The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere (1206.6008v3)

Abstract: Based on the construction by Hosomichi, Seong and Terashima we consider N=1 supersymmetric 5D Yang-Mills theory with matter on a five-sphere with radius r. This theory can be thought of as a deformation of the theory in flat space with deformation parameter r and this deformation preserves 8 supercharges. We calculate the full perturbative partition function as a function of r/g^2, where g is the Yang-Mills coupling, and the answer is given in terms of a matrix model. We perform the calculation using localization techniques. We also argue that in the large N-limit of this deformed 5D Yang-Mills theory this matrix model provides the leading contribution to the partition function and the rest is exponentially suppressed.

• The paper demonstrates that localization techniques can simplify the derivation of the perturbative partition function for supersymmetric 5D Yang-Mills theory on S^5.
• It employs a matrix model formulation integrating vector and hypermultiplet contributions to reveal the theory’s large N asymptotic behavior and UV finiteness.
• The study connects 5D gauge theories to 6D models, offering insights into non-perturbative dynamics and potential scaling laws in higher-dimensional frameworks.

An Analysis of The Perturbative Partition Function of Supersymmetric 5D Yang-Mills Theory on the Five-Sphere

The paper by Källén, Qiu, and Zabzine presents a comprehensive paper of the perturbative partition function associated with five-dimensional (5D) supersymmetric Yang-Mills theory, focusing on formulations over a five-sphere $S^5$. This work explores an intricate domain of theoretical physics, offering insights into the implications of higher-dimensional gauge theories and their connection to six-dimensional (6D) models.

A pivotal aspect of this analysis is the foundation laid by localization techniques, which allow the computation of partition functions in complex gauge theories to be simplified by reducing the functional integrals to finite-dimensional integrals over the localization locus. The authors adopt this robust approach to derive the perturbative partition function of a deformed 5D Yang-Mills theory that preserves eight supercharges and contains matter in a representation $R$.

The results are articulated in terms of a matrix model that exhibits dependencies on parameters such as the Yang-Mills coupling $g_{YM}$ and the radius $r$ of $S^5$. This formulation necessitates sophisticated mathematical tools including the use of Atiyah-Singer index theorem to evaluate one-loop determinants—a method inspired by prior works such as those of Pestun.

Key Results and Numerical Aspects

The matrix model derived in this exploration provides a precise perturbative partition function, notably encapsulating terms such as:

$$Z = \int[d\phi] e^{-\frac{8\pi^2r}{g_{YM}^2} \text{Tr}(\phi^2)} \text{det}_{Ad}\left( \sin(i\pi\phi) \right) e^{2f(i\phi)}$$

where $\phi$ represents a dimensionless matrix, and $f(y)$ is a function involving polylogarithms and ζ-function regularization. The partition function explicitly incorporates contributions from vector multiplets and hypermultiplets, elegantly demonstrating a rich interplay between algebraic structures and topological constructs.

A notable achievement of this paper lies in elucidating the asymptotic behavior of the perturbative partition function in the large $N$-limit. In this regime, instantonic contributions become exponentially suppressed—a noteworthiness that resonates with earlier conjectures concerning the emergence of instanton sectors and UV finiteness in 5D theories.

Theoretical and Practical Implications

The findings presented possess profound implications for the understanding of non-renormalizable theories and their viable embedding in a finite framework. The interconnection proposed by the authors between five-dimensional gauge theories and the elusive six-dimensional (2,0) superconformal theories breathes new life into the quest for a unified description across dimensions, potentially accommodating the degrees of freedom of both theories.

From a practical perspective, the exploration of matrix models in this context sheds light on the thermodynamic properties of these gauge theories, with potential ramifications for the paper of their universal behavior and free energy scaling. In alignment with established 3D localization results, the 5D partition function solicits further exploration to unravel $N^{3/2}$ scaling laws, presaging new developments in non-perturbative string theory formulations.

The authors also acknowledge unsolved challenges, such as determining non-perturbative partition functions influenced by instantons and reconciling their matrix model results with higher-dimensional theories. These issues underscore the inherent complexity and scope for future exploration in this nuanced domain of theoretical physics.

In conclusion, this paper offers an incisive discourse on the perturbative partition function in 5D supersymmetric Yang-Mills theories, charting a course for deeper investigations into the mathematical and physical underpinnings of multidimensional superconformal theories. Through adept utilization of localization, the research exemplifies how contemporary techniques can surmount barriers posed by non-renormalizability and dimensional extrapolations.