flop — Wall-crossing formulas and flop construction

flop — Wall-crossing formulas and flop construction#

Wall-crossing formula implementations for birational flops.

Provides functions to transform intersection numbers and second Chern class across a wall in the extended Kahler cone, and a convenience function to create a new CalabiYauLite representing the flopped phase.

flop_phase(cy_lite, curve, gv_series, label=None)#

Create a new CalabiYauLite by flopping across a wall.

Computes the effective GV invariants from the GV series, then applies the wall-crossing formula to the intersection numbers and second Chern class to produce a new phase.

The Kahler cone, Mori cone, and other cone data are NOT copied because they change under the flop (computed in Phase 3). The GV invariants are also not set because transforming them requires CYTools Invariants manipulation (Phase 3 / INTG-01).

Parameters:

  • cy_lite (CalabiYauLite) – The source phase.

  • curve (array_like) – Curve class \(\mathcal{C}_a\) being flopped.

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

  • label (str, optional) – Phase label for the new CalabiYauLite.

Returns:

CalabiYauLite – A new phase with transformed intersection numbers and c2.

wall_cross_c2(c2, curve, gv_eff_1)#

Transform second Chern class across a wall-crossing.

Applies the wall-crossing formula for the second Chern class when flopping a curve \(\mathcal{C}\):

\[c'_a = c_a + 2 \, n^{\mathrm{eff,1}}_{\mathcal{C}} \, \mathcal{C}_a\]

where \(n^{\mathrm{eff,1}}_{\mathcal{C}}\) is the linear effective GV invariant.

See arXiv:2212.10573 below Eq. (2.7) and Eq. (4.4).

Parameters:

  • c2 (numpy.ndarray) – Second Chern class \(c_2 \cdot D_a\).

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

  • gv_eff_1 (int) – Linear effective GV invariant \(n^{\mathrm{eff,1}}_{\mathcal{C}}\).

Returns:

numpy.ndarray – Transformed second Chern class \(c'_a\).

wall_cross_intnums(int_nums, curve, gv_eff_3)#

Transform intersection numbers across a wall-crossing.

Applies the wall-crossing formula for triple intersection numbers when flopping a curve \(\mathcal{C}\):

\[\kappa'_{abc} = \kappa_{abc} - n^{\mathrm{eff,3}}_{\mathcal{C}} \, \mathcal{C}_a \, \mathcal{C}_b \, \mathcal{C}_c\]

where \(n^{\mathrm{eff,3}}_{\mathcal{C}}\) is the cubic effective GV invariant.

See arXiv:2212.10573 below Eq. (2.7) and Eq. (4.4).

Parameters:

  • int_nums (numpy.ndarray) – Triple intersection numbers \(\kappa_{abc}\).

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

  • gv_eff_3 (int) – Cubic effective GV invariant \(n^{\mathrm{eff,3}}_{\mathcal{C}}\).

Returns:

numpy.ndarray – Transformed intersection numbers \(\kappa'_{abc}\).