Generalized Schur partition functions and RG flows (2506.13764v1)

Abstract: We revisit a double-scaled limit of the superconformal index of ${\cal N}=2$ superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, $\hat {\cal Z}(q,\alpha)$, has a standard $q$-expansion with coefficients depending on a continuous parameter $\alpha$. The Schur index is a special case with $\alpha=1$. Through explicit computations we argue that this partition function is an invariant of certain mass deformations and vacuum expectation value (vev) deformations of the SCFT. In particular, two SCFTs residing in different corners of the same Coulomb branch, satisfying certain restrictive conditions, have the same partition function with a non-trivial map of the parameters, $\hat {\cal Z}_1(q,\alpha_1)=\hat {\cal Z}_2(q,\alpha_2(\alpha_1))$. For example, we show that the Schur index of all the SCFTs in the Deligne-Cvitanovi\'c series is given by special values of $\alpha$ of the partition function of the SU(2) $N_f=4$ ${\cal N}=2$ SQCD.

• The paper introduces a continuous parameter α in the Schur partition function to invariantly trace deformations across SCFTs.
• It demonstrates nontrivial mappings between SCFTs within the Deligne-Cvitanovi series using generalized partition functions.
• The approach enriches RG flow analysis in N=2 theories, potentially streamlining computational methods for complex gauge landscapes.

Generalized Schur Partition Functions and RG Flows: A Synopsis

This paper by Anirudh Deb and Shlomo S. Razamat explores the properties and applications of generalized Schur partition functions within the framework of ${\cal N}=2$ superconformal field theories (SCFTs). The authors extend the traditional Schur index by introducing a continuous parameter $\alpha$ into the partition function, denoted as $\hat {\cal Z}(q,\alpha)$. This generalization allows the partition function to act as an invariant across certain mass and vacuum expectation value (vev) deformations, setting the stage for analyzing RG flows across different SCFTs residing within distinct corners of the same Coulomb branch.

Key Findings and Numerical Results

Through this research, the authors make a compelling case for the parameter $\alpha$ serving as a critical link between different SCFTs. In particular, they demonstrate that two SCFTs embedded within unique zones of the Coulomb branch can share identical partition functions under a non-trivial parametric mapping $$\hat {\cal Z}_1(q,\alpha_1)=\hat {\cal Z}_2(q,\alpha_2(\alpha_1))$$.

Exploring this mapping further, the paper unveils that the Schur index for all SCFTs within the Deligne-Cvitanovi series can be expressed in terms of the generalized partition function of the $SU(2)$ $N_f=4$ ${\cal N}=2$ SQCD. Some notable mappings demonstrated are:

• $$\hat {\cal Z}_{\mathfrak{d}_4}(q,\frac{1}{5}) = {\cal Z}_{\mathfrak{a}_0}(q,1)$$
• $$\hat {\cal Z}_{\mathfrak{d}_4}(q,\frac{1}{2}) = {\cal Z}_{\mathfrak{a}_2}(q,1)$$

These mappings are pivotal in exhibiting that generalized Schur partition functions encapsulate the Schur indices for the entire Deligne series under specific values of $\alpha$.

Implications and Future Developments in AI

The work carries rich implications for theoretical physics, particularly in extending the applicability and computational ease of deriving Schur indices in complex SCFT landscapes. This robustness enriches the understanding of RG flow dynamics and interconnectedness across various SCFT models. From a broader perspective, such formal expansions could significantly benefit future AI implementations seeking to optimize computations or explore novel quantum field theoretical landscapes through machine learning algorithms rooted in these comprehensive frameworks.

Conclusion

Deb and Razamat's exploration into generalized Schur partition functions marks a noteworthy progression in understanding ${\cal N}=2$ SCFTs. By leveraging the continuous parameter $\alpha$, they provide deeper insights into RG flows and establish a unified interpretative model for SCFTs across diverse branches. This work not only advances theoretical pursuits but also sets the course for innovations in computational methodologies, potentially inspiring future research in four-dimensional supersymmetry and its applications in AI environments.