ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY

1992 ◽  
Vol 07 (18) ◽  
pp. 4293-4311 ◽  
Author(s):  
ASHOK DAS ◽  
WEN-JUI HUANG ◽  
SHIBAJI ROY
Keyword(s):  
Current Algebra ◽  
Hamiltonian Structure ◽  
Operator Product ◽  
Curvature Condition ◽  
Supergravity Theory ◽  
Kdv Equations ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Graded Lie Algebra ◽  
The One

The N=2 fermionic extensions of the KdV equations are derived from the zero curvature condition associated with the graded Lie algebra of OSp(2/2). These equations lead to two bi-Hamiltonian systems, one of which is supersymmetric. We also derive the one-parameter family of N=2 supersymmetric KdV equations without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for the corresponding 2D supergravity theory. This current algebra is then used to obtain the operator product expansions of various fields of this theory.

ZERO CURVATURE CONDITION AND 2D GRAVITY THEORIES

1992 ◽  
Vol 07 (15) ◽  
pp. 3447-3472 ◽  
Author(s):  
A. DAS ◽  
W.-J. HUANG ◽  
S. ROY
Keyword(s):  
Current Algebra ◽  
2D Gravity ◽  
Integrable Model ◽  
Integrable Models ◽  
Operator Product ◽  
Curvature Condition ◽  
Kdv Hierarchy ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Gravity Theories

We propose interpreting the zero curvature condition associated with an integrable model as an anomaly equation. This can lead to the WZWN action and the associated current algebra quite readily and clarifies further the connections found between the integrable models and 2D gravity theories. We analyze, in detail, the cases SL (2, R) (KdV hierarchy), OSp (2/1) (sKdV hierarchy) and SL (3, R) (Boussinesq hierarchy) and obtain the operator product expansions of the appropriate fields. We also make some observations on the generalization of our method to SL (n, R).

scholarly journals SELF-DUALITY AND THE SUPERSYMMETRIC KdV HIERARCHY

1993 ◽  
Vol 08 (15) ◽  
pp. 1399-1406 ◽  
Author(s):  
ASHOK DAS ◽  
C. A. P. GALVÃO
Keyword(s):  
Kdv Equation ◽  
Curvature Condition ◽  
Four Dimensions ◽  
Yang Mills ◽  
Kdv Hierarchy ◽  
Self Duality ◽  
Kdv Equations ◽  
Duality Condition ◽  
Graded Lie Algebra ◽  
Supersymmetric Kdv Equation

We show how the supersymmetric KdV equation can be obtained from the self-duality condition on Yang-Mills fields in four dimensions associated with the graded Lie algebra OSp(2/1). We also obtain the hierarchy of SUSY KdV equations as well as the s-KdV equations from such a condition. We formulate the SUSY KdV hierarchy as a vanishing curvature condition associated with the U(1) group and show how an Abelian self-duality condition in four dimensions can also lead to these equations.

scholarly journals THE ZERO CURVATURE FORMULATION OF TB, sTB HIERARCHY AND TOPOLOGICAL ALGEBRAS

1996 ◽  
Vol 11 (16) ◽  
pp. 1317-1329 ◽  
Author(s):  
ASHOK DAS ◽  
SHIBAJI ROY
Keyword(s):  
Hamiltonian Structure ◽  
Scaling Limit ◽  
Topological Algebras ◽  
Supersymmetric Extension ◽  
Curvature Condition ◽  
Zero Curvature ◽  
Close Relationship ◽  
Topological Field ◽  
Relationship Of ◽  
Classical Integrable

A particular dispersive generalization of long water wave equation in (1+1) dimensions, which is important in the study of matrix models without scaling limit, known as two-Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group SL (2, R) ⊗ U (1). The supersymmetric extension of the two-Boson (sTB) hierarchy has similarly been derived from the zero curvature condition associated with the gauge supergroup OSp (2|2). Topological algebras arise naturally as the second Hamiltonian structure of these classical integrable systems, indicating a close relationship of these models with 2-D topological field theories.

scholarly journals Monodromy defects in free field theories

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Lorenzo Bianchi ◽  
Adam Chalabi ◽  
Vladimír Procházka ◽  
Brandon Robinson ◽  
Jacopo Sisti
Keyword(s):  
Entanglement Entropy ◽  
Free Field ◽  
Operator Product ◽  
Dirac Fermions ◽  
Maxwell Theory ◽  
Conformal Field ◽  
Defect Operator ◽  
Crucial Information ◽  
The One ◽  
Group Flow

Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free $$ \mathcal{N} $$ N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.

scholarly journals CLASSICAL N=1 and N=2 SUPER W ALGEBRAS FROM A ZERO-CURVATURE CONDITION

1994 ◽  
Vol 09 (03) ◽  
pp. 383-398 ◽  
Author(s):  
FRANÇOIS GIERES ◽  
STEFAN THEISEN
Keyword(s):  
Superconformal Algebra ◽  
Order System ◽  
Curvature Condition ◽  
First Order ◽  
Zero Curvature

Starting from superdifferential operators in an N=1 superfield formulation, we present a systematic prescription for the derivation of classical N=1 and N=2 super W algebras by imposing a zero-curvature condition on the connection of the corresponding first-order system. We illustrate the procedure on the first nontrivial example (beyond the N=1 superconformal algebra) and also comment on the relation with the Gelfand-Dickey construction of W algebras.

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

scholarly journals Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

1997 ◽  
Vol 12 (34) ◽  
pp. 2623-2630 ◽  
Author(s):  
H. Aratyn ◽  
C. Rasinariu ◽  
A. Das
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Space Structure ◽  
Hermitian Symmetric Space ◽  
Curvature Condition ◽  
Lax Equation ◽  
Abelian Gauge ◽  
Graded Algebras ◽  
Zero Curvature

We generalize the Drinfeld–Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl (n+1,n), providing the Hermitian symmetric space structure.

Braid monodromies on proper curves and pro-ℓ Galois representations

2011 ◽  
Vol 11 (1) ◽  
pp. 161-188 ◽  
Author(s):  
Naotake Takao
Keyword(s):  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Mapping Class ◽  
Algebraic Number Fields ◽  
Universal Family ◽  
Hyperbolic Curve ◽  
Graded Lie Algebra ◽  
The One ◽  
First Time

AbstractLetCbe a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space ofCto the one attached toC, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group onCwithnstrings does not depend onn, even ifn= 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

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