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Lie algebroids, Lie groupoids and TFT
We construct the moduli spaces associated to the solutions of equations of
motion (modulo gauge transformations) of the Poisson sigma model with target
being an integrable Poisson manifold. The construction can be easily extended
to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid
one can associate a BF-like topological field theory which localizes on the
space of algebroid morphisms, that can be seen as a generalization of flat
connections to the groupoid case. We discuss the finite gauge transformations
and discuss the corresponding moduli spaces. We consider the theories both
without and with boundaries.Comment: 21 page
Omni-Lie Superalgebras and Lie 2-superalgebras
We introduce the notion of omni-Lie superalgebra as a super version of the
omni-Lie algebra introduced by Weinstein. This algebraic structure gives a
nontrivial example of Leibniz superalgebra and Lie 2-superalgebra. We prove
that there is a one-to-one correspondence between Dirac structures of the
omni-Lie superalgebra and Lie superalgebra structures on subspaces of a super
vector space.Comment: 14page
Categorified central extensions, \'etale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is
the Lie algebra of a Lie group, fails in infinite dimensions. The modern
account on this phenomenon is the integration problem for central extensions of
infinite-dimensional Lie algebras, which in turn is phrased in terms of an
integration procedure for Lie algebra cocycles.
This paper remedies the obstructions for integrating cocycles and central
extensions from Lie algebras to Lie groups by generalising the integrating
objects. Those objects obey the maximal coherence that one can expect.
Moreover, we show that they are the universal ones for the integration problem.
The main application of this result is that a Mackey-complete locally
exponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie
2-group in the sense that there is a natural Lie functor from certain Lie
2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic
Lie algebra.Comment: 34 pages, essentially revised, to appear in Adv. Mat
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