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Gauge Invariance of the Type IIB Tilded Field Strengths

Physical scope

Conditional local calculation for smooth potentials and gauge parameters on a source-free patch; no large-gauge, brane, or boundary action claim.

Statement

For the transformations in Type IIB Potential Gauge Transformations, with $d H_{3} = 0$ , the four field strengths $H_{3}, G_{1}, {\tilde{G}}_{3}, {\tilde{G}}_{5}$ defined by Type IIB Two Derivative Supergravity Pseudoaction are invariant.

Statement references

Type IIB Potential Gauge Transformations

Type IIB Two Derivative Supergravity Pseudoaction

Facts & Assumptions

Given: A smooth source-free patch; smooth potentials $B_{2}, C_{0}, C_{2}, C_{4}$ and smooth gauge parameters $ζ_{1}, Λ_{1}, Λ_{3}$ ; $d H_{3} = 0$ ; the transformations and field-strength definitions above.

[F1]

Type IIB Two Derivative Supergravity Pseudoaction: $H_{3} = d B_{2}$ , $G_{1} = d C_{0}$ , $G_{3} = d C_{2}$ , $G_{5} = d C_{4}$ , ${\tilde{G}}_{3} = G_{3} - C_{0} H_{3}$ , and ${\tilde{G}}_{5} = G_{5} - \frac{1}{2} H_{3} \land C_{2} + \frac{1}{2} G_{3} \land B_{2}$ .

[F2]

Type IIB Potential Gauge Transformations: $δ B_{2} = d ζ_{1}$ , $δ C_{0} = 0$ , $δ C_{2} = d Λ_{1}$ , and $δ C_{4} = d Λ_{3} - \frac{1}{2} H_{3} \land Λ_{1} + \frac{1}{2} G_{3} \land ζ_{1}$ .

[F3]

The exterior derivative squares to zero and The exterior derivative is a graded derivation: $d^{2} = 0$ and $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β$ .

[F4]

The wedge product is associative and graded commutative: exchanging forms of degrees $r, s$ contributes $(- 1)^{r s}$ .

Proof

1.1F1F2F3algebra

From [F1]–[F3], $H_{3}^{'} = d (B_{2} + d ζ_{1}) = H_{3} + d^{2} ζ_{1} = H_{3}$ and $G_{1}^{'} = d C_{0}^{'} = G_{1}$ . Also $G_{3}^{'} = d (C_{2} + d Λ_{1}) = G_{3}$ ; since $C_{0}^{'} = C_{0}$ and $H_{3}^{'} = H_{3}$ , it follows that ${\tilde{G}}_{3}^{'} = G_{3}^{'} - C_{0}^{'} H_{3}^{'} = {\tilde{G}}_{3}$ .

1.2F1F2F3algebra

Using $d G_{3} = d^{2} C_{2} = 0$ and $d H_{3} = 0$ , [F2]–[F3] give $G_{5}^{'} - G_{5} = d δ C_{4} = \frac{1}{2} H_{3} \land d Λ_{1} - \frac{1}{2} G_{3} \land d ζ_{1}$ . The term $d (d Λ_{3})$ vanishes; $d (H_{3} \land Λ_{1}) = - H_{3} \land d Λ_{1}$ and $d (G_{3} \land ζ_{1}) = - G_{3} \land d ζ_{1}$ by the odd degree of the three-forms.

2.1F1F2F4step 1.2algebra∎

Substituting these changes and [F2] into [F1], ${\tilde{G}}_{5}^{'} - {\tilde{G}}_{5} = (G_{5}^{'} - G_{5}) - \frac{1}{2} H_{3} \land d Λ_{1} + \frac{1}{2} G_{3} \land d ζ_{1} = 0$ by step 1.2. The degree-two and degree-three factors commute by [F4], so the cancellation preserves the displayed wedge order. Thus all four strengths are invariant. The proof uses only local exact gauge shifts and does not address large-gauge or boundary terms.

Depends on

Dependency tree · 2 levels

Sources

M. J. D. Hamilton, The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation, arXiv:1607.00327v3