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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:

Now we are in a position to analyse the D3-brane action, and for simplicity but with no loss of generality, without scalar supergravity backgrounds. Let be a bosonic brane coordinatization in D = 10 flat target bulk space and its fermionic partner given by the Majorana-Weyl spinor index with N = 2 SUSY index . The D3 action for the brane coordinates and worldvolume gauge field must have Kappa symmetry and we require N = 2 SUSY. Hence:

where

with being the Pauli matrices and act on the N = 2 SUSY indices. The 1-form defined by:

and

By use of exterior differential forms on the bulk, with an RR pull-back 2-form and 4-form , we get:

and

where is given by:

with . To check whether the Gaillard-Zumino condition is met, we must calculate the first 2 terms of the condition:

so, is given by:

where I have made an explicit use of the determinant formula for the four-by-four matrix:

and by Hodge duality, can be derived as:

and by the GKP-Witten relation for the D3-brane action:

one gets

and by conjugation, one derives the essential identity:

A few remarks are in order now on the bosonic truncation of the D3-brane action. Note in the above equation, the right-hand-side vanishes completely, and so the Lagrangian transforms accordingly as:

The supersymmetry situation under the Gaillard-Zumino condition can now be considered for the matter field contributions. For a D3-brane, the matter fields transform as:

and hence we get:

while noting that the Majorana-Weyl fermions  and transform as group doublet and the gauge variation of the total Lagrangian with respect to matter fields transforms as:

where is the Hodge dual of , with:

and the Poincaré invariance of D3-branes transfers to and induces a relation on the differential forms:

Combining the last 3 equations, we get:

And so the D3-brane self-duality and Poincaré invariance are satisfied by the Gaillard-Zumino condition. To second-quantize the D3-brane action, one must lift the duality by an introduction of a dilaton and axion living on constant background fields on the bulk space. The redefined Lagrangian, via D3-brane 4-dimensional worldvolume topological tori-throating becomes:

and by such tori redefinition, we obtain:

with:

Now, the super D3-brane action in type IIB SUGRA background with varying dilaton and axion has an SL(2, Z) self-duality, which, given the general Dp-action:

where the Dirac-Born-Infeld action is given by:

and the Wess-Zumino action is given by:

with the DBI and WZ terms:

necessitates that we introduce the Lagrangian multiplier term:

In order to derive the self-duality of the super D3-brane action in a type IIB supergravity background, where we have:

we substitute the self-dual vector potential  in:

thus allowing us to derive the action:

where we have:

and

with

Now we solve the Euler-Lagrange equations for  and plug the solution in the action .

In the Lorentz frame with:

and  block-diagonal:

thus allowing us to derive the action:

in the local Lorentz frame as:

Solutions to the EM for  and  give us:

and

with:

Plugging them in the local Lorentz frame action above, we get the dual action:

with

Now in light of the following dilaton, axion, and form-potentials:

and

our dual action is given by:

where and are stand-ins, respectively, for:

and

By dualization, then under the transformations above of the dilaton, axion, and form-potentials, as well as:

we get an equivalence between:

and

To show that the super D3-brane action satisfies an self-duality in a Type IIB SUGRA background, one introduces the axio-dilatonic variable:

Then the following transformations:

are expressible as:

By holomorphicity and the modularity of the axio-dilaton, we get an  doublet:

that transforms according to a D7/M2 monodromy. To derive the axion-shift:

corresponding to the  member:

one shows that in the super D-string action the super D3-brane action:

where the Dirac-Born-Infeld part is given by:

and the Wess-Zumino one by:

is invariant under the following transformations:

and

It is clear now that the super D3-brane action:

and the type-IIB Lagrangian multiplier term:

and the constraints they impose, are invariant under the first set of transformations and up to a topological term, under the second set.

We define now the dual field strength  by the Hodge antisymmetric tensor construction:

where we have:

Since we are working with the Levi-Civita symbol in 4 dimensions, it follows that:

transforms in accordance with:

for a monodromy action:

given by:

A sufficient condition for the  invariance of the field equation and the invariance of the world-volume energy-momentum tensor is for the Gaillard-Zumino duality equation:

to hold for any  in , where is the world-volume Abelian gauge field variation. We can now show that the D3-brane action on type IIB supergravity background described by the action:

under the above  constraints satisfies the Gaillard-Zumino duality equation under  duality and  spinor-action.

We can now write the super D3-brane Lagrangian in terms of component fields as:

where:

with:

and . It follows that the dual field strength defined by:

is given by:

The  transformations for:

acting on the fields and k-forms are thus given by:

as well as

and the  spinor-rotation is given by:

which necessitate the invariance of the Type IIB SUGRA constraints. We then find that is given by:

The Gaillard-Zumino duality constraint now dictates that equation:

is satisfied for arbitrary variations of , , and . Therefore their coefficients identically vanish. By appropriate substitutions, we get the following three variant equations:

and

The first two equations are trivially satisfied for  and  in light of the DBI-identity:

which holds by virtue of the fact that the dilaton  is contained in  in the form of

which in turn implies that we have a -reduction equation:

Thus we have established that self-duality and  spinor-action imply that the super D3-brane action:

in the most general Type IIB SUGRA background satisfies the Gaillard-Zumino duality condition and exhibits exact self-duality. As we shall see in upcoming posts, this plays an essential role in how one geometrically derives the Standard Model of physics from F-theory.