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Building a connection through obstruction; relating gauge gravity and string theory
Cartwright, Casey
Cartwright, Casey
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2016
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Gauge theories of internal symmetries, e.g.\ the strong and electroweak forces of the Standard Model, have a geometric description in terms of standard fiber bundles. It is tempting then to ask if the remaining force, gravitation, has a description as a gauge theory. The answer is yes, however unlike the internal symmetries of the Standard Model, the story is not so simple. There are dozens of renditions of gravitational gauge theory and no standard fiber bundle description. The main issue in the construction of gravitational gauge theory is the inclusion of translational symmetry. While the Lorentz group, like internal symmetries, acts only at each point, the translational symmetry shifts points in spacetime. For this reason a gauge theory of gravity requires a somewhat more sophisticated fiber bundle known as a composite fiber bundle. When constructing gauge theories of internal symmetries it is easy to take certain topological conditions for granted, like orientability or the ability to define spinors. However it is known that there exist spaces which do not have the properties required to define sensible field theories. Although we may take these topological properties for granted when constructing gauge theories of internal symmetries we haven't had evidence yet to expect we can do the same for gravitational gauge theory. By studying the geometry of the composite bundle formalism underlying viable gauge theories of gravity we have found previously unappreciated subbundles of the primary bundle. We were able to identify these subbundles as the spacetime bundles we would expect to be created by a gauge theory of gravity. Remarkably, the origin of these subbundles leads to the natural inclusion of expected, and unexpected, topological conditions. While the overall bundle used for gravitation is $P(M,ISO(1,3))$, i.e.\ a principal Poincar\'e bundle over a space $M$, the Poincar\'e group ($ISO(1,3)$) can be viewed as a bundle in its own right $ISO(\reals^4,SO(1,3))$. Thinking of the fiber space itself as yet another bundle leads to consideration of two primary bundles $P(E,SO(1,3))$ and $E(M,\reals^4,ISO(1,3),P(M,ISO(1,3)))$. The split of the total bundle $P(M,ISO(1,3))$ into the two bundles $E$ and $P(E,SO(1,3))$ however requires the existence of a global section of the bundle $E$. Such a global section is guaranteed to exist by a theorem of Kobayashi and Nomizu. However it is interesting to investigate the topology of the bundle space $E$ and hence of $P(E,SO(1,3))$. The requirement of the global section leads to the definition of a bundle $Q(M,SO(1,3))\subset P(M,ISO(1,3))$ which can be identified as the frame bundle of spacetime. Its associated bundle, $Q(M,SO(1,3))\times_{SO(1,3)} \reals^4$ where $\times_{SO(1,3)}$ denotes a specific quotient of the product space $Q(M,SO(1,3))\times\reals^4$ by the group $SO(1,3)$, can then be identified as the tangent bundle. The existence of a global section of $E$ leads to topological conditions on the induced spacetime bundles. Using cohomology with compact support one can show that global sections of $E$ descend to global sections of $Q$ and force the Stiefel-Whitney, Euler and first fractional Pontryagin classes of the spacetime bundles to be trivial. Furthermore the triviality of these characteristic classes is equivalent to the condition that the base space $M$ admit a string structure. Each characteristic class has an interpretation as an obstruction to the creation of a global structure or a topological attribute of the bundle. For the composite bundle formulation the obstructions are to orientablility, parallelizablility, global sections, and conditions related to stable causality and string structures. Similar to the case of a supersymmetric point particle, where the parallelizability of the base manifold determines whether there will be a global anomaly encountered during quantization, whether a manifold admits a string structure will determine if a global anomaly will be encountered in the process of quantization of extended degrees of freedom. This implies that the topological aspects of gravitational gauge theories automatically accommodate the consistent introduction of extended degrees of freedom. This path to structures associated with extended degrees of freedom is in contrast to the typical route, i.e.\ demanding a consistent quantum theory of gravitation. Here the need for such structures arises classically from demanding that gravitation be realized from a geometrically supported gauge principle.
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