sh-Lie Algebras Induced by Gauge Transformations

Abstract

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when 'gauge parameters' act in a field dependent way. Such symmetries appear in several field theories, most notably in a 'Poisson induced' class due to Schaller and Strobl [SS94] and to Ikeda[Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of 'particles of spin > 2 led Berends, Burgers and van Dam [Bur85, BBvD84, BBvD85] to study 'field dependent parameters' in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-lie algebra (L ∞ -algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries.

A Note on Gauge Systems from the Point of View of Lie Algebroids

“…Indeed, in some way or the other, this is known to most people familiar with the Batalin-Vilkovisky construction, see for instance [9]. Related considerations have appeared for instance in [10]. Note that the off-shell description gives rise to an sh-Lie algebroid, while L-stage reducible gauge theories are L-Lie algebroids.…”

On the structure of quantum L∞ algebras

“…Furthermore we equip all L n products with an upper index from the set {T, W, ǫ, η} that denotes in which of the four subspaces of X the image of the higher product L n is located. 5 Notice that when expanding the fraction β we get an infinite series with terms at any order in . Separating the different powers of g in different L g n products, as usually done in loop L∞ algebras, see (3.21), is therefore not illuminating in this example.…”

“…For the first time, they actually appeared in the context of higher spin gauge theories [2] and were also discussed in the mathematics literature (see e.g. [3][4][5][6]). Motivated by the study of field theory truncations of string field theory [7], the authors of [8] argued that the symmetry and the action of any consistent perturbative gauge symmetry is controlled by an L ∞ algebra.…”

A Riccati type PDE for light-front higher helicity vertices

“…(v) In [8] Taronna tries to synthesize much of the accumulated knowledge of cubic (and quartic) interactions into a comprehensive scheme based on a general construction of FDA's (see for instance [53]) and L ∞ methods in the spirit of [54][55][56][57]. See also [58,59] for the present author's attempts along such lines.…”