- The paper establishes an explicit reduction method from 4d superconformal indices to 3d partition functions using elliptic-to-hyperbolic function transformations.
- It demonstrates how transformations of q-hypergeometric integrals reveal dualities in 3d N=2 SYM and Chern-Simons theories with SP(2N) gauge groups.
- The work implies that 4d SCIs may underlie 3d PF properties, promoting the discovery of new dualities and linking extremization procedures across dimensions.
The paper "From 4d Superconformal Indices to 3d Partition Functions" by Dolan, Spiridonov, and Vartanov addresses the transition from four-dimensional (4d) superconformal indices (SCIs) to three-dimensional (3d) partition functions (PFs). The authors elucidate a systematic methodology for deriving 3d PFs from 4d SCIs by means of elliptic and hyperbolic special functions, particularly focusing on q-hypergeometric integrals and the double sine function.
The primary objective of the paper is to establish an explicit connection between the SCIs of 4d supersymmetric field theories and the PFs of 3d theories. The paper exhibits how duality relations and partition functions in 3d field theories can be effectively derived by reducing the SCIs of 4d supersymmetric Yang-Mills (SYM) theories. Particularly, it demonstrates that transformations of elliptic hypergeometric integrals can yield hyperbolic q-hypergeometric integrals, which function as PFs for 3d dual theories.
A significant part of the paper involves exploring examples, specifically examining 3d N=2 SYM and Chern-Simons (CS) theories with SP(2N) gauge groups. The work reveals how 4d theories specified by SCIs, associated with complex elliptic hypergeometric integrals, can be systematically reduced to 3d theories characterized by simpler hyperbolic functions. This reduction not only preserves the duality previously identified within the SCIs but also potentially predicts new dualities in three dimensions due to the extensive set of known transformation formulae for elliptic integrals.
The findings have notable implications both practically and theoretically. The paper highlights how SCIs of 4d theories may be perceived as more fundamental constructs than PFs of 3d theories, suggesting that the properties of 3d theories could be inherently inherited from their 4d counterparts. Practically, the methodology promotes uncovering a wide spectrum of previously unidentified 3d dualities by leveraging relationships established within 4d SCIs. Theoretically, it suggests congruence between extremization procedures in three and four dimensions, such as Z-extremization and a-maximization, respectively.
Consequently, the paper supports the conjecture that q-hypergeometric integrals serve foundational roles similar to those of elliptic integrals related to integrable models. The reduction technique proposed provides an efficient path to analyze 3d theories and their duals, enhancing understanding of strong-weak dualities and informing further insights into 3d quantum systems, especially with non-trivial CS levels.
This work is significant in advancing the discourse on 4d-3d theoretical connections, implying future developments in formulating a comprehensive framework for exploring such transitions and their physical ramifications in field theory analysis. The authors' contribution to formalizing these reductions opens pathways both for direct analytical applications and broader theoretical contemplations in the field of supersymmetric field theories.
