
## 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 $\phi^{(0)}$
1. #### Problem 1.1.

Does the holomorphic anomaly equation "integrate" over elliptic fibrations?
• ### Eta-products and root systems and holomorphic anomaly equation

As it turns out mock modular forms (such as eta-products) do not satisfy the holomorphic anaomaly equation.

#### Problem 1.2.

Do they satisfy other differential equations?
Also see problem hae-cy.
• #### Problem 1.3.

Investigate the Holomorphic anomaly equation (HAE) in fibered Calabi-Yau 3-folds and BPS counting.
For example, does the HAE “integrate” over elliptic fibrations or K3 fibers?

Cite this as: AimPL: Gromov-Witten invariants and number theory, available at http://aimpl.org/gromwitnumthry.