# Does Type IIB string theory feature D(-1) and D9 branes?

I'm slightly confused as to how one deduces the D-brane content of a string theory by studying the p-form fields in the effective field theory.

In type IIB supergravity one has:

The Kalb-Ramond 2-form potential, which couples to 1-dimensional objects, namely just F-strings / NS1-branes, and the S-dual is a 6-form potential coupling to NS5-branes.

The additional p-form potentials $A_0$, $A_2$, $A_4$, which couple to D(-1), D1, D3 branes and dually to D7, D5, again D3 branes.

It seems pretty natural to me that the zero-form $A_0$ should couple to 0-manifolds in spacetime and so to instantonic D(-1)-branes. Also, there is no trace of the space-filling D9 in this counting.

Instead I've always heard about the brane content of type IIB as going D1, D3, D5, D7, D9.

My intuition behind this would be that type IIB has D(-1)-branes but people find it excessively formal to call them branes, and that there are independent ways of showing the D9 exists, even if it does not couple to a 10-form.

• Yes, it features both, and your intuition is correct. Jun 11, 2021 at 4:32

The best way to systematically determine the "fundamental" brane content is to classify the relevant cocycles on extended supersymmetry algebras that serve as the WZW-terms ($\kappa$-symmetry) terms for the relevant Green-Schwarz sigma-models for these branes.

For type IIB this was done in:

Makoto Sakaguchi, "IIB-Branes and New Spacetime Superalgebras", JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

see section "2. Chevalley-Eilenberg Cohomology Classification of IIB-branes".

There the author reports finding non-trivial cocycles for D1, D3, D5, D7 and D9-branes (cocycle equations 2.17-2.21).

By the way, the analogous analysis for type IIA is in section 6.1 of

C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, "The geometry of branes and extended superspaces", Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137).