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Dp Brane Dirac Charge Product

Physical scope

Arithmetic charge-unit identity for complementary electric/magnetic Type II Dp and D(6-p) branes in the stated source normalization; the Dirac lattice is a separate postulate, global RR flux quantization is not derived, and p=3 requires a self-dual-field treatment.

Statement

For each integer $p$ with $0 \leq p \leq 6$ , the source-normalized charge units satisfy the arithmetic identity $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}, 2 κ_{0}^{2} = (2 π)^{7} α^{' 4},$ and hence $2 κ_{0}^{2} μ_{p} μ_{6 - p} = 2 π$ . For $p \neq 3$ , the adopted RR Dirac lattice makes this the minimal nonzero pairing $n = 1$ for mutually electric and magnetic Dp and D $(6 - p)$ charges. At $p = 3$ this remains an arithmetic identity only: the adopted lattice postulate expressly excludes the self-dual Type IIB D3 case, whose physical charge interpretation needs a separate self-dual-field framework.

Statement References

Polchinski, Chaudhuri, and Johnson, Lecture IV §4.1

Facts & Assumptions

Given: The charge and gravitational-coupling normalizations for an integer $0 \leq p \leq 6$ ; when $p \neq 3$ , also assume a mutually electric/magnetic complementary RR pair and the adopted Dirac lattice.

[F1]

Type II Dp Brane DBI WZ Action: $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ is the Dp RR charge unit used in the adopted action. This arithmetic use depends only on the explicit charge normalization.

[F2]

Supergravity Conventions And Normalizations: $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ .

[A1]

RR Electric Magnetic Dirac Pairing Convention: for source-normalized complementary Dp and D $(6 - p)$ units with $0 \leq p \leq 6$ and $p \neq 3$ , $μ_{p} μ_{6 - p} / α_{p} = 2 π n$ , with $α_{p} = (2 κ_{0}^{2})^{- 1}$ and $n \in Z$ ; the self-dual $p = 3$ case is excluded from this adopted lattice.

Proof

1.1givenF1algebra

Apply [F1] to $p$ and $6 - p$ . Their product is $μ_{p} μ_{6 - p} = (2 π)^{- p - (6 - p)} α^{' - ((p + 1) + (7 - p)) / 2} = (2 π)^{- 6} α^{' - 4}$ because the two exponents add to $8$ .

2.1givenF2A1step 1.1algebra∎

Multiplying step 1.1 by [F2] gives $2 κ_{0}^{2} μ_{p} μ_{6 - p} = (2 π)^{7} α^{' 4} (2 π)^{- 6} α^{' - 4} = 2 π$ . If $p \neq 3$ , [A1] applies and this realizes $n = 1$ , the minimal nonzero integer pairing. If $p = 3$ , [A1] supplies no lattice premise, so only the arithmetic identity follows. The computation is dimensionless and does not derive a Dirac lattice or global RR flux quantization.

Dp Brane Dirac Charge Product

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources