Perturbatively Flat Vacua

Perturbatively flat vacua (PFVs) are a mechanism for achieving exponentially small values of the flux superpotential \(|W_0| \ll 1\), as required by the KKLT scenario (see Moduli Stabilisation). The idea, introduced in [1903.00596], is to choose quantized fluxes such that all perturbative contributions to \(W_{\text{flux}}\) vanish exactly, leaving only exponentially suppressed worldsheet instanton corrections. This note follows the review in [2512.17095], sections 6.2 and 6.4.

The PFV mechanism

We split the period vector into polynomial and exponential contributions,

\[ \vec{\Pi} = \vec{\Pi}_{\text{poly}} + \vec{\Pi}_{\text{exp}}\;, \]

so that the flux superpotential decomposes as

(1)\[ W_{\text{flux}} = W_{\text{poly}} + W_{\text{exp}}\;, \]

where

\[ W_{\text{poly}} = \vec{\Pi}_{\text{poly}}^T \cdot \Sigma \cdot (\vec{f} - \tau \vec{h})\;, \quad W_{\text{exp}} = \vec{\Pi}_{\text{exp}}^T \cdot \Sigma \cdot (\vec{f} - \tau \vec{h})\;. \]

The goal is to engineer flux choices for which \(\langle W_{\text{poly}} \rangle = 0\) in the vacuum, so that \(|W_0| \sim |\langle W_{\text{exp}} \rangle| \ll 1\).

Flux ansatz and Diophantine conditions

Working with the LCS prepotential (12), we make the flux ansatz

(2)\[ \vec{f} = (R_0, R_a, 0, M^a)^\top\;, \quad \vec{h} = (0, K_a, 0, 0^a)^\top\;, \]

with all entries integer-valued and \(a = 1, \ldots, h^{2,1}\). The polynomial superpotential becomes

(3)\[ W_{\text{poly}}(z^a, \tau) = \frac{1}{2} M^a \widetilde{\kappa}_{abc} z^b z^c - \tau K_a z^a + (R_a - \tilde{a}_{ab} M^b) z^a + \left(R_0 - \frac{M^a \tilde{c}_a}{24}\right)\;, \]

while the exponential part is

(4)\[ W_{\text{exp}}(z^a) = -\frac{1}{(2\pi)^2} \sum_{\tilde{\mathbf{q}} \in \mathcal{M}_{\widetilde{X}}} \mathscr{N}_{\tilde{\mathbf{q}}} \, \tilde{\mathbf{q}}_a M^a \, \text{Li}_2\!\left(e^{2\pi i \, \tilde{\mathbf{q}}_a z^a}\right)\;. \]

To cancel the constant and linear terms in (3), we impose the Diophantine conditions

(5)\[ R_a = \tilde{a}_{ab} M^b \in \mathbb{Z}\;, \quad R_0 = \frac{M^a \tilde{c}_a}{24} \in \mathbb{Z}\;. \]

These can only be satisfied for particular choices of \(M^a\) for which the right-hand sides take integer values.

Flat direction and PFV conditions

With (5) imposed, the polynomial superpotential simplifies to

\[ W_{\text{poly}}(z^a, \tau) = \frac{1}{2} N_{ab} z^a z^b - \tau K_a z^a\;, \]

where \(N_{ab} \coloneqq \kappa_{abc} M^c\). Assuming \(N_{ab}\) is invertible, \(\partial_{z^a} W_{\text{poly}} = 0\) is solved along the one-dimensional locus

(6)\[ z^a = p^a \tau\;, \quad p^a \coloneqq N^{ab} K_b\;. \]

One further requires \(K_a p^a = K_a N^{ab} K_b = 0\), which ensures \(W_{\text{poly}}\) and all its derivatives vanish along (6).

In summary, the PFV conditions are:

(7)\[ \det N \neq 0\;, \quad \vec{p} \in \mathcal{K}_{\widetilde{X}}\;, \quad K_a p^a = 0\;, \quad \tilde{a}_{ab} M^b \in \mathbb{Z}\;, \quad \tilde{c}_a M^a \in 24\mathbb{Z}\;. \]

Racetrack stabilisation

Along the flat direction \(z^a = p^a \tau\), the flux superpotential reduces to \(W_{\text{exp}}\) in (4), which takes the form

(8)\[ W_{\text{eff}}(\tau) = c \left(e^{2\pi i p^1 \tau} + A \, e^{2\pi i p^2 \tau}\right) + \ldots\;, \]

where \(c\) and \(A\) depend on \(\vec{M}\) and \(\vec{K}\) but not on \(\tau\). When \(|p^1 - p^2| \ll p^2\), the two exponential terms compete — this is the racetrack mechanism [hep-th/0404257].

Solving \(\partial_\tau W_{\text{eff}} = 0\) gives

\[ \langle \tau \rangle = \frac{1}{2\pi i} \frac{1}{p^1 - p^2} \ln\!\left(-A \frac{p^2}{p^1}\right)\;, \]

and the stabilised superpotential is

\[ W_{\text{eff}}(\langle \tau \rangle) = c \, \frac{p^2 - p^1}{p^2} \left(-A \frac{p^2}{p^1}\right)^{\frac{p^1}{p^1 - p^2}}\;, \]

which is small precisely when \(|p^1 - p^2| \ll p^2\).

Example: degree-18 hypersurface

For the degree-18 hypersurface in \(\mathbb{CP}_{[1,1,1,6,9]}\) [hep-th/9309013] with \(h^{2,1} = 2\) (on the \(\mathbb{Z}_6 \times \mathbb{Z}_{18}\)-invariant locus) and \(Q_{\text{D3}} = 138\), one can choose

\[\begin{split} \vec{M} = \begin{pmatrix} -16 \\ 50 \end{pmatrix}\;, \quad \vec{K} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\;, \end{split}\]

yielding \(Q_{\text{flux}} = 124\) and a PFV racetrack minimum at

\[ \langle \tau \rangle = 6.856\, i\;, \quad \langle z^1 \rangle = 2.742\, i\;, \quad \langle z^2 \rangle = 2.057\, i\;, \]

with \(|W_0| = 2.037 \times 10^{-8}\).

Conifold PFVs

The PFV mechanism can be extended to stabilise moduli near a conifold singularity, engineering a warped Klebanov-Strassler throat in a compact flux compactification [2004.10740].

Analytic continuation near the conifold

Conifold singularities arise at loci in \(\mathcal{M}_{\text{cs}}(X)\) where a collection of three-cycles shrink to zero volume. The conifold modulus \(z_{\text{cf}} = \tilde{\mathbf{q}}_{\text{cf}} \cdot \mathbf{z}\) controls the size of the shrinking cycle.

Near \(z_{\text{cf}} \to 0\), the instanton prepotential must be analytically continued. For a nilpotent conifold class with GV invariant \(n_{\text{cf}} = \mathscr{N}_{\tilde{\mathbf{q}}_{\text{cf}}}\), one finds

(9)\[ \mathcal{F}(z_{\text{cf}}, z^\alpha) = n_{\text{cf}} \frac{z_{\text{cf}}^2}{4\pi i} \ln(-2\pi i \, z_{\text{cf}}) + \sum_{n=0}^{\infty} \frac{\mathcal{F}^{(n)}(z^\alpha)}{n!} z_{\text{cf}}^n\;, \]

where the logarithmic term encodes the characteristic conifold monodromy and the coefficients \(\mathcal{F}^{(n)}(z^\alpha)\) depend on the polynomial prepotential and the remaining (bulk) instanton corrections evaluated at \(z_{\text{cf}} = 0\).

Bulk and conifold superpotential

With quantized fluxes \(\vec{f} = (P_0, P_a, 0, M^a)^\top\) and \(\vec{h} = (0, K_a, 0, 0^a)^\top\), the superpotential expands at leading order as

(10)\[ W(z^\alpha, z_{\text{cf}}, \tau) = W_{\text{bulk}}(z^\alpha, \tau) + z_{\text{cf}} \, W^{(1)}(z^\alpha, z_{\text{cf}}, \tau) + \mathcal{O}(z_{\text{cf}}^2)\;. \]

The bulk superpotential \(W_{\text{bulk}}\) takes the same form as (3) but with a shifted constant term \(\tilde{c}'_a = \tilde{c}_a + n_{\text{cf}} \delta_{a,1}\). The linear coefficient is

(11)\[ W^{(1)} = -M \frac{n_{\text{cf}}}{2\pi i} \left(\ln(-2\pi i \, z_{\text{cf}}) - 1\right) - \tau K + \widetilde{\kappa}_{1a\gamma} M^a z^\gamma + \ldots\;, \]

where \(M \coloneqq M^1\) and \(K \coloneqq K_1\).

Exponential hierarchy from the conifold

The \(F\)-flatness condition for \(z_{\text{cf}}\) is satisfied at

(12)\[ \langle |z_{\text{cf}}| \rangle = \frac{1}{2\pi} \exp\!\left(-\frac{2\pi K'}{(g_s M) \, n_{\text{cf}}}\right)\;, \]

where

(13)\[ K' = K - g_s \widetilde{\kappa}_{1a\beta} M^a \text{Im}(z^\beta)\;. \]

Provided \(K'/M > 0\), the conifold modulus is stabilised at an exponentially small value, giving rise to a warped throat region. The D3-brane charge stored in the throat is

\[ Q_{\text{flux}}^{\text{throat}} = K' M > 0\;. \]

Importantly, \(K'\) differs from the naive product \(K \cdot M\) by corrections from the bulk moduli (13) — a crucial effect absent in the non-compact Klebanov-Strassler solution.

The bulk moduli are stabilised via the conifold PFV conditions:

(14)\[ \det N \neq 0\;, \quad \vec{p} \in \mathcal{K}_{\text{cf}}\;, \quad K_\alpha p^\alpha = 0\;, \quad \tilde{a}_{\alpha b} M^b \in \mathbb{Z}\;, \quad \tilde{c}'_a M^a \in 24\mathbb{Z}\;, \]

with \(N_{\alpha\beta} = M^a \kappa_{a\alpha\beta}\) and \(p^\alpha = N^{\alpha\beta} K_\beta\) running over the bulk indices \(\alpha\) only. Along \(z^\alpha = p^\alpha \tau\), the remaining flat direction is lifted by the bulk racetrack superpotential

(15)\[ W_{\text{bulk}}^{\text{eff}}(\tau) = -\frac{1}{(2\pi)^2} \sum_{\tilde{\mathbf{q}} \neq \tilde{\mathbf{q}}_{\text{cf}}} \mathscr{N}_{\tilde{\mathbf{q}}} \, \tilde{\mathbf{q}}_a M^a \, \text{Li}_2\!\left(e^{2\pi i \, \tilde{\mathbf{q}}_\alpha p^\alpha \tau}\right)\;. \]

Implementation in jaxvacua

bounded_fluxes.enumerate_fluxes([n_sample, ...])	Main flux enumeration algorithm (Algorithm 1 of arXiv:2501.03984).
bounded_fluxes.sample_bounded_fluxes([...])	Stochastic flux search guided by the eigenvalue bounds of arXiv:2501.03984.

The conifold period computation is handled through the jaxvacua.conifold subpackage and the freezer module jaxvacua.freezer, which implements the light-field EFT for conifold vacua.