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

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

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

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

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

[1]

(with R. J. Duffin) Impedance synthesis without use of transformers, J. Appl. Phys., 20 (1949), 816.

[8]

On torsion in Lie groups, Proc. NAS, 40 (1954), 586-588.

[9]

Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248-261.

[12]

(with H. Samelson) The cohomology ring of $G/T$ , Proc. NAS, 41 (1955), 490-493.

[14]

On the iteration of closed geodesics and the Sturm Intersection theory, Comm. Pure Appl. Math. IX (1956), 171-206.

[15]

Homogeneous vector bundles, Ann. of Math. 66 (1957), 933-935.

[24]

The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313-337.

[33]

(with M. F. Atiyah and A. Shapiro) Clifford modules, Topology 3 (1965), 3-38.

[37]

The index theorem for homogeneous differential operators, in: Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, Princeton, (1964), 167-186.

[38]

(with S. Chern) Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Mathematics 114 (1964), 71-112.

[41]

Vector fields and characteristic numbers, Mich. Math. Jour. 14 (1967), 231-244.

[42]

(with M. F. Atiyah) A Lefschetz fixed point formula for elliptic complexes: I, Ann. of Math. 86 (1967), 374-407.

[43]

A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967), 311-330.

[44]

(with M. F. Atiyah) A Lefschetz fixed point formula for elliptic complexes: II, Ann. of Math. 88 (1968), 451-491.

[51]

On a topological obstruction to integrability, in: Global Analysis, Proceedings of Symposia in Pure Math. XVI (1970), 127-131.

[71]

(with G. Segal) The cohomology of the vector fields on a manifold, Topology 16 (1977), 285-298.

[81]

(with M. F. Atiyah) The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308 (1982), 524-615.

[82]

(with M. F. Atiyah) The moment map and equivariant cohomology, Topology 23 (1984), 1-28.

[91]

(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.

[20]

In memoriam Sumner B. Myers, Mich. Math. Jour., 5 (1958), 1-4.

[79]

(with L. Tu) Differential Forms in Algebraic Topology, Springer-Verlag, (1982), 1-331.

[94]

Georges de Rham: 1901-1990, Notices Amer. Math. Soc. 38 (1991), 114-115.

[95]

Stable bundles revisited, Surveys in Differential Geometry, 1 (1991), 1-18.

[96]

On E. Verlinde's formula in the context of stable bundles, Internat. J. Modern Phys. A 6 (1991), 2847-2858.

[97]

Nomination for Stephen Smale, Notices Amer. Math. Soc., 38 (1991), 758-760..

[98]

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.

[99]

Topological aspects of loop groups, in: Topological Quantum Field Theories and Geometry of Loop Spaces (Budapest, 1989), World Sci. Publishing, (1992), 65-80.

[100]

For the Chern volume, in: Chern--a Great Geometer of the Twentieth Century, Internat. Press, (1992), 106-108.

[101]

Reflections on the theme of the poster, in: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish (1993), 125-135.

[102]

Luncheon talk and nomination for Stephen Smale, in: From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, (1993), 67-72.

[103]

(with C. Taubes) On the self-linking of knots, J. Math. Phys. 35 (1994), 5247-5287.

[104]

On invariants of manifolds, in: Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. of Math. Stud., 137, Princeton Univ. Press, (1995), 29-39.

[105]

Configuration spaces and imbedding invariants, Turkish J. Math. 20 (1996), 1-17.

[106]

Configuration spaces and imbedding problems, in: Geometry and Physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., 184, Dekker (1997), 135-140.

[107]

Critical point theory in mathematics and in mathematical physics, Turkish J. Math. 21 (1997), 9-40.

[108]

(with A. Cattaneo) Integral invariants of $3$ -manifolds, J. Differential Geom. 48 (1998), 91-133.

[109]

An introduction to equivariant cohomology, in: Quantum Field Theory: Perspective and Prospective, Kluwer Academic Publishers, (1999), 35-57.

[110]

(with A. Cattaneo) Integral invariants of $3$ -manifolds, II, to appear in J. Differential Geom.