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Counting BPS States via Holomorphic Anomaly Equations

Shinobu Hosono

TL;DR

This work develops a holomorphic anomaly framework for a rational elliptic surface $ rac{1}{2}$K3 to compute genus g Gromov-Witten invariants via an affine $E_8$ Weyl symmetry. It constructs a generating function $Z_{g;n}$ organized into Weyl-orbits, derives orbit-based decompositions, and uses vanishing/BPS constraints to fix holomorphic ambiguities, yielding explicit BPS numbers $n_g(eta)$ that are integral and compatible with the Gopakumar–Vafa picture. A key contribution is the explicit link between the local BCOV holomorphic anomaly equation and the HST equation in the local limit, including detailed genus-one and genus-two data for the half-K3 and for $old P^1 imesold P^1$ models. The results provide both new enumerative predictions for rational elliptic surfaces and a clearer bridge between holomorphic anomaly techniques in local mirror symmetry and the counting of BPS states.

Abstract

We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).

Counting BPS States via Holomorphic Anomaly Equations

TL;DR

This work develops a holomorphic anomaly framework for a rational elliptic surface K3 to compute genus g Gromov-Witten invariants via an affine Weyl symmetry. It constructs a generating function organized into Weyl-orbits, derives orbit-based decompositions, and uses vanishing/BPS constraints to fix holomorphic ambiguities, yielding explicit BPS numbers that are integral and compatible with the Gopakumar–Vafa picture. A key contribution is the explicit link between the local BCOV holomorphic anomaly equation and the HST equation in the local limit, including detailed genus-one and genus-two data for the half-K3 and for models. The results provide both new enumerative predictions for rational elliptic surfaces and a clearer bridge between holomorphic anomaly techniques in local mirror symmetry and the counting of BPS states.

Abstract

We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).

Paper Structure

This paper contains 16 sections, 6 theorems, 86 equations.

Table of Contents

  1. Introduction and Main results
  2. Generating function and affine $E_8$ Weyl orbits
  3. Notations
  4. Root system
  5. Root system defined in $H^2(S,\bold Z)$
  6. $Z_{g;n}$ and orbit decompositions
  7. The multiplicity functions $P_\lambda$
  8. Theta function $\Theta_{E_8}$
  9. Orbit decomposition and BPS numbers
  10. Vanishing conditions
  11. Orbit decomposition $\mathbf{n\leq2}$
  12. BPS numbers $n_g(\beta)$
  13. Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation
  14. BCOV holomorphic anomaly equation
  15. $\mathbf{S=\frac{1}{2}K3}$
  16. ...and 1 more sections

Key Result

Proposition 2.1

Theorems & Definitions (11)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Proposition 3.2
  • Definition 4.1
  • Proposition 4.2
  • ...and 1 more