The Whitham Deformation of the Dijkgraaf-Vafa Theory

We discuss the Whitham deformation of the effective superpotential in the Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we derive the Whitham equation for the period, which governs flowings of branch points on the Riemann surface. By studying the hodograph solution to the Whitham equation it is shown that the effective superpotential in the DV theory is realized by many different meromorphic differentials. Depending on which meromorphic differential to take, the effective superpotential undergoes different deformations. This aspect of the DV theory is discussed in detail by taking the N=1^* theory. We give a physical interpretation of the deformation parameters.Comment: 35pages, 1 figure; v2: one section added to give a physical interpretation of the deformation parameters, one reference added, minor corrections; v4: minor correction

Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.Comment: Final version to be published in Commun. Math. Phys. . A new section is added and devoted to Conclusion and discussion, where, in particular, a possible relation with the generating function of the absolute Gromov-Witten invariants on CP^1 is commented. Two references are added. Typos are corrected. 32 pages. 4 figure

Factorization of Seiberg-Witten Curves and Compactification to Three Dimensions

We continue our study of nonperturbative superpotentials of four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, broken to N=1 due to a classical superpotential. In a previous paper, hep-th/0304061, we discussed how the low-energy quantum superpotential can be obtained by substituting the Lax matrix of the underlying integrable system directly into the classical superpotential. In this paper we prove algebraically that this recipe yields the correct factorization of the Seiberg-Witten curves, which is an important check of the conjecture. We will also give an independent proof using the algebraic-geometrical interpretation of the underlying integrable system.Comment: laTeX, 14 pages, uses AMSmat

On the Baryonic Branch Root of N=2 MQCD

We investigate the brane exchange in the framework of N=2 MQCD by using a specific family of M fivebrane configurations relevant to describe the baryonic branch root. An exchange of M fivebranes is realized in the Taub-NUT geometry and controlled by the moduli parameter of the configurations. This family also provides two different descriptions of the root. These descriptions are examined carefully using the Taub-NUT geometry. It is shown that they have the same baryonic branch and are shifted each other by the brane exchange.Comment: LaTeX, 25 pages, 7 figures, references adde

The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS_2 x S^2

We study the near horizon limit of a four dimensional extreme rotating black hole. The limiting metric is a completely nonsingular vacuum solution, with an enhanced symmetry group SL(2,R) x U(1). We show that many of the properties of this solution are similar to the AdS_2 x S^2 geometry arising in the near horizon limit of extreme charged black holes. In particular, the boundary at infinity is a timelike surface. This suggests the possibility of a dual quantum mechanical description. A five dimensional generalization is also discussed.Comment: 21 page

Toda Lattice Hierarchy and Generalized String Equations

String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33), (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references are update

On Effective Superpotentials and Compactification to Three Dimensions

We study four dimensional N=2 SO/SP supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice. The connection between vacua of SO(2N) and SO(2kN-2k+2) can also be seen in this framework. Similar analysis can also be applied for SO(2N+1) and SP(2N).Comment: 18 pages, latex file, v2: typos corrected, refs adde

Supersymmetry Flows, Semi-Symmetric Space Sine-Gordon Models And The Pohlmeyer Reduction

We study the extended supersymmetric integrable hierarchy underlying the Pohlmeyer reduction of superstring sigma models on semi-symmetric superspaces F/G. This integrable hierarchy is constructed by coupling two copies of the homogeneous integrable hierarchy associated to the loop Lie superalgebra extension f of the Lie superalgebra f of F and this is done by means of the algebraic dressing technique and a Riemann-Hilbert factorization problem. By using the Drinfeld-Sokolov procedure we construct explicitly, a set of 2D spin \pm1/2 conserved supercharges generating supersymmetry flows in the phase space of the reduced model. We introduce the bi-Hamiltonian structure of the extended homogeneous hierarchy and show that the two brackets are of the Kostant-Kirillov type on the co-adjoint orbits defined by the light-cone Lax operators L_\pm. By using the second symplectic structure, we show that these supersymmetries are Hamiltonian flows, we compute part of the supercharge algebra and find the supersymmetric field variations they induce. We also show that this second Poisson structure coincides with the canonical Lorentz-Invariant symplectic structure of the WZNW model involved in the Lagrangian formulation of the extended integrable hierarchy, namely, the semi-symmetric space sine-Gordon model (SSSSG), which is the Pohlmeyer reduced action functional for the transverse degrees of freedom of superstring sigma models on the cosets F/G. We work out in some detail the Pohlmeyer reduction of the AdS_2xS^2 and the AdS_3xS^3 superstrings and show that the new conserved supercharges can be related to the supercharges extracted from 2D superspace. In particular, for the AdS_2xS^2 example, they are formally the same.Comment: V2: Two references added, V3: Modifications in section 2.6, V4: Published versio

Gauge Theory Description of D-brane Black Holes: Emergence of the Effective SCFT and Hawking Radiation

We study the hypermultiplet moduli space of an N=4, U(Q_1)xU(Q_5) gauge theory in 1+1 dimensions to extract the effective SCFT description of near extremal 5-dimensional black holes modelled by a collection of D1- and D5-branes. On the moduli space, excitations with fractional momenta arise due to a residual discrete gauge invariance. It is argued that, in the infra-red, the lowest energy excitations are described by an effective c=6, N=4 SCFT on T^4, also valid in the large black hole regime. The ``effective string tension'' is obtained using T-duality covariance. While at the microscopic level, minimal scalars do not couple to (1,5) strings, in the effective theory a coupling is induced by (1,1) and (5,5) strings, leading to Hawking radiation. These considerations imply that, at least for such black holes, the calculation of the Hawking decay rate for minimal scalars has a sound foundation in string theory and statistical mechanics and, hence, there is no information loss.Comment: 24 pages, LaTeX, very minor changes, to appear in Nucl. Phys.

N=1 G_2 SYM theory and Compactification to Three Dimensions

We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice based on the dual of the affine G_2 Lie algebra. At extrema of the superpotential the Seiberg-Witten curve typically factorizes, and we study the algebraic equations underlying this factorization. For U(N) theories the factorization was closely related to the geometrical engineering of such gauge theories and to matrix model descriptions, but here we will find that the geometrical interpretation is more mysterious. Along the way we give a method to compute the gauge theory resolvent and a suitable set of one-forms on the Seiberg-Witten curve. We will also find evidence that the low-energy dynamics of G_2 gauge theories can effectively be described in terms of an auxiliary hyperelliptic curve.Comment: 27 pages, late