A local index theorem for families of $$\bar \partial $$ -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

1991 ◽  
Vol 137 (2) ◽  
pp. 399-426 ◽  
Author(s):  
L. A. Takhtajan ◽  
P. G. Zograf
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Index Theorem ◽  
Local Index ◽  
Kähler Metric ◽  
Local Index Theorem ◽  
Kahler Metric

scholarly journals Local index theorem for orbifold Riemann surfaces

2018 ◽  
Vol 109 (5) ◽  
pp. 1119-1143 ◽  
Author(s):  
Leon A. Takhtajan ◽  
Peter Zograf
Keyword(s):  
Riemann Surfaces ◽  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

A local index theorem for non K�hler manifolds

1989 ◽  
Vol 284 (4) ◽  
pp. 681-699 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

scholarly journals Local index theorem for families.

1988 ◽  
Vol 35 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Harold Donnelly
Keyword(s):  
Index Theorem ◽  
Local Index ◽  
Local Index Theorem

NONUNITAL SPECTRAL TRIPLES ASSOCIATED TO DEGENERATE METRICS

2004 ◽  
Vol 16 (01) ◽  
pp. 125-146
Author(s):  
A. RENNIE
Keyword(s):  
Finite Dimension ◽  
Index Theorem ◽  
Smooth Manifolds ◽  
Spectral Triples ◽  
Local Index ◽  
Dimension Spectrum ◽  
Spectrum Hypothesis ◽  
Local Index Theorem ◽  
K Theory

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

scholarly journals LIOUVILLE THEORY: WARD IDENTITIES FOR GENERATING FUNCTIONAL AND MODULAR GEOMETRY

1994 ◽  
Vol 09 (25) ◽  
pp. 2293-2299 ◽  
Author(s):  
LEON A. TAKHTAJAN
Keyword(s):  
Moduli Spaces ◽  
Correlation Functions ◽  
Riemann Surfaces ◽  
Central Charge ◽  
Perturbation Expansion ◽  
Liouville Theory ◽  
Energy Tensor ◽  
Ward Identities ◽  
Generating Functional ◽  
Local Index

We continue the study of quantum Liouville theory through Polyakov’s functional integral,1,2 started in Ref. 3. We derive the perturbation expansion for Schwinger’s generating functional for connected multi-point correlation functions involving stress-energy tensor, give the “dynamical” proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in Ref. 3. We show that conformal Ward identities for these correlation functions contain such basic facts from Kähler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, Kähler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry.4

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