Tannakian classification of equivariant principal bundles on toric varieties

Let $X$ be a complete toric variety equipped with the action of a torus $T$ and $G$ a reductive algebraic group, defined over an algebraically closed field $K$. We introduce the notion of a compatible $\Sigma$--filtered algebra associated to $X$, generalizing the notion of a compatible $\Sigma$--filtered vector space due to Klyachko, where $\Sigma$ denotes the fan of $X$. We combine Klyachko's classification of $T$--equivariant vector bundles on $X$ with Nori's Tannakian approach to principal $G$--bundles, to give an equivalence of categories between $T$--equivariant principal $G$--bundles on $X$ and certain compatible $\Sigma$--filtered algebras associated to $X$, when the characteristic of $K$ is $0$.Comment: 22 pages, revised version, to appear in Transform. Group

A classification of equivariant principal bundles over nonsingular toric varieties

We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.Comment: 14 page

On equivariant Serre problem for principal bundles

Let $E_G$ be a $\Gamma$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $\Gamma$, where $G$ and $\Gamma$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\Gamma$--space with a dense orbit, and $x_0 \in X$ is a $\Gamma$--fixed point. We show that if $\Gamma$ is reductive, then $E_G$ admits a $\Gamma$--equivariant isomorphism with the product principal $G$--bundle $X \times_{\rho} E_G(x_0)$, where $\rho\,:\, \Gamma \, \longrightarrow\, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety.Comment: References added. To appear in the International Journal of Mathematic

Restriction theorems for Higgs principal bundles

AbstractWe prove analogues of Grauert–Mülich and Flennerʼs restriction theorems for semistable principal Higgs bundle over any smooth complex projective variety

Notes on Melonic O(N)q−1O(N)^{q-1} Tensor Models

It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group $O(N)^{q-1}$ agrees with the large $N$ limit of the SYK model. In these notes we investigate aspects of the dynamics of the $O(N)^{q-1}$ theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has $(q-1)\frac{N^2}{2}$ new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like $E \ln E$ (i.e. faster than Hagedorn) up to energies of order $N^2$. The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order $1/\ln N$. We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in $J/m$.Comment: 60 pages, 7 figure

On stability of tangent bundle of toric varieties

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.Comment: Revised version. To appear in Proc. Indian Acad. Sci. Math. Sc