gv — Gopakumar-Vafa invariant utilities

GV series computation and curve classification.

Provides functions for extracting Gopakumar-Vafa series from a CYTools Invariants object, computing effective GV invariants, and classifying curves as potent or nilpotent.

gv_eff(gv_series)

Compute effective GV invariants from a GV series.

The effective GV invariants are weighted sums over the series of multiples:

\[n^{\mathrm{eff},p}_{\mathcal{C}} = \sum_k k^p \, n^0_{k[\mathcal{C}]}\]

This function returns the \(p=1\) (linear) and \(p=3\) (cubic) effective invariants, which enter the wall-crossing formulas for \(c_2\) and \(\kappa_{abc}\) respectively.

See arXiv:2212.10573 below Eq. (2.7).

Parameters:

gv_series (list[int]) – GV series \([n^0_{[\mathcal{C}]}, n^0_{2[\mathcal{C}]}, \ldots]\).

Returns:

tuple[int, int] – (gv_eff_1, gv_eff_3) — the linear and cubic effective GV invariants.

Raises:

ValueError – If gv_series is empty.

gv_series(gv_invariants, curve)

Extract the GV series for multiples of a curve class.

Queries the CYTools Invariants object for \(n^0_{k[\mathcal{C}]}\) with \(k = 1, 2, 3, \ldots\), stopping when the result is None (degree exceeds computed range).

See arXiv:2303.00757 Section 2 for the GV extraction procedure.

Parameters:

• gv_invariants (cytools.Invariants) – A CYTools Invariants object with a gv(curve) method.

• curve (array_like) – Curve class \([\mathcal{C}]\) in the Mori cone basis.

Returns:

list[int] – GV series \([n^0_{[\mathcal{C}]}, n^0_{2[\mathcal{C}]}, \ldots]\). Empty if the first query returns None.

is_nilpotent(gv_series)

Check if a curve is nilpotent based on its GV series.

A curve is nilpotent if the last computed GV invariant in the series is zero, indicating that the series has terminated (or is terminating) and the curve’s contribution to the prepotential is finite.

See arXiv:2212.10573 Section 3.1 for the potent/nilpotent classification.

Parameters:

gv_series (list[int]) – GV series \([n^0_{[\mathcal{C}]}, n^0_{2[\mathcal{C}]}, \ldots]\).

Returns:

bool – True if the curve is nilpotent.

is_potent(gv_series)

Check if a curve is potent based on its GV series.

A curve is potent if the last computed GV invariant in the series is nonzero, indicating that the series has not yet terminated and the curve contributes unboundedly to the prepotential.

See arXiv:2212.10573 Section 3.1 for the potent/nilpotent classification.

Parameters:

gv_series (list[int]) – GV series \([n^0_{[\mathcal{C}]}, n^0_{2[\mathcal{C}]}, \ldots]\).

Returns:

bool – True if the curve is potent.