Gluing Nekrasov Partition Functions

Abstract

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.

Little strings and T-duality

“…Finally, the self-dual string elliptic genera in 6d SCFTs turn out to be related to other interesting observables, such as the superconformal indices [33][34][35][36][37][38]. It will be interesting to see if the elliptic genera for little strings also find similar interesting applications.…”

Gauge theories with 16 supersymmetries on spheres

“…σ I = +1 I = 6, 7 19) where m serves as the hypermultiplet mass. In the case that I, J has one index from the pair 6, 7 and one index from the pair 8, 9, the linear combination α I + α J is unchanged from the maximally supersymmetric case and the third terms cancel.…”

“…By making liberal use of the identities in the appendix, we can further establish 18) which restricts the forms to be (3, 0) or (0, 3). Analogously we can define the superpartner of Φ 19) with similar properties as Φ.…”

“…Following [18] and [19] the calculation of the superdeterminant (5.37) for generic toric v can be reduced to counting holomorphic functions on the metric cone over S 7 , which is just C 4 . Let us sketch the logic here and later we present a different approach for the calculation of (5.37) for the case of the round sphere.…”

“…The calculation of the Kohn-Rossi cohomology and the decomposition into eigenspaces for T 4 can be reduced to the study of homolomorphic functions on the metric cone, C 4 . The seven dimensional calculation is a straightforward generalization of the 5D case and further details can be found in [18] and [19]. The final answer is given by the following matrix model…”