scholarly journals Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

2016 ◽  
Vol 13 (04) ◽  
pp. 1650042 ◽  
Author(s):  
Sumanto Chanda ◽  
Partha Guha ◽  
Raju Roychowdhury
Keyword(s):  
Vacuum Einstein Equation ◽  
Yang Mills ◽  
Gravitational Instanton ◽  
Self Duality ◽  
Bianchi Ix ◽  
Euler Top ◽  
Integrable Dynamical Systems ◽  
Halphen System ◽  
Special Case ◽  
Dual Case

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

SELF-DUAL SUPERGRAVITY AND SUPERSYMMETRIC YANG-MILLS THEORY COUPLED TO GREEN-SCHWARZ SUPERSTRING

1994 ◽  
Vol 09 (17) ◽  
pp. 3077-3101 ◽  
Author(s):  
HITOSHI NISHINO
Keyword(s):  
Shell Structure ◽  
Riemann Tensor ◽  
The Self ◽  
Special Significance ◽  
Yang Mills ◽  
Gravitational Instanton ◽  
Self Duality ◽  
Tensor Multiplet ◽  
Kappa Symmetry ◽  
Previous Series

We present the canonical set of superspace constraints for self-dual supergravity, a “self-dual” tensor multiplet and a self-dual Yang-Mills multiplet with N=1 supersymmetry in the space-time with signature (+,+, −, −). For this set of constraints, the consistency of the self-duality conditions on these multiplets with supersymmetry is manifest. The energy-momentum tensors of all the self-dual “matter” multiplets vanish, to be consistent with the self-duality of the Riemann tensor. In particular, the special significance of the “self-dual” tensor multiplet is noted. This result fills the gap left over in our previous series of papers, with respect to the consistent couplings among the self-dual matter multiplets. We also couple these nontrivial backgrounds to a Green-Schwarz superstring σ model, under the requirement of invariance under fermionic (kappa) symmetry. The finiteness of the self-dual supergravity is discussed, based on its “off-shell” structure. A set of exact solutions for the “self-dual” tensor and self-dual Yang-Mills multiplets for the gauge group SL(2) on a self-dual gravitational instanton background is given, and its consistency with the Green-Schwarz string σ model is demonstrated.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

scholarly journals Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Guido Festuccia ◽  
Anastasios Gorantis ◽  
Antonio Pittelli ◽  
Konstantina Polydorou ◽  
Lorenzo Ruggeri
Keyword(s):  
Vector Field ◽  
Extended Supersymmetry ◽  
General Framework ◽  
Gauge Theories ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Explicit Result ◽  
Yang Mills ◽  
Self Duality ◽  
Witten Theory

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

CONFORMALLY COVARIANT DRESSING TRANSFORMATIONS FOR SUPER SELF-DUALITY EQUATIONS IN D=4

1989 ◽  
Vol 04 (10) ◽  
pp. 971-982
Author(s):  
J. AVAN
Keyword(s):  
Linear System ◽  
Gauge Transformations ◽  
Four Dimensions ◽  
Yang Mills ◽  
Self Duality ◽  
Loop Algebras ◽  
Field Dependent

A set of conformally covariant dressing transformations is constructed for the supersym-metric N=3 self-duality equations in four dimensions, using the associated covariant linear system. They form a closed, 5+6-index algebra, up to field-dependent gauge transformations, containing the previously known loop algebras as a particular subset. This construction generalizes the formerly built set of conformally covariant DT for ordinary self-dual Yang-Mills.

Self-duality in four-dimensional Riemannian geometry

1978 ◽  
Vol 362 (1711) ◽  
pp. 425-461 ◽  
Keyword(s):  
Gauge Group ◽  
Riemannian Geometry ◽  
Complex Analysis ◽  
Three Dimensional ◽  
The Self ◽  
Full Detail ◽  
Yang Mills ◽  
Self Duality ◽  
Compact Gauge Group

We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.

scholarly journals NONCOMMUTATIVE EXTENDED WAVES AND SOLITON-LIKE CONFIGURATIONS IN N = 2 STRING THEORY

2003 ◽  
Vol 18 (26) ◽  
pp. 4889-4931 ◽  
Author(s):  
MATTHIAS IHL ◽  
SEBASTIAN UHLMANN
Keyword(s):  
Field Theory ◽  
String Theory ◽  
String Field Theory ◽  
Plane Waves ◽  
Equation Of Motion ◽  
Lax Pair ◽  
The Self ◽  
Yang Mills ◽  
Self Duality ◽  
Wave Solutions

The Seiberg–Witten limit of fermionic N = 2 string theory with nonvanishing B-field is governed by noncommutative self-dual Yang–Mills theory (ncSDYM) in 2+2 dimensions. Conversely, the self-duality equations are contained in the equation of motion of N = 2 string field theory in a B-field background. Therefore finding solutions to noncommutative self-dual Yang–Mills theory on ℝ2,2 might help to improve our understanding of nonperturbative properties of string (field) theory. In this paper, we construct nonlinear soliton-like and multi-plane wave solutions of the ncSDYM equations corresponding to certain D-brane configurations by employing a solution generating technique, an extension of the so-called dressing approach. The underlying Lax pair is discussed in two different gauges, the unitary and the Hermitian gauge. Several examples and applications for both situations are considered, including Abelian solutions constructed from GMS-like projectors, noncommutative U(2) soliton-like configurations and interacting plane waves. We display a correspondence to earlier work on string field theory and argue that the solutions found here can serve as a guideline in the search for nonperturbative solutions of nonpolynomial string field theory.

MULTI-INSTANTON SOLUTIONS IN EIGHT-DIMENSIONAL CURVED SPACE

1991 ◽  
Vol 06 (05) ◽  
pp. 409-415 ◽  
Author(s):  
A. REŞIT DÜNDARER
Keyword(s):  
Duality Relation ◽  
Winding Number ◽  
Curved Space ◽  
Yang Mills ◽  
Action Integral ◽  
Self Duality ◽  
Abelian Field ◽  
Mills Field

Starting from an SO(8)± Yang-Mills field Fab, a four-index antisymmetric, non-Abelian field Fabcd is defined. It is shown that this field satisfies a self-duality relation in eight-dimensional curved space and that its action integral is proportional to n, the winding number of the mapping from S8 onto S8 given by the octonionic transformation x→w=xn.

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