Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.001.0001
- Subject:
- Mathematics, Geometry / Topology
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 ...
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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? This book asserts 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. The 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.Less
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? This book asserts 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. The 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.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0004
- Subject:
- Mathematics, Geometry / Topology
This chapter sets up a highly complicated proposition (the fundamental proposition 4.13), which is to be proven by an elaborate induction on four parameters. It is organized as follows. Section 4.2 ...
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This chapter sets up a highly complicated proposition (the fundamental proposition 4.13), which is to be proven by an elaborate induction on four parameters. It is organized as follows. Section 4.2 sets up the considerable notational and language conventions needed to state the fundamental Proposition 4.13; it then states the fundamental Proposition and explains how the main algebraic propositions are special cases of it. It also explains the induction using four parameters by which the fundamental proposition will be proven. Section 4.3 distinguishes three cases I, II, III of the hypothesis of Proposition 4.13 and claim three Lemmas, 4.16, 4.19, and 4.24, which correspond to these three cases. Finally, Section 4.4 proves that these three lemmas imply Proposition 4.13. It also asserts certain useful technical lemmas that are also used in the subsequent chapters in this book.Less
This chapter sets up a highly complicated proposition (the fundamental proposition 4.13), which is to be proven by an elaborate induction on four parameters. It is organized as follows. Section 4.2 sets up the considerable notational and language conventions needed to state the fundamental Proposition 4.13; it then states the fundamental Proposition and explains how the main algebraic propositions are special cases of it. It also explains the induction using four parameters by which the fundamental proposition will be proven. Section 4.3 distinguishes three cases I, II, III of the hypothesis of Proposition 4.13 and claim three Lemmas, 4.16, 4.19, and 4.24, which correspond to these three cases. Finally, Section 4.4 proves that these three lemmas imply Proposition 4.13. It also asserts certain useful technical lemmas that are also used in the subsequent chapters in this book.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0005
- Subject:
- Mathematics, Geometry / Topology
This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the assumption of ...
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This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the assumption of Proposition 4.13. This study involves many complicated calculations and also the appropriate use of the inductive assumption of Proposition 4.13.Less
This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the assumption of Proposition 4.13. This study involves many complicated calculations and also the appropriate use of the inductive assumption of Proposition 4.13.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0006
- Subject:
- Mathematics, Geometry / Topology
This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the ...
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This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the inductive assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.Less
This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the inductive assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0007
- Subject:
- Mathematics, Geometry / Topology
This chapter takes up the proof of Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. The chapter recalls that Lemma 4.24 applies when all ...
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This chapter takes up the proof of Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. The chapter recalls that Lemma 4.24 applies when all tensor fields of minimum rank μ in (4.3) have all μ of their free indices being nonspecial. We recall the setting of Case B in Lemma 4.24: Let M > 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 all μ-tensor fields in (4.3) that have at least one factor T₁ containing M free indices; let M' ≤ M be the maximum number of free indices that can belong to the same factor, other than T₁. The setting of Case B in Lemma 4.24 is when M < 2. The chapter recalls that in the setting of case B the claim of Lemma 4.24 coincides with the claim of Proposition 4.13. To derive Lemma 4.24, the authors use all the tools developed in the Chapter 6, most importantly the grand conclusion, but also the two separate equations that were added to derive the grand conclusion.Less
This chapter takes up the proof of Lemma 4.24 in Case B; this completes the inductive step of the proof of the fundamental proposition 4.13. The chapter recalls that Lemma 4.24 applies when all tensor fields of minimum rank μ in (4.3) have all μ of their free indices being nonspecial. We recall the setting of Case B in Lemma 4.24: Let M > 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 all μ-tensor fields in (4.3) that have at least one factor T₁ containing M free indices; let M' ≤ M be the maximum number of free indices that can belong to the same factor, other than T₁. The setting of Case B in Lemma 4.24 is when M < 2. The chapter recalls that in the setting of case B the claim of Lemma 4.24 coincides with the claim of Proposition 4.13. To derive Lemma 4.24, the authors use all the tools developed in the Chapter 6, most importantly the grand conclusion, but also the two separate equations that were added to derive the grand conclusion.
