We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of ${\cal N}=2$, $SU(2)$ super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the $τ$ function of Painleve $\rm III$.
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.
Omega-deformation of the Seiberg-Witten curve is known to be written in terms of the qq-character, namely the trace of a specific operator acting in a Hilbert space spanned by certain Young diagrams. We define a differential form acting on this space and establish two discretised versions of the Seiberg-Witten expressions for the periods and related relations for the prepotential.
We show that the Seiberg-Witten map for a noncommutative gauge theory involves a noncommutative 1-cocycle. The cocycle condition enforces a consistency requirement, which has been previously derived.
A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg-Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg-Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.
We present a generic derivation of the WDVV equations for 6d Seiberg-Witten theory, and extend it to the families of bi-elliptic spectral curves. We find that the elliptization of the naive perturbative and nonperturbative 6d systems roughly "doubles" the number of moduli describing the system.
The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the complex curve and generating differential. The first derivatives of prepotential or quasiclassical tau-function over the extra times, extending the Toda chain, are expressed through the multiple integrals of the Seiberg-Witten one-form. We derive the corresponding quasiclassical Virasoro constraints, discuss the functional formulation of the problem and propose …
We prove Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree $c+2$, with $c=-{1/4}(7χ+11σ)$, for four-manifolds obeying some mild conditions, where $χ$ and $σ$ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactific…
Perturbing the Seiberg-Witten curves for N=2 U(N_c) and SU(N_c) super Yang-Mills theory with N_f<N_c flavours with a mass term for the adjoint field completely lifts the quantum vacuum degeneracy. The generated N=1 effective superpotential can be obtained from the factorization formulae of Seiberg-Witten curves with matter. We show that the Affleck-Dine-Seiberg superpotential emerges. Moreover it appears additive with respect to the classical superpotential for the meson superfields, as expected from the Intriligator-Leigh-Seiberg linearity principle.
This is a very brief review of relations between Seiberg-Witten theories and integrable systems with emphasis on the perturbative prepotentials presented at the E.S.Fradkin Memorial Conference.
All order Seiberg--Witten maps of gauge parameter, gauge field and matter fields are given as a closed recursive formula. These maps are obtained by analyzing the order by order solutions of the gauge consistency and equivalence conditions as well as by directly solving Seiberg-Witten differential equations. The explicit third order non-abelian and fourth order abelian Seiberg-Witten maps of gauge parameter and gauge field are also presented.
Elementary MAPLE calculations are used to support the claim of hep-th/9906240 that the ratios of theta-functions, associated with the Seiberg-Witten complex curves, provide Poisson-commuting Hamiltonians which describe the dual of the original Seiberg-Witten integrable system.
The central charge for the Seiberg-Witten low-energy effective Action is computed using Noether supercharges. A reliable method to construct supersymmetric Noether currents is presented.
We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a formal derivation of Witten's conjecture and its generalization, making use of an infinite dimensional version of the abelian localization theorem.
It is argued that dimensional reduction of Seiberg-Witten map for a gauge field induces Seiberg-Witten maps for the other noncommutative fields of a gauge invariant theory. We demonstrate this observation by dimensionally reducing the noncommutative N=1 SYM theory in 6 dimensions to obtain noncommutative N=2 SYM in 4 dimensions. We explicitly derive Seiberg-Witten maps of the component fields in 6 and 4 dimensions. Moreover, we give a general method to define the deformed supersymmetry transformations that leaves the actions invariant after performing the Seiberg-Witten maps.
This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models.
In this paper we construct Seiberg-Witten maps for superfields on canonically deformed N=1, d=4 Minkowski and Euclidean superspace. On Minkowski superspace we show that the Seiberg-Witten map is not compatible with locality, (anti)chirality and supersymmetry at the same time. On Euclidean superspace we show that there exists a local, chiral and supersymmetric Seiberg-Witten map for chiral superfields if we take the noncommutativity parameter to be selfdual, and a local, antichiral and supersymmetric Seiberg-Witten map for antichiral superfields if we take the noncommutativity parameter to be a…
It is shown how Seiberg-Witten equations can be obtained by means of Fedosov deformation quantization of endomorphism bundle and the corresponding theory of equivalences of star products. In such setting, Seiberg-Witten map can be iteratively computed for arbitrary gauge group up to any given degree with recursive methods of Fedosov construction. Presented approach can be also considered as a generalization of Seiberg-Witten equations to Fedosov type of noncommutativity.
We show how SU(2) $N_f=4$ Seiberg-Witten prepotentials are derived form $\hat{\mathfrak{sl}}_{2,k}$ ($k\to2$) four-point conformal blocks via considering Langlands duality.
A BRST-cohomological analysis of Seiberg-Witten maps and results on gauge anomalies in noncommutative Yang-Mills theories with general gauge groups are reviewed.
We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.
We consider 5d Sp(1) gauge theory with $E_{N_f+1}$ global symmetries based on toric(-like) diagram constructed from (p,q)-web with 7-branes. We propose a systematic procedure to compute the Seiberg-Witten curve for generic toric-like diagram. For $N_f=6,7$ flavors, we explicitly compute the Seiberg-Witten curves for 5d Sp(1) gauge theory, and show that these Seiberg-Witten curves agree with already known $E_{7,8}$ results. We also discuss a generalization of the Seiberg-Witten curve to rank-N cases.
We consider quantum electrodynamics in noncommutative spacetime by deriving a $θ$-exact Seiberg-Witten map with fermions in the fundamental representation of the gauge group as an expansion in the coupling constant. Accordingly, we demonstrate the persistence of UV/IR mixing in noncommutative QED with charged fermions via Seiberg-Witten map, extending the results of Schupp and You [1].
We obtain the leading derivative corrections to an expression for the Seiberg-Witten map given by Banerjee and Yang and show how they affect the noncommutative deformation of the Maxwell action, as well as the matter coupling in noncommutative emergent gravity.
Direct evaluation of the Seiberg-Witten prepotential is accomplished following the localization programme suggested some time ago. Our results agree with all low-instanton calculations available in the literature. We present a two-parameter generalization of the Seiberg-Witten prepotential, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
Talk presented by the second author at the Inaugural Coference of the Asia Pacific Center for Theoretical Physics, Seoul, June 1996. The purpose of this note is to give a resume of the Seiberg-Witten theory in the simplest possible mathematical terms.
We show that a previous paper of Freund describing a solution to the Seiberg-Witten equations has a sign error rendering it a solution to a related but different set of equations. The non-$L^2$ nature of Freund's solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg-Witten equations.
A topological gauge invariant lagrangian for Seiberg-Witten monopole equations is constructed. The action is invariant under a huge class of gauge transformations which after BRST fixing leads to the BRST invariant action associated to Seiberg-Witten monopole topological theory. The supersymmetric transformation of the fields involved in the construction is obtained from the nilpotent BRST algebra.
We derive Seiberg-Witten like equations encoding the dynamics of N=2 ADE quiver gauge theories in presence of a non-trivial Omega-background along a two dimensional plane. The epsilon-deformed prepotential and the chiral correlators of the gauge theory are extracted from difference equations that can be thought as a non-commutative (or quantum) version of the Seiberg-Witten curves for the quiver.
In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be governed by the N=2 Green-Schwarz string.
By analysing the work of Campolattaro we argue that the second Seiberg-Witten equation over the Spin^c_4 manifold, i.e., F^+_{ij}=< M,S_ij M >, is the generalization of the Campolattaro's description of the electromagnetic field tensor F^{μν} in the bilinear form F^{μν}=\barΨ S^{μν}Ψ. It turns out that the Seiberg-Witten equations (also the perturbed Seiberg-Witten equations) can be well understood from this point of view. We suggest that the second Seiberg-Witten equation can be replaced by a nonlinear Dirac-like Equation. We also derive the spinor representation of the connection on the asso…
We briefly review the Whitham hierarchies and their applications to integrable systems of the Seiberg-Witten type. The simplest example of the N=2 supersymmetric SU(2) pure gauge theory is considered in detail and the corresponding Whitham solutions are found explicitely.
Electric-magnetic duality allows to calculate the central charges of N=2 supersymmetric theories with massless hypermultiplets as derivatives of simple modular forms. The procedure reproduces the Seiberg-Witten results for N_f=0,2,3 in a uniform way, but indicates open problems for N_f=1.
We present a cohomological method for obtaining the non-Abelian Seiberg-Witten map for any gauge group and to any order in theta. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator.
We study the equivalence/duality between various non-commutative gauge models at the classical and quantum level. The duality is realised by a linear Seiberg-Witten-like map. The infinitesimal form of this map is analysed in more details.
With the help of the Seiberg-Witten map for photons and fermions we define a theta-deformed QED at the classical level. Two possibilities of gauge-fixing are discussed. A possible non-Abelian extension for a pure theta-deformed Yang-Mills theory is also presented.
The modified Seiberg-Witten monopole equations are presented in this letter. These equations have analytic solutions in the whole 1+3 Minkowski space with finite energy. The physical meaning of the equations and solutions are discussed here.
The instanton-induced effective vertex is derived for N=2 supersymmetric QCD (SQCD) with arbitrary mass matter hypermultiplets for the case of SU(2). The leading term of the low energy effective lagrangian obtained from this vertex agrees with one-instanton effective term of the Seiberg-Witten result.
We present a review of the results on the associativity algebras and WDVV equations associated with the Seiberg-Witten solutions of N=2 SUSY gauge theories. It is mostly based on the integrable treatment of these solutions. We consider various examples of the Seiberg-Witten solutions and corresponding integrable systems and discuss when the WDVV equations hold. We also discuss a covariance of the general WDVV equations.
Starting from the Seiberg-Witten solution of N=2 SQCD with the U(N) gauge group and N_f quark flavors we construct the so-called μ-dual 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 (here 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< N_f-N. We show that the μ-dual theory matches the Seiberg dual in the zero vacua.
We investigate instanton expansions of partition functions of several toric E-string models using local mirror symmetry and elliptic modular forms. We also develop a method to obtain the Seiberg--Witten curve of E-string with arbitrary Wilson lines with the help of elliptic functions.
We analyze the noncommutative two-dimensional Wess-Zumino-Witten model and its properties under Seiberg-Witten transformations in the operator formulation. We prove that the model is invariant under such transformations even for the noncritical (non chiral) case, in which the coefficients of the kinetic and Wess-Zumino terms are not related. The pure Wess-Zumino term represents a singular case in which this transformation fails to reach a commutative limit. We also discuss potential implications of this result for bosonization.
In this paper we construct the Seiberg-Witten maps for superfields on the theta-theta deformed superspaces with N=(1/2,0) and N=(1/2,1/2) supersymmety. We show that on the N=(1/2,0) deformed superspace there is no Seiberg-Witten map for antichiral superfields which is at the same time antichiral, local and which preserves the N=(1/2,0) supersymmetry. Solutions which break these requirements are presented. On the N=(1/2,1/2) deformed superspace we show that for the chiral gauge parameter, and therefore also for the chiral matter field, there is no chiral Seiberg-Witten map. Some other possible …
These notes provide an introductory exposition of the Seiberg-Witten gauge theory. They collect the material presented in a series of seminars given by the author at the University of Milano.
We construct an explicit solution of the Seiberg-Witten map for a linear gauge field on the non-commutative plane. We observe that this solution as well as the solution for a constant curvature diverge when the non-commutativity parameter theta reaches certain event horizon in the theta-space. This implies that an ordinary Yang-Mills theory can be continuously deformed by the Seiberg-Witten map into a non-commutative theory only within one connected component of the theta-space.
We find perfect agreements on the genus-one correction to the prepotential of SU(2) Seiberg-Witten theory with N_f=2, 3 between field theoretical and Dijkgraaf-Vafa-Penner type matrix model results.
Seiberg-Witten maps and a recently proposed construction of noncommutative Yang-Mills theories (with matter fields) for arbitrary gauge groups are reformulated so that their existence to all orders is manifest. The ambiguities of the construction which originate from the freedom in the Seiberg-Witten map are discussed with regard to the question whether they can lead to inequivalent models, i.e., models not related by field redefinitions.
A topological quantum field theory is introduced which reproduces the Seiberg-Witten invariants of four-manifolds. Dimensional reduction of this topological field theory leads to a new one in three dimensions. Its partition function yields a three-manifold invariant, which can be regarded as the Seiberg-Witten version of Casson's invariant. A Geometrical interpretation of the three dimensional quantum field theory is also given.
BFYM on commutative and noncommutative ${\mathbb{R}}^4$ is considered and a Seiberg-Witten gauge-equivalent transformation is constructed for these theories. Then we write the noncommutative action in terms of the ordinary fields and show that it is equivalent to the ordinary action up to higher dimensional gauge invariant terms.
Noncommutative gauge theories defined via Seiberg-Witten map have desirable properties that theories defined directly in terms of noncommutative fields lack, covariance and unrestricted choice of gauge group and charge being among them, but nonperturbative results in the deformation parameter θare hard to obtain. In this article we use a θ-exact approach to study UV/IR mixing in a noncommutative quantum electrodynamics (NCQED) model defined via Seiberg-Witten map. The fermion contribution of the one loop correction to the photon propagator is computed and it is found that it gives the same UV/…
Noncommutative Chern-Simons theory can be classically mapped to commutative Chern-Simons theory by the Seiberg-Witten map. We provide evidence that the equivalence persists at the quantum level by computing two and three-point functions of field strengths on the commutative side and their Seiberg-Witten transforms on the noncommutative side to the first nontrivial order in perturbation theory.
Open Wilson lines are known to be the observables of noncommutative gauge theory with Moyal-Weyl star product. We generalize these objects to more general star products. As an application we derive a formula for the inverse Seiberg-Witten map for star products with invertible Poisson structures.
These lectures are devoted to the low energy limit of \N2 SUSY gauge theories, which is described in terms of integrable systems. A special emphasis is on a duality that naturally acts on these integrable systems. The duality turns out to be an effective tool in constructing the double elliptic integrable system which describes the six-dimensional Seiberg-Witten theory. At the same time, it implies a series of relations between other Seiberg-Witten systems.
We examine the Seiberg-Witten treatment of N=2 super Yang-Mills theory, and note that in the strong coupling region of moduli space, some massive particle excitations appear to have negative norm. We discuss the significance of our observation.
