Actions, equations of motion and boundary conditions for Chern-Simons Branes

Actions for extended objects (branes) based on Chern-Simons and Transgression forms for space-time groups were introduced in [1, 2] by H. Nishino and myself, aiming to build a framework in which previous string and brane models could be embedded, endowing them with desirable additional features. Those models are background independent gauge systems for the space-time groups. The background is not fixed and given a priori, but participates of the dynamics of the extended objects, which are sources of the gauge fields. The actions of the branes have the same mathematical form than that of the background, in what we called ”brane-background democracy”.

The above mentioned work built on, and was inspired by, several previous lines of research:

1. 1.

   It introduces extended objects in a gauge invariance way in Chern-Simons (super)gravity theories [3, 4, 5, 6, 7, 8, 9], which had the goal of extending General Relativity to describe gravity as a gauge theory for some space-time (super)group.

2. 2.

   It generalizes Dixon et al. (DDS) models [10] for the coupling of extended objects to non abelian gauge fields (heterotic branes) to space-time groups and their supersymmetric extensions (treated in the same footing as other possible gauge groups). It differs from the DDS models in that the background described is described by Chern-Simons/Transgression gravity or supergravity, in a way formally analogous to the extended objects themselves, and not by standard gravity or supergravity; and in the fact that our models allow both sets of gauge fields involved to be non pure gauge.

3. 3.

   It was motivated by Green’s construction [11] of the superstring action as a gauged Wess-Zumino-Witten (WZW) model for a supergroup.

4. 4.

   It was also motivated by the Chern-Simons supergravity theory for the supergroup $OSp(32,1)$ proposed by Troncoso and Zanelli [12, 13] and its conjectured relationship with eleven dimensional supergravity[14], and by Horava’s work [15] suggesting a possible relation between a Chern-Simons supergravity with gauge group $OSp(32,1)\times OSP(32,1)$ and M-theory [16, 17, 18].

In the following years there were several works exploring diverse aspects of models of Chern-Simons branes [19, 20, 21, 22, 23, 24, 25, 26], including explicit solutions and conserved charges.

In Ref.[1] we suggested that a model of this class may be useful for of a non perturbative description of String Theory, or M-theory. I still believe that possibility is the main reason to study these models. Some of the motivations to do so, and the features that distinguish them from other extended objects models are:

1. 1.

   As stated in Ref.[1]¹¹1The quote is [1]: ”A heuristic motivation for the kind of models presented here is that if there is some underlying pregeometric theory in which the de Rham complex get physical content through some kind of antighost field variables, we must expect the effective action to be a sum of pieces of diverse dimensions of the form considered here. We believe that this simple observation is the key to understand the existence of extended objects in a fundamental theory of nature.”, considering an action that is a sum of integrals of differential forms of different orders but similar mathematical form ²²2One may think of the action as a sort of generalized integral of a polynomial, or truncated expansion (as befits to Grassmann variables), on the differentials $dx^{\mu}$. can be understood as a way to give physical content to the differential structure of the manifold, as General Relativity had given physical content to the metric structure, giving another way to justify the possible existence of fundamental extended objects. The frequent question ”what are strings made of?” is often answered with the question ”what are particles made of?”, somehow implying the first question is moot, yet from our point of view the answer may be a key to the foundations of String Theory.

2. 2.

   The only dynamical fields are gauge fields, although some of them are fermionic in the case of a gauge supergroup, providing the fermions required for a connection with Superstrings. The p-form fields that would be associated to those occurring in String Theories are composite, and given by Chern-Simons forms of our gauge fields, what means that the anomalous gauge transformations and Bianchi identities of those fields are built-in. The notion that the fundamental degrees of freedom of String Theory may not be the apparent ones has been widely accepted for quite some time, see for instance Ref.[27]³³3Quoting from this Reference: ”In my opinion, the basic challenge in string theory is not, as sometimes said, to ’understand non-perturbative processes in string theory’. In fact, the basic problem does not lie in the quantum domain at all. The basic problem, now and probably for many years to come, is to understand the classical theory properly.” And ”So the central task is to (i) find the right degrees of freedom; (ii) find the right invariance group; and (iii) formulate the right lagrangian.”. In the field theory case it has been found that full gauge invariance (instead of quasi-invariance) leads from Chern-Simons forms to Transgressions, which automatically dictates the correct boundary terms to regularize Noether conserved charges and black hole thermodynamics [28, 29, 30]. This can be taken as evidence for requiring a gauge invariant background independent theory being the right path to follow.

3. 3.

   These theories are already first quantized, analogous to strings on non-flat backgrounds. They differ from those string models in that the background is described by Chern-Simons (super)gravity instead of standard (super)gravity. It is remarkable that the above mentioned theory with supergroup $OSp(32,1)$ of Troncoso and Zanelli [12, 13] has been shown by Izaurieta and Rodríguez [31] to reduce in certain limit to a theory very similar to standard eleven dimensional supergravity [14], with corrections given by a expansion in powers of the parameter used in the limit. It seems possible that with further analysis of the correspondences between the fields of both theories, and a suitable choice of the invariant trace involved, the correspondence may be more precise, and the additional terms could be coherent with anomalous M-theoretic corrections from dualities [18].

4. 4.

   In a sort of holographic property, if the gauge fields are pure gauge then only degrees of freedom at the boundaries of the branes and the space-time remain.

The plan of this Letter is the following: I review the issue of the equations of motion and boundary conditions, both in the case of generic and pure gauge connections, and the symmetries of these theories. I discuss some additional conditions, leading to desirable properties on these models. These include the inclusion of the kinetic term, added at the boundary and its coefficient relative to the bulk Chern-Simons term in the bosonic case, independently of the argument based in kappa-symmetry given in [1], as well as the exclusion of additional possible bulk terms. Afterwords I consider the particular case or sub-sector of the theory that makes contact with the Bosonic String in some limit. Finally I discuss the existence of Dirac quantization conditions on the tensions or coupling constants of the branes.

I consider actions for extended objects (branes) with dynamical variables given by gauge fields given as differential 1-forms $A=A_{\mu}^{A}G_{A}dx^{\mu}$ valued on the gauge group $\mathcal{G}$ algebra with generators $G_{A}$, with $\mu=0,...,D$, with $D$ even, the embedding coordinates of each brane $X_{(d+1)}^{\mu}(\xi_{(d+1)}^{i})$, where $\xi_{(d+1)}^{i}$, with $i=0,...,d$, with $d$ even, are the coordinates of the $d$ brane, and the intrinsic metrics defined at the boundaries of the branes $\gamma_{(d)rs}$, with $r,s=0,...,d-1$. The space-time in which the branes are embedded can be consider itself as a brane, and is itself described by an action of the same class as the actions of the branes, therefore its coordinates are $\xi_{(D+1)}^{\mu}\equiv x^{\mu}$). While the branes themselves may or may not have boundaries, the boundaries of the branes have no boundary. The actions considered, introduced in Ref.[1], are given as a sum of terms of diverse dimensions corresponding to the various branes and the space-time in which these branes are embedded

where the Transgression action part is given by

with the Transgression forms given by

There $\Delta A=A-\overline{A}$, $A_{t}=tA+(1-t)\overline{A}$ and $F_{t}=dA_{t}+A_{t}^{2}$. The brackets $<...>$ denote symmetrized (super)traces on the algebra of the gauge (super)group, corresponding to symmetric invariant tensors $g_{A_{1}...A_{d}}\equiv<G_{A_{1}}...G_{A_{d}}>$. These symmetrized traces or invariant tensors, for which the simplest choice is $<...>=STr[...]$, are not unique. Through all this work we use a notation where the exterior (wedge) product of differential forms is implied $\alpha_{p}\wedge\beta_{q}\equiv\alpha_{p}\beta_{q}$.

The Kinetic parts of the action has support on the boundaries of the branes, if such boundaries exist, and are of the form

where $\Delta A_{i}=\Delta A_{\mu}^{A}\frac{\partial X_{(d)}^{\mu}}{\partial\xi_{(d)}^{i}}G_{A}$.

The AdS group in $D+1$ dimensions $SO(D-1,2)$, which is also the Conformal group in $D$ dimensions, corresponds to linear transformations that leave invariant the metric $\eta_{AB}$, $A,B=0,...,D+1$, with zero off-diagonal entries and diagonal elements $(-1,+1,+1,...,+1,-1)$. The algebra of the generators $J_{AB}$ of the AdS group is

Splitting the indices into $a,b=0,...,d$ and $D+1\equiv*$, and defining the Lorentz generators $J_{ab}$ and the linear momentum generator $P_{a}=\frac{J_{a*}}{l}$, where $l$ is a real and positive parameter with the dimension of length called tha ”AdS radius”, we get

In la limit $l\rightarrow\infty$, known as Inonu-Wigner contraction, this algebra turns into the Poincaré group algebra in $D+1$ dimensions.

The invariant trace of two generators is

It is possible to show that the bosonic string with Polyakov Action is obtained as part of the space of solutions of Chern-Simons branes in a background Chern-Simons AdS gravity in the limit of large $l$, by proceeding as follows.

We consider gauge group elements of the form $g(x)=\exp[-\mathcal{V}^{a}(x)P_{a}]$, where the gauge parameters $\mathcal{V}^{a}(x)$ are some function of the space-time point, instead of the generic $g(x)=\exp[-\mathcal{V}^{a}(x)P_{a}+\mathcal{W}^{ab}(x)J_{ab}]$ . Notice that $\mathcal{V}^{a}P_{a}=\frac{\mathcal{V}^{a}}{l}J_{a*}$, where the $J_{a*}$ are dimensionless, and therefore the $\mathcal{V}^{a}$ must have the dimension of length. Assuming the exponent to be small, which amounts to require $\mathcal{V}^{a}\ll l$, we can consider only the first terms of the expansions for the group element $g=I+\mathcal{V}^{a}P_{a}+\frac{1}{2}\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+...$, its inverse $g^{-1}=I-\mathcal{V}^{a}P_{a}+\frac{1}{2}\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+...$, its exterior derivative $dg=d\mathcal{V}^{a}P_{a}+\frac{1}{2}d\mathcal{V}^{a}\mathcal{V}^{b}P_{a}P_{b}+\frac{1}{2}\mathcal{V}^{a}d\mathcal{V}^{b}P_{a}P_{b}+...$ and the Maurer-Cartan form $g^{-1}dg=d\mathcal{V}^{a}P_{a}+\frac{1}{2l^{2}}d\mathcal{V}^{a}\mathcal{V}^{b}J_{ab}+...$, with leading order $\mathcal{V}/l$. Considering the form of the gauge potential in Chern-Simons AdS gravity in terms of the vielbein $e^{a}=e^{a}_{\mu}dx^{\mu}$ and the spin connection $\omega^{ab}=\omega^{ab}_{\mu}dx^{\mu}$, $A=e^{a}P_{a}+\frac{\omega^{ab}}{2}J_{ab}$, then the pure gauge potential associated to the Maurer-Cartan form $A^{(pure~{}gauge)}=g^{-1}dg$ implies $e^{a}=d\mathcal{V}^{a}$ and $\omega^{ab}=-\frac{1}{4l^{2}}\mathcal{V}^{[a}d\mathcal{V}^{b]}$. Notice that the vielbein part is the one of leading order in the expansion in powers of $\mathcal{V}^{a}/l$, even considering, as we must, $e^{a}/l$. We make a choice of coordinates such that $\frac{\partial\mathcal{V}^{a}}{\partial x^{\mu}}=\delta^{a}_{\mu}+...$, or $\mathcal{V}^{a}=\delta^{a}_{\mu}x^{\mu}+...$, valid in a neighbourhood of $x^{\mu}=0$ such that $x^{\mu}\ll l$. For this coordinate choice the vielbein is $e^{a}_{\mu}=\delta^{a}_{\mu}+...$.