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We continue to study doubled aspects of algebroid structures equipped with the C-bracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant algebroids are constructed by the analogue of the Drinfeld double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space-time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some algebroids. The doubled structures of twisted brackets and those associated with group manifolds are briefly discussed.
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The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.
The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfeld double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfeld double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,tilde{L})$ in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for $(L,tilde{L})$ be a Lie bialgebroid. The analysis provides a foundation toward the coquecigrue problem for the gauge symmetry in DFT.
In this paper we discuss two constructions of an effective field theory starting from a local interaction functional. One relies on the well-established graphical combinatorics of the BPHZ algorithm to renormalize divergent Feynman amplitudes. The other, more recent and due to Costello, relies on an inductive construction of local counterterms that uses no graphical combinatorics whatsoever. We show that these two constructions produce the same effective field theory.
Equivalence between algebraic structures generated by parastatisticstriple relations of Green (1953) and Greenberg -- Messiah (1965), and certain orthosymplectic $mathbb{Z}_2times mathbb{Z}_2$-graded Lie superalgebras is found explicitly. Moreover, it is shown that such superalgebras give more complex para-Fermi and para-Bose systems then ones of Green -- Greenberg -- Messiah.
Lecture notes for the course Batalin-Vilkovisky formalism and applications in topological quantum field theory given at the University of Notre Dame in the Fall 2016 for a mathematical audience. In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.
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