In this post, I shall analyze certain relations between holomorphic properties of Dp-brane actions and SO(2)-duality of type IIB supergravity. Specifically, I will show that the super D3-brane action:

in Type IIB SUGRA background satisfies the Gaillard-Zumino duality condition and exhibits exact self-duality. Dp-branes are p + 1 dimensional Ramond-Ramond charged dynamical hypersurfaces that open strings end on and admit perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions. Naturally, for 4-D spacetime physics, D3 branes are especially important for string-phenomenology due to mirror symmetry on Calabi-Yau 3-folds where they holomorphically wrap Fukaya super-Lagrangians. The D3-brane effective action in the NS5-brane geometry, given that it satisfies D3-brane self-duality and Poincaré invariance, is given by:

with

being the D3-brane tension, and , are the RR-4 and RR-2 exterior forms, and generally, the DBI action is:

and the D-brane WZ action is given by:

In order for the effective action to be integrable with fields in 2nd-quantized form, one must work under the Gaillard-Zumino Condition: that is – using 8-loop counterterms with superspace torsion:

where is the superfield torsion. One starts with a Lagrangian:

in D = 4, with a dependence on a gauge field strength , metric , and matter field . So, we now have:

and the Hodge dual components for the tensor are given by:

The Gaillard-Zumino condition is an infinitesimal duality transformation of and and fermionic transformation given by:

Now, the Lagrangian must transform as:

and one has an transformation given by , , and so the Lagrangian is given by:

and by D3-brane self-duality, it follows that:

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