Skip to main content
eScholarship
Open Access Publications from the University of California

Scholarly Works (13 results)

  • Article
  • Peer Reviewed
Faculty (2014)
The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same invariant arises as the obstruction to smooth boundary asymptotics to the Yamabe problem of finding a metric in a conformal class with constant scalar curvature. We use conformal geometry tools to describe and compute the asymptotics of the Yamabe problem on a conformally compact manifold and thus produce higher order hypersurface conformal invariants that generalise the Willmore invariant. We give a holographic formula for these as well as variational principles for the lowest lying examples.
    • Thesis
    • Peer Reviewed
    UC Davis Electronic Theses and Dissertations (2021)

    The relationship between the boundary of a manifold and its interior is important for studying many problems in science, as it allows us to predict the behavior of certain problems that can be modeled by partial differential equations. We study bulk-boundary relationships for conformal manifolds. A key tool for analyzing conformal manifolds is tractor calculus. By comparing the conformal structure in the interior with that of the boundary, we provide a complete hypersurface tractor calculus and develop a conformally-invariant characterization of the extrinsic curvature of the embedded hypersurface. These tools provide a characterization of families of conformal manifolds with boundaries that are of particular interest to physicists: so-called Poincare--Einstein manifolds and Willmore manifolds. Furthermore, we produce a series of conformally-invariant hypersurface operators and curvatures in boundary dimension four and discuss generalizations of these objects to arbitrary dimension.

      Cover page: On the Extrinsic Geometry of Conformally Embedded Hypersurfaces
      • Article
      • Peer Reviewed
      Faculty (2016)
      The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential operators and in particular, describe how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we develop new higher dimensional analogues of the Willmore energy for embedded surfaces. This complements recent progress on the existence and construction of such functionals.
        • Article
        • Peer Reviewed
        Faculty (2015)
        We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruction density. For even dimensional hypersurfaces it is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solution to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers determine an explicit holographic formula for the obstruction density.
          • Article
          • Peer Reviewed
          Faculty (2002)
          We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality between quaternionic self-adjoint and real symmetric matrix integrals. The asymptotic expansions of these integrals are given in terms of summations over topologies of compact surfaces, both orientable and non-orientable, for all genera and an arbitrary positive number of marked points on them. We show that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) have exactly the same graphical expansion term by term (when appropriately normalized),except that the contributions from non-orientable surfaces with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials. Indeed, we show that this duality is equivalent to Poincare duality of graphs drawn on a compact surface. Another application of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.
            • Article
            • Peer Reviewed
            Faculty (2005)
            The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for the actions and gauge invariances of massive, partially massless and massless (for both bose and fermi statistics) higher spins on constant curvature backgrounds. These are formulated in terms of a minimal set of covariant, unconstrained, fields rather than towers of auxiliary fields. Partially massless gauge transformations are shown to arise as degeneracies of the flat, massless gauge transformation in one dimension higher. Moreover, our results and calculus offer a considerable simplification over existing techniques for handling higher spins. In particular, we show how theories of arbitrary spin in dimension d can be rewritten in terms of a single scalar field in dimension 2d where the d additional dimensions correspond to coordinate differentials. We also develop an analogous framework for spinor-tensor fields in terms of the corresponding superalgebra.
              • Article
              • Peer Reviewed
              Faculty (2016)
              We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincare--Einstein structure, this result recovers Branson's Q-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.
                • Article
                • Peer Reviewed
                Faculty (2016)
                We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.
                  • Thesis
                  • Peer Reviewed
                  UC Davis Electronic Theses and Dissertations (2021)

                  We analyze the two variable series invariant for knot complements originating from a categorification of the SU(2) WRT invariant of closed oriented 3-manifolds. We are especially interested in examining the conjectured hbar-expansion property and the q-holomonic property of the series invariant through an example of a satellite knot, namely, a cabling of the figure eight knot, which has more than twenty crossings. This cable knot result provides nontrivial evidence for the conjectures and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we investigate the conjectured relation between the series invariant and the ADO invariants at roots of unity. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for particular ADO invariants of a class of torus knots obtained from the series invariant for complement of a knot. Additionally a one parameter deformation of ADO3 invariants of torus knots is provided, which unifies the three original ADO3 formulas into one formula. We also present a one variable series for closed plumbed 3-manifolds associated with a type I Lie superalgebra osp(2j2) that categorifies the CGP invariant.

                    Cover page: Topics in Quantum Topology: 3-Manifolds, BPS Series and Categorification
                    • Article
                    • Peer Reviewed
                    Faculty (2010)
                    Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six component vector subject to a certain first order covariant constancy condition at every point in four dimensional spacetime). These results suggest a six dimensional description of four dimensional physics, a viewpoint promulgated by the two times physics program of Bars. The Fefferman--Graham construction relies on a triplet of operators corresponding, respectively to a curved six dimensional light cone, the dilation generator and the Laplacian. These form an sp(2) algebra which Bars employs as a first class algebra of constraints in a six-dimensional gauge theory. In this article four dimensional gravity is recast in terms of six dimensional quantum mechanics by melding the two times and tractor approaches. This "parent" formulation of gravity is built from an infinite set of six dimensional fields. Successively integrating out these fields yields various novel descriptions of gravity including a new four dimensional one built from a scalar doublet, a tractor vector multiplet and a conformal class of metrics.
                      Top