Self-dual Gravity Research Articles

Exact solutions for self-dual Yang-Mills and self-dual tensor multiplets on gravitational instanton background

We give exact solutions for recently developed N = 1 locally supersymmetric self-dual gauge theories in 2 + 2 dimensions. We give the exact solutions for an SL(2) self-dual Yang-Mills multiplet and what we call “self-dual tensor multiplet” on the gravitational instanton background by Eguchi and Hanson. We use a general method to get an SL(2) self-dual Yang-Mills solution from any known self-dual gravity solutions. Our result is the first example of exact solutions for the coupled system of these N = 1 locally supersymmetric self-dual multiplets in 2 + 2 dimensions, which is supposed to have strong significance for integrable models in lower dimensions upon appropriate dimensional reduction or truncation. We also inspect the consistency of our exact solutions as a background for the N = 2 superstring coupled to the Wess-Zumino-Witten term in σ-model formulation.

Kleinian singularities and the ground ring of c = 1 string theory

We investigate the nature of the ground ring of c = 1 string theory at the special ADE points in the c = 1 moduli space associated to discrete subgroups of SU(2). The chiral ground rings at these points are shown to define the ADE series of singular varieties introduced by Klein. The non-chiral ground rings relevant to closed-string theory are 3 real dimensional singular theory are 3 real dimensional singular varieties obtained as U(1) quotients of the kleinian varieties. The unbroken symmetries of the theory at these points are the volume-preserving diffeomophisms of these varieties. The theory of kleinian singularities has a close relation to that of complex hyperKähler surfaces, or gravitational instantons. We speculate on the relevance of these instantons and of self-dual gravity in c = 1 string theory.

SU(∞) (super)gauge theories and self-dual (super)gravity

It is shown that SU(∞) self-dual Yang–Mills equations on a four-dimensional flat Euclidean background are equivalent to Plebanski’s second heavenly equation for the self-dual gravitational background, which is associated with the cotangent space of a suitable Riemann surface. The Riemann surface in question is nothing but the internal two-dimensional surface required in the formulation of SU(∞) self-dual Yang–Mills theories. Such equivalence only occurs when a dimensional reduction has taken place. After a discussion of Euclidean supersymmetries the results are generalized to self-dual super-Yang–Mills theories and self-dual supergravity.

Equivariance in topological gravity

We present models of topological gravity for a variety of moduli space conditions. In four dimensions, we construct a model for self-dual gravity characterized by the moduli condition R + μν =0, and in two dimensions we treat the case of constant scalar curvature. Details are also given for both flat and Yang-Mills type moduli conditions in arbitrary dimensions. All models are based on the same fundamental multiplet which conveniently affords the construction of a complete hierarchy of observables. This approach is founded on a symmetry algebra which includes a local vector supersymmetry, in addition to a global BRST-like symmetry which is equivariant with respect to Lorentz transformations. Different moduli spaces are accommodated through the choice of additional multiplets, and we explicitly construct these together with associated invariant actions.

GAUGE THEORY FORMULATION OF THE c=1 MATRIX MODEL: SYMMETRIES AND DISCRETE STATES

We present a nonrelativistic fermionic field theory in two dimensions coupled to external gauge fields. The singlet sector of the c=1 matrix model corresponds to a specific external gauge field. The gauge theory is one-dimensional (time) and the space coordinate is treated as a group index. The generators of the gauge algebra are polynomials in the single particle momentum and position operators and they form the group [Formula: see text]. There are corresponding Ward identities and residual gauge transformations that leave the external gauge fields invariant. We discuss the realization of the residual symmetries in the Minkowski time theory and conclude that the symmetries generated by the polynomial basis are not realized. We motivate and present an analytic continuation of the model which realizes the group of residual symmetries. We consider the classical limit of this theory and make the correspondence with the discrete states of the c=1 (Euclidean time) Liouville theory. We explain the appearance of the SL(2) structure in [Formula: see text]. We also present all the Euclidean classical solutions and the classical action in the classical phase space. A possible relation of this theory to the N=2 string theory and also self-dual Einstein gravity in four dimensions is pointed out.

Volume-preserving diffeomorphisms in integrable deformations of self-dual gravity

A group of volume-preserving diffeomorphisms in 3D turns out to play a key role in an Einstein-Maxwell theory whose Weyl tensor is self-dual and whose Maxwell tensor has an algebraically general anti-self-dual part. This model was first introduced by Flaherty and recently studied by Park as an integrable deformation of self-dual gravity. A twisted volume form on the corresponding twistor space is shown to be the origin of volume-preserving diffeomorphisms. An immediate consequence is the existence of an infinite number of symmetries as a generalization of w 1 + ∞ symmetries in self-dual gravity. A possible relation to Witten's 2D string theory is pointed out.

4-D SELF-DUAL INSTANTONS FROM 2-D SIGMA MODEL

4-d self-dual gravity and Yang-Mills theories are identified with 2-d sigma models taking values in infinite dimensional groups. This allows us to view 4-d self-dual theories as "large N limits" of 2-d conformal field theories. We also find a possible "integrable deformation" of 4-d self-dual gravity leading to the Einstein-Maxwell system and discuss its implication in the context of higher dimensional integrability.

2D SIGMA MODEL APPROACH TO 4D INSTANTONS

4D self-dual theories are proposed to generalize 2D conformal field theory. We identify 4D self-dual gravity as well as self-dual Yang-Mills theory with 2D sigma models valued in infinite-dimensional gauge groups. It is shown that these models possess infinite-dimensional symmetries with associated algebras—“CP1 extensions” of respective gauge algebras of 2D sigma models—which generalize the Kac-Moody algebra as well as W∞. We address various issues concerning 2D sigma models, twistors and sheaf cohomology. An attempt to connect 4D self-dual theories with 2D conformal field theory is made through sl (∞) Toda field theory.

Topological four-dimensional self-dual gravity.

A theory of topological gravity with an independent vierbein and spin connection is formulated in four space-time dimensions. Topological symmetries, as well as local Lorentz transformations and diffeomorphisms, are fixed, following the formalism of Batalin and Vilkovisky. The gauge-fixing conditions of the topological symmetries are chosen in such a way that self-duality is preserved for both curvature and torsion two-forms. It is shown that these gauge-fixing conditions have nontrivial classical solutions. Nontrivial observables are also constructed.

Integrable deformation of self-dual gravity

We obtain a linear system whose integrability is the vanishing of scalar curvatures of Kähler spaces or equivalently, Einstein-Maxwell equations with a self-(anti-self)-dual Weyl tensor and an algebraically general anti-self (self)-dual Maxwell tensor. This generalizes the linear system for the self-dual gravity (Ricci flat Kähler spaces) and, in some sense, defines an “integrable deformation” of the 4D self-dual gravity.

Geometry of n = 2 strings

We study various aspects of N = 2 critical strings. The most important aspect of these theories is that they provide a consistent quantum theory of self-dual gravity in four dimensions. We discuss the geometrical aspects of the vacua and their relation to twistor space, W ∞, harmonic superspace and the superstring world-sheet. We find many indications that suggest that many of the results valid for physics in two dimensions have analogs in the effective four-dimensional theory of N = 2 strings.

A 4D analogue of KPZ (I)

A four-dimensional analogue of residual η- and ϵ-symmetries in the light-cone gauge formulation of 2D gravity à la Knizhnik- Polyakov-Zamolodchikov is considered. We argue that residual symmetries in the metric for self-dual gravity by Plebanski et al. precisely correspond to this. We also show that the η-symmetry is nothing but a Kähler gauge symmetry in the gauge used by Alekseev and Shatashvili for the derivation of the induced 2D gravity action.

Hamiltonian structure, (anti-)self-adjoint flows in the KP hierarchy and the W 1 + ∞ and W ∞ algebras

The extended conformal W N algebras are known to be related to the generalized KdV hierarchies through their second hamiltonian structure. In this letter we discuss the relationship between the large- N limits of the W N algebras and the KP hierarchy which contains all generalized KdV hierarchies. We show that the Poisson bracket corresponding to the hamiltonian structure found by Watanabe for the KP hierarchy is isomorphic to the classical (or centerless) W 1+∞ algebra, and it contains a subalgebra which is isomorphic to the W ∞ algebra. Moreover, the usual generators of W 1 + ∞ and W ∞ are explicitly expressed in terms of the KP currents, and are shown to relate in a simple way to certain KP flows satisfying a sort of (anti-)self-duality. Our results not only clarify the underlying algebraic structure of the KP hierarchy, but also hint about a possible relationship between the latter and 4D self-dual Yang-Mills equations or gravity.

W∞ AND ANOMALIES OF SELF-DUAL EINSTEIN THEORIES

Recently it has been demonstrated that self-dual Einstein Euclidean instantons possess an infinite-dimensional group of symmetries which contain the standard w∞. Since w∞ has a central extension only in its w2 subalgebra one may claim that there is an anomaly in the symmetry group of these instantons which is described by a 3-dimensional w2 effective action. Thus one has a bosonization of the 3/2 spin anomaly of the Eguchi–Hanson instantons. In analogy to w∞ we construct the 3-dimensional effective action for the Lone Star W∞ algebra. This suggests that there is a quantum deformation of the instantons that contributes to the effective action of 4-dimensional self-dual gravity. We comment on this possibility.

A hamiltonian structure of KP hierarchy, W 1+∞ algebra, and self-dual gravity

It is well known that the second hamiltonian structures for the ( N−1)th generalized KdV hierarchy (of A n type) are isomorphic to the classical W N algebra for finite N(⩾2). Here we construct an analogue of the first structures for the KP hierarchy in its pseudo-differential operator formalism L = D+D −1ν 1+D −2ν 2+D −3ν 3+… , and show that the corresponding Poisson brackets are given by the W 1+∞ algebra with no central charge. Its implication to the quantization of 4D self-dual gravity is briefly discussed.

Induced 4D self-dual quantum gravity: W algebraic approach

The authors consider the functional integral quantization of the equation which describes a development of cross sections in a twistor space corresponding to self-dual gravity. At the classical level the system permits a local area-preserving diffeomorphism as its symmetry. Assuming Winfinity algebra at the quantized level, they derive an effective action for this equation via the coadjoint orbit method. The result consists of an infinite sum of two-dimensional effective Lagrangians including infinitesimal Polyakov two-dimensional (light-cone gauge) gravity as its first term. The higher order terms are the result of the central extensions of Winfinity algebra associated with higher conformal-spin operators.

Self-dual 2-forms and gravity

A Palatini-type formulation of gravity coupled to matter and supergravity is given, in which the gravitational variables are a trio of self-dual 2-forms and an SL(2,C) connection. The action is polynomial in all the fields. This framework is shown to be the natural covariantization of Ashtekar's canonical formalism (1988), and is used to find the general vacuum solution of the four initial value constraints associated with spacetime diffeomorphisms in that formalism.

SELF-DUALITY AND N=2 STRING MAGIC

We consider strings with an N=2 local superconformal symmetry on the worldsheet. The critical dimension for this theory is four (two complex dimensions) with the signature (2, 2). A Kähler function giving rise to self-dual gravity is the only physical degree of freedom of this theory. Some miraculous symmetries are observed corresponding to the exchange of worldsheet and target moduli. The open and heterotic versions of this string theory correspond to self-dual Yang-Mills fields coupled to self-dual gravity in four dimensions.

Self-dual gravity as a large- N limit of the 2D non-linear sigma model

The 4D self-dual gravity is shown to be equivalent to a large- N limit of the 2D non-linear sigma model with Wess-Zumino terms only. Infinite dimensional symmetries of the self-dual Einstein equations are derived and shown to form a closed algebra which includes W ∞—a large- N limit of Zamolodchikov's W N algebra. These are interpreted in terms of Penrose's twistor theory.

Self-dual massive gravity

The first-order action for self-dual linearized massive gravity in D=3 is given. Also two master actions relevant for this spin-2 case are given, explaining the equivalence between the self-dual formulation and the topological massive one. The evaluation of the vacuum amplitude in the presence of an external current shows in a covariant complementary way the propagation of a unique, parity sensitive, spin-2, massive excitation.