Witten Curve Research Articles

Free fermion and Seiberg–Witten differential in random plane partitions

A model of random plane partitions which describes five-dimensional N = 1 supersymmetric SU ( N ) Yang–Mills is studied. We compute the wave functions of fermions in this statistical model and investigate their thermodynamic limits or the semi-classical behaviors. These become of the WKB type at the thermodynamic limit. When the fermions are located at the main diagonal of the plane partition, their semi-classical wave functions are obtained in a universal form. We further show that by taking the four-dimensional limit the semi-classical wave functions turn to live on the Seiberg–Witten curve and that the classical action becomes precisely the integral of the Seiberg–Witten differential. When the fermions are located away from the main diagonal, the semi-classical wave functions depend on another continuous parameter. It is argued that they are related with the wave functions at the main diagonal by the renormalization group flow of the underlying gauge theory.

The off-shell M5-brane and non-perturbative gauge theory

M5-branes wrapping a holomorphic curve in a Calabi–Yau manifold can be used to construct four-dimensional N = 1 gauge theories. In this paper we will consider M5-brane configurations corresponding to N = 2 theories broken to N = 1 by a superpotential for the adjoint scalar field. These M5-brane configurations can be obtained by lifting suitable intersecting brane configurations in type IIA, or equivalently by T-dualizing IIB configurations with branes and/or fluxes. We will show that turning on non-trivial expectation values for the glueball superfields corresponds to non-holomorphic deformations of the M5-brane. We compute the superpotential and show it agrees with that computed by Dijkgraaf and Vafa. Several aspects of the gauge theory, such as the appearance of non-holomorphic one-forms with integer periods on the Seiberg–Witten curve, have a natural interpretation from the M5-brane point of view. We also explain the interpretation of the superpotential in terms of the twisted ( 2 , 0 ) theory living on the fivebrane.

Factorization of Seiberg–Witten curves with matter from random matrix models

In this paper we review the Dijfgraaf–Vafa conjecture for U(Nc) super Yang–Mills theories with matter in the adjoint and fundamental representation. Secondly, we discuss a proposal to derive the effective superpotential in terms of the mesonic superfields and show how the Affleck–Dine–Seiberg non-perturbative superpotential arises directly from the matrix model. Finally, we review how the random matrix model can be used to obtain factorization data of Seiberg–Witten curves with matter.

Matrix-model description of [formula omitted] gauge theories with non-hyperelliptic Seiberg–Witten curves

Using matrix-model methods we study three different N=2 models: U( N)× U( N) with matter in the bifundamental representation, U( N) with matter in the symmetric representation, and U( N) with matter in the antisymmetric representation. We find that the (singular) cubic Seiberg–Witten curves (and associated Seiberg–Witten differentials) implied by the matrix models, although of a different form from the ones previously proposed using M-theory, can be transformed into the latter and are thus physically equivalent. We also calculate the one-instanton corrections to the gauge-coupling matrix using the perturbative expansion of the matrix model. For the U( N) theories with symmetric or antisymmetric matter we use the modified matrix-model prescription for the gauge-coupling matrix discussed in our paper: Cubic curves from matrix models and generalized Konishi anomalies ( hep-th/0303268). Moreover, in the matrix model for the U( N) theory with antisymmetric matter, one is required to expand around a different vacuum than one would naively have anticipated. With these modifications of the matrix-model prescription, the results of this paper are in complete agreement with those of Seiberg–Witten theory obtained using M-theory methods.

Calculating gluino condensates in [formula omitted] SYM from Seiberg–Witten curves

We determine the gluino condensate 〈Trλ2〉 in the pure N=1 super-Yang–Mills theory (SYM) for the classical gauge groups SU(r+1), SO(2r+1), USp(2r) and SO(2r), by deforming the pure N=2 SYM theory with the adjoint scalar multiplet mass, following the work by Finnell and Pouliot, and Ritz and Vainshtein. The value of the gluino condensate agrees in all cases with what was found in the weak coupling instanton calculation.

EFFECTIVE SUPERPOTENTIALS FOR SO/Sp WITH FLAVOR FROM MATRIX MODELS

We study matrix models related to SO/Sp gauge theories with flavors. We give the effective superpotentials for gauge theories with arbitrary tree level superpotential up to first instanton level. For quartic tree level superpotential, we obtained exact one-cut solution. We also derive Seiberg–Witten curve for these gauge theories from matrix model argument.

Explicit factorization of Seiberg–Witten curves with matter from random matrix models

Within the Dijkgraaf–Vafa correspondence, we study the complete factorization of the Seiberg–Witten curve for U( N c ) gauge theory with N f < N c massive flavors. We obtain explicit expressions, from random matrix theory, for the moduli, parametrizing the curve. These moduli characterize the submanifold of the Coulomb branch where all monopoles become massless. We find that the matrix model reveals some nontrivial structures of the gauge theory. In particular the moduli are additive with respect to adding extra matter and increasing the number of colors.

Prepotential and instanton corrections in Script N = 2 supersymmetric SU(N1) × SU(N2) Yang Mills theories

In this paper we analyse the non-hyperelliptic Seiberg-Witten curves derived from M-theory that encode the low energy solution of N=2 supersymmetric theories with product gauge groups. We consider the case of a SU(N_1)xSU(N_2) gauge theory with a hypermultiplet in the bifundamental representation together with matter in the fundamental representations of SU(N_1) and SU(N_2). By means of the Riemann bilinear relations that hold on the Riemann surface defined by the Seiberg--Witten curve, we compute the logarithmic derivative of the prepotential with respect to the quantum scales of both gauge groups. As an application we develop a method to compute recursively the instanton corrections to the prepotential in a straightforward way. We present explicit formulas for up to third order on both quantum scales. Furthermore, we extend those results to SU(N) gauge theories with a matter hypermultiplet in the symmetric and antisymmetric representation. We also present some non-trivial checks of our results.

Seiberg-Witten Curve for &lt;i&gt;E&lt;/i&gt;-String Theory Revisited

We discuss various properties of the Seiberg–Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg–Witten curve for the E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R 4 × T 2 . It has a manifest affineE8 global symmetry with modulus τ and E8 Wilson line parameters {mi}, i = 1,2, . . . ,8 which are associated with the geometry of the rational elliptic surface. When the radii R5, R6 of the torus T 2 degenerate R5, R6 → 0, E-string curve is reduced to the known Seiberg–Witten curves of four- and five-dimensional gauge theories. In this paper we first study the geometry of rational elliptic surface and identify the geometrical significance of the Wilson line parameters. By fine tuning these parameters we also study degenerations of our curve corresponding to various unbroken symmetry groups. We also find a new way of reduction to four-dimensional theories without taking a degenerate limit of T 2 so that the SL(2, Z) symmetry is left intact. By setting some of the Wilson line parameters to special values we obtain the four-dimensional SU(2) Seiberg–Witten theory with 4 flavors and also a curve by Donagi and Witten describing the dynamics of a perturbed N = 4 theory.

SO(N) superpotential, Seiberg–Witten curves and loop equations

We consider the exact superpotential of N=1 super-Yang–Mills theory with gauge group SO(N) and arbitrary tree-level polynomial superpotential of one adjoint Higgs field. A field-theoretic derivation of the glueball superpotential is given, based on factorization of the N=2 Seiberg–Witten curve. Following the conjecture of Dijkgraaf and Vafa, the result is matched with the corresponding SO(N) matrix model prediction. The verification involves an explicit solution of the first non-trivial loop equation, relating the spherical free energy to that of the non-orientable surfaces with topology RP2.

EXCEPTIONAL STRING: INSTANTON EXPANSIONS AND SEIBERG–WITTEN CURVE

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 determine the Seiberg–Witten curve of E-string with the help of elliptic functions.

W( E10) symmetry, M-theory and Painlevé equations

The Weyl group symmetry W( E k ) is studied from the points of view of the E-strings, Painlevé equations and U-duality. We give a simple reformulation of the elliptic Painlevé equation in such a way that the hidden symmetry W( E 10) is manifestly realized. This reformulation is based on the birational geometry of the del Pezzo surface and closely related to Seiberg–Witten curves describing the E-strings. The relation of the W( E k ) symmetry to the duality of M-theory on a torus is discussed on the level of string equations of motion.

WHITHAM DEFORMATIONS OF SEIBERG–WITTEN CURVES FOR CLASSICAL GAUGE GROUPS

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg–Witten curve for the [Formula: see text] SUSY Yang–Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the [Formula: see text] affine Toda system. Our construction, too, uses fractional powers of the superpotential W(x) that characterizes the curve. We also consider the u-plane integral of topologically twisted theories on four-dimensional manifolds X with [Formula: see text] in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.

Deconfinement at the Argyres–Douglas point in SU(2) gauge theory with broken supersymmetry

We consider chiral condensates in SU(2) gauge theory with broken N =2 supersymmetry. The matter sector contains an adjoint multiplet and one fundamental flavor. Matter and gaugino condensates are determined by integrating out the adjoint field. The only nonperturbative input is the Affleck–Dine–Seiberg (ADS) superpotential generated by one instanton plus the Konishi anomaly. These results are consistent with those obtained by the `integrating in' procedure, including a reproduction of the Seiberg–Witten curve from the ADS superpotential. We then calculate monopole, dyon, and charge condensates using the Seiberg–Witten approach. We show that the monopole and charge condensates vanish at the Argyres–Douglas point where the monopole and charge vacua collide. We interpret this phenomenon as a deconfinement of electric and magnetic charges at the Argyres–Douglas point.

Elliptic models and M-theory

We give a unified analysis of four-dimensional elliptic models with N=2 supersymmetry and a simple gauge group, and their relation to M-theory. Explicit calculations of the Seiberg–Witten curves and the resulting one-instanton prepotential are presented. The remarkable regularities that emerge are emphasized. In addition, we calculate the prepotential in the Coulomb phase of the (asymptotically-free) Sp(2 N) gauge theory with N f fundamental hypermultiplets of arbitrary mass.

Picard–Fuchs equations and Whitham hierarchy in N=2 supersymmetric SU(r+1) Yang–Mills theory

In general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of N=2 supersymmetric Yang–Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of SU(r+1) gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic one-form of the Whitham hierarchy on the Seiberg–Witten curve is represented by a finite linear combination of meromorphic one-forms associated with these Whitham times, but it turns out that there are various differential relations among these differentials. Since these relations can be written only in terms of the Seiberg–Witten one-form, their consistency conditions are found to give the Picard–Fuchs equations for the Seiberg–Witten periods.

One-instanton prepotentials from WDVV equations in N=2 supersymmetric SU(4) Yang–Mills theory

Prepotentials in N=2 supersymmetric Yang–Mills theories are known to obey nonlinear partial differential equations called Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations. In this paper, the prepotentials at the one-instanton level in N=2 supersymmetric SU(4) Yang–Mills theory are studied from the standpoint of WDVV equations. Especially, it is shown that the one-instanton prepotentials are obtained from WDVV equations by assuming the perturbative prepotential and by using the scaling relation as a subsidiary condition but are determined without introducing the Seiberg–Witten curve. In this way, various one-instanton prepotentials which satisfy both WDVV equations and the scaling relation can be derived, but it turns out that among them there exist one-instanton prepotentials which coincide with the instanton calculus.

One-instanton predictions of Seiberg–Witten curves for product groups

One-instanton predictions for the prepotential are obtained from the Seiberg–Witten curve for the Coulomb branch of N=2 supersymmetric gauge theory for the product group ∏ n=1 m SU(N n) with a massless matter hypermultiplet in the bifundamental representation (N n, N ̄ n+1) of SU( N n )×SU( N n+1 ) for n=1 to m−1 , together with N 0 and N m+1 matter hypermultiplets in the fundamental representations of SU( N 1) and SU( N m ) respectively. The derivation uses a generalization of the systematic perturbation expansion about a hyperelliptic curve developed by us in earlier work.

$E_n$ Jacobi forms and Seiberg–Witten curves

We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.

Children’s drawings from Seiberg–Witten curves

We consider N = 2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called “dessins d’enfants” or “children’s drawings” on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(Q/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg–Witten theory is useful because gauge theory criteria used to distinguish branches of N = 1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck’s program of classifying dessins into Galois orbits and the physics problem of classifying phases of N = 1 gauge theories.