jaxvacua.conifold.coniLCS_prepotential.F_coni_per

F_coni_per(self, X, conj=False)

Computes the conifold part \(F_{\mathrm{conifold}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) in terms of the periods.

Details

The conifold part \(F_{\mathrm{conifold}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\) can be expressed in terms of the periods \(X^I=(X^0,X^i)\) as

\[F_{\mathrm{conifold}}(X) = \frac{n_{\mathrm{cf}}}{(2\pi\mathrm{i})^2}\, (X^{\mathrm{cf}})^2\, \log\left (\frac{X^{\mathrm{cf}}}{X^0}\right )\, .\]

Here, \(X^{\mathrm{cf}}\) is the period associated to the conifold curve. The conifold period can be identified as the period which vanishes at the conifold locus. The conifold period can be obtained from the other periods by applying a suitable basis transformation. The number of conifolds is denoted by \(n_{\mathrm{cf}}\). The conifold curve can be specified when initialising the class. The conifold curve is stored in the attribute conifold_curve. The conifold curve is expressed in the basis of the Mori cone generators of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). The conifold curve can also be specified indirectly by providing a basis transformation. The basis transformation is applied to the periods such that the first period corresponds to the conifold period: \(X^1 = X^{\mathrm{cf}}\). The basis transformation can be specified when initialising the class. If no basis transformation is provided, the identity is assumed. The basis transformation is stored in the attribute basis_change.

Parameters:

• X (Array) – Values of periods.

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

Returns:

complex – Value of the conifold part \(F_{\mathrm{conifold}}\) of the LCS prepotential \(F_{\mathrm{LCS}}\).

Return type:

complex