Related papers: Quantum BV $\mathcal{L}_{\infty}$-algebras I: Deri…
Let us consider a Lie (super)algebra $G$ spanned by $T_{\alpha}$ where $T_{\alpha}$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{\alpha_1...\alpha_k}$ that determines a homology class of the Lie algebra…
We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures…
In the foundational logical framework of homotopy-type theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of non-perturbative cohomological…
Quillen showed how to describe the homotopy theory of simply-connected rational spaces in terms of differential graded Lie algebras. Here we survey a generalization of Quillen's results that describes the $v_n$-periodic localizations of…
We discuss $w_\infty$ and $sl_q(2)$ symmetries in multiple Chern-Simons theory on a torus. It is shown that these algebraic structures arise from the dynamics of the non-integrable phases of the Chern-Simons fields. The generators of these…
Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of…
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…
We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras,…
We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative…
Recent developments of Batalin-Vilkovisky (BV) formalism and related geometry are reviewed. Mathematical structures of BV formalism are summarized as a Q-manifold and a QP-manifold. Lie algebras, Lie algebroids and other higher algebroids…
It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum…
This paper is about algebro-geometrical structures on a moduli space $\CM$ of anomaly-free BV QFTs with finite number of inequivalent observables or in a finite superselection sector. We show that $\CM$ has the structure of F-manifold -- a…
It is believed that any classical gauge symmetry gives rise to an L$_\infty$ algebra. Based on the recently realized relation between classical ${\cal W}$ algebras and L$_\infty$ algebras, we analyze how this generalizes to the quantum…
In this work we report a homological perturbation calculation to construct effective theories of topological quantum mechanics on $\mathbb{R}_{\geqslant 0}$. Such calculation can be regarded as a generalization of Feynman graph computation.…
This paper studies averaging algebras, say, associative algebras endowed with averaging operators. We develop a cohomology theory for averaging algebras and justify it by interpreting lower degree cohomology groups as formal deformations…
Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner…
A A BV algebra and a QP-structure of degree 3 is formulated. A QP-structure of degree 3 gives rise to Lie algebroids up to homotopy and its algebraic and geometric structure is analyzed. A new algebroid is constructed, which derives a new…
The homotopy Lie or $L_{\infty}$ algebra encoding Yang-Mills theory is the tensor product of a color Lie algebra with the kinematic $C_{\infty}$ algebra. We derive this $C_{\infty}$ algebra, via homotopy transfer, from a strict operator…
It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of…
We develop representation theoretic techniques to construct three dimensional non-semisimple topological quantum field theories which model homologically truncated topological B-twists of abelian Gaiotto-Witten theory with linear matter.…