The structure of correlation functions in single field inflation

Many statistics available to constrain non-Gaussianity from inflation are simplest to use under the assumption that the curvature correlation functions are hierarchical. That is, if the n-point function is proportional to the (n-1) power of the two-point function amplitude and the fluctuations are small, the probability distribution can be approximated by expanding around a Gaussian in moments. However, single-field inflation with higher derivative interactions has a second small number, the sound speed, that appears in the problem when non-Gaussianity is significant and changes the scaling of correlation functions. Here we examine the structure of correlation functions in the most general single scalar field action with higher derivatives, formalizing the conditions under which the fluctuations can be expanded around a Gaussian distribution. We comment about the special case of the Dirac-Born-Infeld action.Comment: 21 Pages, fixed typo in arXiv title; added referenc

Poincare-Birkhoff-Witt Theorems

We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics. Along the way, we compare modern techniques used to establish such results, for example, the Composition-Diamond Lemma, Groebner basis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research Institute Proceeding

Finite groups acting linearly: Hochschild cohomology and the cup product

When a finite group acts linearly on a complex vector space, the natural semi-direct product of the group and the polynomial ring over the space forms a skew group algebra. This algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics. Its Hochschild cohomology predicts various Hecke algebras and deformations of the orbifold. In this article, we investigate the ring structure of the Hochschild cohomology of the skew group algebra. We show that the cup product coincides with a natural smash product, transferring the cohomology of a group action into a group action on cohomology. We express the algebraic structure of Hochschild cohomology in terms of a partial order on the group (modulo the kernel of the action). This partial order arises after assigning to each group element the codimension of its fixed point space. We describe the algebraic structure for Coxeter groups, where this partial order is given by the reflection length function; a similar combinatorial description holds for an infinite family of complex reflection groups.Comment: 30 page