Theoretical Physics

1. The Exclusive Constructions of
Chern-Simons Branes
BOR
IS STOYANOV
DAR
K MODULI INSTITUTE
BR
ANE HEPLAB
SUGR
A INSTITUTE

2. Contents

Introduction

T
ransgression Actions for Branes

The Equations of Motion

Special Choices for The Action

Brane Modified Chern-Simons Actions

Conclusion

3. Introduction to Chern-Simons Branes
The model corresponds to a system of branes of diverse dimensionalities with
Chern-Simons actions for a supergroup, embedded in a background described
also by a Chern-Simons action. The model treats the background and the branes
on an equal footing, providing a ’brane-target space democracy’. Here we suggest
some possible extensions of the original model, and disscuss its equations of
motion, as well as the issue of currents and charges carried by the branes.
We also disscuss the relationship with M-theory and Superstring theory.
R
ecently we suggested a way to introduce fundamental supersymmetric extended
objects on CSS, inspired on the model for the coupling of branes to Yang-Mills
fields, which brought together these approachs. There is a model of that class
which has at least two of the superstring theories (IIA and IIB) as sectors of
its phase space and it describe branes with actions of the same form that the
one describing the background, which is a CSS with the branes acting as sources
for the gauge superfields, and interacting with that background. Also it is
plausible that standard supergravity approximately describes some regime of the
theory. It was then argued that a model of the kind considered might correspond
to the quantum effective action of M-theory.

4. T
ransgression Actions for Branes
The spacetime 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. The actions considered are given as a sum of terms of diverse
dimensions corresponding to the various branes and the spacetime in which
these branes are embedded
where the T
ransgression action part is given by
The Kinetic parts of the action has support on the boundaries of the branes,
if such boundaries exist, and are of the form
We review the issue of the equations of motion and boundary conditions in the case
of generic and pure gauge connections, and the symmetries of these theories.

5. The Equations of Motion
Under variations of the differential 1-forms the general variation of the
T
ransgression part of the action is
The variations of the kinetic terms are
We will need that using Dirac deltas we can rewrite
with Lagrangian of the form

6. The Equations of Motion
If the field strengths (or curvatures) are non zero, the equations of motion
corresponding to the variation of diferential 1-forms are
where the bulk, boundary and
kinetic currents are

7. The Equations of Motion
The corresponding equations of motion are
with the currents in the forms

8. Case of Vanishing Field Strengths
In that case the bulk contribution to the variation of the action vanishes
identically, while the variations of the gauge fields at the boundary are not the
general ones but only those compatible with the pure gauge conditions, which in
turn yields weaker equations of motion at the boundary. The required variations
of the various terms of the action are obtained integrating by parts and
discarding total derivatives
where the gauge and world-volume covariant derivatives are presented with the
Christoffel symbol corresponding to the world-volume metrics
The boundary equations associated to variations of both the metric and the
embedding coordinates are also the same, except for the fact that now the
gauge field strengths vanish and the gauge potentials are pure gauge.

9. The Equations of Motion
T
o determine the equations of motion associated to the embedding coordinates
of the branes we need that the variation of the T
ransgression and Kinetic
actions under the variation of the coordinates are
where the general consruction include

10. The Equations of Motion
Considering that the brane coordinates are the same at the same point of the
boundary of the brane, we obtain the equations of motion for the bulk
At the boundary, for those values of the brane coordinates corresponding to
free boundary conditions we obtain the conditions/equations of motion
These equations of motion only imply a local proportionality between both members
of the previous equation, but in that dimension the action is invariant under local
rescaling of the metric. The action is not invariant under gauge transformations for
which only one of the gauge fields transforms, or both of them transform with
different gauge group elements, but these variations yield only boundary terms,
therefore the action is said to be quasi-invariant under these transformations.

11. Special Choices for The Action
The requirement on the form of the action mentioned, which in the pure gauge case
only boundary degrees of freedom remain, and the related condition that under
gauge transformations of the gauge potentials the action must be quasi-invariant,
exclude not only terms of the form but also kinetic terms with support in the bulk.
For the special choices the variation of the action under variations of the
differential 1-forms are
It is interesting that the same relative coefficient between the kinetic and
transgression parts resulted from the requirement of kappa symmetry.

12. Special Choices for The Action
In two dimensions there are two choices of the exterior (wedge) product of
differential forms that result in special properties for the action, particularly
the decoupling of left and right movers, and Weyl invariance of the action.
In the pure gauge case of vanishing field strengths we have
For instance, the equations of motion obtained, in the case if there are no other
overlapping branes, but Weyl invariance allows to choose the world-volume metric
to be flat, what makes the Christoffel symbols zero. The reason of Weyl invariance,
the metric is an auxiliary variable with no local degrees of freedom, which can be
locally reduced to the flat metric.

13. Special Choices for The Action
The equations can be interpreted as the equations found before in the case
there are no boundaries or branes but now with source terms given by currents
carried by the branes. Then we can write the variation
or in the form
with the current
where we include

14. Special Choices for The Action
The additional brane current is
Concerning the equations of motion corresponding to extremize the action
under variations of the embedding functions it is convenient to write
where we separated the bulk and boundary contributions to the lagrangian.
If we add the kinetic terms there would be an extra term to the current located on
the boundaries of the branes, and extra terms in the Euler-Lagrange equations.
The equations of motion of the auxiliary metrics in the kinetic terms are algebraic.
These equations are related to General R
elativity and the standard Supergravity
constructions in diverse spacetime dimensions.

15. Case of Vanishing Bulk T
erm
A well defined action principle requires a well defined action, with well defined
fundamental fields (dynamical variables), and suitable boundary conditions such
that the variation of the action yields the sum of a bulk term, which vanishes as
a result of the field equations, plus a boundary term that vanishes as a result
of both the field equations and the boundary conditions. Then we have
and on the other hand
therefore the equation which proves our assertion is
The action with two natural choices present itself for the dynamical variables.

16. The Brane Modified Chern-Simons Actions
We emphasize that the presence of D-branes necessarily modifies those CS
terms; by extension through duality, M-branes in 11D and NS5-branes
in 10D must also modify them. The CS term modifications were we give a new,
simple, physically-motivated derivation in the theory, which we can apply to the
10D supergravities. The action is
with Chern-Simons terms
so we consider
the variation

17. The Brane Modified Chern-Simons Actions
Excluding local sources, the action of the bosonic sector in IIB supergravity is
We can add D-brane currents to the previous equation by adding Dirac brane
currents to the field strengths and shifting the action by
We note that the action is sometimes written with an additional CS term
In the absence of branes, this term is a total derivative and does not
contribute to the equations of motion.

18. The Brane Modified Chern-Simons Actions
At this point, the IIB supergravity action has become
It is important to understand the invariance of IIB supergravity action under
gauge transformations of the potentials, in contrast to the usual presentation
in the absence of branes. All together, we have
We note the appearance of the usual CS term and also the Dirac brane
modification, both of which are a consequence of transgression terms
in the field strength and in the divergence of the Dirac brane current.
Finally, there are two new terms involving Dirac brane currents for
electrically and magnetically charged branes.

19. The Brane Modified Chern-Simons Actions
We discuss the well-known relation between D8-branes and the massive IIA
supergravity in light of our new action. We examine the role that Dirac brane
currents play in generating the couplings of the massive IIA theory, including
terms proportional to the mass parameter in D-brane WZ actions. We find
Once again, the modified CS term has precisely the correct coefficients to
ensure that the EOM can be written in terms of the gauge-invariant field
strengths. Of course, this fact is related to gauge invariance of the action
The gauge invariance ensures that the integral over the potentials is well defined.

20. The Brane Modified Chern-Simons Actions
The latter sum is the Chern-Simons action. The field strength is defined
Gauge invariance places constraints on these constants. Consider first gauge
invariance of the field strength with gauge transformationse
At times we will consider formal sums of forms of different ranks and products
of such sums, integrating over a supermanifold of some particular dimension
picks out only the form of that rank. We can also see this constraint in the
requirement that the Bianchi identity be written in terms of gauge-invariant
variables. Exterior derivatives without accents are spacetime derivatives,
hatted exterior derivatives are along physical branes.
We will be forced to consider the gauge invariance of the potentials, this is
of course related to our discussion of monopole branes and the Dirac brane
formalism.

21. The Brane Modified Chern-Simons Actions
The gauge variation of the action in the special case is
Meanwhile, the equation of motion is
An important issue is understanding the supergravity action in the presence of
both D-branes and NS5-branes, particularly the “extra” CS terms sometimes
included in the IIB and IIA supergravity actions and naturally is topological
except in the presence of both D5- and NS5-branes.

22. Conclusion
Actions for extended objects based on T
ransgression and Chern-Simons forms for
spacetime groups and supergroups provide a gauge theoretic framework in which to
embed previously studied String and Brane actions, extending them in interesting ways
that may be useful in a future non perturbative formulation of String Theory.
In the following years there were several works exploring diverse aspects of models
of Chern-Simons branes including explicit solutions and conserved charges.
Having a single action with a doubling of the field content, as it is naturally
the case by construction in the T
ransgressions, with one of the fields (which may
be chosen as a ”vacuum”) regulating the other, may be suggestive of a wider
conceptual framework where a dynamical mechanism that introduce scales in
an originally scale free theory is built in from the start.
The class of theories considered in this scientific presentation has several attractive
properties that merit further investigation.
These theories are already first quantized, analogous to strings on nonflat
backgrounds. They differ from those string models in that the background is
described by Chern-Simons supergravity instead of the standard formulations
of higher-dimensional supergravity.

23. THANK YOU