7D supersymmetric Yang-Mills on a 3-Sasakian manifold

Abstract

In this paper we study 7D maximally supersymmetric Yang-Mills on a specific 3-Sasakian manifold that is the total space of an SO(3)-bundle over CP 2 . The novelty of this example is that the manifold is not a toric Sasaki-Einstein manifold. The hyperkähler cone of this manifold is a Swann bundle with hypertoric symmetry and this allows us to calculate the perturbative part of the partition function of the theory. The result is also verified by an index calculation. We also discuss a factorisation of this result and compare it with analogous results for S 7 .

Supersymmeric Gauge Theory on Curved 7‐Branes

“…It would be interesting to compare the results of such computations to the results of pure SYM localization. [8,20,52] We have used the notion of a kappa-symmetric E7-brane purely as a tool to construct the field theories; similar to the standard procedure of coupling to off-shell supergravity backgrounds, our E7-brane solutions do not solve the equations of motion. However, the violation lies purely in the equation of motion for F 4 ; it may prove worthwhile to turn on fields we have set to zero by hand to see if this can be remedied.…”

7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

“…While these manifolds are also Sasaki-Einstein, they are not necessarily toric and so different techniques have to be used. This paper generalises the 'proof-of-concept' calculation in [10] to arbitrary 7D hypertoric 3-Sasakian manifolds, which we now describe.…”

“…This calculation used that the hyper-Kähler cone of this manifold had hypertoric symmetry. This paper is a continuation of the work in [10] and here we derive a closed form answer for the perturbative partition function for arbitrary 3-Sasakian manifolds whose hyper-Kähler cones are hypertoric. The answer is stated in terms of a special function that enumerate integer lattice points in a cone determined by hypertoric data, similar in spirit to how the generalised sine functions count points in cones determined by toric data in the toric Sasaki-Einstein case.…”

“…2 In [6] some initial steps were taken towards understanding the role such a 3-Sasakian structure play for localisation calculations on S 7 , which in addition to being a toric Sasaki-Einstein manifold is also 3-Sasakian. In [10] a localisation calculation was performed for a specific 7D 3-Sasakian manifold that is not toric in the Sasaki-Einstein sense. This calculation used that the hyper-Kähler cone of this manifold had hypertoric symmetry.…”

“…The next example is the 3-Sasakian manifold X associated to the Swann bundle. We first use the machinery of section 4.2 to redo the computation done in [10], but in much more streamlined fashion.…”