Multi-oriented props and homotopy algebras with branes

Abstract

We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of k linear subspaces in that space, k being the number of extra directions (if k = 0 this structure recovers an ordinary prop); symplectic vector spaces equipped with k Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context of this paper Manin triples are precisely symplectic Lagrangian representations of the 2-oriented generalization of the classical operad of Lie algebras. In a sense, the theory of multi-oriented props provides us with a far reaching strong homotopy generalization of Manin triples type constructions.The homotopy theory of multi-oriented props can be quite non-trivial (and different from that of ordinary props). The famous Grothendieck-Teichmüller group acts faithfully as homotopy non-trivial automorphisms on infinitely many multioriented props, a fact which motivated much the present work as it gives us a hint to a non-trivial deformation quantization theory in every geometric dimension d ≥ 4 generalizing to higher dimensions Drinfeld-Etingof-Kazhdan's quantizations of Lie bialgebras (the case d = 3) and Kontsevich's quantizations of Poisson structures (the case d = 2). 1 2 ... k and defines rules for multi-oriented prop compositions via contractions along admissible multi-oriented subgraphs. However, it is much less evident how to transform that more or less standard rules into non-trivial and interesting representations (i.e. examples) -the intuition from the theory of ordinary (wheeled) props does not help much. Adding new k directions to each edges of a decorated graph of an ordinary prop can be naively understood as extending that ordinary prop into a 2 k -coloured one, but then the requirement that the new directions on graph edges do not create "wheels" (that is, closed directed paths of edges with respect to any of the new orientations) kills that naive picture immediately -the elements of the set of 2 k new colours start interacting with each other in a non-trivial way. We know which structure distinguishes ordinary props (the ones with no wheels in the given single orientation, i.e. the ones which are 1-oriented in the terminology of this paper) from the ordinary wheeled props (that is, 0-oriented 1directed props in the terminology of this paper) in terms of representations in, say, a graded vector space V -it is the 1 dimension of V. In general, a wheeled prop can have well-defined representations in V only in the case dim V < ∞ as graphs with wheels generate the trace operation V × V * → K which explodes in the case dim V = ∞; this phenomenon explains the need for 1-oriented props. How to explain the need for all (or some) of the extra k directions to be oriented? Which structure on a graded vector space V can be used to...

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

“…On the other side, multi-oriented graph complexes play a key role in the deformation theory of multioriented props recently invented by Sergei Merkulov in [7] and to appear in [8]. In his papers Merkulov gives a meaning to the extra directions, and provides interesting applications and representations of multi-oriented props.…”

A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

“…This graded geometric description of the deformation theory of Lie bialgebroids and generalisations thereof will be instrumental in formulating associated universal models in Section 4. The class of universal models introduced in this note involves multi-oriented graphs, as introduced in [63] and studied in [39] in the context of multi-oriented props and their representations on homotopy algebras with branes. The main definitions and results regarding the cohomology of multi-oriented graph complexes are reviewed in Section 3.…”

“…The aim of this section is to review the definition and main results regarding multi-oriented graph complexes and their cohomology, as introduced and studied in [63,64,39] (cf. also [40] for a review).…”

“…30 See Section 5 of [39] and Section 7 of [40] for reviews. 31 Where O(g) denotes the completed symmetric algebra associated with the graded vector space g. 32 See [56] and [57] for the cases o = 0, 1 respectively and [39] for a general statement. 33 Here, fcGC d stands for the usual Kontsevich graph complex of connected undirected graphs.…”

“…33 Here, fcGC d stands for the usual Kontsevich graph complex of connected undirected graphs. 34 See [56,10] for the first arrow and [39] for a general statement. 35 The first Betti number is defined as b := k − N + 1 for a connected graph with N vertices and k edges.…”