Noncommutative gauge and gravity theories and geometric Seiberg–Witten map

Abstract

We give a pedagogical account of noncommutative gauge and gravity theories, where the exterior product between forms is deformed into a $$\star$$ ⋆ -product via an abelian twist (e.g. the Groenewold–Moyal twist). The Seiberg–Witten map between commutative and noncommutative gauge theories is introduced. It allows to express the action of noncommutative Einstein gravity coupled to spinor fields in terms of the usual commutative action with commutative fields plus extra interaction terms dependent on the noncommutativity parameter.

Poisson electrodynamics with charged matter fields

“…With some caution, one can view L(A) as the Lie algebra of the infinite-dimensional Lie group G(A). Regarding the elements of A as scalar fields and replacing the pointwise product of functions with the ⋆-product above, it is not hard to formulate noncommutative counterparts of various field-theoretical models [6][7][8][9].…”

“…A glance is enough to recognize in formulas (2.16) and (2.19) the leading terms of the Seiberg-Witten map for a rank-one gauge theory [5,8,32]. By the very logic, these formulas should arise as the semi-classical limit of the Seiberg-Witten map relating the abelian gauge transformations δ ε A = dε with non-abelian ones (2.6).…”

“…It was shown that the lowenergy effective field theory on a single D-brane has a consistent deformation to a noncommutative gauge theory. This result triggered an explosion of research on various aspects of noncommutative field theory, which is reviewed in [6][7][8][9]. In particular, it became clear that the hopes pinned on noncommutativity were too optimistic: the simplest field-theoretical models on Weyl-Moyal spaces still suffer from divergences due to the phenomenon of UV/IR mixing [10,11].…”

All-Russian scientific conference “Regional dimension of the Russian economic reforms” (Novosibirsk, September 16-17, 2021)

“…With some caution, one can view L(A) as the Lie algebra of the infinite-dimensional Lie group G(A). Regarding the elements of A as scalar fields and replacing the pointwise product of functions with the * -product above, it is not hard to formulate noncommutative counterparts of various field-theoretical models, see [1][2][3][4] for review.…”

“…A glance is enough to recognize in formulas (1.16) and (1.19) the leading terms of the Seiberg-Witten map for a rank-one gauge theory [1], [4], [8]. By the very logic, these formulas should arise as the semi-classical limit of the Seiberg-Witten map relating the abelian gauge transformations δ ε A = dε with non-abelian ones (1.6).…”