A topologically twisted index for three-dimensional supersymmetric theories (1504.03698v3)

Abstract: We provide a general formula for the partition function of three-dimensional $\mathcal{N}=2$ gauge theories placed on $S^2 \times S^1$ with a topological twist along $S^2$, which can be interpreted as an index for chiral states of the theories immersed in background magnetic fields. The result is expressed as a sum over magnetic fluxes of the residues of a meromorphic form which is a function of the scalar zero-modes. The partition function depends on a collection of background magnetic fluxes and fugacities for the global symmetries. We illustrate our formula in many examples of 3d Yang-Mills-Chern-Simons theories with matter, including Aharony and Giveon-Kutasov dualities. Finally, our formula generalizes to $\Omega$-backgrounds, as well as two-dimensional theories on $S^2$ and four-dimensional theories on $S^2 \times T^2$. In particular this provides an alternative way to compute genus-zero A-model topological amplitudes and Gromov-Witten invariants.

• The paper introduces a comprehensive index for chiral states in 3D supersymmetric theories on S²×S¹ by implementing a topological twist and localization techniques.
• It employs advanced methods including supersymmetric localization, contour integration, and the Jeffrey-Kirwan residue to accurately evaluate partition functions.
• The results provide new insights into non-perturbative effects and gauge theory dualities, paving the way for extensions to other dimensions and complex manifolds.

A Topologically Twisted Index for Three-Dimensional Supersymmetric Theories

The paper by Benini and Zaffaroni presents a comprehensive paper of the partition function of three-dimensional $N=2$ gauge theories on $S^2 \times S^1$, incorporating a topological twist on $S^2$. This work introduces a significant general formula viewed as an index for chiral states under the influence of background magnetic fields. The novelty of their approach lies in utilizing the supersymmetry-preserving background and localization technique to extract exact results for such theories.

Summary of Findings

The authors develop the partition function, which maps a complex structure using meromorphic forms in scalar zero-modes. The approach requires summing over magnetic fluxes and computing residues associated with the meromorphic forms. Importantly, the dependence on background magnetic fluxes and fugacities for the global symmetries manifests clearly.

Methodology and Techniques

• Supersymmetric Localization: This technique is central to the analysis, enabling the exact evaluation of path integrals by reducing them to sums over well-defined BPS configurations.
• Contour Integration: A crucial step involves contour integration in the complex plane, specifically identifying and summing residues from meromorphic functions at magnetic fluxes.
• Jeffrey-Kirwan Residue: For nontrivial gauge group structures (rank $r$), the work employs advanced mathematical tools such as the Jeffrey-Kirwan residue to manage multi-dimensional integrals.

Implications and Applications

The results have broad implications, especially in the domain of non-perturbative effects and dualities within three-dimensional gauge theories. The formulated topologically twisted index provides a new tool for probing these aspects. For example, the work contributes evidence towards understanding Aharony and Giveon-Kutasov dualities by enabling explicit calculations of partition functions for classes of dual theories.

Moreover, the paper lays a foundation for generalizations into different dimensions and geometries, such as S^2 in two dimensions and S^2 × T^2 in four dimensions. This opens pathways to potentially novel intersections with string theory and complex Chern-Simons theories.

Speculations on Future Developments

Benini and Zaffaroni's work suggests exciting future lines of inquiry, including more refined analyses incorporating angular momentum and further exploration of complex Chern-Simons theory connections. The expansion to non-simply connected manifolds and their associated indices could uncover new theoretical insights and mathematical structures.

Overall, this meticulous and methodical exploration enriches our understanding of supersymmetric quantum field theories, offering powerful techniques and results applicable to both theoretical pursuits and practical demonstrations, such as elucidating dualities and topological properties inherent in these theories. The mathematical rigor and clever manipulation of physical observables embodied in their formulation represent a notable advancement in the paper of three-dimensional supersymmetric theories.