nLab anti D-brane

Idea

An anti-D-brane is the higher dimensional analog for D-branes of what antiparticles are for fundamental particles.

In perturbative string theory the strings stretching between a D-brane and an anti-D-brane have a tachyon mode. The analog of Sen's conjecture for this case is the statement that the tachyon potential energy is precisely the energy density of the brane and that the condensation of the tachyon mode witnesses the annihiliation of the brane/anti-brane pair. (Sen 98)

Relation to K-theory

In general there are $n$ D-branes and $n'$ anti D-branes coinciding, carrying Chan-Paton gauge fields $V_{brane}$ (of rank $n$) and $V_{\text{anti-brane}}$, respectively, yielding a pair of vector bundles

$$(V_{\text{brane}}, V_{\text{anti-brane}}) \,.$$

Such pairs are also called virtual vector bundles.

Now branes annihilate with anti-branes if they have exact opposite D-brane charge, which here means that they carry the same Chan-Paton vector bundle. In other words, pairs as above of the special form $(W,W)$ are equivalent to pairs of the form $(0,0)$.

$$(W,W) \sim 0 \,.$$

More generally, since there is arbitrary brane/anti-brane pair creation/annihilation, the actual net Chan-Paton charge of coincident branes and anti-branes is the equivalence class of $(V_{\text{brane}}, V_{\text{anti-brane}})$ under the equivalence relation which is generated by the relation

$$(V_{\text{brane}} \oplus W, V_{\text{anti-brane}} \oplus W) \;\sim\; (V_{brane}, V_{anti-brane})$$

for all complex vector bundles $W$ (Witten 98, Section 3).

For a fixed brane worldvolume $X$, the additive abelian group of such equivalence classes of virtual vector bundles is called the topological K-theory of $X$, denoted $K(X)$.

This is one of the arguments which suggest that the true home of the gauge field on multiple D-branes is in generalized cohomology theory called topological K-theory. It follows that also the RR-fields are in K-theory (Moore-Witten 00).

References

Anti D-branes

The version of Sen's conjecture for brane/anti-brane annihilation is due to

• Ashoke Sen, Tachyon Condensation on the Brane Antibrane System, JHEP 9808:012,1998 (arXiv:hep-th/9805170)

Textbook accounts on anti-D-branes include

• Koji Hashimoto, section 5.3.1 of D-brane, Springer 2012

The relation between brane/anti-brane annihilation and the topological K-theory nature of D-brane charge is due to

• Edward Witten, section 3 of D-Branes And K-Theory, JHEP 9812:019,1998 (arXiv:hep-th/9810188)

and the argument that this implies that also the RR-fields are in K-theory is due to

• Gregory Moore, Edward Witten, p. 6 of Self-Duality, Ramond-Ramond Fields, and K-Theory, JHEP 0005:032 (2000) (arXiv:hep-th/9912279)

Review of this is in

• Edward Witten, Overview Of K-Theory Applied To Strings, Int.J.Mod.Phys.A16:693-706,2001 (arXiv:hep-th/0007175)

Anti M-branes

Similarly for lifts to M-branes:

anti-M2-branes:

• Mohammad Garousi, A proposal for M2-brane-anti-M2-brane action, Phys. Lett.B686:59-63, 2010 (arXiv:0809.0381)

…

anti-M5-branes:

• Seiji Terashima, footnote 2 on section 4 of On M5-branes in N=6 Membrane Action, JHEP0808:080,2008 (arXiv:0807.0197)