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Witten's rigidity theorem

Let $ E$ and $ F$ be vector bundles over a compact manifold $ M$ . If a differential operator $ D: \Gamma (E) \to \Gamma (F)$ is elliptic, then $ \ker D$ and $ \operatorname{coker}D$ are finite-dimensional vector spaces and we can define the index of $ D$ to be the virtual vector space

$\displaystyle \index D = \ker D - \operatorname{coker}D.

Now suppose a Lie group $ G$ acts on $ M$ , and $ E$ and $ F$ are $ G$ -equivariant vector bundles over $ M$ . Then $ G$ acts on $ \Gamma (E)$ by

$\displaystyle (g.s)(x)= g.(s(g^{-1}.x)),

for $ g \in G$ , $ s\in \Gamma(E), x\in M$ . The $ G$ -action is said to preserve the differential operator $ D$ if the actions of $ G$ on $ \Gamma (E)$ and $ \Gamma (F)$ commute with $ D$ . In this case $ \ker D$ and $ \operatorname{coker}D$ are representations of $ G$ , and so $ \index D$ is a virtual representation of $ G$ . We say that the operator $ D$ is rigid if its index is a multiple of the trivial representation of dimension $ 1$ . The rigidity of $ D$ means that any nontrivial irreducible representation of $ G$ in $ \ker D$ occurs in $ \operatorname{coker}D$ with the same multiplicity and vice versa.

If the multiple $ m$ is positive, then $ m.1= 1\oplus \dots \oplus 1$ is the trivial representation of dimension of $ m$ . If $ m$ is negative, the $ m.1$ is a virtual representation and the rigidity of $ D$ implies that the trivial representation $ 1$ occurs more often in $ \operatorname{coker}D$ than in $ \ker D$ .

For a circle action on a compact oriented Riemannian manifold, it is well known that the Hodge operator $ d+d*: \Omega^{\text{even}} \to
\Omega^{\text{odd}}$ and the signature operator $ d_s=d+d^*:\Omega^+ \to
\Omega^-$ are both rigid.

An oriented Riemannian manifold of dimension $ n$ has an atlas whose transition functions take values in $ \operatorname{SO}(n)$ . The manifold is called a spin manifold if it is possible to lift the transition functions to the double cover $ \operatorname{Spin}(n)$ of $ \operatorname{SO}(n)$ .

Inspired by physics, Witten discovered infinitely many rigid elliptic operators on a compact spin manifold with a circle action. They are typically of the form $ d_s \otimes R$ , where $ d_s$ is the signature operator and $ R$ is some combination of the exterior and the symmetric powers of the tangent bundle. In [91] Bott and Taubes found a proof, more accessible to mathematicians, of Witten's results, by recasting the rigidity theorem as a consequence of the Atiyah-Bott fixed point theorem.

The idea of [91] is as follows. To decompose a representation, one needs to know only its trace, since the trace determines the representation. By assumption, the action of $ G$ on the elliptic complex $ D: \Gamma (E) \to \Gamma (F)$ commutes with $ D$ . This means each element $ g$ in $ G$ is an endomorphism of the elliptic complex. It therefore induces an endomorphism $ g^*$ in the cohomology of the complex. But $ H^0 = \ker D$ and $ H^1=\operatorname{coker}D$ . The alternating sum of the trace of $ g^*$ in cohomology is precisely the left-hand side of the Atiyah-Bott fixed point theorem. It then stands to reason that the fixed point theorem could be used to decompose the index of $ D$ into irreducible representations.

Papers of Raoul Bott discussed in this article

(with R. J. Duffin) Impedance synthesis without use of transformers, J. Appl. Phys., 20 (1949), 816.
On torsion in Lie groups, Proc. NAS, 40 (1954), 586-588.
Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248-261.
(with H. Samelson) The cohomology ring of $ G/T$ , Proc. NAS, 41 (1955), 490-493.
On the iteration of closed geodesics and the Sturm Intersection theory, Comm. Pure Appl. Math. IX (1956), 171-206.
Homogeneous vector bundles, Ann. of Math. 66 (1957), 933-935.
The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313-337.
(with M. F. Atiyah and A. Shapiro) Clifford modules, Topology 3 (1965), 3-38.
The index theorem for homogeneous differential operators, in: Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, Princeton, (1964), 167-186.
(with S. Chern) Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Mathematics 114 (1964), 71-112.
Vector fields and characteristic numbers, Mich. Math. Jour. 14 (1967), 231-244.
(with M. F. Atiyah) A Lefschetz fixed point formula for elliptic complexes: I, Ann. of Math. 86 (1967), 374-407.
A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967), 311-330.
(with M. F. Atiyah) A Lefschetz fixed point formula for elliptic complexes: II, Ann. of Math. 88 (1968), 451-491.
On a topological obstruction to integrability, in: Global Analysis, Proceedings of Symposia in Pure Math. XVI (1970), 127-131.
(with G. Segal) The cohomology of the vector fields on a manifold, Topology 16 (1977), 285-298.
(with M. F. Atiyah) The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1982), 524-615.
(with M. F. Atiyah) The moment map and equivariant cohomology, Topology 23 (1984), 1-28.
(with C. Taubes) On the rigidity theorem of Witten, Jour. of the Amer. Math. Soc. 2 (1989), 137-186.

Update to the Bibliography of Raoul Bott

Raoul Bott's Bibliography in his Collected Papers [B5] is not complete. This update completes the Bibliography as of June 2000.

In memoriam Sumner B. Myers, Mich. Math. Jour., 5 (1958), 1-4.
(with L. Tu) Differential Forms in Algebraic Topology, Springer-Verlag, (1982), 1-331.
Georges de Rham: 1901-1990, Notices Amer. Math. Soc. 38 (1991), 114-115.
Stable bundles revisited, Surveys in Differential Geometry, 1 (1991), 1-18.
On E. Verlinde's formula in the context of stable bundles, Internat. J. Modern Phys. A 6 (1991), 2847-2858.
Nomination for Stephen Smale, Notices Amer. Math. Soc., 38 (1991), 758-760..
On knot and manifold invariants, in: New Symmetry Principles in Quantum Field Theory (Cargèse, 1991), NATO Adv. Sci. Inst. Ser. B Phys. 295, Plenum, (1992), 37-52.
Topological aspects of loop groups, in: Topological Quantum Field Theories and Geometry of Loop Spaces (Budapest, 1989), World Sci. Publishing, (1992), 65-80.
For the Chern volume, in: Chern--a Great Geometer of the Twentieth Century, Internat. Press, (1992), 106-108.
Reflections on the theme of the poster, in: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish (1993), 125-135.
Luncheon talk and nomination for Stephen Smale, in: From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, (1993), 67-72.
(with C. Taubes) On the self-linking of knots, J. Math. Phys. 35 (1994), 5247-5287.
On invariants of manifolds, in: Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. of Math. Stud., 137, Princeton Univ. Press, (1995), 29-39.
Configuration spaces and imbedding invariants, Turkish J. Math. 20 (1996), 1-17.
Configuration spaces and imbedding problems, in: Geometry and Physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., 184, Dekker (1997), 135-140.
Critical point theory in mathematics and in mathematical physics, Turkish J. Math. 21 (1997), 9-40.
(with A. Cattaneo) Integral invariants of $ 3$ -manifolds, J. Differential Geom. 48 (1998), 91-133.
An introduction to equivariant cohomology, in: Quantum Field Theory: Perspective and Prospective, Kluwer Academic Publishers, (1999), 35-57.
(with A. Cattaneo) Integral invariants of $ 3$ -manifolds, II, to appear in J. Differential Geom.

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