New topological field theories in two dimensions

Abstract

It is shown that two(1 + 1)-dimensional (2D) free Abelian-and self-interacting non-Abelian gauge theories (without any interaction with matter fields) belong to a new class of topological field theories. These new theories capture together some of the key features of Witten-and Schwarz type of topological field theories because they are endowed with symmetries that are reminiscent of the Schwarz type theories but their Lagrangian density has the appearance of the Witten type theories. The topological invariants for these theories are computed on a 2D compact manifold and their recursion relations are obtained. These new theories are shown to provide a class of tractable field theoretical models for the Hodge theory in two dimensions of flat (Minkowski) spacetime where there are no propagating degrees of freedom associated with the 2D gauge boson. *

Abelian 2-form gauge theory: special features

“…This feature is similar to the 2D free Abelian and self-interacting non-Abelian gauge theories which are topological in nature [33,39,40]. (ii) There exist twelve sets of (anti-)BRST and (anti-)co-BRST invariant quantities for the Lagrangian density (2.1) that obey the recursion relations that are reminiscent of such relationships in the context of exact TFTs.…”

“…This fact should be contrasted with the case of 2D free Abelian gauge theory, for which, the signs are same in the equations corresponding to (3.2) and (3.6) [33]. This peculiarity happens because the duality in 4D and 2D are different [48][49][50].…”

“…Because of the existence of the above mapping, the analogue of the Hodge decomposition theorem can be defined in the quantum Hilbert space of states [25]. In fact, all the above cited properties of this section are common to the 2D one-form free Abelian (and self-interacting non-Abelian) as well as 4D free Abelian 2-form gauge theories [25,[31][32][33][34][35][36] except for the sign difference in the inverse relationship (3.6). To be more precise, the analogues of (3.2) and (3.6) for the 2D theories bear the same signs on the r.h.s.…”

“…density and symmetric energy momentum tensor which could be expressed as the sum of the (anti-)BRST and (anti-)co-BRST anti-commutators [33,[38][39][40][41]. The geometrical interpretations for all the symmetry transformations and corresponding conserved charges (as generators) have been discussed in the framework of superfield formulation (see, e.g., [38][39][40][41][42][43] for details).…”

“…The latter 2D theories are also exact TFTs because, the existence of (anti-)BRST and (anti-)co-BRST symmetries in the theory, turns out to be responsible for gauging out both the propagating degrees of freedom of 2D photon (and/or 2D gluon) (see, e.g., [31][32][33]37] for details). In the recent past, there has been some interest in studying almost TFTs (or quasi-TFTs) which involve constraints that leave out only a finite number of degrees of freedom in the theory [37,46].…”