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c-extremization from toric geometry

Antonio Amariti, Luca Cassia, Silvia Penati

TL;DR

This work derives a geometric, toric-diagram-based formulation for the 2d central charge $c_r$ of $(0,2)$ SCFTs arising from the partial topological twist of 4d toric quiver theories on a curved Riemann surface. By identifying the 2d mixing parameters with toric perfect-matching charges and equating fluxes to PM data, the authors express $c_r$ in terms of triple-dacetroid determinants of the toric diagram and then extremize to obtain the exact 2d central charge. The framework is validated across a broad set of examples, including smooth and singular horizons (where perimeter-point handling is required) and demonstrates that baryonic symmetries can mix with the 2d R-current, mirroring but generalizing known 4d phenomena. The results deepen the connection between 2d c-extremization and 4d a-maximization, and suggest holographic links to volumes of internal manifolds, with several avenues for further exploration in AdS$_3$/CFT$_2$ holography and toric geometry techniques.

Abstract

We derive a geometric formulation of the 2d central charge $c_r$ from infinite families of 4d $\mathcal{N}=1$ superconformal field theories topologically twisted on constant curvature Riemann surfaces. They correspond to toric quiver gauge theories and are associated to D3 branes probing five dimensional Sasaki-Einstein geometries in the AdS/CFT correspondence. We show that $c_r$ can be expressed in terms of the areas of the toric diagram describing the moduli space of the 4d theory, both for toric geometries with smooth and singular horizons. We also study the relation between a-maximization in 4d and c-extremization in 2d, giving further evidences of the mixing of the baryonic symmetries with the exact R-current in two dimensions.

c-extremization from toric geometry

TL;DR

This work derives a geometric, toric-diagram-based formulation for the 2d central charge of SCFTs arising from the partial topological twist of 4d toric quiver theories on a curved Riemann surface. By identifying the 2d mixing parameters with toric perfect-matching charges and equating fluxes to PM data, the authors express in terms of triple-dacetroid determinants of the toric diagram and then extremize to obtain the exact 2d central charge. The framework is validated across a broad set of examples, including smooth and singular horizons (where perimeter-point handling is required) and demonstrates that baryonic symmetries can mix with the 2d R-current, mirroring but generalizing known 4d phenomena. The results deepen the connection between 2d c-extremization and 4d a-maximization, and suggest holographic links to volumes of internal manifolds, with several avenues for further exploration in AdS/CFT holography and toric geometry techniques.

Abstract

We derive a geometric formulation of the 2d central charge from infinite families of 4d superconformal field theories topologically twisted on constant curvature Riemann surfaces. They correspond to toric quiver gauge theories and are associated to D3 branes probing five dimensional Sasaki-Einstein geometries in the AdS/CFT correspondence. We show that can be expressed in terms of the areas of the toric diagram describing the moduli space of the 4d theory, both for toric geometries with smooth and singular horizons. We also study the relation between a-maximization in 4d and c-extremization in 2d, giving further evidences of the mixing of the baryonic symmetries with the exact R-current in two dimensions.

Paper Structure

This paper contains 13 sections, 82 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Review
  3. Toric quiver gauge theories
  4. Twisted compactification
  5. $c_{r}$ from toric geometry
  6. Reading the 2d central charge from the toric diagram
  7. Examples
  8. Singular horizons and lattice points lying on the perimeter
  9. $L^{222}$
  10. $L^{131}$
  11. Mixing of the baryonic symmetries
  12. Further directions
  13. Mixing parameters for dP$_2$ and dP$_3$

Figures (21)

  • Figure 1: Quiver and dimer of the dP$_1$ model. In quiver (a) the number of arrows on the straight lines indicates the number of fields connecting two nodes.
  • Figure 2: Toric diagram and tiling for the dP$_1$ model. In figure (a) primitive normal vectors $w_I$ of the toric diagram are also indicated and the different colors clarify their relation with the zig-zag paths in the dimer, figure (b). This is useful for reading the R-charges of the fields in terms of the charges of the zig-zag paths or of the PM.
  • Figure 3: Dimer, zig-zag paths and toric diagram of $S^5$
  • Figure 4: Dimer, zig-zag paths and toric diagram of $T^{1,1}$
  • Figure 5: Quiver of the dP$_2^{(I)}$ model
  • ...and 16 more figures