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We review recent progress regarding the double scaled Sachdev-Ye-Kitaev model and other $p$-local quantum mechanical random Hamiltonians. These models exhibit an expansion using chord diagrams, which can be solved by combinatorial methods.…
Let $\alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $\alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all…
Recent work has shown that two-dimensional non-linear $\sigma$-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces. Here we extend this idea to…
This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal…
We solve the topological Poisson Sigma model for a Poisson-Lie group $G$ and its dual $G^*$. We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target. The resolution of…
We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two-dimensional model we construct is a supersymmetric relative of…
Auxiliary field techniques have recently gained interest in four-dimensional non-linear electrodynamics and two-dimensional integrable sigma models. In these settings, coupling a suitable ``seed'' theory to auxiliary fields provides a…
These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of…
The (2+1)-dimensional analog self-dual gravity which is obtained via spacetime dimension reduction of the (3+1)-dimensional Holst action without reducing the internal gauge group is studied. A Chern-Simons formulation for this theory is…
We consider the concept of a generalised manifold in the O(d,d) setting, i.e., in double geometry. The conjecture by Hohm and Zwiebach for the form of finite generalised diffeomorphisms is shown to hold. Transition functions on overlaps are…
We present a general formalism for computing the Hodge dual of differential forms in arbitrary dimensions subject to a spherical constraint. This problem arises naturally in Kaluza-Klein compactifications, where sphere reductions demand…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that…
We revisit the proposal that coupling two six-dimensional holomorphic Chern-Simons theories generates gaugings throughout the twistor-space diamond relating 6d hCS, 4d self-dual Yang-Mills, 4d Chern-Simons, and 2d integrable models. In…
The surface charges associated with $p$-form gauge fields in the Bondi patch of $D$-dimensional Minkowski spacetime are computed. We show that, under the Hodge duality between the field strengths of the dual formulations, electric-like…
We propose a symmetry law for a doublet of different form fields, which resembles gauge transformations for matter fields. This may be done for general Lie groups, resulting in an extension of Lie algebras and group manifolds. It is also…
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded…
Using bicomplex formalism we construct generalizations of Fordy-Kulish systems of matrix nonlinear Schroedinger equations on two-dimensional space-time in two respects. Firstly, we obtain corresponding equations in three space-time…
We give a global formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields for the generalized situation when the "duality structure" of the Abelian gauge theory is described by a flat symplectic vector…
We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold $X$ to an…
We study the Coulomb-Higgs duality of N=2 supersymmetric Abelian Chern-Simons theories in 2+1 dimensions, by compactifying dual pairs on a circle of radius R and comparing the resulting N=(2,2) theories in 1+1 dimensions. Below the…