Abstract
I propose a self-dual deformation of the classical phase space of lattice Yang--Mills theory, in which both the electric and magnetic fluxes take value in the gauge Lie group. A local construction of the deformed phase space requires the machinery of "quasi-Hamiltonian spaces" by Alekseev et al., which is here reviewed. The results is a full-fledged finite-dimensional and gauge-invariant phase space, whose self-duality properties are largely enhanced in (3+1) spacetime dimensions. This enhancement is due to a correspondence with the moduli space of an auxiliary non-commutative flat connection living on a Riemann surface defined from the lattice itself, which in turn equips the duality between electric and magnetic fluxes with a neat geometrical interpretation in terms of a Heegaard splitting of the space manifold. Finally, I discuss the consequences of the proposed deformation on the quantization of the phase space, its quantum gravitational interpretation, as well as its relevance for the construction of (3+1) dimensional topological field theories with defects.
Highlights
The nonperturbative study of Yang-Mills theories and quantum gravity often appeals to a latticelike regularization
On the lattice, magnetic fluxes are encoded in Lie group variables, whereas electric fluxes are encoded in Lie-algebra variables
This distinction is relevant since the first live in a curved compact space, while the latter live in a noncompact linear space. This leads to noncommutative electric flux operators—even at the gaugecovariant level—with a discrete spectrum, while magnetic flux operators keep enjoying a continuum spectrum and commute among themselves
Summary
The nonperturbative study of Yang-Mills theories and quantum gravity often appeals to a latticelike regularization. This framework was developed in the late 1990s mostly by Alekseev et al [1,2,3,4] Their principal motivation was to provide a finite-dimensional construction of the symplectic structure on the moduli space of flat connection on a Riemann surface [1,5,6,7,8], generalizing the work of Fock and Rosly [9] to the compact group case. [1]—of the statement that the deformed (reduced) phase space of lattice Yang-Mills theory on Γ is naturally isomorphic to the moduli space of an auxiliary flat G-connection on SΓ This third part contains many technical details and is not necessary to follow the rest of the discourse. The first two regard the relation between the (quasi)symplectic and (quasi-)Poisson frameworks, Appendixes A and B, while the last one discusses the relation between the Poisson (non) commutativity of the holonomies in the quasi-Poisson framework with the failure of the Jacobi identity
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