Instanton Calculus and SUSY Gauge Theories on ALE Manifolds

Three dimensional N=2 gauge theories with arbitrary gauge group and fundamental flavors are engineered from degenerations of Calabi-Yau four-folds. We show how Coulomb and Higgs branches emerge in the geometric picture. The analysis of instanton generated superpotentials unravels interesting aspects of the five-brane effective action in M theory.

In this letter we discuss the relation of four-dimensional, $N=2$ supersymmetric string backgrounds to integrable models. In particular we show that non-K\"ahlerian gravitational backgrounds with one $U(1)$ isometry plus non-trivial antisymmetric tensor and dilaton fields arise as the solutions of the Liouville equation or, for the case of vanishing central charge deficit, as the solutions of the continual Toda equation. When performing an Abelian duality transformation, a pa...

We construct a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology R^2*ALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. Our solutions have two Killing spinors and thus preserve N=2 supersymmetry. They are expressed in terms of a quasi harmonic function H (the ``warp factor''), whose properties we study in...

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.

We use the recursive structure of the compactification of the instanton moduli space of N=2 Super Yang-Mills theory with gauge group SU(2), to construct, by inductive limit, a universal moduli space which includes all the multi-instanton moduli spaces. Furthermore, with the aim of understanding the field theoretic structure of the strong coupling expansion, we perform the Borel sum which acts on the parameter defining such a universal moduli space.

N=1^* gauge theories are believed to have fractional instanton contributions in the confining vacua. D3 brane probe computations in gravitation dual of large-N N=2^* gauge theories point to the absence of such contributions in the low energy gauge dynamics. We study fractional instantons in N=2 SU(2) Yang-Mills theory from the field theoretical perspective. We present new solutions to the Seiberg-Witten SU(2) monodromy problem with the same perturbative asymptotic, a massless...

We study N=4 supersymmetric Yang-Mills (SYM) theory with gauge group SU(2) compactified to three dimensions on a circle of circumference beta. The eight fermion terms in the effective action on the Coulomb branch are determined exactly, for all beta, assuming the existence of an interacting Spin(8) invariant fixed point at the origin. The resulting formulae are manifestly invariant under the SL(2,Z) duality of four dimensional N=4 SYM and lead to interesting quantitative pred...

It is well-known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equations from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self...

We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$ gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of $\mathbb R^4$, we derive a differential equation for the Nekrasov's partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev e...

Exact solutions to the low-energy effective action of the four-dimensional, N=2 supersymmetric gauge theories with matter (including N=2 super-QCD) are discussed from the three different viewpoints: (i) instanton calculus, (ii) N=2 harmonic superspace, and (iii) M theory. The emphasis is made on the foundations of all three approaches and their relationship.