Gromov-Witten invariants and number theory

1. Holomorphic anomaly equation

Brief introduction to the holomorphic anomaly equation taken from Albrecht Klemm’s Slides on on Omega backgrounds and generalized holomorphic anomaly equation at STring-math in June 2011. http://www.math.upenn.edu/StringMath2011/notes/Klemm_StringMath2011_talk.pdf, See also the paper by Bershadski, Cecotti, Ooguri, and Vafa 1993.

The B-model definition of the $F^g(a)=F^{(0,g)}(a)$ is given by \[ F^g(a)=\int_{\bar{\mathcal{M}}_g} \langle \prod_{k=1}^{3g-3} \beta^k \bar{\beta}^k \rangle_g \cdot [dm \wedge d\bar{m}], \] The contraction of the coordinates $m_k, \ \bar{m}_k$ with the genus $g$worldsheet correlator of \[ \beta^k = \int_{\Sigma_g} G^- \mu^k, \ \ \ \bar{\beta^k}=\int_{\Sigma_g} G^- \bar{\mu}^k \] gives a real $6g-6$ form on the compactified moduli space $\bar{\mathcal{M}}_g$ of the $g$ Riemann surface $\Sigma_g$.

An infinitessimal anholomorphic perturbation \[ \S(t_i, \bar{t}_i) = S(t_i) + \bar(t)^i\int_{\Sigma_g} \bar{\mathcal{O}_i^{(2)}}, \] with \[ \bar{\mathcal{O}_i^{(2)}} = \{ G_0^+, [\bar{G}_0^+, \bar{\mathcal{O}_i^{(0)}}]\}dz d\bar{z} \] corresponds to an insertion of exact forms. The deformation receives contributions from the boundaries. This leads to the Holomorphic Anomaly Equation ( Bershadski, Cecotti, Ooguri, and Vafa 1993) \[ \bar{\partial}_iF^g = \frac{1}{2}\bar{C}_i^{jk}(D_jD_kF^{g-1} + \sum_{h=0}^{g-1} D_jF^hD_kF^{g-h}), \ \ \ g>1 \] Defining for $g\geq 1$ \[ F^{(n,g)}(t)=\int_{\bar{\mathcal{M}}_g} \langle \mathcal{O}^n \prod_{k=1}^{3g-3} \beta^k \bar{\beta}^k \rangle_g \cdot [dm \wedge d\bar{m}], \] and for $g=0$ \[ F^{(n+1,0)} = \langle \phi^{(0)}(0)\phi^{(0)}(1)\phi^{(0)}(\infty)\mathcal{O}^n \rangle_{g=0} \] where the field operator $\mathcal{O}$ should come from integration a 2-form over the Riemann surface, i.e. \[ \mathcal{O} = \int_{\Sigma_g}\phi^{(2)}, \] and $\phi^{(2)}$ emerges as usual from the descending equation from $\phi^{(0)}$