Finite generation and holomorphic anomaly equation for equivariant Gromov-Witten invariants of $K_{\mathbb{P}^1\times\mathbb{P}^1}$

Abstract

In this paper, we prove finite generation property and holomorphic anomaly equation for the equivariant Gromov-Witten theory ofTwisted I fucntion 4 2.3. Quantum differential equation 4 2.4. Relations among the entries of matrixes A 1 and A 2 6 3. R matrix computations 8 3.1. QDE for R matrix 8 4. Finite generation property of F g 11 5. Holomorphic anomaly equation 12 5.1. Derivatives of the full R matrix 12 5.2. Finite generation property 13 6. Oscillatory integral and Feymann diagram representation of R matrix 14 References 18

Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

“…Holomorphic anomaly equations are predicted for the Gromov-Witten theory of Calabi-Yau manifolds by string theory [6]. In the last years, this structure was proven in various geometries, such as for elliptic orbifold projective lines [64], elliptic curves [78], formal elliptic curves [95], local P 2 [42,20], local P 1 × P 1 [40,94] relative (P 2 , E) [13], C 3 /Z 3 [43,20], toric Calabi-Yau 3-folds [22,23,25,26], the formal quintic 3-fold [44], the quintic 3-fold [29,19], and (partially) elliptic fibrations [79] and K3 fibrations [41]. Conjecture C is maybe the first instance where a general holomorphic anomaly equation is considered in higher dimensions.…”