We construct N = 2 supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. It turns out that for every fixed point one can allocate either instanton or anti-instanton contributions to the partition function, and that this is compatible with supersymmetry. The equivariant Donaldson-Witten theory is a special case of our construction. We present a unified treatment of Pestun's calculation on S 4 and equivariant Donaldson-Witten theory by generalizing the notion of self-duality on manifolds with a vector field. We conjecture the full partition function for a N = 2 theory on any 4D manifold with a Killing vector. Using this new notion of self-duality to localize a supersymmetric theory is what we call "Pestunization". C Transverse elliptic problems in 4D and 5D 48 D Full cohomological complex 52 E Supergravity background 53

In this paper we summarise the localisation calculation of 5D super Yang-Mills on simply connected toric Sasaki-Einstein (SE) manifolds. We show how various aspects of the computation, including the equivariant index, the asymptotic behaviour and the factorisation property are governed by the combinatorial data of the toric geometry. We prove that the full perturbative partition function on a simply connected SE manifold corresponding to an n-gon toric diagram factorises to n copies of perturbative Nekrasov partition function. This leads us to conjecture the full partition function as gluing n copies of full Nekrasov partition function. This work is a generalisation of some earlier computation carried out on Y p,q manifolds, whose moment map cone has a quadrangle and our result is valid for manifolds whose moment map cones have pentagon base, hexagon base, etc. The algorithm we used for dealing with general cones may also be of independent interest.

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Abstract:We construct 4D N = 2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S 2 × S 2 . These theories are constructed through the dimensional reduction along a non-trivial U(1)-fiber of 5D theories on toric SasakiEinstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun's famous result on S 4 .

Abstract. We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of SLr(Z)-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.

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We study properties of the full partition function for the U(1) 5D N = 2 * gauge theory with adjoint hypermultiplet of mass M . This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function G C 2 associated with a certain moment map cone C. The answer exhibits a curious SL(4, Z) modular property. Finally, we propose a set of rules to construct the partition function that resembles the calculation of 5d supersymmetric partition function with the insertion of defects of various co-dimensions.

The problems of the standard model are briefly reviewed and the motivations for introducing supersymmetry are discussed. Two realistic supersymmetric models; the Minimal SuperSymmetric Model, MSSM, and its proposed extension NMSSM are introduced briefly and then some details of the Higgs sector of NMSSM are stated and some constraints on parameters are described. I then explore the Higgs boson masses and couplings for some interesting scenarios, including a few different ways of taking the limit where NMSSM reduces to MSSM.