Showing papers on "Partition function (quantum field theory) published in 2015"
TL;DR: In this article, it was shown that the partition function of the Rademacher series is log-concave for all polylogarithmic values of the Hardy-Ramanujan partition function.
Abstract: We prove that the partition function $$p(n)$$
is log-concave for all
$$n>25$$
. We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer’s estimates on the remainders of the Hardy–Ramanujan and the Rademacher series for
$$p(n)$$
.
114 citations
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TL;DR: In this article, duality walls are used to encode the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories, supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.
Abstract: We propose an explicit description of duality walls which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.
88 citations
TL;DR: In this article, the supersymmetric partition function of the six-dimensional (2, 0) theory of type A====== N −1 on S.............. 1 × S.............. 5 was computed in the presence of both codimension two and codeimension four defects.
Abstract: We compute the supersymmetric partition function of the six-dimensional (2, 0) theory of type A
N −1 on S
1 × S
5 in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of W
N
algebras and affine Lie algebras of type A
N −1.
80 citations
TL;DR: In this article, it was shown that the one-loop partition function of threedimensional flat gravity, computed along the lines originally developed for the anti-de Sitter case, reproduces characters of the BMS3 group.
Abstract: In this note we point out that the one-loop partition function of threedimensional flat gravity, computed along the lines originally developed for the anti-de Sitter case, reproduces characters of the BMS3 group.
64 citations
TL;DR: In this article, the partition function of a random polymer model with gamma-distributed weights was shown to be asymptotic in terms of Whittaker functions, and its law can be expressed as the smallest eigenvalue in the Laguerre unitary ensemble.
Abstract: We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppalainen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.
63 citations
08 Dec 2015
TL;DR: In this article, the third order mock theta function ω(q) (resp. ν(−q)) was shown to be the generating function for repeated partitions with repeated distinct parts such that each odd part is less than twice the smallest part.
Abstract: The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(−q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function ϕ(q). Congruences for the smallest parts partition function(s) associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal number theorem are also obtained.
60 citations
TL;DR: In this article, a new type IIB 5-brane description for the E-string theory is introduced, which is the world-volume theory on the M5brane probing the end of the world M9brane.
Abstract: We introduce a new type IIB 5-brane description for the E-string theory which is the world-volume theory on the M5-brane probing the end of the world M9-brane. The E-string in the new realization is depicted as spiral 5-branes web equipped with the cyclic structure which is key to uplifting to six dimensions. Utilizing the topological vertex to the 5-brane web configuration enables us to write down a combinatorial formula for the generating function of the E-string elliptic genera, namely the full partition function of topological strings on the local 1/2 K3 surface.
58 citations
TL;DR: In this paper, the Frobenius formula is used to evaluate characters associated with certain induced representations of the centrally extended BMS3 group, and a delta function localizes the integral to a single point, allowing us to obtain an exact result.
Abstract: Using the Frobenius formula, we evaluate characters associated with certain induced representations of the centrally extended BMS3 group. This computation involves a functional integral over a coadjoint orbit of the Virasoro group; a delta function localizes the integral to a single point, allowing us to obtain an exact result. The latter is independent of the specific form of the functional measure, and holds for all values of the BMS3 central charges and all values of the chosen mass and spin. It can also be recovered as a flat limit of Virasoro characters.
58 citations
TL;DR: In this paper, exact WKB methods were applied to the partition function of pure ϵi-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence.
Abstract: We apply exact WKB methods to the study of the partition function of pure $$ \mathcal{N}=2 $$
ϵi-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in ϵ2/ϵ1 (i.e. at large central charge) and expansion in ϵ1. We find corrections of the form ~ exp $$ \left[-\frac{\mathrm{SW}\;\mathrm{periods}}{\upepsilon_1}\right] $$
to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of ϵ1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.
57 citations
TL;DR: For k-colorings it is proved that for even k, in a tree nonuniqueness region (which corresponds to k < Δ) there is no FPRAS, unless NP = RP, to approximate the number of colorings for triangle-free Δ-regular graphs.
Abstract: A remarkable connection has been established for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree Δ undergoes a phase transition that coincides with the statistical physics uniqueness/nonuniqueness phase transition on the infinite Δ-regular tree. Despite this clear picture for 2-spin systems, there is little known for multispin systems. We present the first analog of this in approximability results for multispin systems.The main difficulty in previous inapproximability results was analyzing the behavior of the model on random Δ-regular bipartite graphs, which served as the gadget in the reduction. To this end, one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). The view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random Δ-regular bipartite graphs. This yields concentration results for any spin system in which one can analyze the maxima of the first moment. The connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest.For k-colorings we prove that for even k, in a tree nonuniqueness region (which corresponds to k
56 citations
TL;DR: In this article, a supersymmetry Ward identity for the partition function of four-dimensional superconformal field theories on S4 is presented, where the ambiguity is realized by a local supergravity counterterm in the underlying SCFT.
Abstract: The partition function of four-dimensional $$ \mathcal{N} $$
= 2 superconformal field theories on S4 computes the exact Kahler potential on the space of exactly marginal couplings [1]. We present a new elementary proof of this result using supersymmetry Ward identities. The partition function is a section rather than a function, and is subject to ambiguities coming from Kahler transformations acting on the Kahler potential. This ambiguity is realized by a local supergravity counterterm in the underlying SCFT. We provide an explicit construction of the Kahler ambiguity counterterm in the four dimensional $$ \mathcal{N} $$
= 2 off-shell supergravity theory that admits S4 as a supersymmetric background.
TL;DR: In this article, the authors define an elliptic deformation of the Virasoro algebra and show that the Nekrasov partition function reproduces the chiral blocks of this algebra.
Abstract: We define an elliptic deformation of the Virasoro algebra. We argue that the $\mathbb{R}^4\times \mathbb{T}^2$ Nekrasov partition function reproduces the chiral blocks of this algebra. We support this proposal by showing that at special points in the moduli space the 6d Nekrasov partition function reduces to the partition function of a 4d vortex theory supported on $\mathbb{R}^2\times \mathbb{T}^2$, which is in turn captured by a free field correlator of vertex operators and screening charges of the elliptic Virasoro algebra.
TL;DR: Hatsuda et al. as mentioned in this paper proposed an exact instanton expansion of ABJM partition function, and showed that it is possible to obtain an exact ABJM expansion of the partition function.
Abstract: Exact instanton expansion of the ABJM partition function Yasuyuki Hatsuda1,∗, Sanefumi Moriyama2,∗, and Kazumi Okuyama3,∗ 1DESY Theory Group, DESY Hamburg, Notkestrasse 85, D-22603 Hamburg, Germany 2Department of Physics, Osaka City University, Osaka 558-8585, Japan 3Department of Physics, Shinshu University, Matsumoto 390-8621, Japan ∗E-mail: yasuyuki.hatsuda@desy.de, moriyama@sci.osaka-cu.ac.jp, okuyama@azusa.shinshu-u.ac.jp
TL;DR: In this article, the authors consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface, and construct these Schramm-Loewner evolution-type measures by the method of localization in path space.
Abstract: We consider some probabilistic and analytic realizations of Virasoro highest-weight representations. Specifically, we consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface. These Schramm–Loewner evolution-type measures are constructed by the method of localization in path space. Their partition function (total mass) is the highest-weight vector of a Virasoro representation, and the action is given by Virasoro uniformization.
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TL;DR: In this article, the authors studied the partition function of four-dimensional N = 1 supersymmetric field theory on T 2 ×S 2 by applying supersymmetry localization, and they showed that the 4D partition function is given by elliptic genus of certain 2D N = (0,2) theory.
Abstract: We study partition function of four-dimensional N = 1 supersymmetric field theory on T 2 ×S 2 . By applying supersymmetry localization, we show that the T 2 ×S 2 partition function is given by elliptic genus of certain two-dimensional N = (0,2) theory. As an application, we discuss a relation between 4d Seiberg duality duality and 2d (0,2) triality, proposed by Gadde, Gukov and Putrov. In other examples, we identify 4d theories giving elliptic genera of K3, M-strings and E-strings. In the example of K3, we find that there are two 4d theories giving the elliptic genus of K3. This would imply new four-dimensional duality.
TL;DR: In this article, it was shown that the one-loop partition function of three-dimensional flat gravity, computed along the lines originally developed for the anti-de Sitter case, reproduces characters of the BMS3 group.
Abstract: In this note we point out that the one-loop partition function of three-dimensional flat gravity, computed along the lines originally developed for the anti-de Sitter case, reproduces characters of the BMS3 group.
Posted Content•
TL;DR: In this paper, exact WKB methods were applied to the partition function of pure N = 2 epsilon-i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence.
Abstract: We apply exact WKB methods to the study of the partition function of pure N=2 epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of epsilon_1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.
TL;DR: In this article, the authors review progress in determining the partition function of the ABJM theory in the large N expansion, including all of the perturbative and non-perturbative corrections.
Abstract: We review recent progress in determining the partition function of the ABJM theory in the large N expansion, including all of the perturbative and non-perturbative corrections. Especially, we will focus on how these exact expansions are obtained from various beautiful relations to Fermi gas system, topological string theory, integrable model and supergroup.
TL;DR: In this article, the authors studied Higgs branch localization of supersymmetric theories placed on compact Euclidean manifolds and analyzed the resulting localization equations in detail on the four-sphere.
Abstract: We study Higgs branch localization of $$ \mathcal{N}=2 $$
supersymmetric theories placed on compact Euclidean manifolds. We analyze the resulting localization equations in detail on the four-sphere and find that in this case the path integral is dominated by vortex-like configurations as well as singular Seiberg-Witten monopoles located at the north and south pole. The partition function is written accordingly.
TL;DR: In this paper, the exact vortex partition function was derived for 2D vortex string worldsheet theory with two Ω-deformation parameters (ϵ 1, ϵ 2), which can be interpreted as a quantized version of twisted F-term equations characterizing the SUSY vacua.
Abstract: We derive the exact vortex partition function in 2d $$ \mathcal{N}=\left(2,2\right) $$
gauge theory on the Ω-background, applying the localization scheme in the Higgs phase. We show that the partition function at a finite Ω-deformation parameter ϵ satisfies a system of differential equations, which can be interpreted as a quantized version of the twisted F-term equations characterizing the SUSY vacua. Using the differential equations derived in this paper, we show the correspondence between the partition function of the two-dimensional vortex string worldsheet theory and the Nekrasov partition function at the root of Higgs branch of the four-dimensional $$ \mathcal{N}=2 $$
theory with two Ω-deformation parameters (ϵ
1, ϵ
2).
TL;DR: The proposed method is a global topological application of the S[B] paradigm for modeling complex systems and is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes.
Abstract: In order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi's model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function $$Z$$Z with a particular functor of topological field theory--the generating function of the Betti numbers of the state manifold of the system--which contains the same global information of the system configurations and of the data set representing them. The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data, is expected to discover hidden n-ary relations among idiotypes and anti-idiotypes. The data topological analysis will select global features, reducible neither to a mere subgraph nor to a metric or vector space. How the immune system reacts, how it evolves, how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational. Within this metaphor, the proposed method turns out to be a global topological application of the S[B] paradigm for modeling complex systems.
TL;DR: In this article, the heat kernel coefficients for selfadjoint extensions of the Laplace operator over the finite line [0, L] were derived and the derivative of the corresponding spectral zeta function at s = 0 (partition function of the quantum field theory) was obtained.
Abstract: Following the seminal works of Asorey–Ibort–Marmo and Munoz–Castaneda–Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint extension of the Laplace operator over the finite line [0, L]. The derivative of the corresponding spectral zeta function at s = 0 (partition function of the corresponding quantum field theory) is obtained. To compute the correct expression for the a
1/2 heat kernel coefficient, it is necessary to know in detail which non-negative selfadjoint extensions have zero modes and how many of them they have. The answer to this question leads us to analyze zeta function properties for the Von Neumann–Krein extension, the only extension with two zero modes.
TL;DR: A new way of rewriting the partition function of scalar field theory on fuzzy complex projective spaces as a solvable multitrace matrix model as a perturbative high-temperature expansion is described.
Abstract: We describe a new way of rewriting the partition function of scalar field theory on fuzzy complex projective spaces as a solvable multitrace matrix model. This model is given as a perturbative high-temperature expansion. At each order, we present an explicit analytic expression for most of the arising terms; the remaining terms are computed explicitly up to fourth order. The method presented here can be applied to any model of hermitian matrices. Our results confirm constraints previously derived for the multitrace matrix model by Polychronakos. A further implicit expectation about the shape of the multitrace terms is however shown not to be true.
TL;DR: In this paper, Russo and Guillermo describe Russo's work in the field of bio-medical data analysis and propose a method to identify the cause of brain cancer in humans.
Abstract: Fil: Russo, Jorge Guillermo. Universidad de Barcelona. Institucio Catalana de Recerca i Estudis Avancats; Espana
TL;DR: The so-called partition function is a sample moment statistic based on blocks of data and it is shown that its behaviour is strongly influenced by the tail of the distribution underlying the data both in independent identically distributed and weakly dependent cases.
Abstract: The so-called partition function is a sample moment statistic based on blocks of data and it is often used in the context of multifractal processes. It will be shown that its behaviour is strongly influenced by the tail of the distribution underlying the data both in independent identically distributed and weakly dependent cases. These results will be used to develop graphical and estimation methods for the tail index of a distribution. The performance of the tools proposed is analysed and compared with other methods by means of simulations and examples.
Journal Article•
TL;DR: In this article, it was shown that it is #BIS-hard to approximate the partition function of any 2-spin system on bipartite graphs supporting phase-correlated spins and unary symmetry breaking.
Abstract: Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #Sat to approximate. We study #BIS in the general framework of two-state spin systems on bipartite graphs. We define two notions, nearly-independent phase-correlated spins and unary symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any 2-spin system on bipartite graphs supporting these two notions. Consequently, we classify the complexity of approximating the partition function of antiferromagnetic 2-spin systems on boundeddegree bipartite graphs.
TL;DR: In this article, a particular solution of the extended r-reduced KP hierarchy, which is specified by a generalized string equation, is given. But this solution is not a generalization to arbitrary of Buryak's recent results of a solution to the extended open KdV hierarchy which corresponds to r = 2.
Abstract: We derive a particular solution of the extended r-reduced KP hierarchy, which is specified by a generalized string equation . The work is a generalization to arbitrary of Buryakʼs recent results of a solution to the extended open KdV hierarchy which corresponds to r = 2.
TL;DR: In this article, an equilibrium partition function for a nonrelativistic fluid is constructed based on light cone reduction, which is known to reduce the Poincare symmetry to Galilean in one lower dimension.
Abstract: We construct an equilibrium partition function for a nonrelativistic fluid and use it to constrain the dynamics of the system. The construction is based on light cone reduction, which is known to reduce the Poincare symmetry to Galilean in one lower dimension. We modify the constitutive relations of a relativistic fluid, and find that its symmetry broken phase—“null fluid” is equivalent to the nonrelativistic fluid. In particular, their symmetries, thermodynamics, constitutive relations, and equilibrium partition function match exactly to all orders in derivative expansion.
TL;DR: In this article, the authors considered the problem of counting the number of l-regular bipartitions of n in the unrestricted partition function p(n) using Ramanujan's two modular equations of degree 7.
Abstract: In this paper, we study the function B
l
(n) which counts the number of l-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, $$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$
In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B
13(n) modulo 3.
TL;DR: In this paper, the authors derived the 1-loop partition function of the handle-body solutions in the AdS$_3$ gravity have been derived some years ago using the heat-kernel and the method of images.
Abstract: The 1-loop partition function of the handle-body solutions in the AdS$_3$ gravity have been derived some years ago using the heat-kernel and the method of images. In the semiclassical limit, such partition function should correspond to the order $O (c^0)$ part in the partition function of dual conformal field theory on the boundary Riemann surface. The higher genus partition function could be computed by the multi-point functions in the Riemann sphere via sewing prescription. In the large central charge limit, to the leading order of $c$, the multi-point function is further simplified to be a summation over the product of two-point functions, which may form links. Each link is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the eigenvalue of the element in the conjugate class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.