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Constant maps in equivariant topological strings and geometric modeling of fluxes

Luca Cassia, Kiril Hristov

TL;DR

This work develops an equivariant generalization of topological strings on toric Calabi–Yau manifolds, centering on constant-map contributions to regulate non-compact volumes and illuminate their role in flux compactifications and M2-brane holography. It introduces a precise dictionary between genus-zero and higher-genus equivariant constant maps and classical geometric data, and shows how these feed into an exact, non-perturbative holographic match with M2-brane partition functions via Laplace-type transforms and mesonic twists. By analyzing a broad set of toric examples and CY cones, the paper demonstrates that the equivariant partition function encodes both two-derivative supergravity data and higher-derivative corrections, with OSV-like reformulations emerging in vanishing-flux backgrounds. The framework provides a unified geometric model that connects topological-string invariants, fluxes, Sasakian geometry, and holographic duals, and it opens a path toward non-perturbative gravity calculations in string/M-theory via exact transformations between λ- and N-ensembles.

Abstract

We study the equivariant generalization of topological strings on toric manifolds, focusing in particular on defining the contributions of constant maps in the genus expansion of the partition function. This approach regularizes the integration over non-compact Calabi-Yau spaces, producing finite results at each order in the expansion, as illustrated by a broad set of explicit examples. Our investigation highlights the geometric modeling of flux compactifications and clarifies the link between the effective supergravity framework and the equivariant topological string formalism, building on recent developments by Martelli and Zaffaroni. We conclude that the connection between topological string theory and supergravity/field theory involves switching between geometric moduli and fluxes, shedding light on the role of ensemble averages in string theory. We propose an exact non-perturbative holographic match with the corresponding M2-brane partition functions, which we test perturbatively at all orders in the gauge group rank $N$ in a companion paper. A special case of our proposal for vanishing flux reformulates the Ooguri-Strominger-Vafa conjecture within the equivariant topological string framework.

Constant maps in equivariant topological strings and geometric modeling of fluxes

TL;DR

This work develops an equivariant generalization of topological strings on toric Calabi–Yau manifolds, centering on constant-map contributions to regulate non-compact volumes and illuminate their role in flux compactifications and M2-brane holography. It introduces a precise dictionary between genus-zero and higher-genus equivariant constant maps and classical geometric data, and shows how these feed into an exact, non-perturbative holographic match with M2-brane partition functions via Laplace-type transforms and mesonic twists. By analyzing a broad set of toric examples and CY cones, the paper demonstrates that the equivariant partition function encodes both two-derivative supergravity data and higher-derivative corrections, with OSV-like reformulations emerging in vanishing-flux backgrounds. The framework provides a unified geometric model that connects topological-string invariants, fluxes, Sasakian geometry, and holographic duals, and it opens a path toward non-perturbative gravity calculations in string/M-theory via exact transformations between λ- and N-ensembles.

Abstract

We study the equivariant generalization of topological strings on toric manifolds, focusing in particular on defining the contributions of constant maps in the genus expansion of the partition function. This approach regularizes the integration over non-compact Calabi-Yau spaces, producing finite results at each order in the expansion, as illustrated by a broad set of explicit examples. Our investigation highlights the geometric modeling of flux compactifications and clarifies the link between the effective supergravity framework and the equivariant topological string formalism, building on recent developments by Martelli and Zaffaroni. We conclude that the connection between topological string theory and supergravity/field theory involves switching between geometric moduli and fluxes, shedding light on the role of ensemble averages in string theory. We propose an exact non-perturbative holographic match with the corresponding M2-brane partition functions, which we test perturbatively at all orders in the gauge group rank in a companion paper. A special case of our proposal for vanishing flux reformulates the Ooguri-Strominger-Vafa conjecture within the equivariant topological string framework.

Paper Structure

This paper contains 33 sections, 3 theorems, 290 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main proposal
  3. Equivariant constant maps
  4. Equivariant constant maps
  5. Topological strings and intersection numbers
  6. Equivariant volumes, toric geometry, and generating functions
  7. g=0
  8. g=1
  9. g>=2
  10. Calabi--Yau cones, Kähler bases, and mesonic twist
  11. Basic toric examples
  12. Complex plane and discrete quotients, Cn/Gamma
  13. Spindle, WPL
  14. Projective space, Pk
  15. Calabi--Yau cones and resolutions
  16. ...and 18 more sections

Key Result

Lemma 2.1

The equivariant volume $\mathrm{vol}_X(t,\epsilon)$ and the generating function $\mathbb{V}_X(\lambda,\epsilon)$ are related by the identity

Figures (2)

  • Figure 1: Conceptual sketch of exact holography from string/M-theory point of view, based on results in Martelli:2023oqkColombo:2023fhuCassia:2025jkr and here. Blue contours signify a "$\lambda$-ensemble" of equivariant geometric moduli, while yellow correspond to $N$ or $G_{\rm N}$-ensembles in field theory and supergravity, respectively. Green arrows show a change of ensemble, i.e. the relation holds via a forward or inverse transform.
  • Figure 2: Schematic diagram of the relations we propose. Blue contour signifies quantities in the $\lambda$-ensemble, while yellow corresponds to the $N$ or $G_\mathrm{N}$-ensemble in field theory and supergravity, respectively ($N$ and $G_\mathrm{N}$ are related directly via AdS/CFT). Green arrows signify a change of ensemble, and we have suppressed additional indices and (equivariant) parameters.

Theorems & Definitions (10)

  • Definition 2.1
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.2
  • Remark