Seiberg-Witten Models Research Articles

Experiments with the WDVV equations for the gluino-condensate prepotential: the cubic (two-cut) case

We demonstrate by explicit calculation that the first two terms in the CIV-DV prepotential for the two-cut case satisfy the generalized WDVV equations, just as in all other known examples of hyper-elliptic Seiberg–Witten models. The WDVV equations are non-trivial in this situation, provided the set of moduli is extended as compared to the Dijkgraaf–Vafa suggestion and includes also moduli, associated with the positions of the cuts (not only with their lengths). Expression for the extra modulus dictated by WDVV equation, however, appears different from a naive expectation implied by the Whitham theory. Moreover, for every value of the “quantum-deformation parameter” 1/g3, we actually find an entire one-parameter family of solutions to the WDVV equations, of which the conventional prepotential is just a single point.

Integrability, Seiberg-Witten models and Picard-Fuchs equations

Expanded version of the author's contribution to the Concise Encyclopaedia of Supersymmetry, eds. J. Bagger, S. Duplij and W. Siegel

On the WDVV equation and M-theory

A wide class of Seiberg-Witten models constructed by M-theory techniques and described by non-hyperelliptic Riemann surfaces are shown to possess an associative algebra of holomorphic differentials. This is a first step towards proving that also these models satisfy the Witten-Dijkgraaf-Verlinde-Verlinde equation. In this way, similar results known for simpler Seiberg-Witten models (described by hyperelliptic Riemann surfaces and constructed without recourse to M-theory) are extended to certain non-hyperelliptic cases constructed in M-theory. Our analysis reveals a connection between the algebra of holomorphic differentials on the Riemann surface and the configuration of M-theory branes of the corresponding Seiberg-Witten model.

Logarithmic conformal field theory and Seiberg-Witten models

The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential λ SW, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge c= c 2,1=−2. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type F D .

Covariance of WDVV equations

The (generalized) WDVV equations for the prepotentials in 2d topological and 4,5d Seiberg-Witten models are covariant with respect to non-linear transformations, described in terms of solutions of associated linear problem. Both time-variables and the prepotential change non-trivially, but period matrix (prepotential's second derivatives) remains intact.

Supersymmetry and the multi-instanton measure

We propose explicit formulae for the integration measure on the moduli space of charge- n ADHM multi-instantons in N = 1 and N = 2 supersymmetric gauge theories. The form of this measure is fixed by its (super) symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n ⩽ 2. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N = 2 results, chiefly: (1) For any number of flavors N F , we provide a closed formula for F n , the n-instanton contribution to the Seiberg-Witten propotential, as a finite-dimensional collective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case N F = 4, this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, τ micro and τ eff. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N = 2 supersymmetric QCD.

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.

Simple derivation of the Picard-Fuchs equations for the Seiberg-Witten models

A closed form of the Picard-Fuchs equations for N=2 supersymmetric Yang-Mills theories with massless hypermultiplet are obtained for classical Lie gauge groups. We consider any number of massless matter in fundamental representation so as to keep the theory asymptotically free.