jaxvacua.conifold.coniLCS_prepotential.F_coniLCS_bulk

F_coniLCS_bulk(self, moduli, conj=False)

Calculates the value of the LCS prepotential in terms of the complex structure moduli \(z^{i}\).

Details

We want to compute the effective bulk superpotential for coni-LCS limits.

\[F_{\text{coni-LCS-bulk}}(z^1,\ldots , z^{h^{1,2}}) = F_{\text{LCS}}(z^1,\ldots , z^{h^{1,2}}) + \dfrac{n_{\text{cf}}\, z^1}{24}\, ,\]

where we always work in a basis in which \(z^1 = z_{\text{cf}}\). At LCS, we can write the prepotential as

\[F_{\text{LCS}}(z^1,\ldots , z^{h^{1,2}})=F_{\text{poly}}(z^1,\ldots , z^{h^{1,2}}) + F_{\text{inst}}(z^1,\ldots , z^{h^{1,2}})\]

Here, the polynomial piece is given by

\[F_{\text{poly}}(z^1,\ldots , z^{h^{1,2}})=\dfrac{1}{6}\kappa_{ijk}\, z^i z^j z^k+\dfrac{1}{2} a_{ij}z^i z^j+b_i\, z^i+\dfrac{\text{i}}{2}\tilde{\xi}\, ,\]

Here, \(\widetilde{\kappa}_{ijk}\) are the triple intersection numbers of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). Further, we defined

\[\begin{split}a_{ij} = \dfrac{1}{2}\begin{cases} \widetilde{\kappa}_{iij} & i\geq j\\[0.3em] \widetilde{\kappa}_{ijj} & i<j \end{cases} \, , \quad b_i = \dfrac{1}{24} \int_{\tilde{D}^i}\, c_2(\widetilde{X})\, , \quad \tilde{\xi}=\frac{\zeta(3)\, \chi(\widetilde{X})}{(2\pi)^3}\, .\end{split}\]

The instanton contributions read

\[F_{\mathrm{inst}}(z) = -\frac{1}{(2\pi\mathrm{i})^3}\, \sum_{q\in\mathcal{M}(\widetilde{X})}\, n_q^{0}\, \text{Li}_3\left (\text{e}^{2\pi \text{i} q_i z^i}\right )\; , \quad \text{Li}_3\left (x\right )=\sum_{m=1}^{\infty}\, \dfrac{x^{m}}{m^{3}}\, .\]

in terms of genus-0 Gopakumar-Vafa (GV) invariants \(n_q^0\) and the 3rd polylogarithm \(\text{Li}_3(x)\). Both pieces are computed in separate functions, see F_LCS_poly() and F_inst() for details.

Warning

The effective description for the bulk theory holds only provided that \(z^1 = z_{\text{cf}}\ll 1\).

Parameters:

• moduli (Array) – Complex structure moduli values.

• conj (bool) – If True, computes the complex conjugate. Defaults to False.

Returns:

complex – Value of the LCS prepotential \(F_{\text{LCS}}\).

Return type:

complex

See also: F_LCS_poly(), F_inst()