Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions

Abstract

a b s t r a c tWe disclose the mathematical structure underlying the gauge field sector of the recently constructed non-abelian superconformal models in six space-time dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge fields. We show that the algebraic consistency constraints governing this system permit to define a Lie 3-algebra, generalizing the structural Lie algebra of a standard Yang-Mills theory to the setting of a higher bundle. Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type operator Q , this higher bundle can be compactly described by means of a Q -bundle; its fiber is the shifted tangent of the Q -manifold corresponding to the Lie 3-algebra and its base the odd tangent bundle of space-time equipped with the de Rham differential. The generalized Bianchi identities can then be retrieved concisely from Q 2 = 0, which encode all the essence of the structural identities. Gauge transformations are identified as vertical inner automorphisms of such a bundle, their algebra being determined from a Q -derived bracket.

The Embedding Tensor, Leibniz–Loday Algebras, and Their Higher Gauge Theories

“…It is the quotient space of V by the vector space of squares S, which we introduced in the previous section. This observation generalizes [15]: to truncate a Lie m-algebra like V • = m i=1 V −i to a Lie n-algebra for some n < m, one removes all the vector spaces V p with p < −n and considers (3.20) where W = V −n / ker(l 1 ) is concentrated in degree -n. We remark in parenthesis that also W := V −n /im(l 1 ) ⊃ V −n / ker(l 1 ) permits a consistent and in general larger truncation to a Lie n-algebra; however, the physically relevant one describing a finite tensor hierarchy is the previous one and we restrict our discussion to this choice therefore.…”

“…Second, we show that every Leibniz algebra gives canonically rise to a higher homotopy version of a Lie algebra: a Lie ∞-algebra [10][11][12]. In fact, it is known that, at least in the absence of scalar fields, every higher gauge theory underlies a structural Lie ∞-algebra, cf, e.g., [13][14][15]. It is thus comforting to know that there is such a canonical Lie ∞-algebra associated to every Leibniz algebra.…”

Brane Wess–Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid

“…Before we begin reviewing material we mention some work which is perhaps not a direct ancestor of this paper but is nevertheless relevant. Tensor hierarchies (different from the EFT one considered here) have been connected to L ∞ -algebras before, by at least two different groups [43,44]. There have been a number of papers proposing derived bracket structures for DFT [21,45,46], two for EGG [17] [16] and one for EFT [19]; in the last two one also finds the EGG/EFT L ∞ -algebra structure respectively in the M-theory case (although none exhibit the L ∞ -algebroid or dg-symplectic structure which is the main point of this paper).…”

Leibniz–Chern–Simons Theory and Phases of Exceptional Field Theory

“…The results of[39] then imply that this algebraic structure forms part of an L8 algebra[40]. See also[41][42][43].We will leave a more detailed discussion of the significance of such algebras in this context for future work.…”