Dualities and loops on squashed $S^3$ (2212.06813v3)

Abstract: We consider $\mathcal{N}=4$ supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these $\frac{2}{3}$-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large $N$ expression for the ABJM partition function.

• The paper shows that 2/3-BPS Wilson and vortex loops remain squashing-independent, streamlining the analysis of partition functions.
• It presents numerical evidence from U(2) and U(3) gauge groups, affirming that mirror dual partition functions match on deformed geometries.
• The study bridges enhanced supersymmetry and mirror symmetry, with implications for non-perturbative corrections in ABJM theories.

Overview of the mx Dualities and Loops on Squashed $S^3$

This paper provides an in-depth analysis of $N=4$ supersymmetric gauge theories on the squashed three-sphere $S^3_b$, focusing on dualities and loop operators. The paper is motivated by supersymmetric localization, which allows for exact calculations of partition functions and loop operator expectation values. The crux of this investigation is to understand how these functions behave when the supersymmetry on $S^3_b$ is enhanced from four to six supercharges.

The authors initially address the preservation of supercharges in the presence of Wilson and vortex loops and demonstrate that certain configurations, specifically the $\frac{2}{3}$-BPS loops, exhibit squashing independence. This non-dependence on the squashing parameter not only simplifies calculations but also bridges results from the round sphere to the squashed configuration, providing a consistent framework for analyzing these gauge theories under modified geometric settings.

A crucial aspect of the work is the exploration of mirror symmetry and dual partition function matchings on the squashed sphere. The authors successfully lift results previously confined to the round sphere to the squashed configuration, uncovering a coherent relation between gauge theories and their mirror duals. Through numerical evaluations, the partition functions of ABJM theory and its dual, the $N=8$ super-Yang-Mills, are shown to match in lower ranks, leading to a conjectured equality for general values of the squashing parameter $b$.

Numerical and Analytical Results

The numerical evaluations carried out on $U(2)$ and $U(3)$ gauge groups illustrate that partition functions of mirror theories, on deformed geometries, remain consistent across a spectrum of parameters, reinforcing the conjectured behaviors. Specifically, at the enhancement points for supercharges on the squashed sphere, all evaluations align closely with those on the round sphere, defying the expectation of squashing dependency.

The authors investigate the regularized behavior of partition functions, noting zero points corresponding to the Lee-Yang zeros identified in prior studies on the round sphere. A key observation in the numerical section is these zeros' response to changes in mass parameters, migrating towards larger values as the squashing parameter increases.

Additionally, the work offers insights into the large $N$ conjecture for ABJM theories, comparing it against all-order perturbative predictions and identifying non-perturbative corrections. These findings open pathways for employing advanced techniques such as those suggested in references [Hwang:2021ulb, Comi:2022aqo] to provide more concrete verifications of these numerical conjectures.

Theoretical and Practical Implications

The theoretical implications of this paper are multifaceted. Firstly, it underscores the potency of supersymmetric localization in tackling complex three-dimensional $N=4$ gauge theories on modified geometric platforms. The seamless lifting of dualities and operator expectations from the round sphere to the squashed sphere signifies a substantial enhancement in our understanding of supersymmetry and gauge theory dynamics under deformation.

Practically, these results prompt a significant step forward in the numerical modeling of these theories, as evident by the distinguished handling of partition function intricacies and the identification of real-world applications for these mathematical constructs in theoretical physics.

Future Directions

The paper paves the way for further exploration of dualities in more complex, squashed, and deformed topologies, potentially extending the methodologies presented here to analyze higher-dimensional gauge theories. Furthermore, a formal proof of the presented conjectures, particularly regarding non-perturbative aspects, would substantially strengthen the robustness of the numerical findings and their applications. This may involve leveraging advanced mathematical tools and collaborating across different theoretical physics domains to embed these results within broader theoretical contexts.

Overall, the findings presented in this paper offer profound insights into the interplay between geometry, duality, and quantum field theory through a comprehensive examination of supersymmetric techniques and numerical validation on squashed geometries.