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Dp Brane NS Two Form Gauge Invariance Of The Invariant Field

Physical scope

Local Abelian worldvolume NS two-form gauge shift on a smooth Dp probe; proves invariance of the DBI combination and the WZ exponential with RR potentials held fixed, not the full RR-potential gauge crosswalk.

Statement

Under the adopted shift $δ B_{2} = d ζ_{1}$ , $δ A = ι^{*} ζ_{1} / (2 π α^{'})$ on a single Abelian Dp-brane, $F = 2 π α^{'} F_{A} - ι^{*} B_{2}$ is invariant. Consequently the DBI determinant and the WZ exponential are unchanged when the RR potential polyform $C$ is held fixed. This does not establish invariance under the simultaneous R4 RR-potential gauge transformations.

Statement References

Polchinski, Chaudhuri, and Johnson, Lecture II §2.5

Facts & Assumptions

Given: A smooth Abelian connection $A$ , a smooth one-form gauge parameter $ζ_{1}$ on the target, and the adopted NS shift on a fixed smooth embedding.

[F1]

Type II Dp Brane DBI WZ Action: $F = 2 π α^{'} F_{A} - ι^{*} B_{2}$ , $F_{A} = d A$ , $δ B_{2} = d ζ_{1}$ , and $δ A = ι^{*} ζ_{1} / (2 π α^{'})$ . This use is restricted to the explicit NS clause with the RR polyform held fixed; it does not use the separate RR gauge-transformation clause.

[F2]

Dp Brane Worldvolume Fields: for a single brane, $A$ is an Abelian connection and its curvature is $F_{A} = d A$ .

[F3]

The exterior derivative commutes with pullback: for every smooth map $ι$ and form $ζ$ , $d (ι^{*} ζ) = ι^{*} (d ζ)$ .

Proof

1.1givenF1F2F3

By [F1]–[F3], $δ F_{A} = d (δ A) = d (ι^{*} ζ_{1}) / (2 π α^{'}) = ι^{*} (d ζ_{1}) / (2 π α^{'})$ .

2.1givenF1step 1.1algebra∎

Substituting step 1.1 and [F1] gives $δ F = 2 π α^{'} δ F_{A} - ι^{*} (δ B_{2}) = ι^{*} (d ζ_{1}) - ι^{*} (d ζ_{1}) = 0$ . Thus both the determinant input and $e^{F}$ are unchanged; holding $C$ fixed makes the displayed WZ integrand unchanged. The proof makes no claim about the separate RR transformation terms.

Dp Brane NS Two Form Gauge Invariance Of The Invariant Field

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources