Showing papers on "Partition function (quantum field theory) published in 2009"

Membranes at Quantum Criticality

Abstract: We propose a quantum theory of membranes designed such that the ground-state wavefunction of the membrane with compact spatial topology Σh reproduces the partition function of the bosonic string on worldsheet Σh. The construction involves worldvolume matter at quantum criticality, described in the simplest case by Lifshitz scalars with dynamical critical exponent z = 2. This matter system must be coupled to a novel theory of worldvolume gravity, also exhibiting quantum criticality with z = 2. We first construct such a nonrelativistic ``gravity at a Lifshitz point'' with z = 2 in D+1 spacetime dimensions, and then specialize to the critical case of D = 2 suitable for the membrane worldvolume. We also show that in the second-quantized framework, the string partition function is reproduced if the spacetime ground state takes the form of a Bose-Einstein condensate of membranes in their first-quantized ground states, correlated across all genera.

Liouville Correlation Functions from Four-dimensional Gauge Theories

Abstract: We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.

Directed polymers and the quantum Toda lattice

Abstract: We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.

Refined bps state counting from nekrasov's formula and macdonald functions

Abstract: It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

A partition function algorithm for interacting nucleic acid strands

Abstract: Recent interests, such as RNA interference and antisense RNA regulation, strongly motivate the problem of predicting whether two nucleic acid strands interact. Motivation: Regulatory non-coding RNAs (ncRNAs) such as microRNAs play an important role in gene regulation. Studies on both prokaryotic and eukaryotic cells show that such ncRNAs usually bind to their target mRNA to regulate the translation of corresponding genes. The specificity of these interactions depends on the stability of intermolecular and intramolecular base pairing. While methods like deep sequencing allow to discover an ever increasing set of ncRNAs, there are no high-throughput methods available to detect their associated targets. Hence, there is an increasing need for precise computational target prediction. In order to predict base-pairing probability of any two bases in interacting nucleic acids, it is necessary to compute the interaction partition function over the whole ensemble. The partition function is a scalar value from which various thermodynamic quantities can be derived. For example, the equilibrium concentration of each complex nucleic acid species and also the melting temperature of interacting nucleic acids can be calculated based on the partition function of the complex. Results: We present a model for analyzing the thermodynamics of two interacting nucleic acid strands considering the most general type of interactions studied in the literature. We also present a corresponding dynamic programming algorithm that computes the partition function over (almost) all physically possible joint secondary structures formed by two interacting nucleic acids in O(n6) time. We verify the predictive power of our algorithm by computing (i) the melting temperature for interacting RNA pairs studied in the literature and (ii) the equilibrium concentration for several variants of the OxyS–fhlA complex. In both experiments, our algorithm shows high accuracy and outperforms competitors. Availability: Software and web server is available at http://compbio.cs.sfu.ca/taverna/pirna/ Contact:cenk@cs.sfu.ca; backofen@informatik.uni-freiburg.de Supplementary information:Supplementary data are avaliable at Bioinformatics online.

On Combinatorial Expansions of Conformal Blocks

Abstract: In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter \epsilon=\epsilon_1+\epsilon_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6\epsilon^2/\epsilon_1\epsilon_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.

On the effective action of confining strings

Abstract: We study the low-energy effective action on confining strings (in the fundamental representation) in SU(N) gauge theories in D space-time dimensions. We write this action in terms of the physical transverse fluctuations of the string. We show that for any D, the four-derivative terms in the effective action must exactly match the ones in the Nambu-Goto action, generalizing a result of Luscher and Weisz for D=3. We then analyze the six-derivative terms, and we show that some of these terms are constrained. For D=3 this uniquely determines the effective action for closed strings to this order, while for D>3 one term is not uniquely determined by our considerations. This implies that for D=3 the energy levels of a closed string of length L agree with the Nambu-Goto result at least up to order 1/L^5. For any D we find that the partition function of a long string on a torus is unaffected by the free coefficient, so it is always equal to the Nambu-Goto partition function up to six-derivative order. For a closed string of length L, this means that for D>3 its energy can, in principle, deviate from the Nambu-Goto result at order 1/L^5, but such deviations must always cancel in the computation of the partition function. Next, we compute the effective action up to six-derivative order for the special case of confining strings in weakly-curved holographic backgrounds, at one-loop order (leading order in the curvature). Our computation is general, and applies in particular to backgrounds like the Witten background, the Maldacena-Nunez background, and the Klebanov-Strassler background. We show that this effective action obeys all of the constraints we derive, and in fact it precisely agrees with the Nambu-Goto action (the single allowed deviation does not appear).

K-theoretic Donaldson invariants via instanton counting

Abstract: In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.

Branes at Quantum Criticality

Abstract: In this paper we propose new non-relativistic p+1 dimensional theory. This theory is defined in such a way that the potential term obeys the principle of detailed balance where the generating action corresponds to p-brane action. This condition ensures that the norm of the vacuum wave functional of p+1 dimensional theory is equal to the partition function of p-brane theory.

Chern-Simons theory on L(p,q) lens spaces and Localization

Abstract: Using localization technique, we calculate the partition function and the expectation value of Wilson loop operator in Chern-Simons theory on general lens spaces L(p,q)(including S2XS1). Our results are consistent with known results.

The 1/2 BPS 't Hooft loops in N=4 SYM as instantons in 2d Yang-Mills

Abstract: We extend the recent conjecture on the relation between a certain 1/8 BPS subsector of 4d N=4 SYM on S^2 and 2d Yang-Mills theory by turning on circular 1/2 BPS 't Hooft operators linked with S^2. We show that localization predicts that these 't Hooft operators and their correlation functions with Wilson operators on S^2 are captured by instanton contributions to the partition function of the 2d Yang-Mills theory. Based on this prediction, we compute explicitly correlation functions involving the 't Hooft operator, and observe precise agreement with S-duality predictions.

Approximating the Permanent with Belief Propagation

Abstract: This work describes a method of approximating matrix per- manents efficiently using belief propagation. We formulate aprobability distribution whose partition function is exactly the permanent, then use Bethe free energy to approximate this partition function. After deriving some speedups to standard belief propagation, the resulting algorithm requires (n 2 ) time per iteration. Finally, we demonstrate the advantages of using this approximation.

Branes at quantum criticality

Abstract: In this paper we propose new non-relativistic p+1 dimensional theory. This theory is defined in such a way that the potential term obeys the principle of detailed balance where the generating action corresponds to p-brane action. This condition ensures that the norm of the vacuum wave functional of p+1 dimensional theory is equal to the partition function of p-brane theory.

Introduction to integral discriminants

Abstract: The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant Jn|r(S) = ∫e−S(x1,...,xn)dx1...dxn. Actually, S-dependence remains the same if e−S in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate Jn|r in a number of non-Gaussian cases. Using Ward identities — linear differential equations, satisfied by integral discriminants — we calculate J2|3,J2|4,J2|5 and J3|3. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.

Coarse graining, fractional moments and the critical slope of random copolymers

Abstract: For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.

Introduction to Integral Discriminants

Abstract: The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if e^{-S} in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate J_{n|r} in a number of non-Gaussian cases. Using Ward identities -- linear differential equations, satisfied by integral discriminants -- we calculate J_{2|3}, J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.

Partition function of the eight-vertex model with domain wall boundary condition

Abstract: We derive the recursive relations of the partition function for the eight-vertex model on an $N\times N$ square lattice with domain wall boundary condition. Solving the recursive relations, we obtain the explicit expression of the domain wall partition function of the model. In the trigonometric/rational limit, our results recover the corresponding ones for the six-vertex model.

On AGT description of N=2 SCFT with N_f=4

Abstract: We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of N=2 SCFT. We focus our attention on the SU(2) theory with N_f=4 flavor symmetry, whose partition function, according to AGT, is given by the Liouville four-point function on the sphere. The gauge theory with N_f=4 is known to exhibit SO(8) symmetry. We explain how the Weyl symmetry transformations of SO(8) flavor symmetry are realized in the Liouville theory picture. This is associated to functional properties of the Liouville four-point function that are a priori unexpected. In turn, this can be thought of as a non-trivial consistency check of AGT conjecture. We also make some comments on elementary surface operators and WZW theory.

Quiver Matrix Model and Topological Partition Function in Six Dimensions

Abstract: We consider a topological quiver matrix model which is expected to give a dual description of the instanton dynamics of topological U(N) gauge theory on D6 branes. The model is a higher dimensional analogue of the ADHM matrix model that leads to Nekrasov's partition function. The fixed points of the toric action on the moduli space are labeled by colored plane partitions. Assuming the localization theorem, we compute the partition function as an equivariant index. It turns out that the partition function does not depend on the vacuum expectation values of Higgs fields that break U(N) symmetry to U(1)^N at low energy. We conjecture a general formula of the partition function, which reduces to a power of the MacMahon function, if we impose the Calabi-Yau condition. For non Calabi-Yau case we prove the conjecture up to the third order in the instanton expansion.

Generating All Partitions: A Comparison Of Two Encodings

Abstract: Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate all ascending compositions. We develop three new algorithms to generate all ascending compositions and compare these with descending composition generators from the literature. We analyse the new algorithms and provide new and more precise analyses for the descending composition generators. In each case, the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. We develop a new formula for the partition function p(n) as part of our analysis of the lexicographic succession rule for ascending compositions.

On AGT conjecture

Abstract: In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain $\mathcal{N}=2$ SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on $\mathcal{N}=2^{*}$ theory, which is the simplest example of AGT relation.

On Equivalence of two Hurwitz Matrix Models

Abstract: In arXiv:0902.2627 a matrix model representation was found for the simplest Hurwitz partition function, which has Lambert curve phi e^{-phi} = psi as a classical equation of motion. We demonstrate that Fourier-Laplace transform in the logarithm of external field Psi converts it into a more sophisticated form, recently suggested in arXiv:0906.1206.

Quiver Matrix Model and Topological Partition Function in Six Dimensions

Abstract: We consider a topological quiver matrix model which is expected to give a dual description of the instanton dynamics of topological U(N) gauge theory on D6 branes. The model is a higher dimensional analogue of the ADHM matrix model that leads to Nekrasov's partition function. The fixed points of the toric action on the moduli space are labeled by colored plane partitions. Assuming the localization theorem, we compute the partition function as an equivariant index. It turns out that the partition function does not depend on the vacuum expectation values of Higgs fields that break U(N) symmetry to U(1)N at low energy. We conjecture a general formula of the partition function, which reduces to a power of the MacMahon function, if we impose the Calabi-Yau condition. For non Calabi-Yau case we prove the conjecture up to the third order in the instanton expansion.

Weight space structure and analysis using a finite replica number in the Ising perceptron

Abstract: The weight space of the Ising perceptron, in which a set of random patterns is stored, is examined using the generating function of the partition function (n) = (1/N)log[Zn] as the dimension of the weight vector N tends to infinity, where Z is the partition function and represents the configurational average. We utilize (n) for two purposes, depending on the value of the ratio α = M/N, where M is the number of random patterns. For α αs, on the other hand, (n) is used to assess the rate function of a small probability that a given set of random patterns is atypically separable by the Ising perceptrons. We show that the analyticity of the rate function changes at α = αGD = 1.245..., which implies that the dominant configuration of the atypically separable patterns exhibits a phase transition at this critical ratio. Extensive numerical experiments are conducted to support the theoretical predictions.

Partition function of the eight-vertex model with domain wall boundary condition

Abstract: We derive the recursive relations of the partition function for the eight-vertex model on an N×N square lattice with domain wall boundary condition. Solving the recursive relations, we obtain the explicit expression of the domain wall partition function of the model. In the trigonometric/rational limit, our results recover the corresponding ones for the six-vertex model.

The partition function of the two-matrix model as an isomonodromic τ function

Abstract: We consider the Itzykson–Zuber–Eynard–Mehta two-matrix model and prove that the partition function is an isomonodromic τ function in a sense that generalizes that of Jimbo et al. [ “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients,” Physica D 2, 306 (1981)]. In order to achieve the generalization we need to define a notion of τ function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement.

Three-coloring statistical model with domain wall boundary conditions: Functional equations

Abstract: We consider the Baxter three-coloring model with boundary conditions of the domain wall type. In this case, it can be proved that the partition function satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.

Weight space structure and analysis using a finite replica number in the Ising perceptron

Abstract: The weight space of the Ising perceptron in which a set of random patterns is stored is examined using the generating function of the partition function $\phi(n)=(1/N)\log [Z^n]$ as the dimension of the weight vector $N$ tends to infinity, where $Z$ is the partition function and $[ ... ]$ represents the configurational average. We utilize $\phi(n)$ for two purposes, depending on the value of the ratio $\alpha=M/N$, where $M$ is the number of random patterns. For $\alpha \alpha_{\rm s}$, on the other hand, $\phi(n)$ is used to assess the rate function of a small probability that a given set of random patterns is atypically separable by the Ising perceptrons. We show that the analyticity of the rate function changes at $\alpha = \alpha_{\rm GD}=1.245 ... $, which implies that the dominant configuration of the atypically separable patterns exhibits a phase transition at this critical ratio. Extensive numerical experiments are conducted to support the theoretical predictions.