Seiberg-Witten Solution Research Articles

On non-commutative super Yang–Mills

We discuss the Seiberg-Witten solution of the non-commutative N=2 U(N) SYM model. The solution is described in terms of the ordinary Seiberg-Witten curve of the SU(N) theory plus an additional free U(1). Hence, at the two-derivative approximation the theory flows to the ordinary commutative theory in the infra-red (k<1/sqrt(theta)). In particular, the center U(1) is free and it decouples from the other U(1)'s. In addition, no UV/IR mixing is found.

Exact results in noncommutative Script N = 2 supersymmetric gauge theories

We study the low-energy dynamics of noncommutative $\N=2$ supersymmetric U(N) Yang-Mills theories in the Coulomb phase. Exact results are derived for the leading terms in the derivative expansion of the Wilsonian effective action. We find that in the infrared regime the U(1) subgroup decouples, and the remaining SU(N) is described by the ordinary commutative Seiberg-Witten solution. IR/UV mixing is present in the U(1), but not in SU(N). Our analysis is based on explicit perturbative and multi-instanton calculations.

Formula omitted] supersymmetric gauge theories with massive hypermultiplets and the Whitham hierarchy

We embed the Seiberg–Witten solution for the low energy dynamics of N=2 super Yang–Mills theory with an even number of massive hypermultiplets into the Whitham hierarchy. Expressions for the first and second derivatives of the prepotential in terms of the Riemann theta function are provided which extend previous results obtained by Gorsky, Marshakov, Mironov and Morozov. Checks in favour of the new equations involve both their behaviour under duality transformations and the consistency of their semiclassical expansions.

Instanton recursion relations for the effective prepotential in [formula omitted]=2 super-Yang–Mills

Linear recursion relations for the instanton corrections to the effective prepotential of N=2 supersymmetric gauge theories with an arbitrary number of hypermultiplets in the fundamental representation of an arbitrary classical gauge group are derived. The construction proceeds from the Seiberg–Witten solutions and the renormalization group type equations for the prepotential. Successive iterations of these recursion relations allow us to simply obtain instanton corrections to arbitrarily high order, which we exhibit explicitly up to 6th order. For gauge groups SU(2) and SU(3), our results agree with previous ones.

Instantons at strong coupling, averaging over vacua, and the gluino condensate

We consider instanton contributions to chiral correlators, such as 〈0| Tr λ 2(x) Tr λ 2(x′)|0〉 , in N=1 supersymmetric Yang–Mills theory with either light adjoint or fundamental matter. Within the former model, extraction of the gluino condensate from a connected 1-instanton diagram, evaluated at strong coupling, can be contrasted with expectations from the Seiberg–Witten solution perturbed to an N=1 vacuum. We observe a numerical discrepancy, coinciding with that observed previously in N=1 SQCD. Moreover, since knowledge of the vacuum structure is complete for softly broken N=2 Yang–Mills, this model serves as a counter example to the hypothesis of Amati et al. that 1-instanton calculations at strong coupling can be interpreted as averaging over vacua. Within N=1 SQCD, we point out that the connected contribution to the relevant correlators actually vanishes in the weakly coupled Higgs phase, despite having a non-zero value through infra-red effects when calculated in the unbroken phase.

Instanton corrections in [formula omitted] supersymmetric theories with classical gauge groups and fundamental matter hypermultiplets

We compute instanton corrections to the low energy effective prepotential of N=2 supersymmetric theories in a variety of cases, including all classical gauge groups and even number of fundamental matter hypermultiplets. To this end, we take profit of a set of first- and second-order equations for the logarithmic derivatives of the prepotential with respect to the dynamical scale expressed in terms of Riemann's theta-function. These equations emerge in the context of the Whitham hierarchy approach to the low-energy Seiberg–Witten solution of supersymmetric gauge theories. Our procedure is recursive and allows to compute the effective prepotential to arbitrary order in a remarkably straightforward way. General expressions for up to three-instanton corrections are given. We illustrate the method with explicit expressions for several cases.

Solutions to the reflection equation and integrable systems for N = 2 SQCD with classical groups

Integrable systems underlying the Seiberg-Witten solutions for the N = 2 SQCD with gauge groups SO( n) and Sp( n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations.

Monopole Equations on 8-Manifolds with Spin(7) Holonomy

We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit non-trivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case.

Implications of exact SUSY gauge couplings for QCD

The phase structure of SUSY gauge theories can be very different from their nonsupersymmetric counterparts. Nonetheless, there is interesting information which might be gleaned from detailed investigation of these theories. In particular, we study the precise meaning of the strong interaction scale $\Lambda$. We ask whether one can meaningfully apply naive dimensional analysis and also ask whether the study of supersymmetric theories can shed light on the apparent discrepancy between the perturbative scale $\Lambda_{QCD}$ and the ``chiral lagrangian'' scale $\Lambda_\chi$. We show that in N=1 supersymmetric Yang Mills theory, ``naive dimensional analysis'' seems to work well, with $\Lambda_\chi$ consistently equal to the scale at which the perturbatively evolved physical coupling becomes of order $4 \pi$. We turn to N=2 theories to understand better the effect of instantons in accounting for the QCD discrepancy between scales. In N=2 supersymmetric SU(2) the instanton corrections are known to all orders from the Seiberg-Witten solution and give rise to a finite scale ratio between the scale at which the perturbatively evolved and ``nonperturbatively evolved'' couplings blow up. Correspondingly, instanton effects are important even when the associated perturbatively evolved gauge coupling only gives $\alpha$ of order 1 (rather than $4 \pi$). We compare the N=2 result to instanton-induced corrections in QCD, evaluated using lattice data and the instanton liquid model, and find a remarkably similar behavior.

${\rm Gr}Łongrightarrow{\rm SW}$: from pseudo-holomorphic curves to Seiberg-Witten solutions

${\rm Gr}Łongrightarrow{\rm SW}$: from pseudo-holomorphic curves to Seiberg-Witten solutions

Seiberg-Witten Theory and Calogero-Moser Systems*

We present a brief account of a series of recent results on twisted and untwisted elliptic Calogero-Moser systems, and on their fundamental role in the Seiberg-Witten solution of gauge theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G.

Remarks on the M5-brane

The fivebrane of M theory — the M5-brane — is an especially interesting object. It plays a central role in a geometric understanding of the Seiberg-Witten solution of N=2 D=4 gauge theories as well as in certain new 6d quantum theories. The low energy effective action is an interacting theory of a (2,0) tensor multiplet. The fact that this multiplet contains a two-form gauge field with a self-dual field strength poses special challenges. Recent progress in addressing those challenges is reviewed.

The beta-function in N=2 supersymmetric Yang-Mills theory

The constraints of N=2 supersymmetry, in combination with several other quite general assumptions, have recently been used to show that N=2 supersymmetric Yang-Mills theory has a low energy quantum parameter space symmetry characterised by the discrete group Γ U (2). We show that if one also assumes the commutativity of renormalization group flow with the action of this group on the complexified coupling constant τ, then this is sufficient to determine the non-perturbative β-function, given knowledge of its weak coupling behaviour. The result coincides with the outcome of direct calculations from the Seiberg-Witten solution.

The WDVV equations in N=2 supersymmetric Yang-Mills theory

We present a simple proof of the WDVV equations for the prepotential of four-dimensional N=2 supersymmetric Yang-Mills theory with all ADE gauge groups. According to our proof it is clearly seen that the WDVV equations in four dimensions have their origin in the associativity of the chiral ring in two-dimensional topological Landau-Ginzburg models. The WDVV equations for the BC gauge groups are also studied in the Landau-Ginzburg framework. We speculate about the topological field theoretic interpretation of the Seiberg-Witten solution of N=2 Yang-Mills theory.

Isomonodromic properties of the Seiberg-Witten solution

The Seiberg-Witten solution of N = 2 supersymmetric SU(2) gauge theories with matter is analyzed as an isomonodromy problem. We show that the holomorphic section describing the effective action can be deformed by moving its singularities on the moduli space while keeping their monodromies invariant. Well-known examples of isomonodromic sections are given by thev correlators of two-dimensional rational conformal field theories - the conformal blocks. The Seiberg-Witten section similarly admits the operations of braiding and fusing of its singularities, which obey the Yang-Bacter and Pentagonal identities, respectively. Using them, we easily find the complete expressions of the monodromies with affine team, and the full quantum numbers of the BPS spectrum. While the braiding describes the quark-monopole transmutation, the fusing implies the superconformal points in the moduli space. In the simplest case of three singularities, the supersymmetric sections are directly related to the conformal blocks of the logarithmic minimal models.

CP, charge fractionalizations and low-energy effective actions in the SU(2) Seiberg-Witten theories with quarks

Several dynamical aspects of the SU(2) Seiberg-Witten models with Nf quark hypermultiplets are explored. We first clarify the meaning of the number of the singularities of the space of vacua. CP properties of the theories are then studied and periodicity of theories in θ with and without bare quark masses is obtained ((4- N f ) π and π respectively). CP noninvariance at a generic point of QMS manifests itself as the electric and quark-number charge fractionalizations for the dyons; we show that the exact Seiberg-Witten solution contains such effects correctly, in agreement with the semiclassical analysis recently made by F. Ferrari. Upon N = 1 perturbation the low-energy effective theories at the singularities display confinement, and in most cases chiral symmetry breaking as a consequence. In one of the vacua for N f = 3 confinement is not accompanied by chiral symmetry breaking: we interpret it as an example of oblique confinement of 't Hooft. We discuss further the consistency of the physical picture found here by studying the effects of soft supersymmetry breaking as well as the behavior of the theory in the N = 1 limit.

WDVV Equations in seiberg-witten theory and associative Algebras

This is a short review of the results on the associativity algebras and WDVV equations found recently for the Seiberg-Witten solutions of N=2 4d SUSY gauge theories. The presentation is mostly based on the integrable treatment of these solutions.

SOFTLY BROKEN N=2 QCD WITH MASSIVE QUARK HYPERMULTIPLETS, I

We present a general analysis of all possible soft breakings of N=2 supersymmetric QCD preserving the analytic properties of the Seiberg–Witten solutions for the SU(2) group with Nf=1,2,3 hypermultiplets. We obtain all the couplings of the spurion fields in terms of properties of the Seiberg–Witten periods which we express in terms of elementary elliptic functions by uniformizing the elliptic curves associated to each number of flavors. We analyze in detail the monodromy properties of the softly broken theory, and obtain them by a particular embedding into a pure gauge theory with a higher rank group. This allows us to write explicit expressions of the effective potential which are close to the exact answer for moderate values of the supersymmetry breaking parameters. The vacuum structures and phases of the broken theories will be analyzed in the forthcoming second part of this paper.

Dual Isomonodromic Problems and Whitham Equations

The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the Jimbo–Miwa–Mori–Sato (JMMS) equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter e. By the method of multi-scale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed and shown to possess a number of properties similar to the Seiberg–Witten solutions of low energy supersymmetric gauge theories.

Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type

It has been known since the beginning of this century that isomonodromic problems --- typically the Painleve transcendents --- in a suitable asymptotic region look like a kind of ``modulation'' of isospectral problem. This connection between isomonodromic and isospectral problems is reconsidered here in the light of recent studies related to the Seiberg-Witten solutions of $N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a typical isomonodromic problem, namely the Schlesinger equation, which is reformulated to include a small parameter $\epsilon$. In the small-$\epsilon$ limit, solutions of this isomonodromic problem are expected to behave as a slowly modulated finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic method. An inverse period map of Seiberg-Witten type turns out to give general solutions of this modulation equation. This construction of general solution also reveals the existence of deformations of Seiberg-Witten type on the same moduli space of spectral curves. A prepotential is also constructed in the same way as the prepotential of the Seiberg-Witten theory.