Self-dual Supersymmetric Yang-Mills Theory Research Articles

Manifestly-invariant action for self-dual supersymmetric Yang-Mills theories coupled to auxiliary supergravity

We propose a new action-formulation for the N=(1,1) multiplet (AμI,λI) of a self-dual (SD) supersymmetric Yang-Mills (YM) theory in D=2+2 dimensions using an auxiliary supergravity multiplet. We first consider non-supersymmetric case with constraint lagrangians that restrict the metric gμν to be (ημν)=diag.(+,+,−,−). We add the kinetic-term of a purely-bosonic YM-field, and show that the YM-field in the D=2+2 space-time allows only SD or anti-SD (ASD) configurations. In other words, the condition of vanishing energy-momentum (EM) tensor restricts the YM-field to be either SD or ASD, but not an admixture of them. Finally, we generalize this result to N=(1,1) supersymmetric YM multiplet (AμI,λI;DI) in D=2+2 with the off-shell supergravity multiplet (eμm,ψμ;S,P,am) as an ‘auxiliary’ multiplet without its kinetic terms. Special constraint lagrangians restrict the supergravity multiplet to be auxiliary, i.e., both the Riemann-tensor and the gravitino field strength to vanish. We show that the eμm and ψμ-field equations produce the vanishings of both the EM-tensor and spinor-current, which in turn lead only to either SD or ASD supersymmetric YM multiplet, as the only Lorentz and gauge-covariant supersymmetric solutions. A broad class of similar action-formulations with auxiliary supergravity multiplets yielding SD or ASD systems in D=2+2 or higher-dimensions may be constructed in a similar fashion.

Supersymmetric electric-magnetic duality inD=3+3andD=5+5dimensions as foundation of self-dual supersymmetric Yang-Mills theory

We present electric-magnetic (EM)-duality formulations for non-Abelian gauge groups with $N=1$ supersymmetry in $D=3+3$ and $5+5$ space-time dimensions. We show that these systems generate self-dual $N=1$ supersymmetric Yang-Mills (SDSYM) theory in $D=2+2$. For a $N=2$ supersymmetric EM-dual system in $D=3+3$, we have the Yang-Mills multiplet $({{A}_{\ensuremath{\mu}}}^{I},{{\ensuremath{\lambda}}_{A}}^{I})$ and a Hodge-dual multiplet $({{B}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}}^{I},{{\ensuremath{\chi}}_{A}}^{I})$, with an auxiliary tensors ${{C}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}}^{I}$ and ${K}_{\ensuremath{\mu}\ensuremath{\nu}}$. Here, $I$ is the adjoint index, while $A$ is for the doublet of $Sp(1)$. The EM-duality conditions are ${{F}_{\ensuremath{\mu}\ensuremath{\nu}}}^{I}=(1/4!){{\ensuremath{\epsilon}}_{\ensuremath{\mu}\ensuremath{\nu}}}^{\ensuremath{\rho}\ensuremath{\sigma}\ensuremath{\tau}\ensuremath{\lambda}}{{G}_{\ensuremath{\rho}\ensuremath{\sigma}\ensuremath{\tau}\ensuremath{\lambda}}}^{I}$ with its superpartner duality condition ${{\ensuremath{\lambda}}_{A}}^{I}=\ensuremath{-}{{\ensuremath{\chi}}_{A}}^{I}$. Upon appropriate dimensional reduction, this system generates SDSYM in $D=2+2$. This system is further generalized to $D=5+5$ with the EM-duality condition ${{F}_{\ensuremath{\mu}\ensuremath{\nu}}}^{I}=(1/8!){{\ensuremath{\epsilon}}_{\ensuremath{\mu}\ensuremath{\nu}}}^{{\ensuremath{\rho}}_{1}\ensuremath{\cdots}{\ensuremath{\rho}}_{8}}{{G}_{{\ensuremath{\rho}}_{1}\ensuremath{\cdots}{\ensuremath{\rho}}_{8}}}^{I}$ with its superpartner condition ${\ensuremath{\lambda}}^{I}=\ensuremath{-}{\ensuremath{\chi}}^{I}$. Upon appropriate dimensional reduction, this theory also generates SDSYM in $D=2+2$. As long as we maintain Lorentz covariance, $D=5+5$ dimensions seems to be the maximal space-time dimensions that generate SDSYM in $D=2+2$. Namely, EM-dual system in $D=5+5$ serves as the Master Theory of all supersymmetric integrable models in dimensions $1\ensuremath{\le}D\ensuremath{\le}3$.

ℵ 0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO( N) symmetry. We call these ℵ 0 (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ 0 supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ 0 globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ 0 supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.

SUPER-LAX OPERATOR EMBEDDED IN SELF-DUAL SUPERSYMMETRIC YANG-MILLS THEORY

We show that the super-Lax operator for N=1 supersymmetric Kadomtsev-Petviashvili (SKP) equation of Manin and Radul in three dimensions can be embedded into recently developed self-dual supersymmetric Yang-Mills theory in 2+2 dimensions, based on general features of its underlying super-Lax equation. The whole hierarchy of the SKP equations of Manin and Radul is generated by geometrical superfield equations of self-dual supersymmetric Yang-Mills theory. The differential geometrical relationship in superspace between the embedding principle of the super-Lax operator and its associated super-Sato equation is clarified. This result provides a good guiding principle for the embedding of other integrable subsystems in the super-Lax equation into the four-dimensional self-dual supersymmetric Yang-Mills theory, which is the consistent background for N=2 superstring theory, and potentially generates other unknown supersymmetric integrable models in lower dimensions.

N=2 SUPERSTRING THEORY GENERATES SUPERSYMMETRIC CHERN-SIMONS THEORIES

We show that the action of self-dual supersymmetric Yang-Mills theory in four dimensions, which describes the consistent massless background fields for N=2 superstring, generates the actions for N=1 and N=2 supersymmetric non-Abelian Chern-Simons theories in three dimensions after some dimensional reductions. Since the latters play important roles for supersymmetric integrable models, this result indicates the fundamental significance of the N=2 superstring theory controlling (possibly all) supersymmetric integrable models in lower dimensions.

Self-dual supersymmetric Yang-Mills theory generates two-dimensional supersymmetric WZNW models

We show that recently formulated four-dimensional self-dual supersymmetric Yang-Mills theory, which is consistent background for open N = 2 superstring, generates two-dimensional N = (1, 1), N = (1, 0) and N = (2, 0) supersymmetric gauge Wess-Zumino-Novikov-Witten σ-models on coset manifolds G/H, after appropriate dimensional reductions. This is supporting evidence for the conjecture that the self-dual supersymmetric Yang-Mills theory will generate lower-dimensional supersymmetric integrable models after dimensional reductions.

Self-dual supersymmetric Yang-Mills theory generates Witten's topological field theory

We show that the recently constructed N=4 supersymmetric self-dual Yang-Mills theory as the consistent background of the N=2 open superstring will generate Witten's topological field theory in two dimensions as a descendant theory after appropriate twisted dimensional reduction/truncations. We also show that this topological field theory further generates supersymmetric Korteweg de Vries equations, SL ( n)-Toda theory and W ∞-gravity in the n→∞ limit. Considering also that this topological field theory is to generate other integrable and topological systems, our resutls give supporting evidence for the conjecture that the N=2 open superstring theory is the underlying “master theory” of integrable and topological systems, via the intermediate N=4 supersymmetric self-dual Yang-Mills theory as its consistent target space-time background.

Supersymmetric integrable systems embedded in supersymmetric self-dual Yang-Mills theory

We perform dimensional reductions of recently constructed self-dual N=2 supersymmetric Yang-Mills theory in 2+2 dimensions into two dimensions. We show that the universal equations obtained in these dimensional reductions can embed supersymmetric integrable systems, such as N=1 and N=2 supersymmetric Korteweg-de Vries equations, N=1 supersymmetric Liouville theory or supersymmetric Toda theory. This is the first supporting evidence for the conjecture that the (2+2)-dimensional self-dual supersymmetric Yang-Mills theory generates supersymmetric integrable systems in lower dimensions.