### Calibrations with gauge field

#### Posted by Robert H.

Assume some closed string background preserving some susy. This comes with at least one Killing spinor whose square can be expanded into harmonic form fields ($J$, $\Omega $, $\varphi $ being prominent examples in the Kähler, Calabi-Yau and G2 cases).

A brane in that background geometry preserves (part of) the super-symmetry if it is calibrated by those forms, that is if that form pulled back to the brane’s world volume is the volume form of that brane. This follows for example from the boundary conditions of the ${S}_{\alpha}$ fields on the string in the GS formulation.

However, on the brane there can be additional fields and I am specifically interested in the gauge field of D-branes. If the brane is a flat brane in a flat background, I know that the BPS condition is some sort of (generalized, non-linear) self-duality of the field-strength and this statement can be shown to be T-dual to the statement about calibrations.

My question now ist: What is known about the combined situation? What is the condition on a curved brane with gauge field in a curved background to preserve some susy?

Posted at April 21, 2005 12:43 PM UTC
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### Re: Calibrations with gauge field

I don’t know the answer to your question. But let me try to guess:

In as far as one can use T-duality to get back to the case of no gauge field, it seems that your question should be answerable in terms of the Buscher rules for T-duality in curved backgrounds.

No?

This follows for example from the boundary conditions of the S α fields on the string in the GS formulation.

I guess I could figure that out myself. But since I am busy with other stuff: Do you have a reference for this?

### Re: Calibrations with gauge field

Urs,

unfortunately, I am interested in the most general situation of a curved brane with a gauge field. Under T-duality (if at all possible) the curvature turns into more gauge field. So I cannot get to a situation without gauge field as far as I can see.

As far as the reference goes, I have to admit I don’t have a good reference where this was clearly stated for the first time. It can probably be traced back to hep-th/9604091 where Green and Gutperle write down the fermionic gluing matrix as the product of the $\gamma$-matrices in the Neumann directions which of course is nothing but the Clifford expression for the calibration form as a bispinor. However, curved branes are not explicitly considered in that paper.

### Re: Calibrations with gauge field

In hep-th/0501088’s equation (11) there is something that looks roughly like this: $M{\gamma}^{I}{M}^{t}={\gamma}^{J}{N}^{\mathrm{IJ}}$ where $M$ is the fermionic gluing matrix and $N=(1-F)/(1+F)$. But what I am looking for is a geometric formulation rather than a statement about gluing matrices.

## Re: Calibrations with gauge field

I don’t know the answer to your question. But let me try to guess:

In as far as one can use T-duality to get back to the case of no gauge field, it seems that your question should be answerable in terms of the Buscher rules for T-duality in curved backgrounds.

No?

I guess I could figure that out myself. But since I am busy with other stuff: Do you have a reference for this?