Celebratio Mathematica

M. Movshev and A. Schwarz: “On maximally supersymmetric Yang–Mills theories,” Nuclear Phys. B 681 : 3 (2004), pp. 324–350. MR 2038191 Zbl 1044.81097 ArXiv hep-th/0311132 article

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, \( L_{\infty} \)- and \( A_{\infty} \)-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 \( (\Omega,\partial) \) of \( (0,k) \)-forms on some supermanifold; the Lie algebra is tensor product of \( (\Omega,\partial) \) and matrix algebra. We construct several other algebras that are quasiisomorphic to \( (\Omega,\partial) \) and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, \( (\Omega,\partial) \) is quasiisomorphic to the algebra \( (B,d) \), constructed by Berkovits. The algebras \( (\Omega_0,\partial) \) and \( (B_0,d) \) obtained from \( (\Omega,\partial) \) 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.

M. Movshev and A. Schwarz: “Algebraic structure of Yang–Mills theory,” pp. 473–523 in The unity of mathematics (Cambridge, MA, 31 August–4 September 2003). Edited by P. Etingof, V. Retakh, and I. M. Singer. Progress in Mathematics 244. Birkhäuser (Boston), 2006. Conference in honor of the ninetieth birthday of I. M. Gelfand. MR 2181815 Zbl 1229.81281 ArXiv hep-th/0404183 incollection

In the present paper we analyze algebraic structures arising in Yang–Mills theory. The paper should be considered as a part of a project started with [2004] and devoted to maximally supersymmetric Yang–Mills theories. In this paper we collected those of our results which are correct without assumption of supersymmetry and used them to give rigorous proofs of some results of [2004]. We consider two different algebraic interpretations of Yang–Mills theory — in terms of \( A_{\infty} \)-algebras and in terms of representations of Lie algebras (or associative algebras). We analyze the relations between these two approaches and calculate some Hochschild (co)homology of algebras in question.

Michael V. Movshev	Related
Vladimir Solomonovich Retakh	Related
Pavel Ilyich Ètingof	Related
Israïl Moiseevich Gelfand	Related
Isadore M. Singer	Related	Volume

M. Movshev, A. Schwarz, and R. Xu: Homology of Lie algebra of supersymmetries. Preprint, 2010. ArXiv 1011.4731 techreport

Michael V. Movshev	Related
Renjun Xu	Related

M. V. Movshev and A. Schwarz: “Maximal supersymmetry,” pp. 175–193 in Supersymmetry in mathematics and physics (Los Angeles, 6–7 February 2010). Edited by S. Ferrara, R. Fioresi, and V. S. Varadarajan. Lecture Notes in Mathematics 2027. Springer (Berlin), 2011. MR 2906343 Zbl 1246.81380 incollection

We have studied supersymmetric and super Poincaré invariant deformations of maximally supersymmetric gauge theories, in particular, of ten-dimensional super Yang–Mills theory and of its reduction to a point. We have described all infinitesimal super Poincaré invariant deformations of equations of motion and proved that all of them are Lagrangian deformations and all of them can be extended to formal deformations. Our methods are based on homological algebra, in particular, on the theory of \( L_{\infty} \) and \( A_{\infty} \)-infinity algebras. In this paper we formulate some of the results we have obtained, but skip all proofs. However, we describe the results of the theory of \( L_{\infty} \) and \( A_{\infty} \) algebras that serve as the main tool in our calculations.

Michael V. Movshev	Related
Rita Fioresi	Related
Veeravalli S. Varadarajan	Related
Sergio Ferrara	Related

M. V. Movshev, A. Schwarz, and R. Xu: “Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra,” Nuclear Phys. B 854 : 2 (2012), pp. 483–503. MR 2844329 Zbl 1229.81117 ArXiv 1106.0335 article

Michael V. Movshev	Related
Renjun Xu	Related

M. V. Movshev and A. Schwarz: “Supersymmetric deformations of maximally supersymmetric gauge theories,” J. High Energy Phys. 2012 : 9 (2012), pp. article no. 136, 77 pages. MR 3044913 Zbl 1397.81390 ArXiv 0910.0620 article

We study supersymmetric and super Poincaré invariant deformations of tendimensional super Yang–Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang–Mills theory and deformations of the reduction to a point. We also discuss how these infinitesimals can be extended to formal deformations. Our methods are based on homological algebra, in particular, on the theory of \( L_{\infty} \) and \( A_{\infty} \) algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.

R. Xu, M. Movshev, and A. Schwarz: “Integral invariants in flat superspace,” Nuclear Phys. B 884 (2014), pp. 28–43. MR 3214872 Zbl 1323.81093 ArXiv 1403.1997 article

Michael V. Movshev	Related
Renjun Xu	Related

M. V. Movshev and A. Schwarz: “Generalized Chern–Simons action and maximally supersymmetric gauge theories,” pp. 327–340 in String-Math 2012 (Bonn, Germany, 16–21 July 2012). Edited by R. Donagi, S. Katz, A. Klemm, and D. R. Morrison. Proceedings of Symposia in Pure Mathematics 90. American Mathematical Society (Providence, RI), 2015. MR 3409803 Zbl 1356.81200 ArXiv 1304.7500 incollection

Michael V. Movshev	Related
David Robert Morrison	Related
Sheldon H. Katz	Related
Albrecht Klemm	Related
Ron Yehuda Donagi	Related

M. Movshev and A. Schwarz: “Quantum deformation of planar amplitudes,” J. High Energy Phys. 2018 : 4 (2018). article no. 121, 20 pages. MR 3801153 Zbl 1390.81613 ArXiv 1711.10053 article

In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells. (This was proven by N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka.) We establish that volume forms on positroids used to express scattering amplitudes can be \( q \)-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.

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Albert S. Schwarz

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