Celebratio Mathematica

V. Mathai, R. B. Melrose, and I. M. Singer: “The index of projective families of elliptic operators,” Geom. Topol. 9 : 1 (2005), pp. 341–373. MR 2140985 Zbl 1083.58021 ArXiv math.DG/0206002 article

Varghese Mathai	Related
Richard Burt Melrose	Related

V. Mathai, R. B. Melrose, and I. M. Singer: “Fractional analytic index,” J. Diff. Geom. 74 : 2 (2006), pp. 265–292. MR 2258800 Zbl 1115.58021 ArXiv math/0402329 article

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier–Douady, 3-class, \( w \), on the manifold, we define the ring of differential operators ‘acting on sections of the projective bundle’ in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e., the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of \( K \)-theory twisted by \( w \); hence the index is a rational number but in general it is not an integer.

Varghese Mathai	Related
Richard Burt Melrose	Related

V. Mathai, R. B. Melrose, and I. M. Singer: “Equivariant and fractional index of projective elliptic operators,” J. Diff. Geom. 78 : 3 (2008), pp. 465–473. MR 2396250 Zbl 1147.58018 ArXiv math/0611819 article

Varghese Mathai	Related
Richard Burt Melrose	Related

V. Mathai, R. B. Melrose, and I. M. Singer: “The index of projective families of elliptic operators: The decomposable case,” pp. 255–296 in From probability to geometry, II: Volume in honor of the 60th birthday of Jean-Michel Bismut. Edited by X. Dai, R. Léandre, X. Ma, and W. Zhang. Astérisque 328. Société Mathématique de France (Paris), 2009. MR 2674880 Zbl 1207.19006 ArXiv 0809.0028 incollection

An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting, i.e. Dixmier–Douady, class is in \[ H^2(X;\mathbb{Z}) \cup H^1(X;\mathbb{Z}) \subset H^3(X;\mathbb{Z}) \] and secondly that the 2-class part is trivialized on the total space of the fibration. One of the features of this special case is that the corresponding Azumaya bundle can be refined to a bundle of smoothing operators. The topological and the analytic ndex of a projective family of elliptic operators associated with the smooth Azumaya bundle both take values in twisted \( K \)-theory of the parameterizing space and the main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern–Weil theory.

Xianzhe Dai	Related
Xiaonan Ma	Related
Varghese Mathai	Related
Richard Burt Melrose	Related
Rémi Léandre	Related
Weiping Zhang	Related
Jean-Michel Bismut	Related

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Isadore M. Singer

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