Supersymmetric Self-dual Yang-Mills Research Articles

Double-copying self-dual Yang-Mills theory to self-dual gravity on twistor space

We construct a simple Lorentz-invariant action for maximally supersymmetric self-dual Yang-Mills theory that manifests colour-kinematics duality. We also show that this action double-copies to a known action for maximally supersymmetric self-dual gravity. Both actions live on twistor space and illustrate nicely the homotopy algebraic perspective on the double copy presented in [1]. This example is particularly interesting as the involved Hopf algebra controlling the momentum dependence is non-commutative and suggests a generalisation to gauged maximally supersymmetric self-dual gravity.

Self-dual Yang-Mills multiplet in three dimensions coupled to supergravity

We couple a recently established N=1 globally supersymmetric self-dual Yang-Mills multiplet in three dimensions to supergravity. This becomes possible due to our previous result on globally supersymmetric formulation based on a compensator multiplet. We further couple the self-dual vector to a supersymmetric {sigma}-model on the coset SO(8,n)/SO(8)xSO(n) via minimal couplings for an arbitrary gauged subgroup H{sub 0} subset of SO(8)xSO(n). A corresponding superspace formulation is also presented.

On supertwistor geometry and integrability in super gauge theory

In this thesis, we report on different aspects of integrability in supersymmetric gauge theories. The main tool of investigation is twistor geometry. In trying to be self-contained, we first present a brief review about the basics of twistor geometry. We then focus on the twistor description of various gauge theories in four and three space-time dimensions. These include self-dual supersymmetric Yang-Mills (SYM) theories and relatives, non-self-dual SYM theories and supersymmetric Bogomolny models. Furthermore, we present a detailed investigation of integrability of self-dual SYM theories. In particular, the twistor construction of infinite-dimensional algebras of hidden symmetries is given and exemplified by deriving affine extensions of internal and space-time symmetries. In addition, we derive self-dual SYM hierarchies within the twistor framework. These hierarchies describe an infinite number of flows on the respective solution space, where the lowest level flows are space-time translations. We also derive infinitely many nonlocal conservation laws.

Interacting fields of arbitrary spin and N ?gr 4 supersymmetric self-dual Yang-Mills equations

Interacting fields of arbitrary spin and N ?gr 4 supersymmetric self-dual Yang-Mills equations

Interacting fields of arbitrary spin and N > 4 supersymmetric self-dual Yang-Mills equations

We show that the self-dual Yang-Mills equations afford supersymmetrisation to systems of equations invariant under global N-extended super-Poincaré transformations for arbitrary values of N, without the limitation ( N ⩽ 4) applicable to standard non-self-dual Yang-Mills theories. These systems of equations provide novel classically consistent interactions for vector supermultiplets containing fields of spin up to N−2 2 . The equations of motion for the component fields of spin greater than 1 2 are interacting variants of the first-order Dirac-Fierz equations for zero rest-mass fields of arbitrary spin. The interactions are governed by conserved currents which are constructed by an iterative procedure. In (arbitrarily extended) chiral superspace, the equations of motion for the (arbitrarily large) self-dual supermultiplet are shown to be completely equivalent to the set of algebraic supercurvature constraints defining the self-dual superconnection.

Action for N=4 supersymmetric self-dual Yang-Mills theory.

The $N=4$ supersymmetric self-dual Yang-Mills theory in a four- dimensional space with signature $(2,2)$ is formulated in harmonic superspace. The on-shell constraints of the theory are reformulated in the equivalent form of vanishing curvature conditions for three gauge connections (one harmonic and two space-time). The constraints are then obtained as variational equations from a superspace action of the Chern-Simons type. The action is manifestly $SO(2,2)$ invariant. It can be viewed as the Lorentz-covariant form of the light-cone superfield action proposed by Siegel.

Supersymmetric KP equation embedded in supersymmetric self-dual Yang-Mills theory

We show that the N = 1 supersymmetric Kadomtsev-Petviashvili (SKP) equation of Manin and Radul can be embedded into recently formulated N = 1 self-dual supersymmetric Yang-Mills theories after appropriate dimensional reduction and truncation, which yield three-dimensional supersymmetric Chern-Simons theories. Based on this result, we also give a conjectural N = 2 SKP equation. Subsequently some exact solutions of these systems including fermionic fields are given.

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 self-dual Yang-Mills and supergravity as background of the Green-Schwarz superstring

We present the Green-Schwarz σ-model coupled to the N=1 supersymmetric Yang-Mills and supergravity in a four-dimensional space—time with the indefinite signature (+, +, −, −). We first confirm the κ-invariance of the Green-Schwarz action, and show that all the β-functions for the backgrounds vanish consistently after the use of their superfield equations. Subsequently, we inspect the supersymmetric self-duality conditions, that have been developed in our previous paper on the Yang-Mills and supergravity backgrounds. Remarkably, the Majorana-Weyl spinor dictating the supersymmetric self-duality conditions is consistent with the couplings of the Green-Schwarz superstring. Such a Green-Schwarz superstring is supposed to be the underlying theory of the supersymmetric self-dual Yang-Mills theory, which is conjectured to generate all exactly soluble supersymmetric systems in lower dimensions.

Majorana-Weyl spinors and self-dual gauge fields in 2+2 dimensions

The properties of Dirac gamma matrices in a four-dimensional space—time with the (2, 2) signature are studied. The basic spinors are classified, and the existence of Majorana-Weyl spinors is noted. Supersymmetry in 2+2 dimensions is discussed, and the existence of the real chiral scalar supermultiplet is discovered. Supersymmetric self-dual Yang-Mills theories and self-dual supergravity model in 2+2 dimensions, that are apparently relevant to integrable systems, are formulated for the first time.

Self-dual supersymmetry and supergravity in Atiyah-Ward space-time

We study supersymmetry and self-duality in a four-dimensional space-time with the signature (2,2), that we call the Atiyah-Ward space-time. Dirac matrices and spinors, in particular Majorana-Weyl spinors, are investigated in detail. We formulate N ⩾ 1 supersymmetric self-dual Yang-Mills theories and self-dual supergravities. An N = 1 “self-dual” tensor multiplet is constructed and a possible ten-dimensional theory that gives rise to the four-dimensional self-dual supersymmetric theories is found. Instanton solutions are given as the zero modes in the N = 2 self-dual Yang-Mills theory. The N = 2 superstrings are conjectured to have no possible counter-terms at quantum level to all orders. These self-dual supersymmetric theories are to generate exactly soluble supersymmetric in lower dimensions.

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.

Extended supersymmetry and self-duality in 2 + 2 dimensions

The N = 2 supersymmetric self-dual Yang-Mills theory and the N = 4 and N = 2 self-dual supergravities in 2 + 2 space-time dimensions are formulated for the first time. These formulations utilize solutions of the Bianchi identities subject to the super-Yang-Mills or supergravity constraints in the relevant N-extended superspace with the space-time signature (2,2). The self-duality condition on the Yang-Mills field strength or on the Riemann tensor is not modified, but is accompanied by the field equations of other superpartners. These self-dual systems are conjectured to generate supersymmetric exactly soluble models in lower dimensions.

Supersymmetric instantons and heterotic solitons

Heterotic solitons are analyzed in terms of the four-dimensional euclidean supersymmetric self-dual Yang-Mills equations. A new set of solutions to the low energy heterotic string field equations is obtained. Its geometrical structure is discussed.

A GENERALIZED BÄCKLUND TRANSFORMATION FOR THE (SUPER SYMMETRIC) SELF-DUAL YANG-MILLS FIELDS

We derive a Bäcklund transformation for the self-dual Yang-Mills fields by using the method of special Riemann problem with one zero, in which the moving poles in the complex parameter plane are incorporated. This BT is more general than a previously given one and is capable of generating in one step monopole or instanton solutions from a vacuum solution. We further extend this result to the case of the supersymmetric self-dual Yang-Mills fields.