249. The fate of unstable gauge flux compactifications

This is a guest post by Ivonne Zavala from the Bethe Center for Theoretical Physics (Bonn). Dmitry.

Let me start by thanking Dmitry for the invitation to write a guest blog entry about my recent paper arXiv:0812.3902. This has been done in collaboration with Cliff Burgess and Susha Parameswaran, and it is about possible end points where unstable monopole configurations can decay to.

Why is this interesting? There is a number of reasons to study these unstable configurations. However, let me just mention that in recent years, flux-supported compactifications, for which various n-form field strengths thread cycles, which are topologically blocked from relaxing to zero, have become very popular. This is because they constitute an attractive mechanism that dynamically stabilises many of the moduli present in most extra-dimensional compactifications. Perhaps the simplest such construction, is the more than 20 years old Salam and Sezgin monopole solution [1]. This threads a Maxwell flux through the extra dimensions in 6D supergravity to stabilise its compactification to .

A not so well known fact about a great many of such monopole configurations, is that they are unstable, when the flux involved arises as a Dirac monopole embedded into a non-Abelian gauge sector. Explicit calculations [2] have shown that sphere-monopole compactifications in anomaly-free supergravity – and their warped braneworld generalizations – are generically unstable, even though the monopole in question carries nontrivial topological charge.

In order to understand this instability, one can go back to pure Yang Mills (YM) theories. There, it is well understood that there exist typically more monopole solutions than there are distinct topological sectors. Topological flux conservation thus suggests, that an unstable monopole decays to the (often unique) stable representative in any topological class [3,4,5]. It is also known that coupling to gravity does not remove the instability [6,7], and requires the geometry to relax as the monopole decays. The fate of such decay has however remained as an open question, which we seek an answer to.

In our search, topology turned out to be an important guide. We start with the simplest system one can consider, building up towards a configuration of six dimensional supergravity with co-dimension two branes.

Let us begin then by looking at Einstein-Yang-Mills (EYM) in 6D, including a positive cosmological constant. The later allows the flux compactification to be a solution to the equations of motion. As in pure YM, the unstable monopole evolves towards the unique stable monopole in the same topological class [5]. As it does so, the geometry adjusts by curving the large 4 dimensions towards the  maximally symmetric AdS or dS, and shrinking/stretching the size of the supported extra-dimensional sphere . These two solutions are possible end points, as can be easily checked from the equations of motion. However, one expects that it is the AdS (with radius of ) the preferred one. This can actually be confirmed via an energetic analysis.  Indeed, by computing the potential energy of the four dimensional effective theory, one finds that it is the AdS solution which lowers the energy with respect to the initial (4D) Minkowski configuration (whereas the dS increases it). Understood this example, let us move on to the supergravity case.

In the supergravity system, the bosonic field content includes a dilaton , Kalb-Ramond two form , with field strength (and hyperscalars , which we can safely ignore in what follows). The situation then becomes more complicated, since as the monopole decays towards its stable cousin following flux conservation, there are more degrees of freedom which can become active as this process occurs.

The starting unstable configuration in the supergravity case, is a monopole with a Maxwell flux thread over the sphere, but now we also have that in the initial solution. As the monopole decays to its stable cousin, one has to take into account that the dilaton equation of motion is sourced by the magnetic flux as well as the three form flux:

An apparently promising endpoint solution which allows the dilaton to stay constant, is one in which the two form becomes active, , compensating for the change in the value of , keeping . This seems yet more attractive given the existence of a supersymmetric solution with precisely non-zero field and constant dilaton [9]. These solutions have geometries , where denotes a one-parameter family of “squashed” 3-spheres.

Is this the endpoint of the evolution away from the unstable monopole? Such a scenario would be very attractive, indicating a dynamic spontaneous compactification wherein the monopole instability triggers one of the large 4 dimensions to roll up into one of the directions in . Better yet, the supersymmetric solution can be obtained formally from the solutions by taking a suitable limit indicating there might be a plausible path through field space leading from the initial unstable configuration to the final supersymmetric one. There are a number of possible objections to the proposal that these solutions represent the endpoint of the monopoles of present interest, however. Not least, the natural way to obtain 4 large directions from is by taking the lone squashed direction to become large. However in this limit the curvature of the large 3 dimensions becomes larger and not smaller. A further obstruction arises because taking this limit requires the value of to be very large, which in turn implies that the radius of the unsquashed directions becomes imaginary! Therefore we seek other options for the decay endpoint.

Having discarded the solution above, it is clear that the dilaton must become active, preventing the system’s relaxation towards a maximally symmetric solution. This is because implies , leading to a breakdown of some of the spacetime symmetries. Finding solutions with nonzero dilaton is in general a nontrivial task. Luckily, things can be highly simplified by making use of an elegant trick [10]. This consists of recasting the dilaton as the volume modulus of fictitious dimensions in a yet-higher dimensional, , non-dilatonic Einstein-Yang Mills theory [10]. One can then use this to map the unstable initial monopole-supported supergravity configuration onto an unstable monopole-supported state in the still-higher dimensional theory. Assuming this EYM system relaxes in the simple maximally-symmetric way tells us its endpoint, and this can then be mapped to determine the endpoint EYM-dilaton configuration that is supported by the final stable state into which the monopole decays in six dimensions.

The final stable endpoint geometry found in this way results to be conformal to . Its nontrivial conformal factor and dilaton break the maximal 4D symmetry, giving rise to a singular geometry for which the dilaton and curvature blow up at a point in the 4D spacetime. One can check as in the simple EYM case, that the configuration nonetheless has a lower potential energy than did the initial one. Moreover, it turns out to be stable, constituting a reasonable candidate for the endpoint of the instability.

So let us now consider a system of co-dimension 2 branes located at the poles of the sphere, where our 4D world is supposed to be localised. One can understand the gravitational effect of such objects by going back to four dimensional gravity. There it is known that co-dimension 2 objects correspond to strings in field theory. When coupled to gravity, one can have strings of cosmic size, or cosmic strings, which produce precisely a conical singularity in spacetime. In the same spirit, we add to the 6 dimensional supergravity configuration, a pair of conical singularities in the extra-dimensional geometry, sourced by a pair of co-dimension 2 branes.

For generic brane tensions, the resulting geometry has a nontrivial warping. It has been shown that these monopole configurations are also unstable in warped brane-world compactifications with positive-tension brane sources [2].

The system at this point seems to become quite involved. However, the oxidation/reduction trick via fictitious dimensions can be applied again, following all the steps performed in the unwarped case. The result is that, although the dilaton changes the dynamics drastically, the presence of branes and warping do not make much difference. The solutions found in this way turn out to be the warped version of the unwarped case above.

Thus, just as in the unwarped case, the non-trivial profile of the dilaton in 4D generates a singular, static, Kasner-like geometry that is conformal to warped , where the radius of the 2-sphere grows with the distance from the singularity. How to interpret the naked timelike singularity to which the instability seems to lead is an important open question; does it signal an inconsistency or does it suggest some new physics beyond any supergravity approximation? One way to resolve the singularity is to pass to the higher dimensional Einstein-Yang Mills theory in (6+n)D, in which case the singularity results from projecting the smooth geometry onto six dimensions (such ideas have been discussed in the literature, e.g. in [11]). However, it can be shown that the final configuration is perturbatively stable, and that the decay results in a finite total energy which is lower (counting gradient and potential contributions) than the initial one.

We have thus found candidate end point solutions where the monopole instability is likely to go. One interesting question, which we leave open, is how a time dependent evolution occurs from the unstable to the stable geometries.

As a final speculative possibility, it is interesting to note that the instability suffered by Yang-Mills sectors in the background of a monopole is the spherical analogue of the Nielsen-Olesen instability that occurs in flat 4D Yang-Mills theory [12]. In that case, it was proposed that condensation of the tachyonic modes leads to the formation of magnetic flux tubes [13], in a rather beautiful imitation of the vortex formation in superconductor physics [14]. That such a dynamics might also be possible in the present case is certainly an interesting speculation.

Literature:

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2. S.L. Parameswaran, S. Randjbar-Daemi and A. Salvio, JHEP 0801 (2008) 051.
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4. R.A. Brandt and F. Neri, Nucl. Phys. B 186 (1981) 84.
5. S.R. Coleman, The Magnetic Monopole Fifty Years Later, Erice Subnuclear 1981:21 (QCD161:I65:1981)
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10. T. Kobayashi and T. Tanaka, Phys. Rev. D 69, 064037 (2004) [arXiv:hep-th/0311197].
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