Mirror symmetry on a circle

Abstract: We investigate the small circle, or high temperature, limit of the supersymmetric index identities for the three-dimensional abelian mirror symmetry of SQED. There exist two possible limits, depending on how the parameters of the theory are scaled with the radius of the circle. In both cases the result is qualitatively similar. One side always reduces to the sphere partition function of a two-dimensional $\mathcal{N}=(2,2)$ gauged linear sigma model (GLSM). The opposite side has a two-fold interpretation, either as the sphere partition function of the Landau--Ginzburg (LG) model that is Hori--Vafa dual to the GLSM, or as a Coulomb gas integral for a correlation function of Liouville or Toda CFT. This approach thus provides a systematic way to generate integral identities between partition functions of GLSMs on the one hand, and partition functions of LG models or CFT Coulomb gas integrals on the other. The latter perspective finds useful applications in the recently proposed 2d/2d correspondence, which relates sphere partition functions of unitary 2d $\mathcal{N}=(2,2)$ theories and correlation functions of non-unitary 2d CFTs that both descend from compactifications of a unitary 4d $\mathcal{N}=2$ SCFT. We present an example based on the $(A_{k-1},A_{N-1})$ Argyres--Douglas theories, where the CFT is a non-unitary minimal model. We also give a purely two-dimensional derivation of the identities obtained in the small circle limit which is inspired by the Kapustin--Strassler piecewise derivation of 3d abelian mirror symmetry.

• The paper establishes explicit integral identities from 3D abelian mirror symmetry that reduce to 2D partition functions under different asymptotic limits.
• It employs supersymmetric localization to connect GLSM and LG mirror dualities, thereby relating 3D supersymmetric indices to 2D CFT correlators via Coulomb gas integrals.
• It introduces a two-dimensional analog of the Kapustin–Strassler dualization, paving the way for extending dualities to more complex non-abelian settings.

Mirror Symmetry on a Circle: Dimensional Reduction and Duality in Supersymmetric QFT

Introduction and Theoretical Framework

The paper "Mirror symmetry on a circle" (2606.13775) systematically studies the high-temperature (small circle) limit of supersymmetric index identities associated with three-dimensional (3d) abelian mirror symmetry for supersymmetric QED (SQED), focusing on the implications for two-dimensional (2d) quantum field theories (QFTs) and their connections to 2d conformal field theory (CFT) correlators. The analysis leverages the machinery of supersymmetric localization to derive non-trivial integral identities between partition functions of 2d $\mathcal{N}=(2,2)$ gauged linear sigma models (GLSMs), their Landau--Ginzburg (LG) mirrors (Hori--Vafa duals), and Coulomb gas representations of CFT correlators, illuminating broad structural features of dualities and correspondences obtained via circle compactification.

A central theme is the study of two asymptotic regimes for the supersymmetric index of 3d gauge theories on $S^2\times S^1$, characterized by different scalings of mass and Fayet--Iliopoulos (FI) parameters with respect to the shrinking $S^1$ radius. In both regimes, mirror symmetry leads, after dimensional reduction, to integral identities between 2d theory partition functions, which can be interpreted both as duality statements and as correspondences between QFT and CFT observables.

Dimensional Reduction and Parameter Scalings

The paper carefully distinguishes two asymptotic limits for the index as the circle shrinks ($\beta \to 0$ with $x^2=e^{-\beta}$):

• Limit 1 (finite mass, vanishing FI): Masses $M$ held fixed, FIs $\eta\sim\beta\tau$ ($\tau$ finite). This limit yields the $S^2$ partition function of a 2d LG model (the Hori--Vafa mirror) or, equivalently, a Coulomb gas integral for a Liouville or Toda CFT correlator.
• Limit 2 (vanishing mass, finite FI): Masses scale as $M\sim\beta m$ ($S^2\times S^1$0 finite), FIs fixed. Here the index reduces to the $S^2\times S^1$1 partition function of a 2d GLSM matching the original 3d gauge theory.

Under the 3d abelian mirror symmetry (e.g., SQED with $S^2\times S^1$2 flavors dual to the affine $S^2\times S^1$3 quiver), these two limits exchange the roles of mass and FI parameters on the two sides, reflecting the canonical mirror map.

Main Technical Results

Integral Identities from Mirror Symmetry

The authors provide explicit derivations of finite, non-trivial integral identities relating partition functions of 2d GLSMs and their LG mirrors in both limits. For the canonical $S^2\times S^1$4-flavor abelian case, the key identities take the form:

• Limit 1:

$S^2\times S^1$5

• Limit 2:

$S^2\times S^1$6

Here, the explicit dependencies on masses, FI parameters, and holonomy variables encode the mappings under mirror symmetry. The analytic structure of these expressions is controlled by the underlying R-charge assignments and supersymmetric vacua.

Connection to CFT (Coulomb Gas Integrals)

The more general novelty arises by relating the above 2d partition functions—particularly those of LG models in Limit 1—to Coulomb gas integral representations for correlation functions in non-unitary 2d CFTs, including Liouville and $S^2\times S^1$7 Toda theory. For instance, the $S^2\times S^1$8 case essentially reproduces the complex beta-function integral (Virasoro--Shapiro amplitude), and the $S^2\times S^1$9-flavor case is shown to match the 1-screening Coulomb gas representation for the $S^1$0-point function.

Crucially, the mapping:

• GLSM FI parameters ⟷ CFT operator positions
• Masses ⟷ CFT momenta is demonstrated explicitly, rendering the physics of 2d mirror symmetry computable in terms of well-understood CFT data.

2d/2d Correspondence and Argyres--Douglas Theories

The study exploits a recently proposed 2d/2d correspondence between partition functions of 2d $S^1$1 SCFTs (arising from twisted circle reduction of 4d $S^1$2 SCFTs) and correlation functions in non-unitary 2d CFTs associated to vertex operator algebras (VOAs) corresponding to the parent 4d theory. Utilizing this framework, the identities presented yield explicit GLSM representatives for 2d theories associated to Argyres--Douglas points (e.g., $S^1$3), allowing calculation of minimal model CFT correlators from supersymmetric partition functions.

Concretely, the sphere partition function $S^1$4 of the class $S^1$5 theory $S^1$6 is equated (up to non-universal factors) with the CFT correlator $S^1$7 in $S^1$8, for suitable choices of vertex operators $S^1$9.

Two-Dimensional Kapustin--Strassler Dualization

A further significant technical contribution is the development of a two-dimensional analog of the Kapustin--Strassler "piecewise" derivation. Originally formulated for 3d dualities, this approach bootstraps higher-flavor mirror symmetry from the $\beta \to 0$0 case via functional Fourier transforms and repeated gauging. Here, the method is generalized to 2d; in essence, all higher $\beta \to 0$1 abelian dualities and the associated integral identities can be reduced to iterations of the fundamental $\beta \to 0$2 block, now interpreted as a $\beta \to 0$3-form symmetry gauging.

Numerical and Contradictory Claims

The paper presents explicit analytic formulae for partition function identities and verifies their equivalence with known (and new) CFT results via Gamma function and hypergeometric identities, with the region of validity controlled by R-charge assignments ensuring convergence of the relevant integrals.

A notable and nontrivial physical assertion is that two seemingly distinct 2d mirror symmetry identities (arising from different scaling regimes of 3d parameters) are actually related and not independent; Limit 2 identities can be derived from those in Limit 1 via contour deformations and applications of the Dotsenko--Fateev method.

Moreover, the work emphasizes—contrary to naive expectations—that such integral identities and their mirror symmetry interpretations persist even in non-compact target space, albeit with careful attention to analytic continuation of the relevant parameters.

Implications and Prospects

The findings have both practical and theoretical implications:

• They provide constructive tools for computing nontrivial correlators in Toda/Liouville CFTs as supersymmetric partition functions of 2d theories, enabling systematic study of the 2d/2d correspondence and physical quantities in non-unitary CFTs.
• The analytic framework furnishes explicit reductions of higher-dimensional dualities to lower dimensions, clarifying the structure of dualities and correspondences beyond simple cases.
• The methods extend naturally to explorations of non-abelian mirror symmetry and their partition function identities, which would yield even more general CFT correlators, including those with multiple screening charges.
• The work facilitates the construction and validation of new dualities—both within 2d QFT and as uplifts to higher dimensions—by recasting intricate field-theoretic questions in the language of special functions and integral transforms.

Outlook

Future directions anticipated include:

• Classification and exploitation of non-abelian 3d mirror symmetry reductions, yielding broader families of 2d dualities and further connections to multi-point CFT correlators.
• Further elucidation of the 2d/2d correspondence via central charge computations and elliptic genus matching, especially for compactifications of Argyres–Douglas and other strongly coupled theories.
• Investigation of the analytic structure and convergence domains of these identities beyond the initial range derived from GLSM constraints, opening new avenues for both physics and mathematics.
• Extension of algorithmic dualization (mirror dualization algorithms), already familiar in 3d gauge theory, into 2d, with applications to understanding modular functors and categorical structures in 2d SCFT and CFT.

Conclusion

This paper establishes a deep technical bridge between 3d abelian mirror symmetry and 2d dualities and correspondences, providing explicit analytic control over the reduction of supersymmetric index identities and their interpretation in terms of both QFT dualities and CFT correlators. The careful dissection of parameter scalings, the explicit construction of partition function identities, and the confirmation of their equivalence to CFT results enhance our understanding of dimensional reduction, duality webs, and the unifying structure of supersymmetric and conformal field theory. The work opens clear pathways toward understanding the exact matching of observables under dualities and correspondences in a wide array of settings, both abelian and non-abelian, unitary and non-unitary.