On noncommutative Nahm transform

Motivated by the recently observed relation between the physics of $D$-branes in the background of $B$-field and the noncommutative geometry we study the analogue of Nahm transform for the instantons on the noncommutative torus.Comment: Latex, 22 p

D-particle bound states and generalized instantons

We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0+1 dimensions of $\mathcal{N}=1$, $D=4,6,10$ super-Yang-Mills theories with gauge group SU(N). The results are: ${1\over{N^{2}}}$ for $D=4,6$, $\sum_{d | N} {1\over{d^{2}}}$ for D=10. We also discuss the D=3 case.Comment: harvmac, 24 pages; v2. references added, typos corrected; v3. one more reference adde

Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on $T^{*}\Sigma$ for \Sigma = {\IC}, {\IC}^{*} or elliptic curve, and on ${\bf C}^{2}/{\Gamma}$ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperk\"ahler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of $D$-branes and string duality.Comment: harvmac, 27 pp. big mode; v2. typos and references correcte

Integrating Over Higgs Branches

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 R^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 Schroedinger system. We discuss some applications of our results.Comment: 32pp. harvmac big mode; v.2 34pp. typos fixed, sections 4.1, 5.2 substantially improve