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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.