Nonperturbative Prepotential in Supersymmetry

• Nonperturbative prepotential is a central object in supersymmetric gauge theory that encodes the exact low-energy dynamics, including all instanton contributions.
• It is derived using methods such as geometric engineering and matrix models, where period integrals relate to the genus-zero free energy.
• The computation involving glueball superfields and flux-induced superpotentials is essential for moduli stabilization and controlled supersymmetry breaking.

A nonperturbative prepotential is a central object in supersymmetric gauge theory, string theory, and quantum integrable systems that encodes the exact low-energy dynamics of moduli spaces—often including all instanton contributions and corrections beyond perturbation theory. Its structure is deeply connected to special Kähler geometry, matrix models, flux compactifications, modularity, and dualities in physics and mathematics. Recent advances have allowed for explicit computation and interpretation of nonperturbative prepotentials using diverse methodologies such as geometric engineering, matrix models, conformal field theory, and lattice gauge theory.

1. Geometric Engineering and Special Geometry

Nonperturbative prepotentials can be geometrically engineered in string theory by compactifying type IIB strings on local Calabi–Yau threefolds. The moduli space of interest typically emerges from the family of complex structures of the Calabi–Yau, described by a Riemann surface (e.g., a hyperelliptic curve) of the form

$y^2 = P_n(x)^2 - f_{n-1}(x)$

where the degree-$n$ polynomial $P_n(x)$ corresponds to non-dynamical parameters (e.g., positions of cuts or "Coulomb branch" data) and $f_{n-1}(x)$ encodes the normalizable, dynamical moduli. The physical "special coordinates" $S^i$ (often glueball superfields) are defined by period integrals: $$S^i = \frac{1}{2\pi i} \oint_{A^i} \lambda, \quad \text{with} \quad \lambda = x\,dy$$ Once these periods are known, the prepotential $\mathcal{F}_0(S)$ is defined such that

$$\Pi^i = \frac{1}{2\pi i} \oint_{B^i} \lambda = \frac{\partial \mathcal{F}_0}{\partial S^i}$$

This structure endows the moduli space with a special Kähler metric: $$g_{ij} = \operatorname{Im} \left( \frac{\partial^2 \mathcal{F}_0}{\partial S^i \partial S^j} \right)$$ The nonperturbative nature of the prepotential enters via the full, resummed series in the glueball fields, incorporating instanton contributions, as opposed to being just a one-loop or perturbative object (0804.4006).

2. Matrix Model Formulation and the CIV--DV Prepotential

A powerful computational approach equates the computation of the prepotential to evaluating the large $N$ limit (planar free energy) of a matrix model. The Dijkgraaf–Vafa (DV) prescription maps the prepotential to the genus-zero free energy $\mathcal{F}_0$ of a matrix model with partition function

$$Z = \int d^{N^2}\Phi\, \exp\left[-\frac{1}{g_s} W(\Phi)\right]$$

where $W(\Phi)$ is a polynomial potential. The "filling fractions" $S^i = g_s N^i$ correspond to the special coordinates above. Expanding $W(x)$ near its critical points and summing Feynman diagrams yields an explicit series for $\mathcal{F}_0(S)$, for example (to quintic order in $S$ as achieved in (0804.4006)). The expansion includes both quadratic and higher-order (nonperturbative) terms: $$\mathcal{F}_0(S) = \frac{1}{2} S^i \tau_{ij} S^j + \sum_{ijk} a_{ijk} S^i S^j S^k + \dots$$ This expansion matches the physically required instanton sum and correctly incorporates nonperturbative effects (0804.4006, Morozov et al., 2010).

3. Flux-Induced Superpotentials and Moduli Stabilization

The type IIB string context allows for moduli stabilization and supersymmetry breaking via 3-form fluxes. Turning on fluxes $G_3 = F_3 - \tau H_3$ supported noncompactly (“at infinity”) yields an effective superpotential (Gukov–Vafa–Witten type): $W_{\text{flux}} = \int G_3 \wedge \Omega$ which, in the local Calabi–Yau, induces (after regularization) a superpotential for the normalizable moduli: $W_{\text{eff}}(S) = \sum_m T^m \Sigma_{m+1}(S)$ with flux parameters $T^m$ and

$$\Sigma_m(S) = \frac{1}{2\pi i m}\oint_{\infty} x^{m-1} \lambda$$

This superpotential perturbs the original $\mathcal{N}=2$ theory, allowing the construction of metastable, nonsupersymmetric vacua upon suitable tuning of the fluxes, as dictated by the Ooguri--Ookouchi--Park (OOP) mechanism (0804.4006).

4. Structure of the Expansion in Glueball Fields

The nonperturbative prepotential is organized as an expansion in the glueball superfields: $$\mathcal{F}_0(S) = \sum_i a_i (S^i)^2 + \sum_{ijk} a_{ijk} S^i S^j S^k + \dots$$ This structure underpins the complete special geometry of the moduli space, including all higher-order quantum corrections, which are directly associated with nonperturbative effects such as instanton sums and nontrivial vacuum dynamics. The leading and subleading terms are critical for accurately computing the moduli space metric and understanding supersymmetry breaking mechanisms (0804.4006).

5. Application of the OOGURI–OOKOUCHI–PARK (OOP) Formalism

When a local superpotential perturbs an $\mathcal{N}=2$ theory, the OOP formalism provides the conditions for the existence of metastable, non-supersymmetric vacua. By expressing the effective superpotential in Kähler normal coordinates around a reference configuration $S^{(0)}$ and tuning the flux parameters so that

$W_{\text{eff}} \sim k_i z^i$

with $z^i$ linearized coordinates, the resulting scalar potential

$$V = g^{ij} (\partial_i W_{\text{eff}})(\partial_j \bar{W}_{\text{eff}})$$

can be engineered to produce a metastable minimum at $S^{(0)}$, stabilized by the nonperturbative corrections encoded in $\mathcal{F}_0(S)$ and the flux-induced $T^m$ parameters. The careful matching of the expansion with the OOP form produces full control over the local dynamics (0804.4006).

6. Embedding in Global Compactifications and Physical Implications

Although the analysis is often performed in a local (“noncompact”) Calabi–Yau setup with fluxes at infinity, it can be embedded in a larger compact Calabi–Yau via a factorization limit. In this setting, flux supported on distant 3-cycles leaks into the local region, acting as if it were noncompact flux at infinity. This construction provides a microscopic string origin for the noncompact flux picture and ensures that the nonperturbative prepotential calculated in the local geometry is physically meaningful for the effective theory arising from global compactifications (0804.4006).

7. Significance, Computation, and Applications

The nonperturbative prepotential computed via the combination of matrix model, period-integral, and geometric engineering methods is essential for:

• Determining the full special Kähler geometry and metric of the moduli space.
• Computing effective superpotentials and moduli stabilization potentials from fluxes.
• Engineering models with controlled supersymmetry breaking (via, e.g., the OOP mechanism).
• Embedding constructions into full string compactifications and studying their global consistency.
• Enabling explicit calculations of higher instanton effects in expanded cases (up to $S^5$ for arbitrary numbers of cuts in multi-cut geometries).

This nonperturbative control is indispensable for both phenomenological applications and the investigation of string dualities, non-supersymmetric vacuum structure, and the precise interplay of algebraic geometry, quantum field theory, and string theory (0804.4006, Morozov et al., 2010).

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Key references: (0804.4006, Morozov et al., 2010)