Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by "conformal" (or Dotsenko-Fateev) matrix models, but in non-Gaussian Dijkgraaf-Vafa phases, where different eigenvalues are integrated along different contours. In such matrix models, the determinant representations and integrability are restored only after a peculiar Fourier transform in the numbers of integrations. From the point of view of conformal blocks, this is Fourier transform w.r.t. the intermediate dimensions and this explains why such quantities are expressed through tau-functions in Miwa parametrization, with external dimensions playing the role of multiplicities. In particular, these determinant representations provide solutions to the Painlevé VI equation. We also explain how this pattern looks in the pure gauge limit, which is described by the Brezin-Gross-Witten matrix model.
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks ha…
We relate Nekrasov partition functions, with arbitrary values of $ε_1,ε_2$ parameters, to matrix models for $β$-ensembles. We find matrix models encoding the instanton part of Nekrasov partition functions, whose measure, to the leading order in $ε_2$ expansion, is given by the Vandermonde determinant to the power $β=-ε_1/ε_2$. An additional, trigonometric deformation of the measure arises in five-dimensional theories. Matrix model potentials, to the leading order in $ε_2$ expansion, are the same as in the $β=1$ case considered in 0810.4944 [hep-th]. We point out that potentials for massive hyp…
We consider the compactifcation of 5d non-simply laced fractional quiver gauge theory constructed in arXiv:1705.04410. In contrast to the simply laced quivers, here two $Ω$-background parameters play different roles, so that we can take two possible Nekrasov-Shatashvili limits. We demonstrate how different quantum integrable systems can emerge from these two limits, using $BC_2$-quiver as the simplest illustrative example for our general results. We also comment possible connections with compactified 3d non-simply laced quiver gauge theory.
We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-ensemble), dS = ydz + O(ε^2) = \sum_p ε^{2p} ρ_β^{(p|1)}(z), where εand βare r…
We briefly discuss the recent claims that the ordinary KP/Toda integrability, which is a characteristic property of ordinary eigenvalue matrix models, persists also for the Dijkgraaf-Vafa (DV) partition functions and for the refined topological vertex. We emphasize that in both cases what is meant is a particular representation of partition functions: a peculiar sum over all DV phases in the first case and hiding the deformation parameters in a sophisticated potential in the second case, i.e. essentially a reformulation of some questions in the new theory in the language of the old one. It is …
The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of $\mathcal{N}=2$ gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal v…
We study three-point functions of single-trace operators in the su(1|1) sector of planar N = 4 SYM borrowing several tools based on Integrability. In the most general configuration of operators in this sector, we have found a determinant expression for the tree-level structure constants. We then compare the predictions of the recently proposed hexagon program against all available data. We have obtained a match once additional sign factors are included when the two hexagon form-factors are assembled together to form the structure constants. In the particular case of one BPS and two non-BPS ope…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites (``W-constraints''), determinantal formulas and continuum limits, taking one kind of models into another. Subtle points and directions of the future research are also discussed.
It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point hbar=2pi, where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular prop…
We develop an integrability-based framework to compute structure constants of two sub-determinant operators and a single-trace non-BPS operator in ABJM theory in the planar limit. In this first paper, we study them at weak coupling using a relation to an integrable spin chain. We first develop a nested Bethe ansatz for an alternating SU(4) spin chain that describes single-trace operators made out of scalar fields. We then apply it to the computation of the structure constants and show that they are given by overlaps between a Bethe eigenstate and a matrix product state. We conjecture that the …
In our previous work, the two-loop integrability of ABJM determinant like operator has been well established. In this paper, we push the integrability to all loop orders. The asymptotic Bethe ansatz equations for ABJM determinant like operator (open string attached on giant graviton) are obtained. In the derivation, the symmetries preserved by the bulk and the boundary played a crucial role. Taking the weak coupling limit and applying appropriate fermionic dualities, we obtained a different set of scalar sector Bethe equations with our previous results. When the "gauge" transformation on Bethe…
We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a defi…
We calculate the four-point function of $1/2$-BPS determinant operators in $\mathcal{N}=4$ SYM at next-to-leading order at weak coupling. We use two complementary methods recently developed for a class of determinant three-point functions: one is based on Feynman diagrams and it extracts perturbative data at finite $N$, while the other one expresses a generic correlator of determinants as the zero-dimensional integral over an auxiliary matrix field. We generalise the latter approach to calculate one-loop corrections and we solve the four-point function in a semi-classical approach at large $N$…
We investigate a deformed matrix model of type 0A theory related to supersymmetric Witten's black hole in two-dimensions, generalization of bosonic model suggested by Kazakov et. al. We find a free field realization of the partition function of the matrix model, which includes Ramond-Ramond perturbations in the type 0A theory. In a simple case, the partition function is factorized into two determinants, which are given by $τ$ function of an integrable system. We work out the genus expansion of the partition function. Holographic relation with the supersymmetric Witten's black hole is checked b…
We review the problem of BPS state counting described by the generalized quiver matrix model of ADHM type. In four dimensions the generating function of the counting gives the Nekrasov partition function and we obtain generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably after integration the pa…
We give analytic expression for the three-point function of three large classical non-BPS operators N=4 Super-Yang-Mills theory at weak coupling. We restrict ourselves to operators belonging to an su(2) sector of the theory. In order to carry out the calculation we derive, by unveiling a hidden factorization property, the thermodynamical limit of Slavnov's determinant.
We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in $\mathcal{N}=4$ SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbi…
Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this $T\bar T$ flow equation one can find a simple expression for both the energy spectrum and the $S$-matrix of the $T\bar T$ deformed theories. Our goal is to find the renormalized Lagrangian of the $T\bar T$ deformed theories. In the context of the $T\bar T$ deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the $S$-matrix factoriz…
Antipodal Self-Duality for Generalized Fishnets from Integrability
SLAC National Accelerator Laboratory, Stanford University（斯坦福大学）；Universität Bonn（波恩大学）；Stanford University（斯坦福大学）；University of Oxford（牛津大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We study the Hermitian supermatrix model involving an external source. We derive the determinantal formula for the supermatrix partition function, and also for the expectation value of the characteristic polynomial ratio, which yields the duality between the characteristic polynomial and the external source with an arbitrary matrix potential function. We also show that the supermatrix integral satisfies the one and two dimensional Toda lattice equations as well as the ordinary matrix model.
We calculate planar tree level one-point functions of non-protected operators in the defect conformal field theory dual to the D3-D5 brane system with k units of the world volume flux. Working in the operator basis of Bethe eigenstates of the Heisenberg XXX_{1/2} spin chain we express the one-point functions as overlaps of these eigenstates with a matrix product state. For k=2 we obtain a closed expression of determinant form for any number of excitations, and in the case of half-filling we find a relation with the Néel state. In addition, we present a number of results for the limiting case o…
We introduce a nonperturbative approach to correlation functions of two determinant operators and one non-protected single-trace operator in planar N=4 supersymmetric Yang-Mills theory. Based on the gauge/string duality, we propose that they correspond to overlaps on the string worldsheet between an integrable boundary state and a state dual to the single-trace operator. We determine the boundary state using symmetry and integrability of the dual superstring sigma model, and write down expressions for the correlators at finite coupling, which we conjecture to be valid for operators of arbitrar…
We consider type 0A matrix model in the presence of a spacelike D brane,localized in matter direction at any arbitrary point. It appears that in order to have an appropriate string/MQM correspondence we must need to impose a constraint on the matrix model side which is equivalent to an operator constraint on the matter part of the boundary state that arises from open string boundary condition. This condition constrains the Hilbert space generated by the macroscopic loop operator but the bulk matrix model remains unaffected, thereby describing a situation parallel to string theory. We have anal…
There has been renewed interest in the path-integral computation of the partition function of AdS3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called ``pro…
Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one …
We develop a novel nonperturbative approach to a class of three-point functions in planar $\mathcal{N}=4$ SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS$_5\times$S$^{5}$. They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coup…
Young diagrams can be parameterized with the help of hook variables, which is well known but never studied in big detail. We demonstrate that this is the most adequate parameterization for many physical applications: from the Schur functions, conventional, skew and shifted, which all satisfy their own kinds of determinant formulas in these coordinates, to KP/Toda integrability and related basis of cut-and-join $\hat W$-operators, which are both actually expressed through the single-hook diagrams. In particular, we discuss a new type of multi-component KP $τ$-functions, Matisse $τ$-functions. W…
Using integrability and analyticity properties of the AdS5/CFT4 Y-system we reduce it to a finite set of nonlinear integral equations. The Z4 symmetry of the underlying coset sigma model, in its quantum version, allows for a deeper insight into the analyticity structure of the underlying Y-functions and T-functions, as well as for their analyticity friendly parameterization in terms of Wronskian determinants of Q-functions. As a check for the new equations, we reproduce the numerical results for the Konishi operator previously obtained from the original infinite Y-system.
We study the matrix product state which appears as the boundary state of the AdS/dCFT set-up where a probe D7 brane wraps two two-spheres stabilized by fluxes. The matrix product state plays a dual role, on one hand acting as a tool for computing one-point functions in a domain wall version of N=4 SYM and on the other hand acting as the initial state in the study of quantum quenches of the Heisenberg spin chain. We derive a number of selection rules for the overlaps between the matrix product state and the eigenstates of the Heisenberg spin chain and in particular demonstrate that the matrix p…
The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain $τ$-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent on the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special $τ$-function satisfying…
