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:

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

with the WZ term:

and the dual 6–form potential:

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

Let us consider the gauge covariantization of the proper-time gauge and the Ramond-Ramond gauge discussed above. The action for the covariant bosonic open string field theory is implicitly defined by the BRST operator :

with respect to the BPZ conjugation-derived inner product, where the string field has the following Fock space expansion:

where the following holds:

for the bosonic case, and:

are the associated space-time fields. We can now write the action as:

and is invariant under the gauge transformation:

with the gauge parameter being a Grassmann string field of

given as:

In terms of and , the gauge transformation is expressible as:

It follows then that:

is gauge invariant. Hence, in terms of , the action:

becomes:

Note that the gauge invariance of each of:

and

entails that the string field, by conformal gauge theory, has the following form:

and we also have:

Thus, our action becomes a sum of two gauge invariant terms:

and

Now, crucially:

is equivalent to a gauge invariant action of massless vector field

and by the metaplecticity of:

it follows that gauge transformation up to level N = 1 is expandible in terms of a gauge parameter as:

We perform now a Virasoro reparametrization of the evolving string surface as a transformation:

with

the wave-function of the string, which in string field theory, must be interpreted as a functional , giving us the functional action:

where the inner product is defined in terms of integrals over the whole string configuration space and the string field kinetic energy operator. By reparametrization invariance, we can derive the following:

and the following relations can be easily checked:

Now, what makes unique contrastively to is its invariance under an large group of extra symmetries in addition to reparametrization implicitly expressed by:

specifically, given by shifts:

and is a type of meta-gauge symmetry acting directly on the metaplectic phase space. Let us study some properties of this metaplectic gauge group as well as .

We expand in terms of the eigenstates of the mass operator:

where the state is annihilated by all . Then the string field functional can be written as:

The kinetic gauge of is given by the action of on new string functionals. At first order we get the following equation of motion:

and since satisfies:

it follows that:

has the property of linearized Yang-Mills gauge invariance and the following transformation laws can be derived:

where is the Chan-Paton field term.

With a 0-level state and an eigenstate of with eigenvalue , we have the following definition for the contravariant form

Now we must define:

satisfying:

with the 0-th projection operator:

Combining, we have at n-th-mass level a Klein-Gordon equation:

which is gauge-invariant, thus the existence of .

Adding the Stueckelberg string fields to the fundamental string field , we have the local Stueckelberg action:

We now define N-th-projection operators:

along with:

with

Now, since we have:

it follows that the Stueckelberg action:

is equivalent to the kinetic metaplectic gauge field action:

Now we introduce a new gauge:

At N=1 gauge field level, we thus have:

which is a hybrid Landau-Stueckelberg gauge. So for

this fixes the gauge invariance of the Stueckelberg action above.

Note that under this condition, since the following holds:

any string field:

satisfies the Landau-Stueckelberg gauge condition given:

with:

This in turn entails that:

holds.

The action for this gauge condition is then:

with:

with the odd/even Grassmann string fields and the projection operator defined implicitly by:

Putting all together, it follows that the action is BRST invariant.

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