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2011

Scalar Curvature for the Noncommutative Two Torus

Fathizadeh, Farzad, Khalkhali, Masoud

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We give a local expression for the {\it scalar curvature} of the noncommutative two torus $ A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor.

This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $\zeta_a(s): = \text{Trace}(a \triangle^{-s})$ at $s=0$ as a linear functional in $a \in C^{\infty}(\mathbb{T}_{\theta}^2)$.
A new, purely noncommutative, feature here is the appearance of the {\it modular automorphism group} from the theory of type III factors and quantum statistical mechanics in the final formula for the curvature.
This formula coincides with the formula that was recently obtained independently by Connes and Moscovici in their recent paper.

Scalar Curvature for the Noncommutative Two Torus · Around

Built on

A. Connes, C ∗ C^{*} -algèbres et géométrie différentielle
1980
Earlier work this paper cites.
P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem
1984
Earlier work this paper cites.
A. Connes, Noncommutative differential geometry
1985
Earlier work this paper cites.
S. Baaj, Calcul pseudo-différentiel et produits croisés de C ∗ C^{*} -algèbres
1988
Earlier work this paper cites.
A. Connes, The action functional in noncommutative geometry
1988
Earlier work this paper cites.

Similar

P. B. Cohen, On the non-commutative torus of real dimension two
1990
Cited alongside, same era.
A. Connes, Noncommutative geometry
1994
Cited alongside, same era.
A. Connes and H. Moscovici, The local index formula in noncommutative geometry
1995
Cited alongside, same era.
A. Chamseddine and A. Connes, The Spectral action principle
1997
Cited alongside, same era.
A. Chamseddine and A. Connes, Scale invariance in the spectral action
2006
Cited alongside, same era.
T. A. Bhuyain and M. Marcolli, The Ricci flow on noncommutative two-tori
Original
Cited in the paper.
P. B. Cohen and A. Connes, Conformal geometry of the irrational rotation algebra
Cited in the paper.
A. Connes and H. Moscovici, Modular curvature for noncommutative two-tori
Cited in the paper.
F. Fathizadeh and M. Khalkhali, The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure
Original
Cited in the paper.

Then

A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives
2008
Later among the works it cites.
A. Connes and H. Moscovici, Type III and spectral triples
2008
Later among the works it cites.
A. Connes and P. Tretkoff, The Gauss-Bonnet theorem for the noncommutative two torus
Original
2011
Closest in time.
A. Connes, On the spectral characterization of manifolds
Original
2088
Closest in time.

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