jaxvacua.conifold.coniLCS_prepotential.F_coniLCS_exp_per

F_coniLCS_exp_per(self, X0, XConi, XPerBulk, conj=False, n=0)

Computes the expansion of the prepotential \(F\) around the conifold point in terms of the periods.

Details

The prepotential \(F\) can be expanded around the conifold point as

\[F(X) = \sum_{n=0}^{\infty}\, \dfrac{1}{n!}\, \dfrac{\partial^n F}{\partial (X^{\mathrm{cf}})^n}\Bigg |_{X^{\mathrm{cf}}=0}\, (X^{\mathrm{cf}})^n\, .\]

Here, \(X^{\mathrm{cf}}\) is the period associated to the conifold curve.

This function returns the coefficient of \((X^{\mathrm{cf}})^n\) in this expansion, i.e. it computes \(\tfrac{1}{n!}\,\partial^n_{\!X^{\mathrm{cf}}} F\big|_{X^{\mathrm{cf}}=0}\). The full coniLCS prepotential is reconstructed in F_coniLCS_series_per() by summing these coefficients up to order nmax.

The individual contributions are:

• Polynomial part — obtained from F_coniLCS_poly_split_per() and

  its derivatives dF_coniLCS_poly_per(), ddF_coniLCS_poly_per(), dddF_coniLCS_poly_per() evaluated at \(X^{\mathrm{cf}}=0\).

• Conifold part — the dilogarithm \(F_{\mathrm{coni}}\) contributes

  Riemann zeta values at integer arguments: \(\zeta(3)\) at \(n=0\), \(\zeta(2)=\pi^2/6\) at \(n=1\), \(3/2\) at \(n=2\), and the Bernoulli numbers \(B_{n-2}\) for \(n \ge 3\).

• Worldsheet instantons — added via F_inst_per_coni() when

  maximum_degree > 0.

Parameters:

• X0 (complex) – Value of period \(X^0\).

• XConi (complex) – Value of conifold period \(X^{\mathrm{cf}}\).

• XPerBulk (Array) – Values of bulk periods \(X^i\).

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

• n (int) – Order of derivative. Defaults to 0.

Returns:

complex – Value of the \(n\)-th derivative of the prepotential \(F\) around the conifold point.

Return type:

complex