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 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 quasiisomorphic to the algebra ( B, d), constructed by Berkovits. The algebras (Ω 0, ∂ ̄ ) and ( B 0, d) obtained from (Ω, ∂ ̄ ) and ( B, d) by means of reduction to a point can be used to give a BV-formulation of IKKT model. We introduce associative algebra SYM as algebra where relations are defined as equations of motion of IKKT model and show that Koszul dual to the algebra ( B 0, d) is quasiisomorphic to SYM.