zbMATH — the first resource for mathematics

Functorial Hodge identities and quantization. (English) Zbl 1031.53065
Let \(\Gamma_M (\Lambda)\) be the space of real differential forms over an oriented manifold \(M\) with non-degenerate metric of arbitrary signature. There are fundamental operators acting on the above space: the degree \(\partial\), the Hodge star opearator \(*\), the exterior derivative \(d\), its formal adjoint \(d^*\) and the Hodge d’Alembertian \( d d^* + d^* d\). Such operators satisfy some algebraic relations, first proved in the positive-definite case by W. V. Hodge [The theory and applications of harmonic integrals, Cambridge University Press (1941; Zbl 0024.39703)], K. Kodaira [Proc. Imp. Acad. Tokyo 20, 186-198 (1944; Zbl 0063.03285)] and P. Bidal and G. de Rham [Comment. Math. Helv. 19, 1-49 (1946; Zbl 0063.00378)].
These relations hold also for arbitrary Kähler manifolds [A. Weil, Comment. Math. Helv. 20, 110-116 (1947; Zbl 0034.35801)] and similar relations are valid also for certain operators on hyper-Kähler manifolds [M. S. Verbitsky, Funct. Anal. Appl. 24, 229-230 (1990; Zbl 0717.53041)].
In the paper the author gives a uniform, abstract derivation of new sets of relations for Riemannian, Kähler and hyper-Kähler geometry showing that the relations he obtains are functorial. The new relations imply the known ones, but they are more general since they describe Lie group rather than Lie algebra actions on differential operators and they hold for arbitrary signature. For each of the three types of geometry, a multiplicative functor from the corresponding category of real, graded, flat vector bundles to the category of infinite-dimensional \({\mathbb Z}_2\)-projective representations of an algebraic structure is defined. The algebraic structure depends only on the type of geometry and not on the manifold, metric or flat vector bundle. Moreover, the author defines new multiplicative numerical invariants of closed Kähler and hyper-Kähler manifolds which are invariant under deformation of the metric.
Reviewer: Anna Fino (Torino)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58A10 Differential forms in global analysis
58A14 Hodge theory in global analysis
Full Text: DOI
[1] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl. 1, 3 – 38. · Zbl 0146.19001 · doi:10.1016/0040-9383(64)90003-5 · doi.org
[2] Bernstein, J. – Lectures on SUSY (notes by P. Deligne and J. Morgan) in ‘Quantum Fields and Strings: a course for mathematicians’, Vol 1 (ed P. Deligne et al) - American Mathematical Society, Providence, RI, 1999.
[3] Bidal, P. and de Rham, G. – Les formes différentielles harmoniques, Comm. Math. Helvetica, vol. 19, 1946, 1-49. · Zbl 0063.00378
[4] J. M. Figueroa-O’Farrill, C. Köhl, and B. Spence, Supersymmetry and the cohomology of (hyper)Kähler manifolds, Nuclear Phys. B 503 (1997), no. 3, 614 – 626. · Zbl 0938.81041 · doi:10.1016/S0550-3213(97)00548-8 · doi.org
[5] W. V. D. Hodge, The theory and applications of harmonic integrals, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1989. Reprint of the 1941 original; With a foreword by Michael Atiyah. · Zbl 0024.39703
[6] Roger Howe, Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons, Applications of group theory in physics and mathematical physics (Chicago, 1982) Lectures in Appl. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1985, pp. 179 – 207. · Zbl 0558.22018
[7] Kodaira, K. – Über die Harmonischen Tensorfelder in Riemannschen Mannigfaltigkeiten I, Proc. Imp. Acad Tokyo, vol. 20, 1944, pp. 186-198. · Zbl 0063.03285
[8] Richard S. Palais, Seminar on the Atiyah-Singer index theorem, With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. · Zbl 0137.17002
[9] M. J. Slupinski, Dual pairs in \?\?\?(\?,\?) and Howe correspondences for the spin representation, J. Algebra 202 (1998), no. 2, 512 – 540. · Zbl 0909.22034 · doi:10.1006/jabr.1997.7279 · doi.org
[10] Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. · Zbl 0584.10010
[11] M. S. Verbitskiĭ, Action of the Lie algebra of \?\?(5) on the cohomology of a hyper-Kähler manifold, Funktsional. Anal. i Prilozhen. 24 (1990), no. 3, 70 – 71 (Russian); English transl., Funct. Anal. Appl. 24 (1990), no. 3, 229 – 230 (1991). · Zbl 0717.53041 · doi:10.1007/BF01077967 · doi.org
[12] Weil, A. – Sur la théorie des formes différentielles attachées à une variété analytique complexe, Comm. Math. Helv., vol. 20, 1947, pp. 110-116. · Zbl 0034.35801
[13] Weil, A. – Variétés Kählériennes. - Hermann, Paris, 1958. · Zbl 0205.51701
[14] Edward Witten, Homework, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 609 – 717. · Zbl 1137.81302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.