Seiberg-Witten Solution Research Articles

Disconnected gauge groups in the infrared

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the global form of the gauge group but not its perturbative dynamics. In this work, we study the Seiberg-Witten solution of theories resulting from the gauging of charge conjugation in 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories with SU(N) gauge group and fundamental hypermultiplets. The basic idea of our procedure is to identify the ℤ2 action at the level of the SW curve and perform the quotient, and it should also be applicable to non-lagrangian theories. We study dynamical aspects of these theories such as their moduli space singularities and the corresponding physics; in particular, we explore the complex structure singularity arising from the quotient procedure. We also discuss some implications of our work in regards to three problems: the geometric classification of 4d SCFTs, the study of non-invertible symmetries from the SW geometry, and the String Theory engineering of theories with disconnected gauge groups.

Exploring the strong-coupling region of SU(N) Seiberg-Witten theory

We consider the Seiberg-Witten solution of pure mathcal{N} = 2 gauge theory in four dimensions, with gauge group SU(N). A simple exact series expansion for the dependence of the 2(N − 1) Seiberg-Witten periods aI(u), aDI(u) on the N − 1 Coulomb-branch moduli un is obtained around the ℤ2N-symmetric point of the Coulomb branch, where all un vanish. This generalizes earlier results for N = 2 in terms of hypergeometric functions, and for N = 3 in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential K = frac{1}{2}{sum}_I Im( overline{a} IaDI), which is single valued on the Coulomb branch. Evidence is presented that K is a convex function, with a unique minimum at the ℤ2N-symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

Revisiting the multi-monopole point of SU(N) mathcal{N} = 2 gauge theory in four dimensions

Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure SU(N) mathcal{N} = 2 gauge theory in four dimensions. At this point N − 1 mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D’Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to SU(2) and SU(3) gauge theories, the large-N results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite N , finding exact agreement with our first calculation.

Resolving spacetime singularities in flux compactifications & KKLT

In flux compactifications of type IIB string theory with D3 and seven-branes, the negative induced D3 charge localized on seven-branes leads to an apparently pathological profile of the metric sufficiently close to the source. With the volume modulus stabilized in a KKLT de Sitter vacuum this pathological region takes over a significant part of the entire compactification, threatening to spoil the KKLT effective field theory. In this paper we employ the Seiberg-Witten solution of pure SU(N) super Yang-Mills theory to argue that wrapped seven-branes can be thought of as bound states of more microscopic exotic branes. We argue that the low-energy worldvolume dynamics of a stack of n such exotic branes is given by the (A1, An−1) Argyres-Douglas theory. Moreover, the splitting of the perturbative (in α′) seven-brane into its constituent branes at the non-perturbative level resolves the apparently pathological region close to the seven-brane and replaces it with a region of mathcal{O} (1) Einstein frame volume. While this region generically takes up an mathcal{O} (1) fraction of the compactification in a KKLT de Sitter vacuum we argue that a small flux superpotential dynamically ensures that the 4d effective field theory of KKLT remains valid nevertheless.

Toward the pole

We study the analytic structure of semiclassical conformal blocks, namely of the 1-point conformal block on the torus and of the 4-point conformal block on the sphere, as functions of the intermediate dimension. We interpret their discontinuities, which can be revealed with the use of a particular resummation procedure, holographically in terms of configurations of geodesics in AdS3. In other words, we study the behavior of the Ω-deformed mathcal{N} = 2 SYM theory in the Nekrasov-Shatashvili limit near those singular points, where naively the W-boson with non-vanishing angular momentum becomes massless. Upon a proper resummation of instanton contributions, these singularities disappear, which is similar to the Seiberg-Witten solution in the undeformed case where there is no massless W-boson. It is shown that the order parameter undergoes a non-analytic behavior near positions of the poles. The jump of the order parameter is interpreted holographically in terms of geodesic networks.

Seiberg-Witten geometry of four-dimensional N=2 SO–USp quiver gauge theories

We apply the instanton counting method to study a class of four-dimensional $\mathcal{N}=2$ supersymmetric quiver gauge theories with alternating $\mathrm{SO}$ and $\mathrm{USp}$ gauge groups. We compute the partition function in the $\Omega$-background and express it as functional integrals over density functions. Applying the saddle point method, we derive the limit shape equations which determine the dominant instanton configurations in the flat space limit. The solution to the limit shape equations gives the Seiberg-Witten geometry of the low energy effective theory. As an illustrating example, we work out explicitly the Seiberg-Witten geometry for linear quiver gauge theories. Our result matches the Seiberg-Witten solution obtained previously using the method of brane constructions in string theory.

Seiberg–Witten Differential via Primitive Forms

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional $\mathcal{N}=2$ SCFT. The Seiberg-Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding Seiberg-Witten differential is given by the Gelfand-Leray form of K. Saito's primitive form. Our result also extends the Seiberg-Witten solution to include irrelevant deformations.

Supersymmetric tools in Yang–Mills theories at strong coupling: The beginning of a long journey

Development of holomorphy-based methods in super-Yang–Mills theories started in the early 1980s and lead to a number of breakthrough results. I review some results in which I participated. The discovery of Seiberg’s duality and the Seiberg–Witten solution of [Formula: see text] Yang–Mills were the milestones in the long journey of which, I assume, much will be said in other talks. I will focus on the discovery (2003) of non-Abelian vortex strings with various degrees of supersymmetry, supported in some four-dimensional Yang–Mills theories and some intriguing implications of this discovery. One of the recent results is the observation of a soliton string in the bulk [Formula: see text] theory with the [Formula: see text] gauge group and four flavors, which can become critical in a certain limit. This is the case of a “reverse holography,” with a very transparent physical meaning.

Seiberg-Witten for Spin(n) with spinors

$$ \mathcal{N}=2 $$ supersymmetric Spin(n) gauge theory admits hypermultiplets in spinor representations of the gauge group, compatible with β ≤ 0, for n ≤ 14. The theories with β < 0 can be obtained as mass-deformations of the β = 0 theories, so it is of greatest interest to construct the β = 0 theories. In previous works, we discussed the n ≤ 8 theories. Here, we turn to the 9 ≤ n ≤ 14 cases. By compactifying the D N (2,0) theory on a 4-punctured sphere, we find Seiberg-Witten solutions to almost all of the remaining cases. There are five theories, however, which do not seem to admit a realization from six dimensions.

5d Higgs branch localization, Seiberg-Witten equations and contact geometry

In this paper we apply the idea of Higgs branch localization to 5d supersymmetric theories of vector multiplet and hypermultiplets, obtained as the rigid limit of $\mathcal{N} = 1$ supergravity with all auxiliary fields. On supersymmetric K-contact/Sasakian background, the Higgs branch BPS equations can be interpreted as 5d generalizations of the Seiberg-Witten equations. We discuss the properties and local behavior of the solutions near closed Reeb orbits. For $U(1)$ gauge theories, we show the suppression of the deformed Coulomb branch, and the partition function is dominated by 5d Seiberg-Witten solutions at large $\zeta$-limit. For squashed $S^5$ and $Y^{pq}$ manifolds, we show the matching between poles in the perturbative Coulomb branch matrix model, and the bound on local winding numbers of the BPS solutions.

Classification of 4d $ \mathcal{N} $ =2 gauge theories

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

DetailingN=1Seiberg’s duality through the Seiberg-Witten solution ofN=2

Starting from the Seiberg-Witten solution of $\mathcal{N}=2$ supersymmetric QCD with the $\mathrm{U}(N)$ gauge group and ${N}_{f}$ quark flavors we construct the so-called $\ensuremath{\mu}$-dual $\mathcal{N}=1$ theory in the $r$ vacua in the regime analogous to that existing to the left of the left edge of the Seiberg conformal window, where $r$ is the number of condensed quarks. The strong-weak coupling duality is shown to exist in the so-called zero vacua which can be found at $r&lt;{N}_{f}\ensuremath{-}N$. We show that the $\ensuremath{\mu}$-dual theory matches the Seiberg dual in the zero vacua.

Deformed $ \mathcal{N}=2 $ theories, generalized recursion relations and S-duality

Abstract We study the non-perturbative properties of $ \mathcal{N}=2 $ super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic ϵ-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov’s approach. These results allow us to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly ϵ-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.

Renormalization effects on the MSSM from a calculable model of a strongly coupled hidden sector

We investigate possible renormalization effects on the low-energy mass spectrum of the minimal supersymmetric standard model (MSSM), using a calculable model of strongly coupled hidden sector. We model the hidden sector by $\mathcal{N}=2$ supersymmetric quantum chromodynamics with gauge group $SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ and ${N}_{f}=2$ matter hypermultiplets, perturbed by a Fayet-Iliopoulos term which breaks the supersymmetry down to $\mathcal{N}=0$ on a metastable vacuum. In the hidden sector the K\"ahler potential is renormalized. Upon identifying a hidden sector modulus with the renormalization scale, and extrapolating to the strongly coupled regime using the Seiberg-Witten solution, the contribution from the hidden sector to the MSSM renormalization group flows is computed. For concreteness, we consider a model in which the renormalization effects are communicated to the MSSM sector via gauge mediation. In contrast to the perturbative toy examples of hidden sector renormalization studied in the literature, we find that our strongly coupled model exhibits rather intricate effects on the MSSM soft scalar mass spectrum, depending on how the hidden sector fields are coupled to the messenger fields. This model provides a concrete example in which the low-energy spectrum of MSSM particles that are expected to be accessible in collider experiments is obtained using strongly coupled hidden sector dynamics.

Quantization of integrable systems and a 2d/4d duality

We present a new duality between the F-terms of supersymmetric field theories defined in two- and four-dimensions respectively. The duality relates N=2 supersymmetric gauge theories in four dimensions, deformed by an Omega-background in one plane, to N=(2,2) gauged linear sigma-models in two dimensions. On the four dimensional side, our main example is N=2 SQCD with gauge group SU(L) and 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.

Microscopic quantum superpotential in 𝒩 = 1 gauge theories

We consider the N=1 super Yang-Mills theory with gauge group U(N), adjoint chiral multiplet X and tree-level superpotential Tr W(X). We compute the quantum effective superpotential W_mic as a function of arbitrary off-shell boundary conditions at infinity for the scalar field X. This effective superpotential has a remarkable property: its critical points are in one-to-one correspondence with the full set of quantum vacua of the theory, providing in particular a unified picture of solutions with different ranks for the low energy gauge group. In this sense, W_mic is a good microscopic effective quantum superpotential for the theory. This property is not shared by other quantum effective superpotentials commonly used in the literature, like in the strong coupling approach or the glueball superpotentials. The result of this paper is a first step in extending Nekrasov's microscopic derivation of the Seiberg-Witten solution of N=2 super Yang-Mills theories to the realm of N=1 gauge theories.

Non-Abelian string junctions as confined monopoles

Various dynamical regimes associated with confined monopoles in the Higgs phase of N=2 two-flavor QCD are studied. The microscopic model we deal with has the SU(2)xU(1) gauge group, with a Fayet-Iliopoulos term of the U(1) factor, and large and (nearly) degenerate mass terms of the matter hypermultiplets. We present a complete quasiclassical treatment of the BPS sector of this model, including the full set of the first-order equations, derivations of all relevant zero modes, and derivation of an effective low-energy theory for the corresponding collective coordinates. The macroscopic description is provided by a CP(1) model with or without twisted mass. The confined monopoles -- string junctions of the microscopic theory -- are mapped onto BPS kinks of the CP(1) model. The string junction is 1/4 BPS. Masses and other characteristics of the confined monopoles are matched with those of the CP(1)-model kinks. The matching demonstrates the occurrence of an anomaly in the monopole central charge in 4D Yang-Mills theory. We study what becomes of the confined monopole in the bona fide non-Abelian limit of degenerate mass terms where a global SU(2) symmetry is restored. The solution of the macroscopic model is known e.g. from the mirror description of the CP(1) model. The monopoles, aka CP(1)-model kinks, are stabilized by nonperturbative dynamics of the CP(1) model. We explain an earlier rather puzzling observation of a correspondence between the BPS kink spectrum in the CP(1) model and the Seiberg-Witten solution.

Multi-Trace Superpotentials vs. Matrix Models

We consider ᵊ9=1 supersymmetric U(N) field theories in four dimensions with adjoint chiral matter and a multi-trace tree-level superpotential. We show that the computation of the effective action as a function of the glueball superfield localizes to computing matrix integrals. Unlike the single-trace case, holomorphy and symmetries do not forbid non-planar contributions. Nevertheless, only a special subset of the planar diagrams contributes to the exact result. In addition, the computation of the superpotential localizes to doing matrix integrals. In view of the results of Dijkgraaf and Vafa for single-trace theories, one might have naively expected that these matrix integrals are related to the free energy of a multi-trace matrix model. We explain why this naive identification does not work. Rather, an auxiliary single-trace matrix model with additional singlet fields can be used to exactly compute the field theory superpotential. Along the way we also describe a general technique for computing the large-N limits of multi-trace Matrix models and raise the challenge of finding the field theories whose effective actions they may compute. Since our models can be treated as ᵊ9=1 deformations of pure ᵊ9=2 gauge theory, we show that the effective superpotential that we compute also follows from the ᵊ9=2 Seiberg-Witten solution. Finally, we observe an interesting connection between multi-trace local theories and non-local field theory.

Perturbative analysis of gauged matrix models

We analyze perturbative aspects of gauged matrix models, including those where classically the gauge symmetry is partially broken. Ghost fields play a crucial role in the Feynman rules for these vacua. We use this formalism to elucidate the fact that non-perturbative aspects of N=1 gauge theories can be computed systematically using perturbative techniques of matrix models, even if we do not possess an exact solution for the matrix model. As examples we show how the Seiberg-Witten solution for N=2 gauge theory, the Montonen-Olive modular invariance for N=1*, and the superpotential for the Leigh-Strassler deformation of N=4 can be systematically computed in perturbation theory of the matrix model/gauge theory (even though in some of these cases the exact answer can also be obtained by summing up planar diagrams of matrix models).

On geometry and matrix models

We point out two extensions of the relation between matrix models, topological strings and N=1 supersymmetric gauge theories. First, we note that by considering double scaling limits of unitary matrix models one can obtain large- N duals of the local Calabi–Yau geometries that engineer N=2 gauge theories. In particular, a double scaling limit of the Gross–Witten one-plaquette lattice model gives the SU(2) Seiberg–Witten solution, including its induced gravitational corrections. Secondly, we point out that the effective superpotential terms for N=1 ADE quiver gauge theories is similarly computed by large- N multi-matrix models, that have been considered in the context of ADE minimal models on random surfaces. The associated spectral curves are multiple branched covers obtained as Virasoro and W-constraints of the partition function.