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5D Super Yang-Mills on $Y^{p,q}$ Sasaki-Einstein manifolds

Jian Qiu, Maxim Zabzine

TL;DR

This work extends localisation techniques for 5D Yang–Mills with matter from $S^5$ to toric Sasaki–Einstein spaces $Y^{p,q}$, deriving the full perturbative partition function in terms of a generalized $S^{\Lambda}$ function and analysing its dependence on the Reeb data and squashing. The authors compute the one-loop determinants using transversally elliptic indices on $Y^{p,q}$, organise the result into a matrix-model-like integral, and extract the large-$N$ behavior for an adjoint hypermultiplet, revealing an $N^3$ scaling controlled by the SE volume ratio $\varrho$. They provide explicit asymptotics for the determinants, showing that the cubic term in the effective potential vanishes for adjoint matter, simplifying the matrix model to a linear-potential form in the large-$N$ limit and yielding a precise $N^3$ free energy. The results connect the geometry of $Y^{p,q}$, SE volumes, and localisation to concrete predictions for 5D SCFTs and their holographic relatives, with clear paths for generalisation to broader toric SE manifolds and inclusion of instanton sectors in future work.

Abstract

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as $Y^{p,q}$ manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large $N$ behaviour for the case of single hypermultiplet in adjoint representation and we derive the $N^3$-behaviour in this case.

5D Super Yang-Mills on $Y^{p,q}$ Sasaki-Einstein manifolds

TL;DR

This work extends localisation techniques for 5D Yang–Mills with matter from to toric Sasaki–Einstein spaces , deriving the full perturbative partition function in terms of a generalized function and analysing its dependence on the Reeb data and squashing. The authors compute the one-loop determinants using transversally elliptic indices on , organise the result into a matrix-model-like integral, and extract the large- behavior for an adjoint hypermultiplet, revealing an scaling controlled by the SE volume ratio . They provide explicit asymptotics for the determinants, showing that the cubic term in the effective potential vanishes for adjoint matter, simplifying the matrix model to a linear-potential form in the large- limit and yielding a precise free energy. The results connect the geometry of , SE volumes, and localisation to concrete predictions for 5D SCFTs and their holographic relatives, with clear paths for generalisation to broader toric SE manifolds and inclusion of instanton sectors in future work.

Abstract

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large behaviour for the case of single hypermultiplet in adjoint representation and we derive the -behaviour in this case.

Paper Structure

This paper contains 30 sections, 236 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Super Yang-Mills Theory on Sasaki-Einstein 5-folds
  3. 5D Yang-Mills theory with matter on $S^5$
  4. Sasaki-Einstein Manifolds
  5. Sasaki Einstein 5-folds and Killing Spinors
  6. Geometry of $Y^{p,q}$ Manifolds
  7. $Y^{p,q}$ from Reduction
  8. Looking for Free $U(1)$ in $Y^{p,q}$
  9. The base of the $U(1)$ fibration
  10. The Reeb that Admits a Sasaki-Einstein Metric
  11. The Cohomological Complex and Localisation
  12. Vector-Multiplet
  13. Hyper-Multiplet
  14. Localisation
  15. Localisation Locus
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The lattice $\Gamma$ for summation for $p=3$ and $q=1$
  • Figure 2: The two lower lines have $m<0$ and the two upper $m\geq0$. And each lattice point on the lines in the first quadrant has multiplicity $1+2i+m(p-q)$, while those in the third quadrant $-2i+m(q-p)-1$.