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$tt^*$ equations, localization and exact chiral rings in 4d N=2 SCFTs

Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas

TL;DR

The paper addresses how to obtain exact coupling-constant dependence of 2- and 3-point functions of chiral primaries in four-dimensional N=2 SCFTs. It develops and applies a four-dimensional tt* framework, combining it with exact Zamolodchikov metrics from S^4 localization to derive non-perturbative results, including a complete solution for SU(2) SCQCD via a semi-infinite Toda chain. It provides perturbative checks up to two loops and discusses SL(2, Z) duality properties, extending the analysis to general SU(N) theories where the chiral-ring structure is richer and more intricate. The work offers a path to exact non-perturbative data for chiral rings, extremal correlators, and potential bootstrap applications in N=2 SCFTs, with clear demonstrations in SU(2) and substantive notes for higher rank cases.

Abstract

We compute exact 2- and 3-point functions of chiral primaries in four-dimensional N=2 superconformal field theories, including all perturbative and instanton contributions. We demonstrate that these correlation functions are nontrivial and satisfy exact differential equations with respect to the coupling constants. These equations are the analogue of the $tt^*$ equations in two dimensions. In the SU(2) N=2 SYM theory coupled to 4 hypermultiplets they take the form of a semi-infinite Toda chain. We provide the complete solution of this chain using input from supersymmetric localization. To test our results we calculate the same correlation functions independently using Feynman diagrams up to 2-loops and we find perfect agreement up to the relevant order. As a spin-off, we perform a 2-loop check of the recent proposal of arXiv:1405.7271 that the logarithm of the sphere partition function in N=2 SCFTs determines the Kähler potential of the Zamolodchikov metric on the conformal manifold. We also present the $tt^*$ equations in general SU(N) N=2 superconformal QCD theories and comment on their structure and implications.

$tt^*$ equations, localization and exact chiral rings in 4d N=2 SCFTs

TL;DR

The paper addresses how to obtain exact coupling-constant dependence of 2- and 3-point functions of chiral primaries in four-dimensional N=2 SCFTs. It develops and applies a four-dimensional tt* framework, combining it with exact Zamolodchikov metrics from S^4 localization to derive non-perturbative results, including a complete solution for SU(2) SCQCD via a semi-infinite Toda chain. It provides perturbative checks up to two loops and discusses SL(2, Z) duality properties, extending the analysis to general SU(N) theories where the chiral-ring structure is richer and more intricate. The work offers a path to exact non-perturbative data for chiral rings, extremal correlators, and potential bootstrap applications in N=2 SCFTs, with clear demonstrations in SU(2) and substantive notes for higher rank cases.

Abstract

We compute exact 2- and 3-point functions of chiral primaries in four-dimensional N=2 superconformal field theories, including all perturbative and instanton contributions. We demonstrate that these correlation functions are nontrivial and satisfy exact differential equations with respect to the coupling constants. These equations are the analogue of the equations in two dimensions. In the SU(2) N=2 SYM theory coupled to 4 hypermultiplets they take the form of a semi-infinite Toda chain. We provide the complete solution of this chain using input from supersymmetric localization. To test our results we calculate the same correlation functions independently using Feynman diagrams up to 2-loops and we find perfect agreement up to the relevant order. As a spin-off, we perform a 2-loop check of the recent proposal of arXiv:1405.7271 that the logarithm of the sphere partition function in N=2 SCFTs determines the Kähler potential of the Zamolodchikov metric on the conformal manifold. We also present the equations in general SU(N) N=2 superconformal QCD theories and comment on their structure and implications.

Paper Structure

This paper contains 30 sections, 171 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Marginal deformations and the chiral ring
  3. The chiral ring of ${\cal N}=2$ theories
  4. Marginal deformations
  5. The exact Zamolodchikov metric from supersymmetric localization
  6. $tt^*$ equations in four-dimensional ${\cal N}=2$ SCFTs
  7. $tt^*$ equations and the connection on the bundles of chiral primaries
  8. Differential equations for 2- and 3-point functions of chiral primaries
  9. Global issues
  10. Solving the $tt^*$ equations
  11. Extremal correlators
  12. ${\cal N}=4$ theories
  13. ${\cal N}=2$ superconformal QCD as an instructive example
  14. Definitions
  15. $SU(2)$ with 4 hypermultiplets
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The diagrams $D_1$ and $D_2$. Solid double lines represent $\varphi$ propagators, dashed double lines correspond to hyperscalars and dashed lines to hyperfermions. $a,b,c$ and $d$ are adjoint gauge indices and $p$ is the incoming momentum.
  • Figure 2: Master integrals appearing in the reduction of the Feynman diagrams of figure \ref{['fig:2diag']}. They are associated to loop integrals consisting of scalar propagators only (i.e. without non-trivial numerators).