On W-gravity in two dimensions

Gerasimov, A.; Levin, A.; Marshakov, A.

doi:10.1016/0550-3213(91)90415-t

Cited by 36 publications

(55 citation statements)

References 24 publications

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“…Since the moduli of these spaces correspond to variation of Hodge structures [9,15], there must exist a description of these structures directly in the higher dimension. This can be done via Higgs bundles in higher dimensions which has been studied by Simpson [28].…”

Section: Discussionmentioning

confidence: 99%

“…The compatibility of this equation with the equations of the KdV hierarchy as well as the differential equation provides restrictions on the allowed Beltrami differentials. In order to relate the W-manifolds to W-gravity, we then show that the higher dimensional Beltrami equation projects to the W-surface as the Beltrami equation which occurs in W-gravity [9,26]. In order to make this correspondence we discuss an old puzzle raised by Di Francesco et al regarding the relationship of KdV flows and W-diffeomorphism which appears in our setting as the relationship of the Beltrami differentials of higher dimensional uniformisation and the Beltrami differentials of W-gravity.…”

Section: Deformation Of Complex Structurementioning

confidence: 99%

“…The possibility of using the KdV times to linearise diffeomorphisms is not new [9,10,11]. However, what is new is the use of the fact that they provide isomonodromic deformations.…”

Section: Introductionmentioning

confidence: 99%

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Higher dimensional uniformisation and W-geometry

Govindarajan

1995

Nuclear Physics B

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We formulate the uniformisation problem underlying the geometry of Wn-gravity using the differential equation approach to W -algebras. We construct Wn-space (analogous to superspace in supersymmetry) as an (n − 1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The Wn-manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n, R) which acts properly discontinuously on a simply connected domain in CP n−1 . The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W -diffeomorphisms to (linear) diffeomorphisms on the Wn-manifold. We discuss how the Teichmüller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the Wn-manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all "generic" W-algebras. *

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Section: Discussionmentioning

confidence: 99%

Section: Deformation Of Complex Structurementioning

confidence: 99%

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Higher dimensional uniformisation and W-geometry

Govindarajan

1995

Nuclear Physics B

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“…Arising of the Gelfand-Dikij-type operator (5.15) in this context together with the W -flows in (5.21) suggests a nontrivial connection with the so-called Wgravity [37] and geometry of generalized Teichmüller spaces.…”

Section: Pure Gauge Theories and Whittaker Vectorsmentioning

confidence: 99%

Gauge theories as matrix models

Marshakov

2011

Theor Math Phys

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The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. The matrix models of Eguchi-Yang type are derived quasiclassically as describing the instantonic contribution to the deformed partition functions of supersymmetric gauge theories. The constructed explicitly exact solution for the case of conformal four-dimensional theory is studied in detail, and some aspects of its relation with the recently proposed logarithmic beta-ensembles are considered. We discuss also the "quantization" of this picture in terms of twodimensional conformal theory with extended symmetry, and stress its difference from common picture of perturbative expansion a la matrix models. Instead, the representation for Nekrasov functions in terms of conformal blocks or Whittaker vector suggests some nontrivial relation with Teichmüller spaces and quantum integrable systems.

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“…In a conformal quantum field theory which realizes Wsymmetry one expects then that the W-currents are no longer holomorphic quantities, the Ward identities arising from W-transformations are anomalous. These questions have been addressed in [17], [18], [19], [20]), in particular detail for the case of the W-structures pertaining to SL (3). In these papers special emphasis was on possible interpretations of W Ward identities in terms of zero curvature conditions and in relation with integrable hierarchies and their Lax pair formulation.…”

Section: Introductionmentioning

confidence: 99%

𝒲-gauge structures and their anomalies: An algebraic approach

Garajeu

Grimm

Lazzarini

1995

Journal of Mathematical Physics

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Starting from flat two-dimensional gauge potentials we propose the notion of W-gauge structure in terms of a nilpotent BRS differential algebra. The decomposition of the underlying Lie algebra with respect to an SL(2) subalgebra is crucial for the discussion conformal covariance, in particular the appearance of a projective connection. Different SL(2) embeddings lead to various W-gauge structures.We present a general soldering procedure which allows to express zero curvature conditions for the W-currents in terms of conformally covariant differential operators acting on the W gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by W-currents, gauge fields and the ghost fields corresponding to W-diffeomorphisms. As illustrations we treat the cases of SL(2) itself and to the two different SL(2) embeddings in SL(3) , viz. the W (1) 3 -and W (2) 3 -gauge structures, in some detail. In these cases we determine algebraically W-anomalies as solutions of the consistency conditions and discuss their Chern-Simons origin.

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On W-gravity in two dimensions

Cited by 36 publications

References 24 publications

Higher dimensional uniformisation and W-geometry

Higher dimensional uniformisation and W-geometry

Gauge theories as matrix models

𝒲-gauge structures and their anomalies: An algebraic approach

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