Moduli Stabilisation

Moduli Stabilisation#

The moduli problem#

A generic Calabi-Yau compactification gives rise to massless scalar fields (moduli) in four dimensions: the complex structure moduli \(z^i\), the axio-dilaton \(\tau\), and the Kähler moduli \(T_\alpha\). These parametrize flat directions of the classical scalar potential and are phenomenologically unacceptable: massless scalars mediate long-range fifth forces, cause time-varying fundamental constants, and produce overabundant non-relativistic matter in the early universe (the cosmological moduli problem).

As described in Type IIB Three-Form Fluxes, three-form fluxes generate the no-scale scalar potential

\[ V_{\rm flux} = e^K K^{a\bar b} D_a W_{\rm flux}\, D_{\bar b}\overline{W}_{\rm flux} \geq 0 \]

that stabilises \(z^i\) and \(\tau\) at the ISD locus. However, the Kähler moduli \(T_\alpha\) (which control the volumes of four-cycles in \(X_3\)) remain flat directions at the classical level due to the no-scale structure. Their stabilisation requires quantum corrections — perturbative or non-perturbative.

KKLT scenario#

The KKLT scenario [hep-th/0301240] achieves complete moduli stabilisation in two steps, followed by an uplifting procedure.

Step 1: Flux stabilisation. Choose fluxes such that the no-scale minimum sits at

\[ W_0 = \langle W_{\rm flux} \rangle \ll 1\;. \]

This fixes \(z^i\) and \(\tau\), leaving the Kähler moduli as flat directions.

Step 2: Non-perturbative superpotential. Non-perturbative effects — from Euclidean D3-brane instantons wrapping four-cycles, or from gaugino condensation on D7-branes — generate a correction

\[ W_{\rm np} = \sum_\alpha A_\alpha\, e^{-a_\alpha T_\alpha}\;, \]

where \(a_\alpha = 2\pi\) for a D3-instanton and \(a_\alpha = 2\pi/N\) for \(SU(N)\) gaugino condensation. The total superpotential is

\[ W = W_0 + W_{\rm np}\;. \]

For a single Kähler modulus \(T = \tau_K + i\theta\) with Kähler potential \(K = -3\log(T + \bar T)\), the scalar potential develops an AdS minimum at

\[ a\tau_K^* \approx -\frac{3}{2}\left(1 + \frac{W_0}{A e^{-a\tau_K^*}}\right)\;. \]

The small \(|W_0|\) ensures \(\tau_K^*\) is in the perturbative regime.

Step 3: Uplifting to de Sitter. The AdS minimum can be lifted to a metastable de Sitter vacuum by introducing a positive energy contribution. The canonical mechanism uses an anti-D3 brane placed at the tip of a Klebanov-Strassler warped throat, contributing

\[ V_{\rm up} \sim \frac{\epsilon^4}{\mathcal{V}^2} \]

where \(\epsilon \ll 1\) is an exponential warping factor determined by the flux quantization in the throat.

Large Volume Scenario (LVS)#

The Large Volume Scenario [hep-th/0502058, hep-th/0505076] stabilises Kähler moduli using \(\alpha'\) corrections to the Kähler potential.

For a Calabi-Yau with at least two Kähler moduli — a large modulus \(\tau_b\) controlling the overall volume \(\mathcal{V} \sim \tau_b^{3/2}\) and a small modulus \(\tau_s\) — the \((\alpha')^3\)-corrected Kähler potential reads

\[ K = -2\log\!\left(\mathcal{V} + \frac{\hat\xi}{2}\right)\;, \]

where the correction coefficient is

\[ \hat\xi = -\frac{\chi(X_3)\,\zeta(3)}{2(2\pi)^3} \]

and is proportional to the Euler characteristic \(\chi(X_3)\).

Including the non-perturbative superpotential \(W = W_0 + A_s\, e^{-a_s T_s}\), the scalar potential takes the LVS form

\[ V_{\rm LVS} \simeq \lambda_s \frac{\sqrt{\tau_s}\,|W_0|^2\,e^{-2a_s\tau_s}}{\mathcal{V}} - \mu_s \frac{|W_0|^2\,a_s\tau_s\,e^{-a_s\tau_s}}{\mathcal{V}^2} + \nu \frac{|W_0|^2\,\hat\xi}{\mathcal{V}^3}\;, \]

where \(\lambda_s, \mu_s, \nu\) are calculable numerical coefficients. This potential has a non-supersymmetric AdS minimum at exponentially large volume

\[ \mathcal{V}_* \sim e^{a_s\tau_s^*}\;, \]

where \(\tau_s^*\) is determined by \(\hat\xi\) and \(A_s\). The parametrically large volume suppresses higher-order corrections, making LVS a controlled approximation.

de Sitter vacua#

Phenomenologically, a positive cosmological constant \(\Lambda > 0\) is required to match observations. Constructing de Sitter vacua in string theory remains an active area of research. The main uplifting mechanisms discussed in the literature include:

  • Anti-D3 branes in warped throats (original KKLT proposal)

  • D-term contributions from gauge fluxes on D7-branes

  • F-term uplifting using matter fields (ISS-type hidden sector)

  • Perturbatively flat vacua (PFVs) with specific \(\alpha'\) and string-loop corrections (see Perturbatively Flat Vacua)

For a comprehensive and pedagogical treatment of these mechanisms, including recently constructed explicit candidate de Sitter vacua, see the TASI lectures by McAllister and Schachner [arXiv:2512.17095] and references therein.

JAXVacua currently focuses on finding SUSY flux vacua (ISD solutions minimising \(V_{\rm flux}\)) and non-SUSY vacua (local minima of the full scalar potential). For implementations see jaxvacua.flux_eft and Non-supersymmetric flux vacua.