On maximally supersymmetric Yang–Mills theories

Abstract

We consider ten-dimensional supersymmetric Yang-Mills theory (10D SUSY YM theory) and its dimensional reductions, in particular, BFSS and IKKT models. We formulate these theories using algebraic techniques based on application of differential graded Lie algebras and associative algebras as well as of more general objects, L∞-and A∞-algebras.We show that using pure spinor formulation of 10D SUSY YM theory equations of motion and isotwistor formalism one can interpret these equations as Maurer-Cartan equations for some differential Lie algebra. This statement can be used to write BV action functional of 10D SUSY YM theory in Chern-Simons form. The differential Lie algebra we constructed * The work of both authors was partially supported by NSF grant No. DMS 0204927 1 is closely related to differential associative algebra (Ω,∂) of (0, k)-forms on some supermanifold; the Lie algebra is tensor product of (Ω,∂) and matrix algebra . We construct several other algebras that are quasiisomorphic to (Ω,∂) and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, (Ω,∂) is quasi-

Operators on Pure Spinor Spaces

“…The fact that pure spinors had a rôle to play in maximally supersymmetric models was recognised early by Nilsson [1] and Howe [2,3]. Pure spinor superfields were developed with the purpose of covariant quantisation of superstrings by Berkovits [4,5,6,7] and the cohomological structure was independently discovered in supersymmetric field theory and supergravity, originally in the context of higher-derivative deformations [8,9,10,11,12,13,14,15]. The present lecture only deals with pure spinors for maximally supersymmetric field theory.…”

“…[5] and ref. [16]) with the six terms in the nominator of the partition function Z(t) = 1 − 10t 2 + 16t 3 − 16t 5 + 10t 6 − t 8 (1 − t) 16 = (1 + t 2 )(1 + 4t + t 2 ) (1 − t) 11 .…”

‐Algebras, the BV Formalism, and Classical Fields

“…We introduce the second‐order Yang–Mills complex by setting [] where ⋆ is the Hodge operator on X . This complex can be given an ‐structure by defining the non‐vanishing products by [] …”

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

“…In particular, the Batalin–Vilkovisky (BV) formalism associates to each classical field theory an ‐algebra and for interacting field theories, this ‐algebra is not merely a differential graded Lie algebra. This fact is well‐known to experts on BV quantisation, see for example [], in particular [], which is based on the earlier work [], or the later works [], but it seems to be much less known in general. The recent paper [] revived interest in the ‐algebras of classical field theories, but only a very partial picture of the categorified structures and their origin was given.…”