Notes for a Quantum Index Theorem

Abstract

We view DHR superselection sectors with finite statistics as Quantum Field Theory analogs of elliptic operators where KMS functionals play the role of the trace composed with the heat kernel regularization. We extend our local holomorphic dimension formula and prove an analogue of the index theorem in the Quantum Field Theory context. The analytic index is the Jones index, more precisely the minimal dimension, and, on a 4-dimensional spacetime, the DHR theorem gives the integrality of the index. We introduce the notion of holomorphic dimension; the geometric dimension is then defined as the part of the holomorphic dimension which is symmetric under charge conjugation. We apply the AHKT theory of chemical potential and we extend it to the low dimensional case, by using conformal field theory. Concerning Quantum Field Theory on curved spacetime, the geometry of the manifold enters in the expression for the dimension. If a quantum black hole is described by a spacetime with bifurcate Killing horizon and sectors are localizable on the horizon, the logarithm of the holomorphic dimension is proportional to the incremental free energy, due to the addition of the charge, and to the inverse temperature, hence to the surface gravity in the Hartle-Hawking KMS state. For this analysis we consider a conformal net obtained by restricting the field to the horizon (``holography''). Compared with our previous work on Rindler spacetime, this result differs inasmuch as it concerns true black hole spacetimes, like the Schwarzschild-Kruskal manifold, and pertains to the entropy of the black hole itself, rather than of the outside system. An outlook concerns a possible relation with supersymmetry and noncommutative geometry.

Only one change to previous version: ref. [62] misnumbering was fixed on request to the cited author

References in corpus (2)

• Algebraic Holography
• The Bisognano-Wichmann Theorem for Charged States and the Conformal Boundary of a Black Hole

Cited by in corpus (31)

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• Noncommutative Spectral Invariants and Black Hole Entropy
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• Spectral triples and the super-Virasoro algebra
• Thermal States in Conformal QFT. II
• Nuclearity and Thermal States in Conformal Field Theory
• Algebraic Supersymmetry: A case study
• Representations of Conformal Nets, Universal C*-Algebras and K-Theory
• On Landauer's principle and bound for infinite systems
• Thermal States in Conformal QFT. I
• N=2 superconformal nets
• Modular Theory, Non-Commutative Geometry and Quantum Gravity
• Modular conjugations in 2D conformal field theory and holographic bit threads
• Natural Energy Bounds in Quantum Thermodynamics
• The Principle of Locality. Effectiveness, fate and challenges
• Non-Equilibrium Thermodynamics and Conformal Field Theory
• The C(X)-algebra of a net and index theory
• Relative Haag Duality for the Free Field in Fock Representation
• The K-homology of nets of C*-algebras
• Computing the Jones index of quadratic permutation endomorphisms of O_2
• Super-KMS functionals for graded-local conformal nets
• The Bisognano-Wichmann Theorem for Charged States and the Conformal Boundary of a Black Hole
• From Operator Algebras to Superconformal Field Theory
• Local-entire cyclic cocycles for graded quantum field nets
• Modular Conjugation and the Implementation of Supersymmetry
• Loop groups and noncommutative geometry
• Uniqueness of purifications is equivalent to Haag duality
• KMS states and the chemical potential for disordered systems
• Some topics in quantum disordered systems