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We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric inn+2dimensions that encodes a conformal class of metrics inndimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric inn+1dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? Deser and Schwimmer asserted that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern-Gauss-Bonnet integrand. This book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants--such as the classical Riemannian invariants and the more recently studied conformal invariants--and the study of global invariants, in this case conformally invariant integrals. Key tools used to establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula.

Conformal invariance has been a spectacularly successful tool in advancing our understanding of the two-dimensional phase transitions found in classical systems at equilibrium. This volume sharpens our picture of the applications of conformal invariance, introducing non-local observables such as loops and interfaces before explaining how they arise in specific physical contexts. It then shows how to use conformal invariance to determine their properties. Moving on to cover key conceptual developments in conformal invariance, the book devotes much of its space to stochastic Loewner evolution (SLE), detailing SLE's conceptual foundations as well as extensive numerical tests. The chapters then elucidate SLE's use in geometric phase transitions such as percolation or polymer systems, paying particular attention to surface effects. As clear and accessible as it is authoritative, this publication is as suitable for non-specialist readers and graduate students alike.

This book contains papers presented by speakers at the AMS-IMS-SIAM Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, held at Mount Holyoke College in June 1992. One group of papers deals with one aspect of conformal field theory, namely, vertex operator algebras or superalgebras and their representations. Another group deals with various aspects of quantum groups. Other topics covered include the theory of knots in three-manifolds, symplectic geometry, and tensor products. This book provides an excellent view of some of the latest developments in this growing field of research.

Let F : M → C P 2 be an isometric immersion of a closed surface in the complex projective plane C P 2 . In this paper, we consider the functional W N ( F ) = ∫ M ( K ¯ ⊥ - K ⊥ ) d M , which is a global conformal invariant. The critical surfaces of W N ( F ) are called normal critical surfaces. We compute the first variation of W N ( F ) . Moreover, we build an index formula for the normal critical surfaces in the spirits of Webster (J Differ Geom 20:463–470, 1984.), Wolfson (J Diff Geom 29:281–294, 1989), Chen-Tian (Geom Funct Anal 7:873–916, 1997) and Han-Li (J Eur Math Soc 12:505–527, 2010). In particular, when the Kähler angle α satisfies cos α ⩽ - 5 3 or - 5 3 ⩽ cos α ⩽ 5 3 or cos α ⩾ 5 3 , M is the normal critical surface if and only if it is the minimal surface with constant Kähler angle α .

The present paper deals with the characterization of a new submersion named semi-invariant conformal submersions with horizontal Reeb vector field from almost contact metric manifolds onto Riemannian metric manifolds which is the generalization of some known submersions on Riemannian metric manifolds. We give important and adequate conditions for such submersions to be totally geodesic and harmonic. Also, a few examples are examined for such submersions endowed with horizontal Reeb vector field.

We prove that if E is a compact subset of the unit disk D in the complex plane, if E contains a sequence of distinct points a n ≠ 0 for n ≥ 1 such that lim n → ∞ a n = 0 and for all n we have | a n + 1 | ≥ | a n | / 2 , and if G = D \\ E is connected and 0 ∈ ∂ G , then there is a constant c > 0 such that for all z ∈ G we have λ G ( z ) ≥ c / | z | where λ G ( z ) is the density of the hyperbolic metric in G .

After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion.