Supersymmetry and Gauge Symmetry Breaking from

Intersecting Branes

A. Giveon, D.K.

hep-th/0703135

Introduction

One way to break supersymmetry in string theory is to start with a supersymmetric background, and add to it a collection of D-branes that does not leave any unbroken supercharges.

The dynamics of such backgrounds is in general a non-trivial problem, which in some cases can be addressed using existing techniques.

Examples:

Non-BPS D-branes and brane-antibrane systems in flat spacetime.

D-branes propagating in the vicinity of NS fivebranes.

Branes and antibranes wrapping cycles of Calabi-Yau manifolds.

D-brane models of chiral symmetry breaking.

In this talk we will discuss a system of this type, which is related to some of the above examples and to recent work on supersymmetry breaking in field theory.

Our main emphasis will be on the classical dynamics of the branes, but we will also comment on quantum corrections.

The fivebrane configuration

We start with a supersymmetric configuration which contains two types of NS5-branes intersecting in 3+1 dimensions:

NS: (012345)

NS’: (012389)

They are placed as follows:

where we used the notation:

One can think of this configuration as a dual description of a non-compact CY manifold.For k=1 (one NS brane), each of the NS-NS’ intersections is dual to a conifold. y1, y2

correspond to resolution parameters of the two conifolds; x2- x1 is the separation between them.

Thus, our discussion is relevant for the study of non-supersymmetric brane systems on Calabi-Yau manifolds.

Adding D4-branes

To break supersymmetry, we add D4-branes (shown in red) as follows:

We would like to analyze the low energy dynamics of the branes as a function of the parameters of the brane configuration, xi , yi , k.

The standard rules give a Yang-Mills theory with gauge group

and fermions in the adjoint representation, whose dynamics is described by N=1 SYM.

Supersymmetry is broken due to the presence of a bifundamental scalar field corresponding to a fundamental string stretched between the two stacks of branes.

For large separation (x2- x1>>ls) it is massive, so one might think that it can be ignored at low energies, but we will see later that this is not always the case.

More generally, we will see that while for small fields the low energy dynamics is supersymmetric, in some regions in parameter space it is important to take into account large field effects when analyzing these systems.

This is clear already at the simplest level of discussion, where the fivebranes are treated as hypersurfaces on which the D-branes can end.

Indeed, the brane configuration we started with can be continuously deformed to:

The gauge symmetry is broken by the reconnection of the branes:

To determine which of the two configurations is the true ground state we need to compare their energies. In the flat space approximation the difference of energies is given by

This is positive for

and negative otherwise.

A couple of useful things to note:

Both brane configurations break supersymmetry; nevertheless, we will argue later that the vacuum is in fact supersymmetric.

Both configurations are classically locally stable in this approximation. Thus the one with higher energy can only decay by tunneling. We will see that this expectation is modified when certain classical corrections to our picture are included.

DBI analysis

The corrections in question are due to the effect of the gravitational potential of the k NS-branes on the D4-branes. It can be studied by analyzing the dynamics of the D-branes in the geometry created by the fivebranes.

The latter is given by the CHS geometry

with

The D-branes are described by the DBI action in this background. This description is accurate at large k, but is known to capture some features of the dynamics exactly for small k as well.

We are looking for a solution in which the D-branes are described by a smooth curve y=y(x), connecting the point (x1,y1) and (x2,y2) . Its shape is obtained by extremizing the DBI action

The solution takes the qualitative form

Its features depend on the values of the parameters of the brane configuration. We next describe its properties for y1=y2=y.

In this case, the parameters of the brane configuration above are related as follows:

There are two distinct regimes: y<l and y>l.

For y<l, the smooth curved solution above only exists for and when it exists its energy is lower than that of the straight brane solution with unbroken gauge symmetry. In this regime, the qualitative behavior of the potential for the D4-branes is

The fact that the straight, unbroken brane configuration is a local maximum of the potential can also be understood by thinking about the dynamics of a fundamental string stretched between the branes and antibranes. Its lowest mode is the open string tachyon, whose mass in the linear dilaton throat of the fivebranes is

To summarize, for y<l the system undergoes a second order phase transition at . The order parameter can be taken to be the vacuum expectation value of the bifundamental tachyon T, which behaves like

The energetics of the branes can be described by the following plot:

For y>l one can repeat the above analysis, and find the following energetics:

In this case the phase transition is first order:

Quantum effects

So far we discussed the dynamics of the branes classically. It turns out that quantum effects lead to interesting modifications. To see that, let’s go back to our original brane configuration and consider it for small x2-x1. In this regime the brane configuration is a small deformation of:

This configuration corresponds to a gauge theory with gauge group

and matter in the following representations:

There are bifundamentals of the gauge group, and an adjoint of U(N2), with the superpotential

The second term is absent in the configuration of the previous slide. It measures the separation of the NS’-branes in the x direction,

The gauge theory description is good for small separation. In that regime one can use it to analyze the vacuum structure. Using standard techniques, one finds that supersymmetry is classically broken, but is restored quantum mechanically. The classical, non-supersymmetric, vacuum becomes a long-lived excited state.

From the perspective of the brane system, this means that the phase diagram contains another line near the origin.

It is an interesting question what happens for separations of order ls and larger. We believe that these lines continue to this regime and the vacuum is supersymmetric, but have not proven that. Indeed, it would be surprising if the supersymmetric ground states found in the field theory analysis ceased to exist for finite x2-x1.

Also, the brane configurations we have been studying are expected to be related by a stringy version of Seiberg duality to

Comments

We see that the ground states we found using the classical DBI analysis are, likely, metastable, with a lifetime that goes to infinity as the string coupling goes to zero.

In some region in parameter space our problem reduces to the one studied in gauge theory by ISS. In other regions one finds a stringy generalization of their system, which exhibits very similar qualitative phenomena.

Much remains to be done to sort out the detailed dynamics of this system. For example, it would be nice to prove directly in string theory that a supersymmetric ground state exists, and to analyze the spectrum and other properties of the classical metastable states.

There are many natural generalizations of this system that can be studied using similar tools.