## Final Report Summary - NPEGST (Non perturbative effects in gauge and string theories)

Project context and objectives

The main objective of the current project was to investigate non-perturbative effects in Super-symmetric Yang-Mills theories from the purely field theoretical point of view as well as from the string theory perspective. In the framework of field theory we have intended also to extend the results concerning super-symmetric Yang-Mills theory on flat space-time to the cases of non-trivial gravitational background. On string theory side the main objective was to investigate stringy instanton effects, which besides purely theoretical significance also have phenomenological implications in all directions significant progress has been achieved during the realisation of the project. The results are presented in six scientific publications which we detail one by one.

1) A system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the $\epsilon_2\rightarrow 0$ limit is derived. It is shown that the prepotential with generic $\epsilon_1$ is directly related to the (rescaled by $\epsilon_1$) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields $\langle tr \phi^J \rangle $ are simple symmetric functions of their column lengths. An entire function whose zeros are determined by the column lengths is introduced. It is shown that this function satisfies a functional equation, closely resembling Baxter's equation in 2d integratable models.

2) We revisit Kaluza-Klein compactification of 11-d supergravity on $S^7/Z_k$ using group theory techniques that may find application in other flux vacua with internal coset spaces. Among the SO(2) neutral states, we identify marginal deformations and fields that couple to the recently discussed world-sheet instanton of Type IIA on $CP^3.

3) We discuss a string model where a conformal four-dimensional N=2 gauge theory receives corrections to its gauge kinetic functions from "stringy" instantons. These contributions are explicitly evaluated by exploiting the localisation properties of the integral over the stringy instanton moduli space. The model we consider corresponds to a setup with D7/D3-branes in type I' theory compactified on T4/Z2 x T2, and possesses a perturbatively computable heterotic dual.

4) Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalised prepotential of ${\cal N}=2$ SYM with $f=0,1,2,3,4$ (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established.

5) We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localisation techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1.

6) We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so-called exotic. In analogy with the Seiberg-Witten theory in 4 D space-time, the prepotential and correlators in the chiral ring are derived via localisation formulae.

The main objective of the current project was to investigate non-perturbative effects in Super-symmetric Yang-Mills theories from the purely field theoretical point of view as well as from the string theory perspective. In the framework of field theory we have intended also to extend the results concerning super-symmetric Yang-Mills theory on flat space-time to the cases of non-trivial gravitational background. On string theory side the main objective was to investigate stringy instanton effects, which besides purely theoretical significance also have phenomenological implications in all directions significant progress has been achieved during the realisation of the project. The results are presented in six scientific publications which we detail one by one.

1) A system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the $\epsilon_2\rightarrow 0$ limit is derived. It is shown that the prepotential with generic $\epsilon_1$ is directly related to the (rescaled by $\epsilon_1$) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields $\langle tr \phi^J \rangle $ are simple symmetric functions of their column lengths. An entire function whose zeros are determined by the column lengths is introduced. It is shown that this function satisfies a functional equation, closely resembling Baxter's equation in 2d integratable models.

2) We revisit Kaluza-Klein compactification of 11-d supergravity on $S^7/Z_k$ using group theory techniques that may find application in other flux vacua with internal coset spaces. Among the SO(2) neutral states, we identify marginal deformations and fields that couple to the recently discussed world-sheet instanton of Type IIA on $CP^3.

3) We discuss a string model where a conformal four-dimensional N=2 gauge theory receives corrections to its gauge kinetic functions from "stringy" instantons. These contributions are explicitly evaluated by exploiting the localisation properties of the integral over the stringy instanton moduli space. The model we consider corresponds to a setup with D7/D3-branes in type I' theory compactified on T4/Z2 x T2, and possesses a perturbatively computable heterotic dual.

4) Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalised prepotential of ${\cal N}=2$ SYM with $f=0,1,2,3,4$ (anti) fundamental or an adjoint hypermultiplets. In all cases the large expectation value limit is derived explicitly. A precise relationship between generic 1-point conformal block on torus and specific 4-point conformal block on sphere is established.

5) We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localisation techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1.

6) We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so-called exotic. In analogy with the Seiberg-Witten theory in 4 D space-time, the prepotential and correlators in the chiral ring are derived via localisation formulae.