Index theory of elliptic operators

M.F. Atiyah and I.M. Singer were awarded the highly prestigious and lucrative 2004 Abel Prize, which is the mathematical equivalent of the Nobel Prize, in recognition of their fundamental and monumental contributions to mathematics and physics, notably Atiyah-Singer index theory.

At the risk of oversimplifying, in 1962, Atiyah and Singer defined the Dirac operator  ∂, on any compact spin manifold M of even dimension. Recall that the analytic index of the Dirac operator is defined to be the integer,

Indexa(∂)  =   dim(ker(∂)) - dim(coker(∂))  ε Z.

Then the Atiyah-Singer index theorem for Dirac operators states that,

Indexa(∂)  =  ∫M Â(M).

Here the right hand side is the A-hat genus of the manifold M, which is a topological invariant. Then the following is a natural question in this context.

Question: Since M Â(M) continues to make sense for non-spin manifolds M, (although however it may not be an integer in these cases) what corresponds to the analytic index in this situation, since the usual Dirac operator does not exist?

Fractional analytic index

In some joint work with R.B. Melrose and I.M. Singer, we propose 2 solutions to the question above, and relate them.

The first solution given in [MMS3] is as follows. We construct a global distributional section of the Clifford algebra bundle on the product space, supported within the diagonal and conormal wrt it, representing the spin Dirac operator - we call this the projective spin Dirac operator. We define its fractional analytic index in an essentially analytic way, and prove an index theorem showing that it equals the A-hat genus. We also prove the fractional index theorem for all projective elliptic operators.

The second solution given in [MMS4] shows that there also always exists a Spin equivariant transversally elliptic Dirac operator on the oriented orthonormal frame bundle of an oriented manifold. The precise relation between the fractional analytic index of the projective Dirac operator and the equivariant index of the associated Spin equivariant transversally elliptic Dirac operator is explained there. We also prove a similar result precisely relating the fractional analytic index of any projective elliptic operator and the equivariant index of the associated equivariant transversally elliptic operator.

For example, the fractional analytic index of the projective spin Dirac operator on the complex projective spaces CP2n is never an integer! Notice that CP2n is never a spin manifold.

Index Theory for projective families of elliptic operators

In the publication [MMS1], we have proved the index theorem for projective families of elliptic pseudodifferential operators, which is a generalization of the renowned Atiyah-Singer index theorem for families of elliptic pseudodifferential operators. Viewed on the parametrizing space, the projective family consists of local families of elliptic operators, which have a cocycle compatibility relation on triple overlaps. The index of such a projective family lies in the twisted K-theory of the parametrizing space. We have also computed the Chern character of the index in terms of characteristic classes. In proving these results, we have had to assume that the twist was a torsion class in the third cohomology of the parametrizing space.

In [MMS2], we made a breakthough, by proving the index theorem for projective families of elliptic pseudodifferential operators, whenever the twist is a decomposable class in the third cohomology of the parametrizing space. This includes many interesting nontorsion classes for the first time. Although the Azumaya bundle is necessarily infinite dimensional in this case, the geometry resolving it is finite dimensional, which is used in an essential way in the proof.

February 2008 - January 2012, Editor in charge of "Global Analysis, Noncommutative Geometry, and the Mathematics of String Theory", Proceedings of the American Mathematical Society.

Recent invited talks in Index Theory

I was an invited speaker at the following conferences:

(invited participant) Perspectives in Mathematics and Physics: a conference celebrating I.M. Singer's 85th birthday, MIT and Harvard University, May 22-24, 2009.

From Wave Propagation to K-theory: a Conference in Honour of the 60th Birthday of Richard Melrose, Stanford University, October 25-26, 2008.

Motives, Quantum Field Theory, and Pseudodifferential Operators, Boston University, June 2-13, 2008.

Geometry and Analysis on Manifolds, at the Chern Institute of Mathematics, Nankai University, Tianjin, China, April 8-14, 2007.

I was invited to the memorable opening ceremony of the impressive new Shiing-shen Chern building of the Nankai Institute of Mathematics, Tianjin, China, which was held during the XXIII International Conference of Differential Geometric methods in Theoretical Physics, August 20-27, 2005. I gave an invited talk on the fractional analytic index based on [MMS3].

I co-organized a major international AMSI workshop entitled, Noncommutative Geometry and Index Theory, which was held at ANU, Canberra, 22 July - 1 August, 2005.

I gave an invited talk on the fractional analytic index based on [MMS3] at the second annual spring institute on noncommutative geometry and operator algebras, May 5-25, 2004, Vanderbilt University, Nashville, Tennessee, USA

References

[MM] V. Mathai and R.B. Melrose,
Geometry of Pseudodifferential algebra bundles and Fourier Integral Operators
51 pages, [1210.0990] [MMS2] V. Mathai, R.B. Melrose and I.M. Singer,
The index of projective families of elliptic operators: the decomposable case,
Astérisque,
328 (2009) 255-296. [0809.0028]

[MMS4] V. Mathai, R.B. Melrose and I.M. Singer,
Equivariant and fractional index of projective elliptic operators,
Journal of Differential Geometry,
78 no.3 (2008) 465-473. [math.DG/0611819]

[MMS3] V. Mathai, R.B. Melrose and I.M. Singer,
Fractional Analytic Index,
Journal of Differential Geometry,
74 no.2 (2006) 265-292. [math.DG/0402329]

[MMS1] V. Mathai, R.B. Melrose and I.M. Singer,
The index of projective families of elliptic operators,
Geometry and Topology,
9 (2005) 341-373. [math.DG/0206002]