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 orientations (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 multi-oriented 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 \(d\ge 4\) 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 \(d=2\)).
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Notes
As the symmetric monoidal category of infinite-dimensional vector spaces is not closed, one must be careful about the definition of the endomorphism prop \({{\mathcal {E}}}nd_V\) in this category, see Sect. 4.1 for details.
Strictly speaking, this is true only in finite dimensions. In infinite dimensions, the subspaces \(W_i^+\) are defined as direct limits of systems of finite-dimensional spaces while their complements \(W^-_i\) always come as projective limits, so their intersection makes sense only at the level of finite-dimensional systems first (it is here where the interpretation of \(W^+\) and \(W^-\) as subspaces of one and the same vector space plays its role), and then taking either the direct or projective limit in accordance with the rule explained in Sect. 4.
The (unordered) tensor product \(\bigotimes _{i\in I} X_i\) of vector spaces \(X_i\) labelled by elements i of a finite set I of cardinality, say, n is defined as the space of \({{\mathbb {S}}}_n\)-coinvariants \( \left( \bigoplus _{\sigma : [n]{\mathop {\longrightarrow }\limits ^{\simeq }} I} X_{\sigma (1)}\otimes X_{\sigma (2)} \otimes \ldots \otimes X_{\sigma (n)}\right) _{{{\mathbb {S}}}_n}.\)
Here, we denote the elements of \([1^+]\) by \(\bar{0}\) and \(\bar{1}\) so that the value \({{\mathfrak {s}}}_i\) of the map \({{\mathfrak {s}}}\) on an element \(i\in I\) is itself a map of sets \({{\mathfrak {s}}}_i:\{\bar{0},\bar{1}\} \rightarrow \{out,in\}\).
Here we use the facts that for any vector space M and any inverse system of finite-dimensional vector spaces \(\{N_i\}\) one has \(\displaystyle \lim _{\longleftarrow } \text {Hom}(N_i,M)\cong \text {Hom}(\lim _{\longrightarrow }N_i, M)\) and \(\displaystyle \lim _{\longleftarrow }\text {Hom}(M, N_i) \cong \text {Hom}(M, \lim _{\longleftarrow }N_i)\), while \(\displaystyle \lim _{\longleftarrow } (N_i\otimes M)\cong (\lim _{\longleftarrow }N_i)\otimes M\) only if M is finite-dimensional. On the other hand, for any direct system \(\{N_i\}\) the equality \(\displaystyle \lim _{\longrightarrow }(M \otimes N_i) \cong M\otimes \lim _{\longrightarrow }N_i\) holds true for any M, while the equality \(\displaystyle \lim _{\longrightarrow }\text {Hom}(M, N_i) \cong \text {Hom}(M,\lim _{\longrightarrow }N_i) \) is true if and only if M is finite-dimensional.
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Acknowledgements
It is a pleasure to thank Assar Andersson, Anton Khoroshkin, Thomas Willwacher and Marko Živković for valuable discussions. I am also grateful to the referee for several very useful comments and suggestions.
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Merkulov, S. Multi-oriented props and homotopy algebras with branes. Lett Math Phys 110, 1425–1475 (2020). https://doi.org/10.1007/s11005-019-01248-x
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DOI: https://doi.org/10.1007/s11005-019-01248-x