Witten Curve Research Articles

Painlevé Kernels and Surface Defects at Strong Coupling

AbstractIt is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between $${\mathcal {N}}=2$$ N = 2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $${{\,\mathrm{O(2)}\,}}$$ O ( 2 ) matrix model. We then show that these eigenfunctions are computed by surface defects in $${{\,\mathrm{SU(2)}\,}}$$ SU ( 2 ) super Yang–Mills in the self-dual phase of the $$\Omega $$ Ω -background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

A-D hypersurface of su(n) 𝒩=2 supersymmetric gauge theory with Nf=2n−2 flavors

In the previous paper, arXiv:2210.16738[hep-th], we found a set of flavor mass relations as constraints such that the [Formula: see text]-deformed [Formula: see text] quiver matrix model restores the maximal symmetry in the massive scaling limit and reported the existence of Argyres–Douglas critical hypersurface. In this paper, we derive the concrete conditions on moduli parameters which maximally degenerates the Seiberg–Witten curve while maintaining the flavor mass relations. These conditions define the A-D hypersurface.

Machine-learning dessins d’enfants: explorations via modular and Seiberg–Witten curves

We apply machine-learning to the study of dessins d’enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg–Witten (SW) curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. The SW curves for those with rational coefficients are also tabulated.

Quantum periods for [formula omitted]SU(2) SQCD around the superconformal point

We study the Argyres–Douglas theories realized at the superconformal point in the Coulomb moduli space of N=2 supersymmetric SU(2) QCD with Nf=1,2,3 hypermultiplets in the Nekrasov–Shatashvili limit of the Omega-background. The Seiberg–Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg–Witten curve, we calculate the quantum corrections to the Seiberg–Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, which is a projective toric orbifold Xk such that Xk∖Xk is a Zk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on Xk with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of Xk. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the Seiberg–Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg–Witten curve with characteristics related to the nontrivial holonomies.

Large [formula omitted] from Seiberg–Witten curve and localization

When N=2 gauge theories are compactified on S4, the large Nc limit then selects a unique vacuum of the theory determined by saddle-point equations, which remains determined even in the flat-theory limit. We show that exactly the same equations can be reproduced purely from Seiberg–Witten theory, describing a vacuum where magnetically charged particles become massless, and corresponding to a specific degenerating limit of the Seiberg–Witten spectral curve where 2Nc−2 branch points join pairwise giving aDn=0, n=1,…,Nc−1. We consider the specific case of N=2SU(Nc) SQCD coupled with 2Nf massive fundamental flavors. We show that the theory exhibits a quantum phase transition where the critical point describes a particular Argyres–Douglas point of the Riemann surface.

Three-particle integrable systems with elliptic dependence on momenta and theta function identities

We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which were introduced in [1,2] and are made from the theta-functions on Jacobians of the Seiberg–Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the two-particle systems are also obtained.

SINGULARITY STRUCTURE OF ${\mathcal N} = 2$ SUPERSYMMETRIC YANG–MILLS THEORIES

In this paper, we consider the case where electrons, magnetic monopoles and dyons become massless. Here, we consider the [Formula: see text] supersymmetric Yang–Mills (SYM) theories with classical gauge groups with a rank r, SU(r+1), SO(2r), Sp(2r) and SO(2r+1) which are studied by Riemann surfaces called Seiberg–Witten curves. We discuss physical singularity associated with massless particles, which can be studied by geometric singularity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particular attention to the cases where mutually nonlocal states become massless (Argyres–Douglas theories), which corresponds to Riemann surfaces degenerating into cusps. We discuss nontrivial topology on the moduli space of the theory, which is reflected as monodromy associated to vanishing 1-cycles. We observe how dyon charges get changed as we move around and through singularity in moduli space.

The Hilbert series of [formula omitted][formula omitted] and [formula omitted] SQCD, Painlevé VI and integrable systems

We present a novel approach for computing the Hilbert series of 4d N=1 supersymmetric QCD with SO(Nc) and Sp(Nc) gauge groups. It is shown that such Hilbert series can be recast in terms of determinants of Hankel matrices. With the aid of results from random matrix theory, such Hankel determinants can be evaluated both exactly and asymptotically. Several new results on Hilbert series for general numbers of colours and flavours are thus obtained in this paper. We show that the Hilbert series give rise to families of rational solutions, with palindromic numerators, to the Painlevé VI equations. Due to the presence of such Painlevé equations, there exist integrable Hamiltonian systems that describe the moduli spaces of SO(Nc) and Sp(Nc) SQCD. To each system, we explicitly state the corresponding Hamiltonian and family of elliptic curves. It turns out that such elliptic curves take the same form as the Seiberg–Witten curves for 4d N=2SU(2) gauge theory with 4 flavours.

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann–Hilbert problem. This Riemann–Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg–Witten curve of the deformed U(1) gauge theory. This solution of the Riemann–Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg–Witten differential are also considered.

Kummer surfaces associated with Seiberg–Witten curves

By carrying out a rational transformation on the base curve C P 1 of the Seiberg–Witten curve for N = 2 supersymmetric pure SU ( 2 ) -gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17 . The isogeny relating the Seiberg–Witten curve for pure SU ( 2 ) -gauge theory to the one for SU ( 2 ) -gauge theory with N f = 2 massless hypermultiplets extends to define a Nikulin involution on each K3 surface in the family. We show that the desingularization of the quotient of the K3 surface by the involution is isomorphic to a Kummer surface of the Jacobian variety of a curve of genus two. We then derive a relation between the Yukawa coupling associated with the elliptic K3 surface and the Yukawa coupling of pure SU ( 2 ) -gauge theory.

Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves

We study the dynamics of a N=2 supersymmetric SU(N) gauge theory with fundamental or adjoint matter in presence of a non trivial Omega-background along a two dimensional plane. The prepotential and chiral correlators of the gauge theory can be obtained, via a saddle point analysis, from an equation which can be viewed as a non commutative version of the "standard" Seiberg and Witten curve.

Period integrals, quantum numbers, and confinement in Susy QCD

We compute the period integrals on degenerate Seiberg—Witten curves for supersymmetric QCD explicitly and also show how these periods determine the changes in the quantum numbers of the states when passing from the weak- to strong-coupling domains in the mass moduli space of the theory. We discuss the confinement of monopoles at a strong coupling and demonstrate that the ambiguities in choosing the path in the moduli space do not affect the physical conclusions on confinement of monopoles in the phase with condensed light dyons.

Direct integration and non-perturbative effects in matrix models

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

On confinement index

The smallest integer t for which the Wilson loop W t fails to exhibit area law is known as the confinement index of a given field theory. The confinement index provides us with subtle information on the vacuum properties of the system. We study the behavior of the Wilson and 't Hooft loops and compute the confinement index in a wide class of N = 1 supersymmetric gauge theories. All possible electric and magnetic screenings are taken into account. The results found are consistent with the θ periodicity, and whenever such a check is available, with the factorization property of Seiberg–Witten curves.

Seiberg–Witten theory and matrix models

We derive a family of matrix models which encode solutions to the Seiberg–Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg–Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi–Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi–Yang matrix model, matrix models for bundles over P 1 , and Chern–Simons matrix models for lens spaces, which arise as various limits of our general result.

Quaternionic Kähler manifolds, constrained instantons and the magic square: I

The classification of homogeneous quaternionic manifolds has been done by Alekseevskii, Wolf et al. using transitive solvable group of isometries. These manifolds are not generically symmetric, but there is a subset of quaternionic manifolds that are symmetric and Einstein. A further subset of these manifolds are the magic square manifolds. We show that all the symmetric quaternionic manifolds including the magic square can be succinctly classified by constrained instantons. These instantons are mostly semilocal, and their constructions for the magic square can be done from the corresponding Seiberg–Witten curves for certain N = 2 gauge theories that are in general not asymptotically free. Using these, we give possible constructions, such as the classical moduli space metrics, of constrained instantons with exceptional global symmetries. We also discuss the possibility of realising the Kähler manifolds in the magic square using other solitonic configurations in the theory, and point out an interesting new sequence of these manifolds in the magic square.

The chiral ring and the periods of the resolvent

The strongly coupled vacua of an N = 1 supersymmetric gauge theory can be described by imposing quantization conditions on the periods of the gauge theory resolvent, or equivalently by imposing factorization conditions on the associated N = 2 Seiberg–Witten curve (the so-called strong-coupling approach). We show that these conditions are equivalent to the existence of certain relations in the chiral ring, which themselves follow from the fact that the gauge group has a finite rank. This provides a conceptually very simple explanation of why and how the strongly coupled physics of N = 1 theories, including fractional instanton effects, chiral symmetry breaking and confinement, can be derived from purely semi-classical calculations involving instantons only.

AMOEBAS AND INSTANTONS

We study a statistical model of random plane partitions. The statistical model has interpretations as five-dimensional [Formula: see text] supersymmetric SU (N) Yang–Mills on ℝ4× S1and as Kähler gravity on local SU (N) geometry. At the thermodynamic limit a typical plane partition called the limit shape dominates in the statistical model. The limit shape is linked with a hyperelliptic curve, which is a five-dimensional version of the SU (N) Seiberg–Witten curve. Amoebas and the Ronkin functions play intermediary roles between the limit shape and the hyperelliptic curve. In particular, the Ronkin function realizes an integration of thermodynamical density of the main diagonal partitions, along one-dimensional slice of it and thereby is interpreted as the counting function of gauge instantons. The radius of S1can be identified with the inverse temperature of the statistical model. The large radius limit of the five-dimensional Yang–Mills is the low temperature limit of the statistical model, where the statistical model is frozen to a ground state that is associated with the local SU (N) geometry. We also show that the low temperature limit corresponds to a certain degeneration of amoebas and the Ronkin functions known as tropical geometry.

Classical spin chains and exact three-dimensional superpotentials

We study exact effective superpotentials of four-dimensional N = 2 supersymmetric gauge theories with gauge group U ( N ) and various amounts of fundamental matter on R 3 × S 1 , broken to N = 1 by turning on a classical superpotential for the adjoint scalar. On general grounds these superpotentials can easily be constructed once we identify a suitable set of coordinates on the moduli space of the gauge theory. These coordinates have been conjectured to be the phase space variables of the classical integrable system which underlies the N = 2 gauge theory. The sought low energy effective superpotential can then be constructed from the conserved quantities in the integrable system. For the gauge theory under study these integrable systems are degenerations of the classical, inhomogeneous, periodic SL ( 2 , C ) spin chain. Ambiguities in the degeneration provide multiple coordinate patches on the gauge theory moduli space. By studying the vacua of these superpotentials in several examples we find that the spin chain provides coordinate patches that parametrize holomorphically the part of the gauge theory moduli space which is connected to the electric (as opposed to magnetic or baryonic) Higgs and Coulomb branch vacua. The baryonic branch root is on the edge of some coordinate patches. As a product of our analysis all maximally confining (non-baryonic) Seiberg–Witten curve factorizations for N f ⩽ N c are obtained, explicit up to one constraint for equal mass flavors and up to two constraints for unequal mass flavors. Gauge theory addition and multiplication maps are shown to have a natural counterpart in this construction. Furthermore it is shown how to integrate in the meson fields in this formulation in order to obtain three- and four-dimensional Affleck–Dine–Seiberg-like superpotentials.