Supersymmetric partition functions on Riemann surfaces (1605.06120v2)

Abstract: We present a compact formula for the supersymmetric partition function of 2d N=(2,2), 3d N=2 and 4d N=1 gauge theories on $\Sigma_g \times T^n$ with partial topological twist on $\Sigma_g$, where $\Sigma_g$ is a Riemann surface of arbitrary genus and $T^n$ is a torus with n=0,1,2, respectively. In 2d we also include certain local operator insertions, and in 3d we include Wilson line operator insertions along $S^1$. For genus g=1, the formula computes the Witten index. We present a few simple Abelian and non-Abelian examples, including new tests of non-perturbative dualities. We also show that the large N partition function of ABJM theory on $\Sigma_g \times S^1$ reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS$_4$ whose horizon has $\Sigma_g$ topology.

• The paper presents a compact formula via supersymmetric localization for exact partition function computations on curved manifolds, including operator insertions.
• It rigorously employs techniques like Jeffrey-Kirwan residues and Bethe Ansatz Equations to test non-perturbative dualities across 2d, 3d, and 4d supersymmetric gauge theories.
• The large N analysis in ABJM theory bridges quantum gauge computations with black hole entropy, providing deep holographic insights into gravitational physics.

Supersymmetric Partition Functions on Riemann Surfaces

The paper ""Supersymmetric partition functions on Riemann surfaces," co-authored by Francesco Benini and Alberto Zaffaroni, presents significant advancements in the computation of supersymmetric partition functions for gauge theories by generalizing their calculation to manifolds in the form of a Riemann surface $\Sigma_g$ and products with tori $T^n$. The investigation spans dimensions two to four, encompassing 2d $N=(2,2)$, 3d $N=2$, and 4d $N=1$ supersymmetric gauge theories. A notable accomplishment of this work is deriving a compact formula for partition functions when topological twists are applied to $\Sigma_g$.

The authors employ sophisticated mathematical tools, including supersymmetric localization, that enable exact computations of partition functions by turning challenging path integrals into more tractable algebraic expressions. An essential aspect of this paper is its capacity to address theories involving local operator insertions in 2d and Wilson line operator insertions in 3d, facilitating a comprehensive examination of different types of operator contributions within complex geometries.

A variety of theoretical and mathematical constructs underpin this research, such as Jeffrey-Kirwan residues, Bethe Ansatz Equations (BAEs), and supersymmetric localization, which together furnish an accurate computation framework. The introduction of BAEs provides new perspectives, enabling checks on novel anisotropies and the elucidation of topological sectors useful for understanding non-perturbative dualities. The derivation is rigorous, providing a detailed formalism in the form of equations that express partition functions in terms of classical, one-loop, and operator contributions.

This work offers several groundbreaking tests of non-perturbative dualities. For instance, in three dimensions, the paper conducts new tests for Aharony duality and the "duality appetizer," unraveling interactions between supersymmetric indices and duality relations. Additionally, it provides a simple verification of Seiberg duality for theories with four dimensions.

Another achievement is the exploration of the large $N$ limit of partition functions within the context of ABJM theory and its comparison to the Bekenstein-Hawking entropy of supersymmetric black holes in $AdS_4$, lending insight into quantum aspects of black hole entropy. The consistent match across genera affirms the robustness of supersymmetric indices in replicating expected entropy counts, thus bridging theoretical calculations with gravitational physics.

The paper establishes a foundational contribution towards the paper of gauge theories in curved spaces with preserved supersymmetries, with implications stretching into areas of quantum field theory, string theories, and beyond. Speculations about future developments include the exploration of higher-dimensional cases, embedding supersymmetric partition functions into the broader landscape of quantum gravity, and applications in AI where topological aspects of data could be modeled through analogous mathematical constructs. The protocols and frameworks devised here broaden our perceptions of mathematical physics, unfolding pathways for interpreting quantum physical phenomena through supersymmetry.

This research stands as a comprehensive articulation of the mathematical discipline required to dissect and utilize the supersymmetric partition functions, heralding future explorations in both theoretical constructs and practical applications such as the derivation of quantum mechanical quantities like entropy from high-dimensional manifold geometries.