17,327 research outputs found
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
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