nLab wrapped brane

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Idea

In physics (string theory), a (factor space of a) brane worldvolume ϕ:ΣX\phi \colon \Sigma \longrightarrow X is said to wrap a cycle cH dim(Σ)(X)c \in H_{\dim(\Sigma)}(X) in spacetime XX when the pushforward ϕ *[Σ]H (X)\phi_\ast [\Sigma] \in H_\bullet(X) of the fundamental class of the manifold (or orbifold) Σ\Sigma is the class, [c][c], of the given cycle in XX. If the pushforward is a multiple of [c][c], then the brane is said to wrap cc multiple times.

Here H H_\bullet is typically taken to be ordinary homology but may also be K-homology (cf. D-brane charge quantization in K-theory) and could in principle by any other generalized homology-theory thought to encode the flux quantization.

In mathematics (algebraic topology), a cycle represented by a manifold this way is said to be Steenrod representable.

Examples

Via the identification of D-brane charge in K-theory, the K-theoretic McKay correspondence formalizes how D-branes wrap the fundamental cycles in the blow-up resultion of an ADE-singularity (Gonzalez-Sprinberg & Verdier 83)

graphics grabbed from Sugimoto 16

References

Last revised on May 8, 2025 at 16:43:57. See the history of this page for a list of all contributions to it.