We study noncommutative field theories at finite temperature to learn more about the degrees of freedom in the non-planar sector of these systems. We find evidence for winding states. At temperatures for which the thermal wavelength is smaller than the noncommutativity scale, there is a drastic reduction of the degrees of freedom in the non-planar sector. In this regime, the non-planar sector has thermodynamics resembling that of a 1 + 1 dimensional field theory.

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A 6-dimensional grand unified theory with the compact space having the topology of a real projective plane, i.e., a 2-sphere with opposite points identified, is considered. The space is locally flat except for two conical singularities where the curvature is concentrated. One supersymmetry is preserved in the effective 4d theory. The unified gauge symmetry, for example SU(5) , is broken only by the non-trivial global topology. In contrast to the Hosotani mechanism, no adjoint Wilson-line modulus associated with this breaking appears. Since, locally, SU(5) remains a good symmetry everywhere, no UV-sensitive threshold corrections arise and SU(5)-violating local operators are forbidden. Doublettriplet splitting can be addressed in the context of a 6d N = 2 super Yang-Mills theory with gauge group SU(6). If this symmetry is first broken to SU(5) at a fixed point and then further reduced to the standard model group in the above non-local way, the two light Higgs doublets of the MSSM are predicted by the group-theoretical and geometrical structure of the model.

Distributional sources of matter on codimension-two and higher branes are only well-defined as regularized objects. Nevertheless, intuition from effective field theory suggests that the low-energy physics on such branes should be independent of any highenergy regularization scheme. In this paper, we address this issue in the context of a scalar field model where matter fields (the standard model) living on such a brane interact with bulk fields (gravity). The low-energy effective theory is shown to be consistent and independent of the regularization scheme, provided the brane couplings are renormalized appropriately at the classical level. We perform explicit computations of the classical renormalization group flows at tree and one-loop level, demonstrate that the theory is renormalizable against codimension-two divergences, and extend the analysis to several physical applications such as electrodynamics and brane localized kinetic terms.