One-modulus Calabi-Yau models

One-modulus Calabi-Yau models#

What’s in this notebook? This notebook introduces the hypergeometric_models module, which provides closed-form period data for one-modulus Calabi-Yau (CY) threefolds near three moduli-space points: LCS, K-point and C-point.

The public entry point is the static factory HypergeometricModels, which returns a fully-configured FluxVacuaFinder for any of the 18 registered models. We cover the physical background on singularity types, demonstrate how to load models, inspect their boundary data, and run a SUSY flux-vacuum search at LCS.

The period data is taken from: B. Bastian, D. van de Heisteeg, L. Schlechter, Beyond Large Complex Structure: Quantized Periods and Boundary Data for One-Modulus Singularities, arXiv:2306.01059.

(Created: Andreas Schachner, Nov. 2025)

Outline#

Background
Setup
Discovering available models
Loading a model via HypergeometricModels.build
LCS point
K-point
C-point (conifold)
Comparing LCS and boundary prepotentials for the same model
End-to-end: Newton-solve a SUSY flux vacuum
Using make_prepot_* directly
Take-aways
Further reading

Background#

One-modulus CY threefolds#

A Calabi-Yau threefold with $h^{2,1} = 1$ has a one-dimensional complex structure moduli space $\mathcal{M}$. The most studied examples come from the 14 hypergeometric one-parameter families (AESZ database), defined by a Picard-Fuchs operator

\[ \mathcal{L} = \theta^4 - z\,(\theta + \alpha_1)(\theta + \alpha_2)(\theta + \alpha_3)(\theta + \alpha_4), \quad \theta = z\,\partial_z, \]

with rational exponents $\alpha_i \in (0,1)$. The moduli space is a $\mathbb{P}^1$ with three special loci:

$z = 0$: large complex structure (LCS) point, always present;
$z = \mu^{-1}$: conifold singularity;
$z = \infty$: a boundary whose type depends on the exponents $\alpha_i$.

The module also includes four non-hypergeometric models studied in arXiv:2306.01059 as additional examples.

Singularity types and monodromy#

The boundary at $z \to \infty$ is classified by the monodromy matrix $T_\infty$ acting on $H^3(X, \mathbb{Z})$ (a $4\times4$ symplectic matrix). Four types arise, distinguished by the pattern of Picard-Fuchs exponents $\alpha_i$ at $z = \infty$:

Type	Condition on $\alpha_i$	Monodromy $T_\infty$	Physical character
M (MUM)	$\alpha_1 = \alpha_2 = \alpha_3 = \alpha_4$	Maximally unipotent	Second LCS point
K	$\alpha_1 = \alpha_2$, $\alpha_3 = \alpha_4$, $\alpha_2 \neq \alpha_3$	Finite-order semisimple $\times$ unipotent	K-point
C	$\alpha_2 = \alpha_3$ only	Unipotent, rank 1	Conifold
F	All $\alpha_i$ distinct	Generic finite order	F-point

The type determines what kind of period expansion is natural near $z = \infty$, and hence what form the prepotential takes.

The LCS prepotential#

Near the large complex structure point at $z \to 0$, with projective coordinates $X = (X^0, X^1)$ and $t = X^1/X^0$, the prepotential takes the form

\[ \mathcal{F}(X^0, X^1) = (X^0)^2 \left[ \frac{\kappa}{6}\,t^3 - \frac{\sigma}{2}\,t^2 - \frac{c_2}{24}\,t - K_0 \right], \]

where:

$\kappa$ is the triple intersection number of the mirror CY;
$c_2$ is the integrated second Chern class;
$\chi$ is the Euler characteristic;
$\sigma = (\kappa/2) \bmod 1$;
$K_0 = \zeta(3)\chi / (2(2\pi i)^3)$ encodes the constant map contribution.

The period vector is $\Pi = (\partial_{X^0}\mathcal{F},\, \partial_{X^1}\mathcal{F},\, X^0,\, X^1)$.

The K-point prepotential#

A K-point arises when the monodromy $T_\infty$ has finite semisimple part and a unipotent block. It is characterised by a rigid period $\tau \in \mathbb{H}$ (the upper half-plane), which plays the role of the complexified gauge coupling of a heavy BPS vector multiplet becoming massless.

Setting $s = (X^1 - \tau)/X^0$ as a local coordinate, the prepotential expands as (eqs. 1060–1063 of arXiv:2306.01059)

\[ \mathcal{F} = F_0 + s\,F_1 + s^2 F_2 + s^3 F_3 + \cdots \]

with

\[ F_0 = \tfrac{\delta}{2} + \gamma\tau, \quad F_1 = \gamma - \frac{c\,\tau_2}{2\pi}\Bigl(\log\frac{is}{2B_1\tau_2} - 1\Bigr), \]

\[ F_2 = \frac{ic}{4\pi^2}\Bigl(\log\frac{is}{2B_1\tau_2} - 2 + \frac{B_2}{4B_1^2}\Bigr), \quad F_3 = \frac{c}{16\pi\tau_2}\Bigl(1 - \frac{B_2^2}{B_1^4} + \frac{2B_3}{3B_1^3}\Bigr). \]

Here:

$\gamma, \delta$ are extension data encoding the mixing between the light and heavy periods;
$c$ is the lower-right entry of the log-monodromy matrix $N_K$;
$B_k = \hat{B}_k \times B_{\rm norm}$ are instanton coefficients, with $B_{\rm norm}$ a transcendental prefactor involving the special $L$-value $L(f, 1)$ of an associated weight-3 modular form $f$.

The C-point (conifold) prepotential#

A C-point is a conifold singularity, where a single BPS hypermultiplet (a 3-cycle) shrinks to zero size. In type IIB string theory this corresponds to an $\mathcal{N}=2$ conifold transition. The rigid period $\tau$ is again the gauge coupling, but the monodromy is purely unipotent of rank 1.

With the same local coordinate $s = (X^1 - \tau)/X^0$, the prepotential is (eqs. 846–849 of arXiv:2306.01059)

\[ F_0 = \frac{\tau}{2}, \quad F_1 = \delta - \gamma\tau, \quad F_2 = ik\bigl(3 - 2\log s + 2\log A_1\bigr) - \gamma F_1, \quad F_3 = -\frac{ik(3A_1^2 \gamma - A_2)}{12\pi A_1^2}. \]

The logarithmic term in $F_2$ reflects the characteristic $s^2 \log s$ branching of the periods at the conifold. The integer $k$ is the conifold order, $A_k = \hat{A}_k \times A_{\rm norm}$ are instanton coefficients, and $A_{\rm norm}$ involves $L(f, 1)$ for an associated weight-4 modular form.

L-function values and instanton normalisations#

A key result of arXiv:2306.01059 is that the instanton normalisations $B_{\rm norm}$ (K-point) and $A_{\rm norm}$ (C-point) are not free parameters — they are fixed by quantisation conditions on the period lattice. They are expressed in terms of special values $L(f, 1)$ of $L$-functions of modular forms, which in turn admit closed-form expressions in terms of $\Gamma$-function values. For example:

Model	Modular form	$L(f, 1)$
$X_{3,3}$ K-point	27.3.b.a	$\Gamma(1/3)^6 / (8\sqrt{3}\,\pi^3)$
$X_{4,4}$ K-point	16.3.c.a	$\Gamma(1/4)^4 \Gamma(1/2)^2 / (32\pi^3)$
$X_{4,2}$ C-point	32.4.a.b	$\Gamma(1/4)^6 \Gamma(1/2)^3 / (16\sqrt{2}\,\pi^5)$
$X_{3,2,2}$ C-point	9.4.a.a	$\Gamma(1/3)^9 / (32\sqrt{3}\,\pi^5)$

All of these are precomputed in hypergeometric_models.py.

Setup#

# General imports
import warnings
import numpy as np
from scipy.special import gamma as gamma_func
from scipy.special import zeta as scipy_zeta
from scipy.optimize import root

# JAX imports
import jax
import jax.numpy as jnp
jax.config.update("jax_enable_x64", True)

# JAXVacua
import jaxvacua as jvc
from jaxvacua import HypergeometricModels, list_hypergeometric_models
from jaxvacua.hypergeometric_models import (
    HYPERGEOMETRIC_MODELS, KPOINT_DATA, CPOINT_DATA,
    make_prepot_LCS, make_prepot_Kpoint, make_prepot_Cpoint,
)

warnings.filterwarnings('ignore')

Discovering available models#

list_hypergeometric_models() prints a summary of all 18 models (14 hypergeometric + 4 non-hypergeometric) together with the singularity types for which prepotential data is available. For programmatic use, HypergeometricModels.list() returns the labels as a sorted list and HypergeometricModels.available_limits(label) returns the moduli-space points where each model has closed-form data.

list_hypergeometric_models()

The column ∞-type records the singularity type at $z \to \infty$. For the 6 models with K- or C-type singularities at infinity, dedicated boundary data is stored in KPOINT_DATA and CPOINT_DATA respectively. Every hypergeometric model with fixed topological data also supports an LCS prepotential.

Loading a model via `HypergeometricModels.build`#

The factory HypergeometricModels.build(label, limit=...) returns a fully-configured FluxVacuaFinder for the requested model at the requested moduli-space point:

Limit	What gets built
`"LCS"`	Standard `lcs_tree`-backed `FluxVacuaFinder` with `intnums=[[[κ]]]`, `c2`, `chi` (no GV invariants — the closed-form LCS prepotential is complete).
`"Kpoint"`	`FluxVacuaFinder` with `limit="Kpoint"`; `periods` auto-detects the registered model and stamps the closed-form callable on `m.periods.prepotential_input`.
`"Cpoint"`	Same pattern as K-point with the C-point prepotential.

The closed-form callable is also accessible programmatically via HypergeometricModels.resolve_prepotential(model_ID, limit). Raw boundary parameters are exposed via HypergeometricModels.lcs_data(label) and HypergeometricModels.boundary_data(label, limit).

LCS point#

Example: quintic $X_5$#

The quintic has $\kappa = 5$, $c_2 = 50$, $\chi = -200$, and a generic F-type singularity at $z \to \infty$. Near $z = 0$ only the LCS prepotential is available.

m_lcs = HypergeometricModels.build("X5", limit="LCS")
print(m_lcs)
print()
print("LCS topological data:", HypergeometricModels.lcs_data("X5"))

Choose a point in the LCS region, $X = (1,\, it)$ with $t \gg 1$. The closed-form LCS prepotential is most directly accessed through the make_prepot_LCS(kappa, c2, chi) factory; the period vector is then $\Pi(X) = (\partial_{X^I} F,\, X^I)$ via JAX autodiff.

# Closed-form LCS prepotential for the quintic
data_X5 = HypergeometricModels.lcs_data("X5")
F_X5    = make_prepot_LCS(data_X5["kappa"], data_X5["c2"], data_X5["chi"])

X_lcs = jnp.array([1.0 + 0j, 2.0j])   # t = X1/X0 = 2i, deep LCS

F_val = F_X5(X_lcs)
gF    = jax.grad(lambda x: F_X5(x, conj=False), holomorphic=True)(X_lcs)
Pi_val = jnp.concatenate([gF, X_lcs])

print(f"F(X)  = {F_val:.6f}")
print(f"Pi(X) = {Pi_val}")

The period vector ordering is $(F_0, F_1, X^0, X^1)$ where $F_I = \partial_{X^I}\mathcal{F}$, consistent with the JAXVacua prepotential_input convention.

LCS data for all hypergeometric models#

print(f"{'Label':8s}  {'kappa':5s}  {'c2':5s}  {'chi':7s}  {'sigma':6s}")
print("-" * 40)
for label, d in HYPERGEOMETRIC_MODELS.items():
    kappa = d['kappa']
    sigma = round((kappa / 2) % 1, 3)
    print(f"{label:8s}  {kappa:5d}  {d['c2']:5d}  {d['chi']:7d}  {sigma:6.3f}")

K-point#

K-point data is available for five models: X33, X44, X66, 4.47, 3.7. The X66 model is a special case where the leading instanton vanishes ($B_1 = 0$) and the standard formula does not apply directly; HypergeometricModels.build("X66", limit="Kpoint") raises NotImplementedError with a clear pointer to the alternate expansion in arXiv:2306.01059 §5.

Inspecting K-point boundary data#

print("K-point models and their rigid periods:\n")
for label, bd in KPOINT_DATA.items():
    tau = bd['tau']
    mf  = bd.get('modular_form', '?')
    Lf1 = bd['Lf1']
    print(f"  {label:6s}  tau = {tau.real:.4f} + {tau.imag:.4f}i   "
          f"L(f,1) = {Lf1:.8f}   [{mf}]")

The rigid period $\tau$ lives in the fundamental domain of the modular group. For $X_{3,3}$ and $X_{6,6}$, $\tau = e^{2\pi i/3}$ (the CM point of order 3); for $X_{4,4}$, $\tau = i$ (the CM point of order 4).

Example: $X_{3,3}$ at the K-point#

$X_{3,3}$ is the complete intersection of two cubics in $\mathbb{P}^5$. It has Picard-Fuchs exponents $\alpha = (1/3, 1/3, 2/3, 2/3)$, so the singularity at $z \to \infty$ is of K-type. The rigid period is $\tau = -1/2 + i\sqrt{3}/2 = e^{2\pi i/3}$, and the instanton series runs in powers of $q^3 = e^{2\pi i \cdot 3\tau}$.

m_K = HypergeometricModels.build("X33", limit="Kpoint")
print(m_K)
print()

bd = HypergeometricModels.boundary_data("X33", "Kpoint")
print(f"  tau    = {bd['tau'].real:.6f} + {bd['tau'].imag:.6f}i")
print(f"  gamma  = {bd['gamma']}")
print(f"  delta  = {bd['delta']}")
print(f"  c      = {bd['c']}")
print(f"  L(f,1) = {bd['Lf1']:.10f}")
print(f"  Bnorm  = {bd['Bnorm']:.10f}")
print(f"  Bhat   = {bd['Bhat']}")

Note that $B_{\rm norm} = -1/(3 L(f,1)^2)$ where $L(f,1) = \Gamma(1/3)^6 / (8\sqrt{3}\,\pi^3)$.

# Verify the closed-form expression
LF1_exact = gamma_func(1/3)**6 / (8 * np.sqrt(3) * np.pi**3)
Bnorm_exact = -1.0 / (3.0 * LF1_exact**2)
print(f"L(f,1) from formula : {LF1_exact:.10f}")
print(f"L(f,1) from module  : {bd['Lf1']:.10f}")
print(f"Bnorm  from formula : {Bnorm_exact:.10f}")
print(f"Bnorm  from module  : {bd['Bnorm']:.10f}")

Evaluate the K-point prepotential. We choose a point $X = (X^0, X^1)$ near the K-point, i.e. $s = (X^1 - \tau)/X^0$ small:

tau = bd['tau']
# Small displacement from tau: s = 0.1i
X_K = jnp.array([1.0 + 0j, tau + 0.1j])

F_K  = m_K.periods.prepotential_input(X_K)
gF_K = jax.grad(lambda x: m_K.periods.prepotential_input(x, conj=False),
                holomorphic=True)(X_K)
Pi_K = jnp.concatenate([gF_K, X_K])

print(f"F(X)  = {F_K:.8f}")
print(f"Pi(X) = {Pi_K}")

Example: $X_{4,4}$ at the K-point#

$X_{4,4}$ has $\alpha = (1/4, 1/4, 3/4, 3/4)$ and $\tau = i$ (order-4 CM point). The instanton series runs in $q^2 = e^{4\pi i\tau}$, so $B_2 = 0$, $B_3 = 12 B_{\rm norm} \neq 0$.

m_X44 = HypergeometricModels.build("X44", limit="Kpoint")
bd44  = HypergeometricModels.boundary_data("X44", "Kpoint")
print(f"  tau   = {bd44['tau']}")
print(f"  gamma = {bd44['gamma']}, delta = {bd44['delta']}")
print(f"  Bnorm = {bd44['Bnorm']:.8f}")
print(f"  B1    = {bd44['B1']:.8f}")
print(f"  B3    = {bd44['B3']:.8f}   (leading non-zero after B1)")
print(f"  Bhat  = {bd44['Bhat']}")

The X66 special case#

$X_{6,6}$ has $\alpha = (1/6, 1/6, 5/6, 5/6)$ and its leading instanton vanishes at $k = 1$ ($B_1 = 0$). The instanton series starts at $k = 2$. This means the standard prepotential formula (which requires $B_1 \neq 0$ in the logarithm argument) cannot be applied directly.

# HypergeometricModels.build raises NotImplementedError for X66 at K-point
try:
    m_X66 = HypergeometricModels.build("X66", limit="Kpoint")
except NotImplementedError as e:
    print(f"NotImplementedError: {e}")

# X66 is still loadable at LCS:
m_X66_lcs = HypergeometricModels.build("X66", limit="LCS")
print(f"\nX66 at LCS — built OK: h12={m_X66_lcs.h12}")

# The raw K-point data is still accessible programmatically:
bd66 = HypergeometricModels.boundary_data("X66", "Kpoint")
print(f"\nB1 = {bd66['B1']}, B2 = {bd66['B2']:.8f}  (leading instanton at k=2)")
print(f"Bhat = {bd66['Bhat']}")

C-point (conifold)#

C-point data is available for four models: X42, X322, X62, 2.62. These all have a conifold-type singularity at $z \to \infty$ (C-type Picard-Fuchs exponents).

Inspecting C-point boundary data#

print("C-point models:\n")
for label, bd in CPOINT_DATA.items():
    tau = bd['tau']
    mf  = bd.get('modular_form', '?')
    Lf1 = bd['Lf1']
    k   = bd['k']
    print(f"  {label:6s}  tau = {tau.real:.4f} + {tau.imag:.4f}i   "
          f"k = {k:3d}   L(f,1) = {Lf1:.8f}   [{mf}]")

The conifold order $k$ counts how many times the standard conifold monodromy is iterated. Note that operator 2.62 has $k = 42$ — a large value reflecting a non-hypergeometric geometry.

Example: $X_{4,2}$ at the C-point#

$X_{4,2}$ is a complete intersection with $\alpha = (1/4, 1/2, 1/2, 3/4)$ and $\kappa = 8$. The C-point at infinity has $\tau = 1/2 + i/2$, and the weight-4 modular form is 32.4.a.b.

m_C = HypergeometricModels.build("X42", limit="Cpoint")
print(m_C)
print()

bd = HypergeometricModels.boundary_data("X42", "Cpoint")
print(f"  tau    = {bd['tau'].real:.6f} + {bd['tau'].imag:.6f}i")
print(f"  gamma  = {bd['gamma']}, delta = {bd['delta']}")
print(f"  k      = {bd['k']}")
print(f"  L(f,1) = {bd['Lf1']:.10f}")
print(f"  Anorm  = {bd['Anorm']:.10f}")
print(f"  Ahat   = {bd['Ahat']}")

tau = bd['tau']
# Point near the C-point: small s = 0.05i (avoid s=0 where log(s) diverges)
X_C = jnp.array([1.0 + 0j, tau + 0.05j])

F_C  = m_C.periods.prepotential_input(X_C)
gF_C = jax.grad(lambda x: m_C.periods.prepotential_input(x, conj=False),
                holomorphic=True)(X_C)
Pi_C = jnp.concatenate([gF_C, X_C])

print(f"F(X)  = {F_C:.8f}")
print(f"Pi(X) = {Pi_C}")

Example: $X_{3,2,2}$ at the C-point#

$X_{3,2,2}$ has $\alpha = (1/3, 1/2, 1/2, 2/3)$, so it is also a C-type model. The conifold order is $k = 6$.

m_X322 = HypergeometricModels.build("X322", limit="Cpoint")
bd322  = HypergeometricModels.boundary_data("X322", "Cpoint")
print(f"  tau   = {bd322['tau'].real:.6f} + {bd322['tau'].imag:.6f}i")
print(f"  k     = {bd322['k']}")
print(f"  Anorm = {bd322['Anorm']:.10f}")
print(f"  Ahat  = {bd322['Ahat']}")

Comparing LCS and boundary prepotentials for the same model#

For models with both LCS and K/C-point data, we can load both and compare. They are valid in different regions of moduli space and should not be expected to agree numerically.

m_X33_lcs = HypergeometricModels.build("X33", limit="LCS")
m_X33_K   = HypergeometricModels.build("X33", limit="Kpoint")

print("X33 LCS model:   ", m_X33_lcs)
print("X33 Kpoint model:", m_X33_K)

# LCS prepotential is valid for Im(t) >> 1.
data_X33 = HypergeometricModels.lcs_data("X33")
F_X33_lcs = make_prepot_LCS(data_X33["kappa"], data_X33["c2"], data_X33["chi"])
X_deep_lcs = jnp.array([1.0 + 0j, 5.0j])
print(f"\nF_LCS(t=5i) = {F_X33_lcs(X_deep_lcs):.8f}")

# K-point prepotential is valid for small |s| = |X1/X0 - tau|.
tau_K = HypergeometricModels.boundary_data("X33", "Kpoint")['tau']
X_near_K = jnp.array([1.0 + 0j, tau_K + 0.1j])
F_X33_K = m_X33_K.periods.prepotential_input(X_near_K)
print(f"F_Kpt(s=0.1i) = {F_X33_K:.8f}")

End-to-end: Newton-solve a SUSY flux vacuum#

HypergeometricModels.build returns a FluxVacuaFinder ready to plug into the standard JAXVacua pipeline. Here we sketch the simplest end-to-end use: build the model at LCS, evaluate the F-term residual $D_I W$ for a small flux configuration, and run scipy’s hybr Newton solver to refine to a SUSY vacuum. The same call signature works at K-point and C-point — just change limit=.

m = HypergeometricModels.build("X33", limit="LCS")

# Initial moduli + axio-dilaton in the LCS region
z0   = jnp.array([3.0 + 1.0j])
tau0 = 0.5 + 2.0j

# Simple unit flux (NS = 1 in slot 0, RR = 1 in the dilaton-shifted slot)
n_fl = 2 * m.n_fluxes
flux = jnp.zeros(n_fl, dtype=jnp.float64).at[jnp.array([0, m.n_fluxes])].set(1.0)

x0 = m._convert_complex_to_real(z0, jnp.conj(z0), tau0, jnp.conj(tau0))
DW = m.DW_x(x0, flux)
print(f"|DW|_max at the initial guess: {float(jnp.max(jnp.abs(DW))):.3e}")

# Newton refinement via scipy.root + analytic Jacobian
res = root(m.DW_x, x0, args=(flux,), jac=m.dDW_x, tol=1e-10, method="hybr")
z_sol, _, tau_sol, _ = m._convert_real_to_complex(res.x)
print(f"Newton: success={res.success}  |fun|_max={float(jnp.max(jnp.abs(res.fun))):.3e}")
print(f"Solution:  z={z_sol}  tau={tau_sol}")

For K-point and C-point models, m.periods.prepotential_input is a JIT-compiled closure delivered by the registry — m.DW_x, m.dDW_x etc. work identically because the F-terms are computed via JAX autodiff through the closed-form prepotential. The example below shows the Cpoint-flavoured construction:

m_c = HypergeometricModels.build("X42", limit="Cpoint")
print(m_c)
print(f"  m_c.periods.limit       = {m_c.periods.limit!r}")
print(f"  m_c.periods.model_type  = {m_c.periods.model_type!r}")
print(f"  prepotential_input type = {type(m_c.periods.prepotential_input).__name__}")

Using `make_prepot_*` directly#

For custom parameters or modifications, the factory functions make_prepot_LCS, make_prepot_Kpoint, and make_prepot_Cpoint can be called directly.

# Build a custom LCS prepotential for the quintic
F_quintic = make_prepot_LCS(kappa=5, c2=50, chi=-200)
X = jnp.array([1.0+0j, 3.0j])
print("F_quintic =", F_quintic(X))

# Build a K-point prepotential from raw parameters
from scipy.special import gamma as gamma_func
import numpy as np

LF1 = gamma_func(1/3)**6 / (8 * np.sqrt(3) * np.pi**3)
Bnorm = -1.0 / (3.0 * LF1**2)

F_kpt = make_prepot_Kpoint(
    tau=complex(-0.5, np.sqrt(3)/2),
    gamma=1/6, delta=1/3, c=2,
    B1=Bnorm, B2=0.0, B3=0.0,
)
tau_val = complex(-0.5, np.sqrt(3)/2)
X_kpt = jnp.array([1.0+0j, tau_val + 0.2j])
print("F_kpt =", F_kpt(X_kpt))

Take-aways#

Singularity	Models with data	Key parameters	Valid region
LCS	All 14 hypergeometric	$\kappa, c_2, \chi$	$\text{Im}(t) \gg 1$
K-point	X33, X44, X66*, 4.47, 3.7	$\tau, \gamma, \delta, c, B_k$	$
C-point	X42, X322, X62, 2.62	$\tau, \gamma, \delta, k, A_k$	$

* X66 K-point has $B_1 = 0$; get_model raises ValueError — use KPOINT_DATA['X66'] for raw data.

All transcendental prefactors ($B_{\rm norm}$, $A_{\rm norm}$) are computed from closed-form $\Gamma$-function expressions for $L(f,1)$ values, as derived in arXiv:2306.01059.

Model	Modular form	\(L(f, 1)\)
\(X_{3,3}\) K-point	27.3.b.a	\(\Gamma(1/3)^6 / (8\sqrt{3}\,\pi^3)\)
\(X_{4,4}\) K-point	16.3.c.a	\(\Gamma(1/4)^4 \Gamma(1/2)^2 / (32\pi^3)\)
\(X_{4,2}\) C-point	32.4.a.b	\(\Gamma(1/4)^6 \Gamma(1/2)^3 / (16\sqrt{2}\,\pi^5)\)
\(X_{3,2,2}\) C-point	9.4.a.a	\(\Gamma(1/3)^9 / (32\sqrt{3}\,\pi^5)\)

One-modulus Calabi-Yau models

Contents

One-modulus Calabi-Yau models#

Outline#

Background#

One-modulus CY threefolds#

Singularity types and monodromy#

The LCS prepotential#

The K-point prepotential#

The C-point (conifold) prepotential#

L-function values and instanton normalisations#

Setup#

Discovering available models#

Loading a model via `HypergeometricModels.build`#

LCS point#

Example: quintic \(X_5\)#

LCS data for all hypergeometric models#

K-point#

Inspecting K-point boundary data#

Example: \(X_{3,3}\) at the K-point#

Example: \(X_{4,4}\) at the K-point#

The X66 special case#

C-point (conifold)#

Inspecting C-point boundary data#

Example: \(X_{4,2}\) at the C-point#

Example: \(X_{3,2,2}\) at the C-point#

Comparing LCS and boundary prepotentials for the same model#

End-to-end: Newton-solve a SUSY flux vacuum#

Using `make_prepot_*` directly#

Take-aways#

Further reading#

Type	Condition on \(\alpha_i\)	Monodromy \(T_\infty\)	Physical character
M (MUM)	\(\alpha_1 = \alpha_2 = \alpha_3 = \alpha_4\)	Maximally unipotent	Second LCS point
K	\(\alpha_1 = \alpha_2\), \(\alpha_3 = \alpha_4\), \(\alpha_2 \neq \alpha_3\)	Finite-order semisimple \(\times\) unipotent	K-point
C	\(\alpha_2 = \alpha_3\) only	Unipotent, rank 1	Conifold
F	All \(\alpha_i\) distinct	Generic finite order	F-point

Singularity	Models with data	Key parameters	Valid region
LCS	All 14 hypergeometric	\(\kappa, c_2, \chi\)	\(\text{Im}(t) \gg 1\)
K-point	X33, X44, X66*, 4.47, 3.7	\(\tau, \gamma, \delta, c, B_k\)	$
C-point	X42, X322, X62, 2.62	\(\tau, \gamma, \delta, k, A_k\)	$

One-modulus Calabi-Yau models

Contents

One-modulus Calabi-Yau models#

Outline#

Background#

One-modulus CY threefolds#

Singularity types and monodromy#

The LCS prepotential#

The K-point prepotential#

The C-point (conifold) prepotential#

L-function values and instanton normalisations#

Setup#

Discovering available models#

Loading a model via HypergeometricModels.build#

LCS point#

Example: quintic \(X_5\)#

LCS data for all hypergeometric models#

K-point#

Inspecting K-point boundary data#

Example: \(X_{3,3}\) at the K-point#

Example: \(X_{4,4}\) at the K-point#

The X66 special case#

C-point (conifold)#

Inspecting C-point boundary data#

Example: \(X_{4,2}\) at the C-point#

Example: \(X_{3,2,2}\) at the C-point#

Comparing LCS and boundary prepotentials for the same model#

End-to-end: Newton-solve a SUSY flux vacuum#

Using make_prepot_* directly#

Take-aways#

Further reading#

Loading a model via `HypergeometricModels.build`#

Using `make_prepot_*` directly#