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The Decomposition of Global Conformal Invariants (AM-182)

The Decomposition of Global Conformal Invariants (AM-182)

Spyros Alexakis
Copyright Date: 2012
Pages: 568
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  • Book Info
    The Decomposition of Global Conformal Invariants (AM-182)
    Book Description:

    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.

    eISBN: 978-1-4008-4272-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Acknowledgments
    (pp. vii-x)
  4. Chapter One Introduction
    (pp. 1-18)

    The aim of the present book is to provide a rigorous proof of a conjecture of Deser and Schwimmer, originally formulated in [21]. This work is a continuation of the previous two papers of the author [2, 3], which established the conjecture in a special case and introduced tools that laid the groundwork for the resolution of the full conjecture. The present volume is complemented by the two papers [4, 6], in which certain techical lemmas that are asserted and used here are proven. Thus this book, together with the papers [2, 3] and [4, 6], provides a confirmation of...

  5. Chapter Two An Iterative Decomposition of Global Conformal Invariants: The First Step
    (pp. 19-70)

    Before outlining our iterative strategy for the proof of the Deser-Schwimmer conjecture, we make a remark on the use of Riemannian invariants in this work. We stress that given any Riemannian invariantF(g), (thought of now in the sense of (1.2)), the associated formal expression$ \sum \nolimits _{l \in L} a_lC^l (g) $isnot unique.¹ However, thisdoes notimpede our strategy: We will always be referring to aspecificlinear combination$ \sum \nolimits _{l \in L} a_lC^l (g) $and in that context we will be referring to the algebraic properties of the complete contractions$ \{C ^l(g)\} _{l \in L} $. Let us now outline the ideas of the present chapter:

    Categories and the notion of...

  6. Chapter Three The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition
    (pp. 71-134)

    In this chapter we prove Proposition 2.8. In a nutshell, the content of this Proposition is the following: Let$ P(g) = \sum \nolimits _{l \in L} a_lC^l (g) $be a Riemannian invariant for which

    a.$ \int \nolimits _{M^n} \, P (g)dV_g $is a global conformal invariant.

    b. Ifσ> 0 is the smallest number of factors among all complete contractions$ C^l(g), l \in L $, and$ L _\sigma \subset L $is their index set, then the complete contractions$ C^l(g), l \in L_\sigma $are all in the form (2.49); i.e., they involve only factors in the form ∇(m)W(iterated covariant derivatives of the Weyl tensor.)

    Then there exists a local conformal invariantW(g) and a divergence of a Riemannian vector field, diviTi(g), such...

  7. Chapter Four A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition
    (pp. 135-210)

    The remainder of this book is devoted to proving the main algebraic propositions 2.28, 3.27 and 3.28. As explained above, this will complete the proof of the Deser-Schwimmer conjecture. We stress that the material in the remaining chapters of this book islogically independentof the preceding chapters. In particular, the main algebraic propositions dealexclusivelywith algebraic properties of theclassicalscalar Riemannian invariants; they make no reference to integration. The author was led to the main algebraic propositions through the strategy that he felt was necessary to solve the Deser-Schwimmer conjecture; however, they can be thought of as...

  8. Chapter Five The Inductive Step of the Fundamental Proposition: The Simpler Cases
    (pp. 211-296)

    The purpose of the present chapter is to prove Lemmas 4.16 and 4.19; Lemma 4.24 is proven in the next chapters, and the two technical Lemmas (4.22 and 4.23) in the Appendix. In view of the results of Chapter 4, this proves the main algebraic propositions 2.28, 3.27 and 3.28 and thus completes our proof of the Deser-Schwimmer conjecture.

    We recall that the three Lemmas 4.16, 4.19, and 4.24 imply Proposition 4.13;¹ this proposition is proven by a multiple induction on four parameters, and Lemmas 4.16, 4.19 and 4.24 establish the inductive step of Proposition 4.13, as proven in Chapter...

  9. Chapter Six The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I
    (pp. 297-360)

    The aim of the present chapter and of the next one is to prove Lemma 4.24. The task of proving this lemma is much harder than the one of proving Lemmas 4.16, 4.19. We partly build on the work of the previous chapter, but also bring in new ideas.

    The main tool that we introduce to derive Lemma 4.24 is anew, less complicated local equation, which is a consequence of (4.3); we call this new equation the grand conclusion; this grand conclusion always holds under the hypotheses of Lemma 4.24. In Case A of Lemma 4.24, the grand conclusion...

  10. Chapter Seven The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II
    (pp. 361-402)

    Our aim in this chapter is to prove Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. We recall that Lemma 4.24 applies when all tensor fields of minimum rankμin (4.3) have allμof their free indices beingnonspecial.¹ We recall the setting of Case B in Lemma 4.24: LetM> 0 stand for the maximum number of free indices that can belong to the same factor, among all tensor fields in (4.3). Then consider allμ-tensor fields in (4.3) that have at least one factorT₁ containing...

  11. A Appendix
    (pp. 403-442)
  12. Bibliography
    (pp. 443-446)
  13. Index of Authors and Terms
    (pp. 447-448)
  14. Index of Symbols
    (pp. 449-449)