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$N = 1$ Constant-Superpotential Potential on a Flat Chiral Patch

Physical scope

One local neutral chiral scalar and an uncharged spectator U(1) in the adopted two-derivative N=1 Einstein-frame model, with no auxiliary-independent scalar-potential term; the example evaluates the scalar potential and makes no string compactification or vacuum-realization claim.

Example

Take one complex chiral coordinate $z \in C$ of mass dimension 1, $M_{P} > 0$ , the trivial Kähler line bundle, Kähler potential $K (z, \overset{ˉ}{z}) = z \overset{ˉ}{z}$ , and constant nonzero superpotential $W (z) = w_{0}$ of mass dimension 3. Couple one spectator $U (1)$ vector with $f = 1$ , zero Killing vector, and zero moment map $P = 0$ . This is local one-field $N = 1$ data in the Kähler-Hodge patch convention; the vector is uncharged and contributes no D-term. Take the scalar potential to be generated solely by eliminating the displayed auxiliary sector, with no additional auxiliary-independent term, and evaluate it using the F/D formula stated below.

Example references

Louis, §9.1, equation (9.2)

Hohl, §8.5.5, equations (8.223)–(8.230)

Four-Dimensional $N = 1$ Kähler–Hodge Line Data

Four-Dimensional $N = 1$ Matter-Coupled Supergravity

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential

Facts & Assumptions

Given: $z \in C$ , $K = z \overset{ˉ}{z}$ , $W = w_{0} \neq 0$ , $M_{P} > 0$ , the trivial line-bundle patch, and the uncharged vector data $f = 1$ , $k = 0$ , $P = 0$ . In this example no auxiliary-independent scalar-potential term is added.

[F1]

Four-Dimensional $N = 1$ Kähler–Hodge Line Data fixes $D_{z} W = \partial_{z} W + (K_{z} / M_{P}^{2}) W$ and the mass dimensions.

[F2]

Four-Dimensional $N = 1$ Matter-Coupled Supergravity adopts the old-minimal $N = 1$ component framework used in this example.

[F3]

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential gives the conditional F/D scalar-potential formula.

Verification

1.1givenF1F3algebra

Since $K_{z} = \overset{ˉ}{z}$ , $\partial_{z} W = 0$ , and $W = w_{0}$ , [F1] gives $D_{z} W = \overset{ˉ}{z} w_{0} / M_{P}^{2}$ . Also $K_{z \overset{ˉ}{z}} = 1$ , so $K^{z \overset{ˉ}{z}} = 1$ ; the metric is positive and the inverse used in [F3] exists.

2.1givenF1F2F3step 1.1algebra∎

Substitution into [F3] gives $V = e^{∣ z ∣^{2} / M_{P}^{2}} (\frac{∣ z ∣^{2} ∣ w_{0} ∣^{2}}{M_{P}^{4}} - \frac{3 ∣ w_{0} ∣^{2}}{M_{P}^{2}}) = e^{∣ z ∣^{2} / M_{P}^{2}} \frac{∣ w_{0} ∣^{2}}{M_{P}^{2}} (\frac{∣ z ∣^{2}}{M_{P}^{2}} - 3)$ , because $P = 0$ removes the D-term. At $z = 0$ , this yields $V (0) = - 3 ∣ w_{0} ∣^{2} / M_{P}^{2}$ . The result has mass dimension four and is a value of the adopted model potential, not a claim about a realized string vacuum.

Remarks

This calculation uses the adopted $N = 1$ framework and scalar-potential carriers on the A page, with the four-dimensional Einstein-frame and Planck normalization of Supergravity Conventions And Normalizations. It remains a conditional example in that low-energy model.

$N = 1$ Constant-Superpotential Potential on a Flat Chiral Patch

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

N=1N=1 Constant-Superpotential Potential on a Flat Chiral Patch

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

$N = 1$ Constant-Superpotential Potential on a Flat Chiral Patch