Gluing Nekrasov partition functions

TL;DR

This work addresses the perturbative partition function for 5D N=1 SYM on simply connected toric Sasaki–Einstein manifolds $X$ by localisation, reducing the calculation to counting holomorphic functions on the CY cone $C(X)$. It shows that the perturbative answer factorises into $n$ copies of Nekrasov partition functions on $\mathbb{C}^2\times S^1$, with toric data fixing the equivariant parameters via the moment-map cone; it also proves the equivalence of the constrained-lattice and cone descriptions and derives the asymptotics at large $N$. The authors conjecture a full nonperturbative partition function obtained by gluing the same Nekrasov blocks and discuss limitations for non-simply connected manifolds and potential 6D interpretations. Overall, the paper provides a geometry-driven factorisation principle for higher-dimensional localisation, linking toric SE data to Nekrasov-building blocks and offering a route toward a complete nonperturbative understanding.

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.

Figures (6)

• Figure 1: The polygon base of a polytope cone. Over the interior of the polygon there is a $T^3$ fibre, but over the faces the $T^3$ degenerates into $T^2$, which further degenerate over the vertices to $S^1$, drawn as the circles in the figure. These circles are the only generic closed Reeb orbits.
• Figure 2: The polytope cone, projected onto the plane $y=1$, depending on the specific case, one of the faces may move off to infinity, that its two neighbouring faces turn parallel. The circles represent the closed Reeb orbits. The right panel is the inward pointing normals of the cone.
• Figure 3:
• Figure 4: Sum between the two blue lines, depicted in the left panel. One can add more lines in between, as in the right panel. The numbers label the normal of each line. The slopes of the lines are not drawn to scale.
• Figure 5: Further division of the $m_2$-$m_3$ plane, by adding lines. The normals of all lines are pointing counterclockwise.

Theorems & Definitions (1)

• Lemma A.1