Mathematics

We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0+1 dimensions of N = 1, D = 4, 6, 10 super-Yang-Mills theories with gauge group SU (N ). The results are: 1 N 2 for D = 4, 6, d|N 1 d 2 for D = 10. We also discuss the D = 3 case.

We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theory. Finally, we define four-dimensional "conformal blocks" and present an analog of the Verlinde formula.

We show that the N = 0 theories on the self-dual D3-branes of Type 0 string theory are in the class of the previously considered tadpole-free orbifolds of N = 4 theory (although they have SO(6) global symmetry) and hence have vanishing beta function in the planar limit to all orders in 't Hooft coupling. Also, all planar amplitudes in this theory are equal to those of N = 4 theory, up to a rescaling of the coupling.

Supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N ) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model which are related to the four dimensional theory compactified on T 2 . A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as T * Gr(N, L) and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models as well as the dynamical spin chains like Hubbard model. We give the gauge-theoretic interpretation of Drinfeld polynomials and Baxter operators. The two-sphere compactifications of the four dimensional N = 2 theories lead to the instanton corrected Bethe equations. We suggest the Yangian, quantum affine, and elliptic algebras are a completely novel kind of symmetry of the (collections of the) interacting quantum field theories. To Prof. T. Eguchi on the occasion of his 60th anniversary a On leave of absence from ITEP, Moscow, Russia 0

We present an explicit construction for the central extension of the group Map(X, G) where X is a compact manifold and G is a Lie group. If X is a complex curve we obtain a simple construction of the extension by the Picard variety Pic(X). The construction is easily adapted to the extension of Aut(E), the gauge group of automorphisms of a nontrivial vector bundle E.

We investigate higher-dimensional analogues of the bc systems of 2D RCFT. When coupled to gauge fields and Beltrami differentials defining integrable holomorphic structures, the bc partition functions can be explicitly evaluated using anomalies and holomorphy. The resulting induced actions generalize the chiral algebras of 2D RCFT to 2n dimensions. Moreover, bc systems in four and six dimensions are closely related

We discuss topological theories, arising from the general N = 2 twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme

We develop some useful techinques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkahler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkahler periods. We also reduce these volumes for a large class of hyperkahler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on IR 4 and ALE manifolds, Hitchin spaces, and moduli spaces of (parabolic) Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the non-linear Schrödinger system. We discuss some applications of our results.

This paper lays groundwork for the detailed study of the non-trivial renormalization group flow connecting supersymmetric fixed points in four dimensions using string theory on AdS spaces. Specifically, we consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The N = 1 superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE N = 2 superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry with a single D3-brane and discuss the near-horizon supergravity solution for a large number N of coincident D3-branes.

Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearence of the new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive.