$\mathcal{N}{=}1$ supersymmetric indices and the four-dimensional A-model (1707.05774v2)

Abstract: We compute the supersymmetric partition function of $\mathcal{N}{=}1$ supersymmetric gauge theories with an $R$-symmetry on $\mathcal{M}4 \cong \mathcal{M}{g,p}\times S^1$, a principal elliptic fiber bundle of degree $p$ over a genus-$g$ Riemann surface, $\Sigma_g$. Equivalently, we compute the generalized supersymmetric index $I_{\mathcal{M}{g,p}}$, with the supersymmetric three-manifold ${\mathcal{M}{g,p}}$ as the spatial slice. The ordinary $\mathcal{N}{=}1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$-model for the abelianized gauge fields on the base $\Sigma_g$. This $A$-model---or $A$-twisted two-dimensional $\mathcal{N}{=}(2,2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^2$ inside $\Sigma_g \times T^2$. Being defined by compactification on the torus, the $A$-model also enjoys natural modular properties, governed by the four-dimensional 't Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.

• The paper derives N=1 supersymmetric partition functions using a 4D A-model framework that sums over Bethe vacua and employs an effective twisted superpotential.
• It presents a compact Bethe vacua formula that offers clear tests of Seiberg duality between electrically and magnetically dual theories.
• The computed generalized indices exhibit modular properties governed by ’t Hooft anomalies, providing new insights into duality and operator algebra.

Supersymmetric Indices and the Four-Dimensional A-Model

The paper "N=1 Supersymmetric Indices and the Four-Dimensional A-Model" by Closset, Kim, and Willett investigates the computation of supersymmetric partition functions for $N=1$ supersymmetric gauge theories. These computations are executed with an $R$-symmetry on a manifold $\mathcal{M}_4 \cong \mathcal{M}_{g,p} \times S^1$, a principal elliptic fiber bundle over a genus-$g$ Riemann surface. This work extends the understanding of supersymmetric indices beyond the round three-sphere by employing a topological twist that yields a two-dimensional $A$-model, or $A$-twisted $N=(2,2)$ gauge theory. This paper provides new tests for Seiberg duality and introduces a novel evaluation formula for the three-sphere index.

The principal contribution lies in deriving the supersymmetric partition functions using $A$-model techniques, focusing on generalized indices as expectation values of canonically-defined surface defects. The $A$-model approach effectively captures modular properties governed by four-dimensional 't Hooft anomalies and offers both a computation technique and a physical perspective on dualities within supersymmetric theories.

Key Results and Methodology

1. Four-Dimensional $A$-Model: The paper constructs a topological $A$-model by examining the infra-red dynamics of $N=1$ gauge theories compactified on a torus. The authors argue that this $A$-model captures the supersymmetric partition functions, specifically detailing interactions in terms of the effective twisted superpotential ($W$) and the effective dilaton ($\Omega$). The $A$-model is pivotal in deriving the generalized indices of four-dimensional theories, viewing them as twisted $N=(0,2)$ indices.
2. Bethe Vacua Formula: The partition function on $\mathcal{M}_{g,p} \times S^1$ is formulated as a sum over the Bethe vacua of the two-dimensional theory. This not only furnishes a compact expression for the partition functions but also establishes a clear framework for investigating Seiberg duality via hierarchies of surface operators and their corresponding dual theories.
3. Generalized Indices and Modularity: The generalized indices computed are shown to respect modular symmetries, inheriting modular transformations from the toroidal compactification inherent to the $A$-model. This suggests the pivotal role of 't Hooft anomalies in determining the behavior of these indices under different large gauge transformations along the fiber.
4. Seiberg Duality Insights: The paper provides substantial evidence supporting Seiberg duality by showing agreement in supersymmetric partition functions across dual theories. The results here do not involve cumbersome ellipsoidal integral identities, instead of employing $A$-model observables and tracing a duality map between Bethe vacua of electrically and magnetically dual theories.

Implications and Future Directions

This work lays foundational steps towards understanding higher dimensions within the field of supersymmetric indices. By offering a methodology tethered in topological quantization, it opens avenues for exploring quantum field theories with complicated dualities. Future research may explore:

• Elaborating connections between the $A$-model and holomorphic blocks through an $\Omega$-deformation.
• Extending these constructs toward further non-abelian dualities, potentially engaging with theories beyond $N=1$.
• Investigating further applications to mathematics, particularly in string theory and mirror symmetry, by leveraging these modular and duality properties.

In summary, this paper rigorously advances supersymmetric theory computation techniques, presenting a framework with robust implications for future theoretical and mathematical physics research. The $A$-model approach presents a cleaner path forward in studying dualities and consequent relations in the algebra of operators in four-dimensional quantum field theories.