Gravitational monopoles, anomalies and ME₈bundles

Abstract

In this paper we try to clarify the physical meaning of the gauge theory that underlies the K-theoretical classification of RR charges in type IIA. Our main tool are the conditions for the cancellation of the Freed-Witten global anomaly when we take into account the effects of a flat and a general B-field. In each case we will see how K-theory captures some eleven dimensional information. In the first case and studying the electric properties of the D6-brane we see an eleven dimensional U(2) gauge symmetry, while the second can be related to an E 8 theory. Moreover, in the reduction from the general to the flat case, we find that the Romans' mass gives the number of unstable intial D9-branes. 1 3-form, C (3) , along the compact direction of Taub-NUT, in a construction that resembles the Julia-Zee dyon, [22].One may think that including this 3-form would result into a noncommutative deformation in the lower dimensional theory. However, as we will see by a reconstruction of the moduli space and in terms of the Nahm-Hitchin-Donaldson (NHD) construction, [23,24,25,26,27], since this field enters in the construction as the "temporal" component of the gauge field, we find that the gauge group for this monopole is SU(2), with no central extension (as should be the noncommutative case [20]).A "spin-from-isospin" construction allows us to see this SU(2) gauge group as one of the two invariant subspaces of the SO(4) group of rotations in Taub-NUT. Moreover, we can see it as the part of the U(2) isometry group acting on the R 3 part of this space, while the extra U(1) acts as motions along the compact NUT direction, i.e. in the one we can identify as the eleventh direction of space-time. Thus we can naturally relate the geometry of Taub-NUT with the geometry of the moduli space of a non-abelian monopole. Moreover, the reconstruction of the moduli space via the NHD construction allows us to identify motions along the compact NUT direction as generators of the electric charge for this gravitational monopole.However, from the point of view of eleven dimensions, the whole gauge group is U(2). This is a reflection of the fact that in our case the "invariant direction" in the NHD construction is physical and, following the usual KK interpretation, there is a U(1) symmetry associated to it. This U(2) structure should be compared with the U(2) gauge group appearing in the ten dimensional K-theory discussion. In fact, as we will see in section 4, we can map one group to the other.This can be understood from the interpretation of the π 3 in K-theory as the characteristic class for the Hopf bundle of the NUT direction over the sphere at infinity surrounding the D-brane, implying that the K-theoretical U(2) gauge group lives in eleven dimensions. As, we will see, a careful analysis of the conditions for the cancellation of global anomalies leads to the same conclusion.The previous analysis is done when the B-field is pure torsion, [28,29,2]. This is perfectly natural for the D6-brane, where this condition is dictated by geometry, howev...

Dualized gravity beyond linear approximation

“…This solution is separately regular on the two patches and differentiable in the region where they are overlapped. The gravitational monopole solutions have also been studied in K-theory [53]. This is done by reconstructing the moduli space for such solutions using the NHD construction.…”

Dualized Gravity beyond Linear Approximation

“…This solution is separately regular on the two patches and differentiable in the region where they are overlapped. The gravitational monopole solutions have also been studied in K-theory [13]. This is done by reconstructing the moduli space for such solutions using the NHD construction.…”