Showing papers on "Partition function (quantum field theory) published in 2019"
23 Jun 2019
TL;DR: It is shown that a high dimensional walk on a weighted simplicial complex mixes rapidly if for every link of the complex, the corresponding localized random walk on the 1-skeleton is a strong spectral expander, and an FPRAS is designed to count the number of bases of any matroid given by an independent set oracle.
Abstract: We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0
147 citations
TL;DR: The Pólya–Jensen criterion for the Riemann hypothesis asserts that RH is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d≥1 and all shifts n, and this criterion is confirmed for all sufficiently large shifts n.
Abstract: In 1927, Polya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ ( s ) at its point of symmetry. This hyperbolicity has been proved for degrees d ≤ 3 . We obtain an asymptotic formula for the central derivatives ζ ( 2 n ) ( 1 / 2 ) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d ≤ 8 . These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.
84 citations
TL;DR: In this article, the authors systematically studied the interesting relations between the quantum elliptic Calogero-Moser system and its generalization, and their corresponding supersymmetric gauge theories, and constructed the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition functions of the corresponding gauge theory.
Abstract: 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.
56 citations
TL;DR: In this article, the rank N magnificent four theory is studied, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter (a super-group U(n|N) gauge theory) on a Calabi-Yau fourfold.
Abstract: We study the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang–Mills theory with matter (a super-group U(N|N) gauge theory) on a Calabi–Yau fourfold. Our theory contains the higher rank Donaldson–Thomas theory of threefolds. We conjecture an explicit formula for the partition function $${\mathcal {Z}}$$, and report on the performed checks. The partition function $${\mathcal {Z}}$$ has a free field representation. Surprisingly, it depends on the Coulomb and mass parameters in a simple way. We also clarify the definition of the instanton measure.
52 citations
TL;DR: In this article, a quasi-modular differential operator acting on the torus partition function is proposed to determine the thermal correlation functions of two-dimensional CFTs with quantum KdV charges.
Abstract: 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 an expression for an arbitrary correlation function which is determined 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.
49 citations
TL;DR: In this paper, Liouville quantum field theory on Riemann surfaces of genus $g\geq 2$ has been studied and it has been shown that it is a conformal field theory.
Abstract: Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $g\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory [Po] (also called 2d bosonic string theory) and to Liouville quantum gravity. More specifically, we show the convergence of Polyakov's partition function over the moduli space of Riemann surfaces in genus $g\geq 2$ in the case of D=1 boson. This is done by performing a careful analysis of the behaviour of the partition function at the boundary of moduli space. An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.
46 citations
TL;DR: In this paper, a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS4/CFT3 correspondence is derived, and the holographically renormalized action may be expressed in terms of the weights of this vector field at its fixed points, together with certain topological data.
Abstract: We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS4/CFT3 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.
34 citations
TL;DR: In this paper, the authors attempt to classify topological phases in 1D interacting non-Hermitian systems and show that the classification of these phases is exactly the same as their Hermitian counterparts.
Abstract: Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.
28 citations
TL;DR: In this article, an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size was developed 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.
Abstract: 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.
26 citations
TL;DR: In this article, the authors considered 4d N = 1 gauge theories with R-symmetry on a hemisphere times a torus and applied localization techniques to evaluate the exact partition function through a cohomological reformulation of...
Abstract: We consider 4d N = 1 gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of ...
25 citations
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TL;DR: In this article, a generalization of the deformation to curved spaces is proposed, where a suitable flow equation for the partition function is defined at the quantum level and a connection to 3D gravity is made.
Abstract: We propose a generalisation of the $T \bar{T}$ deformation to curved spaces by defining, and solving, a suitable flow equation for the partition function. We provide evidence it is well-defined at the quantum level. This proposal identifies, for any CFT, the $T \bar{T}$ deformed partition function and a certain wavefunction of 3d quantum gravity. This connection, true for any $c$, is not a holographic duality --- the 3d theory is a "fake bulk." We however emphasise that this reduces to the known holographic connection in the classical limit. Concretely, this means the deformed partition function solves exactly not just one global equation, defining the $T \bar{T}$ flow, but in fact a local Wheeler-de Witt equation, relating the $T \bar{T}$ operator to the trace of the stress tensor. This also immediately suggests a version of the $T \bar{T}$ deformation with locally varying deformation parameter. We flesh out the connection to 3d gravity, showing that the partition function of the deformed theory is precisely a 3d gravity path integral. In particular, in the classical limit, this path integral reproduces the holographic picture of Dirichlet boundary conditions at a finite radius and mixed boundary conditions at the asymptotic boundary. Further, we reproduce known results in the flat space limit, as well as the large $c$ $S^2$ partition function, and conjecture an answer for the finite $c$ $S^2$ partition function.
TL;DR: In this article, the authors studied 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 partition function that can be salvaged.
Abstract: 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.
TL;DR: The thermal partition function of JT gravity in asymptotically Euclidean $AdS_2$ background using the matrix model description was studied by Saad, Shenker and Stanford.
Abstract: 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.
04 Sep 2019
TL;DR: In this paper, a pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions is presented, illustrated in ferromagnetism, but can be applied to other physical systems.
Abstract: This article presents a pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. The method is illustrated in ferromagnetism, but can be applied to other physical systems.
TL;DR: In this paper, the authors study the limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide.
Abstract: Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Koscaz and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\mathbb{C}^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.
TL;DR: In this paper, the partition function and expectation values of Wilson loop operators in Chen-Simons theory on general lens spaces L(p,q) (including S2 × S1) were calculated using localization technique.
Abstract: Using localization technique, we calculate the partition function and expectation values of Wilson loop operators in Chen-Simons theory on general lens spaces L(p,q) (including S2 × S1). Our results are consistent with known results.
TL;DR: In this paper, an expression for the spectral density in terms of the light spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1, was proposed.
Abstract: 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 $j
eq 0$. 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.
TL;DR: In this paper, the authors studied the 3D gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several K¨ahler moduli.
Abstract: 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 K¨ahler moduli. We propose a multi-cover formula that relates the 3d BPS world- volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants.
TL;DR: In this paper, the authors studied the finite temperature effects of a mass dimension one fermionic field, sometimes called Elko field, and the equilibrium partition function was calculated by means of the imaginary ti...
Abstract: This work studies the finite temperature effects of a mass dimension one fermionic field, sometimes called Elko field. The equilibrium partition function was calculated by means of the imaginary ti...
TL;DR: Recently, a new partition function associated with Ramanujan's third order mock theta function ω ( q ) was discovered, and subsequently its overpartition analogue was introduced as discussed by the authors.
TL;DR: In this article, the authors identify the Kontsevich-penner matrix integral with the isomonodromic tau function for finite size n and derive the string and Dilaton equations via a purely Riemann-Hilbert approach.
Abstract: We identify the Kontsevich–Penner matrix integral, for finite size n, with the isomonodromic tau function of a $$3\times 3$$
rational connection on the Riemann sphere with n Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann–Hilbert boundary value problem, we can pass to the limit $$n\rightarrow \infty $$
(at a formal level) and identify an isomonodromic system in terms of Miwa variables, which play the role of times of a KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann–Hilbert approach. The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formulae for the correlators of this matrix model in terms of the solution of the Riemann Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.
TL;DR: In this article, the authors extended the Gross-Witten-wadia unitary one-matrix model by the logarithmic potential, which is a way to study Argyres-Douglas type theory.
Abstract: We continue to study the matrix model of the $N_f =2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlev\'{e} system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomial and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.
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TL;DR: An FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides of the bipartition is given.
Abstract: We give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides $(L,R)$ of the bipartition. This includes, among others, the biregular case when $\lambda=1$ (approximating the number of independent sets of $G$) and $\Delta_R \geq 7\Delta_L \log(\Delta_L)$. Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. As a consequence of the method, we also prove that the hard-core model on such graphs exhibits exponential decay of correlations by utilizing connections between the cluster expansion and joint cumulants.
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TL;DR: The complexity/partition function relation is utilized to study the complexity of the thermofield double state of extended SYK models for various conditions and the relation between complexity growth rate and black hole phase transition is discussed.
Abstract: We examine the complexity/volume conjecture and further investigate the possible connections between complexity and partition function. The complexity/volume 2.0 states that the complexity growth rate $\mathcal{\dot{C}}\sim PV$. In the standard statistics, there is a fundamental relation among $PV$, the grand potential $\Omega$ and the partition function $\mathcal{Z}$. By using this relation, we are able to construct an ansatz between complexity and partition function. The complexity/partition function relation is then utilized to study the complexity of the thermofield double state of extended SYK models for various conditions. The relation between complexity growth rate and black hole phase transition is also discussed.
TL;DR: In this article, the authors discuss congruences satisfied by the partition function $p(n)$ when $k$ is a rational number, which corresponds to the case when $n = 1.
Abstract: Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discuss congruences satisfied by $p_{k}(n)$ when $k$ is a rational number.
TL;DR: A random matrix model with both pairwise and non-pairwise contracted indices is considered, and the properties of the wave function of a toy model closely related to a tensor model in the Hamilton formalism are investigated.
Abstract: We consider a random matrix model with both pairwise and non-pairwise contracted indices. The partition function of the matrix model is similar to that appearing in some replicated systems with random tensor couplings, such as the p-spin spherical model for the spin glass. We analyze the model using Feynman diagrammatic expansions, and provide an exhaustive characterization of the graphs which dominate when the dimensions of the pairwise and (or) non-pairwise contracted indices are large. We apply this to investigate the properties of the wave function of a toy model closely related to a tensor model in the Hamilton formalism, which is studied in a quantum gravity context, and obtain a result in favor of the consistency of the quantum probabilistic interpretation of this tensor model.
TL;DR: It is shown how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture.
Abstract: We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would refute the Unique Games Conjecture.
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TL;DR: In this article, it was shown that the partition function of quantum JT gravity is equivalent to the partition functions of a Maass-Laplace operator of large weight acting on non-compact, infinite area, hyperbolic Riemann surfaces of arbitrary genus.
Abstract: We show that the partition function of quantum Jackiw-Teitelboim (JT) gravity, including topological fluctuations, is equivalent to the partition function of a Maass-Laplace operator of large -- imaginary -- weight acting on non-compact, infinite area, hyperbolic Riemann surfaces of arbitrary genus. The resulting spectrum of this open quantum system is semiclasically exact and given by a regularized Selberg trace formula, namely, it is expressed as a sum over the lengths of primitive periodic orbits of these hyperbolic surfaces. By using semiclassical techniques, we compute analytically the spectral form factor and the variance of the Wigner time delay in the diagonal approximation. We find agreement with the random matrix theory (RMT) prediction for open quantum chaotic systems. Our results show that full quantum ergodicity is a distinct feature of quantum JT gravity.
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TL;DR: In this article, it was shown that strong spatial mixing on the Bethe lattice is not equivalent to strong mixing on all graphs of a graph of maximum degree, provided only that the graph has a complex neighborhood of the region in parameter space corresponding to mixing.
Abstract: We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite $\Delta$-regular tree) implies strong spatial mixing on all graphs of maximum degree $\Delta$ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model
We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function This extension allows us to give the first deterministic FPTAS for counting the number of $q$-colorings of a graph of maximum degree $\Delta$ provided only that $q\ge 2\Delta$ This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo We also give an improved version of this result for triangle-free graphs
TL;DR: In this article, a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework is presented, which is based on a power series expansion of the quantum partition function in its off-diagonal terms.
Abstract: We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.