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of the corresponding gauge theory. This is equivalent to the introduction of certain co-dimension two defects. We next generalize our construction to the folded instanton partition function obtained through the so-called "gauge origami" construction and precisely obtain the corresponding characteristic polynomial for the doubled version, named the elliptic double Calogero-Moser (edCM) system., Comment: 36 pages, added references

The disk partition function of certain 3d N=2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kahler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several Kahler moduli. We propose a multi-cover formula that relates the 3d BPS world-volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants., 25p, v2: references added

We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at given base-wrapping can be computed from the knowledge of the genus 0 Gromov-Witten invariants at this base-wrapping, and the partition function at lower base-wrappings. For the class of geometries leading to the higher rank E-strings, this leads to the result stated in the opening sentence., Comment: 40 pages

We develop an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size. The method is based on the algebro-geometric approach to the resolution of Bethe ansatz equations initiated in a previous work, and on further ingredients introduced in the present paper. The latter include rational Q-system, primary decomposition, algebraic extension and Galois theory. Using this approach, we probe new structures in the solution space of the Bethe ansatz equations which enable us to boost the efficiency of the computation. As an application, we study the zeros of the partition function in a partial thermodynamic limit of M × N tori with N ≫ M. We observe that for N → ∞ the zeros accumulate on some curves and give a numerical method to generate the curves of accumulation points., Journal of High Energy Physics, 2019 (3), ISSN:1126-6708, ISSN:1029-8479

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation $2\tau=g_s(1-b) $ between single K\"ahler parameter $\tau$, string coupling constant $g_s$ and refinement parameter $b$, where $b=\frac{5}{3},\frac{5}{2},3,4,6$ for $G_2, F_4, E_6, E_7, E_8$, respectively. The non-zero BPS invariants $N^d_{J_L,J_R}$ ($d$ - degree) are $N^2_{0,\frac{1}{2}}=1, N^{11}_{0,1}=1$. Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory. Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line $Exc$ with Vogel's parameters satisfying $\gamma=2(\alpha+\beta)$. This line contains points, corresponding to the all exceptional groups. The same results are obtained for $F$ line $\gamma=\alpha+\beta$ (containing $SU(4), SO(10)$ and $E_6$ groups), with the non-zero $N^2_{0,\frac{1}{2}}=1, N^{7}_{0,1}=1$. In both cases refinement parameter $b$ ($=-\epsilon_2/\epsilon_1$ in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by $b=-\beta/\alpha$., Comment: 8 pages

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m+1,2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level., 56 pages, 1 figure; v2, references added & updated

The duality symmetries of the STU-model of Sen and Vafa are very restrictive. This is utilized to determine the holomorphic function that encodes its two-derivative Wilsonian effective action and its couplings to the square of the Weyl tensor to fifth order in perturbation theory. At fifth order some ambiguities remain which are expected to resolve themselves when proceeding to the next order. Subsequently, a corresponding topological string partition function is studied in an expansion in terms of independent invariants of $S$, $T$ and $U$, with coefficient functions that depend on an effective duality invariant coupling constant $u$, which is defined on a Riemann surface $\mathbb{C}$. The coefficient function of the invariant that is independent of $S$, $T$ and $U$ is determined to all orders by resummation. The other functions can be solved as well, either algebraically or by solving differential equations whose solutions have ambiguities associated with integration constants. This determination of the topological string partition function, while interesting in its own right, reveals new qualitative features in the result for the Wilsonian action, which would be difficult to appreciate otherwise. It is demonstrated how eventually the various ambiguities are eliminated by comparing the results for the effective action and the topological string. While we only demonstrate this for the leading terms, we conjecture that this will hold in general for this model., Comment: 39 pages; v2: various text improvements

We consider the $T\bar{T}$ deformation of two dimensional Yang--Mills theory on general curved backgrounds. We compute the deformed partition function through an integral transformation over frame fields weighted by a Gaussian kernel. We show that this partition function satisfies a flow equation which has been derived previously in the literature, which now holds on general backgrounds. We connect ambiguities associated to first derivative terms in the flow equation to the normalization of the functional integral over frame fields. We then compute the entanglement entropy for a general state in the theory. The connection to the string theoretic description of the theory is also investigated., Comment: 17 pages, no figures, references updated, minor corrections and expanded discussions in v3

We consider the $S^2 \times_q S^1$ supersymmetric index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ when $M_3$ is a plumbed 3-manifold. We engineer an effective description of $T[M_3]$ from the expression of the homological block for plumbed 3-manifolds as a $D^2 \times_q S^1$ partition function of a 3d $\mathcal{N}=2$ theory $T[M_3]$ with a boundary condition. We check that the supersymmetric index for such a $T[M_3]$ is invariant under the 3d Kirby moves., Comment: 21 pages, 3 figures, v2 minor revisions, published version

We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS$_4$/CFT$_3$ correspondence. Such solutions are equipped with a supersymmetric Killing vector, and we show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk. We illustrate our general formula with a number of explicit examples, in which exact dual field theory computations are also available, which include supersymmetric Taub-NUT and Taub-bolt type spacetimes, as well as black hole solutions. Our simple topological formula also allows us to write down the action of any solution, provided it exists., Comment: 43 pages

In [arXiv:1805.05057 [hep-th]],[arXiv:1812.00811 [hep-th]], the partition function of the Gross-Witten-Wadia unitary matrix model with the logarithmic term has been identified with the $\tau$ function of a certain Painlev\'{e} system, and the double scaling limit of the associated discrete Painlev\'{e} equation to the critical point provides us with the Painlev\'{e} II equation. This limit captures the critical behavior of the $su(2)$, $N_f =2$ $\mathcal{N}=2$ supersymmetric gauge theory around its Argyres-Douglas $4D$ superconformal point. Here, we consider further extension of the model that contains the $k$-th multicritical point and that is to be identified with $\hat{A}_{2k, 2k}$ theory. In the $k=2$ case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model., Comment: 15 pages

We relate the heat kernel and quasinormal mode methods of computing the 1-loop partition function of arbitrary spin fields on a rotating (Euclidean) BTZ background using the Selberg zeta function associated with ℍ3/ℤ, extending (arXiv:1811.08433) [1]. Previously, Perry and Williams [2] showed for a scalar field that the zeros of the Selberg zeta function coincide with the poles of the associated scattering operator upon a relabeling of integers. We extend the integer relabeling to the case of general spin, and discuss its relationship to the removal of non-square-integrable Euclidean zero modes.

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process., 17 pages, no figures

It is shown that the generating function of $\mathscr{N}=2$ topological strings, in the heterotic weak coupling limit, is identified with the partition function of a six-dimensional Melvin background. This background, which corresponds to an exact CFT, realises in string theory the six-dimensional $\varOmega$-background of Nekrasov, in the case of opposite deformation parameters $\epsilon_1=-\epsilon_2$, thus providing the known perturbative part of the Nekrasov partition function in the field theory limit. The analysis is performed on both heterotic and type I strings and for the cases of ordinary $\mathscr{N}=2$ and $\mathscr{N}=2^*$ theories., Comment: 28 pages

We apply differential geometry methods to the computation of the anomaly-induced hydrodynamic equilibrium partition function. Implementing the imaginary-time prescription on the Chern-Simons effective action on a stationary background, we obtain general closed expressions for both the invariant and anomalous part of the partition function. This is applied to the Wess-Zumino-Witten action for Goldstone modes, giving the equilibrium partition function of superfluids. In all cases, we also study the anomaly-induced gauge currents and energy-momentum tensor, providing explicit expressions for them., 46 pages, LaTeX. v2: new subsection 5.3 included, discussion expanded, typos removed, and references added. It matches the version published in Journal of High Energy Physics

We derive the partition function of a non-relativistic quantum string which its ends are allowed to freely slide on the two angled straight solid rods. We first derive the classical solution of the model and then use it to derive the partition function utilizing the path integral method. We show that the vacuum energy is sum of the Luscher potential plus a term which depends on the relative angle between rods., 8 pages, 2 figures. Updated version of published article. Some typos corrected

We refine a previous proposal for obtaining the multi-instanton partition function from the supersymmetric index of the 1d supersymmetric gauge theory on the worldline of D0-branes. We provide examples where the refinements are crucial for obtaining the correct result., 30 pages, 6 figures. Typos corrected

We study the phase diagram of q-deformed Yang-Mills theory on S-2 at non-zero theta-angle using the exact partition function at finite N. By evaluating the exact partition function numerically, we find evidence for the existence of a series of phase transitions at non-zero theta-angle as conjectured in [hep-th/0509004]., Article, JOURNAL OF HIGH ENERGY PHYSICS.(4):59(2018)