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Admissible nonperturbative sectors contributing to a moduli superpotential

Physical scope

Conditional semiclassical Euclidean-brane or four-dimensional gauge-dynamics contributions to a specified compactification superpotential; zero modes, orientifold projection, flux lifting, anomalies, global axion period, determinants, gauge group, and thresholds are model data, not universal consequences.

Definition

For a specified four-dimensional $N = 1$ compactification effective field theory and a holomorphic field patch with dimensionless coordinates $(Z^{1}, \dots, Z^{n}, T)$ , where $T$ is a retained complexified cycle-volume coordinate and the $Z^{I}$ are the remaining retained coordinates, a semiclassical Euclidean-brane contribution is a candidate superpotential term $A (Z, T) e^{- S_{i n s t} (Z, T)}$ only when the instanton measure leaves exactly the two universal fermion zero modes integrated by $d^{2} θ$ ; every additional neutral or charged zero mode is absent or explicitly lifted or saturated; $A$ is nonzero and holomorphic on the patch with $[A] = {m a s s}^{3}$ ; $S_{i n s t}$ is dimensionless, with its volume and axion dependence retained in $(Z, T)$ ; $S_{i n s t}$ and the prefactor obey the gauged symmetries; and the axion charge is compatible with the separately established global period. The special form $A (Z) e^{- a T}$ requires a separately established action $S_{i n s t} = a T$ with dimensionless $a$ and a prefactor independent of $T$ ; the general term above makes neither restriction. A D7 gaugino-condensation term $A (Z) e^{- a T}$ additionally requires the actual low-energy gauge group, matter thresholds, and gauge-kinetic normalization to establish the relevant pure or mass-deformed $N = 1$ dynamics and dimensionless coefficient $a$ , fixed using that convention and the dual Coxeter number. For this displayed D7 form the holomorphic gauge coupling and threshold matching must also put all remaining $T$ dependence into $e^{- a T}$ . These are sector-specific admissibility conditions, not a universal instanton theorem or a claim that rigidity alone suffices.

Definition references

Witten, §2

Görlich, Kachru, Tripathy, and Trivedi, §§1, 5.2, and 6

Baumann and McAllister, §§6.4–6.5

Facts & Assumptions

Given: A fixed compactification, a specified four-dimensional $N = 1$ EFT patch, declared gauge symmetries and axion normalization, and one of the stated semiclassical or low-energy gauge sectors.

[F1]

Compactification-derived four-dimensional moduli EFT data supplies the declared chiral field chart, superpotential data, gauge functions, gauging, and approximation domain; the O3/O7 specialization used on that item retains its declared local-coordinate, kinetic-action, and approximation premises. The specified gauging and Hamiltonian lift are EFT input data, not an instanton existence result.

[A1]

The local instanton measure is modeled by Berezin integration over fermionic collective coordinates, with $\int d η 1 = 0$ and $\int d η η = 1$ . The two universal modes are the superspace coordinates integrated by $d^{2} θ$ .

[A2]

The Euclidean action $S_{i n s t} (Z, T)$ and determinant prefactor $A (Z, T)$ are holomorphic on the full declared patch, $A$ is nonzero there, and the total term has the declared gauge transformation and quantized axion charge.

[A3]

For a D7 condensation contribution, the actual low-energy four-dimensional gauge theory and all matter thresholds have been checked in the selected compactification; no general theorem about arbitrary divisors is assumed.

Proof

Given: The EFT patch [F1], the semiclassical measure convention [A1], the holomorphy and symmetry inputs [A2], and, for the D7 case, the model-specific gauge data [A3].

1.1givenA1algebra

Let $η^{1}, η^{2}$ be the two universal fermionic collective coordinates. Their Berezin integral selects precisely the coefficient of $η^{1} η^{2}$ , the superspace $d^{2} θ$ contribution. If another independent fermionic zero mode $ξ$ is unsaturated, the integral factors by $\int d ξ 1 = 0$ , so that configuration contributes no superpotential term under the stated measure. A lifting or insertion can change this conclusion only when its coupling is explicitly included in the measure.

2.1givenF1A2step 1.1algebra

Since the exponential of a holomorphic function is holomorphic and the product of holomorphic functions is holomorphic, $A (Z, T) e^{- S_{i n s t} (Z, T)}$ is holomorphic on the patch. It is an admissible contribution only when it has the required superpotential transformation under the supplied gauge and Kähler-line lift and its axion charge belongs to the independently supplied charge lattice; neither property follows from holomorphy or from local zero-mode counting.

3.1givenF1A2A3step 2.1∎

In the D7 case, [A3] supplies an actual low-energy gauge sector and threshold matching, so a proposed condensate term may be included only with its gauge-kinetic convention and group-dependent coefficient fixed from that sector. These inputs are model-specific and are not consequences of the geometry or zero-mode assumptions in [F1]–[A2]. Therefore this definition does not infer condensation from divisor rigidity or assert a universal mechanism.

Depends on

Compactification-derived four-dimensional moduli EFT data