Continuing from my last post where I discussed the triangular interplay between string-string duality, string field theory, and the action of Dp/M5-branes, here I shall discuss Stueckelberg string fields and derive the BRST invariance of the Landau-Stueckelberg action. Recalling that the action of M-theory in the Witten gauge is:

with the kappa symmetry term, the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:

and the worldsheet action:

being the sum of three terms:

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

Something deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, and *not* a three-dimensional theory

*not*a three-dimensional theory

which is precisely the long wavelength limit behavior of the **open** membrane:

the gauge fields are defined in terms of fields that live on 10-dimensional boundaries of **M**-theory

**M**-theory

In the **closed** membrane case:

the gauge fields are defined in terms of **11**-dimensional fields

**11**-dimensional fields

which brought us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

with:

for and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

with:

and the momentum operators are given by:

Since open string end-points are topologically glued to Dp-branes, open strings must have inequivalent quantum states and thus, the string field has to carry the gauge group indices of :

where are the generators of the SU(N) group, with . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

which yields the field theory action:

BRST-invariantly as:

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

with the dilaton, the curvature of a (p + 1)–form gauge field:

where the two–index NS/NS tensor and the dual six-index heterotic five–brane tensor are given by:

and

Now we have the general form of a 10-D p-brane solution:

with:

and:

with

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

is:

Hence, the M2-brane solution is:

squaring the field strength gives the following M5-brane solution:

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

From the string-string duality above and , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

…