jaxvacua.conifold.coniLCS_prepotential.F_inst_per_coni

F_inst_per_coni(self, X0, XPerBulk, conj=False, n=0)

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

Details

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

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

Here the sum is performed over all effective curve classes \(q\in\mathcal{M}(\widetilde{X})\) in the Mori cone \(\mathcal{M}(\widetilde{X})\) of the mirror dual manifold \(\widetilde{X}\). Here, the \(n_q^{0}\) are the genus-0 Gopakumar-Vafa (GV) invariants which can be computed systematically using methods described in hep-th/9308122.

The infinite sum appearing in the poly-logarithm \(\text{Li}_3\) can be rewritten to arrive at

\[\sum_{q\in\mathcal{M}(\widetilde{X})}\, n_q^{0}\, \text{Li}_3\left (\text{e}^{2\pi \text{i} q_i X^i / X^0}\right ) = \sum_{q\in\mathcal{M}(\widetilde{X})}\, N_q\, \text{e}^{2\pi \text{i} q_i X^i / X^0}\]

in terms of genus-0 Gromov-Witten (GW) invariants \(N_q\). We typically work with the latter to simplify the calculation. The relation between the two types of invariants is more explicitly given by

\[N_q = \sum_{d|q}\, \dfrac{1}{d^3}\, n_{q/d}^{0}\, .\]

Here, the sum runs over all divisors \(d\) of the curve class \(q\). The \(N_q\) are typically rational numbers, while the \(n_q^{0}\) are integers. The curve classes \(q\) are specified in a basis of the Mori cone \(\mathcal{M}(\widetilde{X})\) of the mirror dual Calabi-Yau threefold \(\widetilde{X}\). The Mori cone is dual to the Kähler cone of \(\widetilde{X}\). The curve classes \(q\) can be expressed in terms of the generators of the Mori cone. The curve classes are also referred to as curve charges in the following. The curve charges are stored in the attribute GW_charges (GV_charges) for the GW (GV) invariants. The corresponding invariants are stored in the attributes GW_inv (GV_inv). The curve charges and invariants are limited to a certain maximum degree \(d=\text{max_deg}\) in the attributes GW_charges_lim, GV_charges_lim, GW_inv_lim, GV_inv_lim. The maximum degree can be specified when initialising the class. The maximum degree is defined with respect to a grading vector which can be specified when initialising the class. If no grading vector is provided, the default grading vector \((1,1,\ldots,1)\) is used. The maximum available degree for the instanton data is stored in the attribute max_available_deg. If the specified maximum degree exceeds the maximum available degree, a warning is raised and the maximum available degree is used instead.

Note

The sum over curve classes is truncated at a certain maximum degree \(d=\text{max_deg}\) in our implementation. This is justified since the instanton contributions are exponentially suppressed at large complex structure. The maximum degree can be specified when initialising the class.

Parameters:

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

• XPerBulk (Array) – Values of bulk periods \(X^i\) (excluding the conifold period).

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

• n (int) – Derivative order with respect to the conifold period. Defaults to 0.

Returns:

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

Return type:

complex

See also: F_coniLCS_series_per()