nLab Narasimhan–Seshadri theorem

Idea

The Narasimhan–Seshadri theorem (Narasimhan-Seshadri 65) identifies certain moduli spaces of flat connections over a (compact) Riemann surface $\Sigma$ with (compact) complex manifolds of stable holomorphic vector bundles over $\Sigma$.

For the special case of line bundles this may be viewed as a special case of the Hodge-Maxwell theorem, also of Deligne’s characterization of the intermediate Jacobian, see there at Examples – Picard variety. The analogue of the theorem for higher dimensional complex manifolds is the Kobayashi-Hitchin correspondence. The special case of that for Kähler manifolds is the Donaldson-Uhlenbeck-Yau theorem.

Statement

An indecomposable Hermitian holomorphic vector bundle $E$ on a Riemann surface $(\Sigma,g)$ is stable precisely if there is a compatibly unitary connection $\nabla$ on $E$ with constant central curvature

$$\star F_\nabla = - \mu(E)$$

equal to (minus) the slope of $E$ (where $\star$ is the Hodge star operator).

e.g. (Evans, p. 2)

(Atiyah-Bott 83 (8.1))

Applications

In Quantum field theory

In gauge field theories such as notably Chern-Simons theory, moduli spaces of flat connections may appear as phase spaces. The Narasimhan–Seshadri theorem then provides a complex structure on these phase spaces and thus a spin^c structure (see there). This is the data required to obatain the geometric quantization by push-forward of the phase space. See also at Hitchin connection.

Related concepts

• Hitchin connection

• quantization of Chern-Simons theory

• equivariant elliptic cohomology

Related theorems

• Harder-Narasimhan theorem

• Koszul-Malgrange theorem

• Donaldson-Uhlenbeck-Yau theorem

References

The original article is

• Mudumbai Narasimhan, C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Annals of Mathematics. Second Series 82 3 (1965) 540-567 [doi:10.2307/1970710, jstor:1970710]

and another proof appeared in

• Simon Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. Volume 18, Number 2 (1983), 269-277. (EUCLID)

A good general survey and re-discussion is in

• Michael Atiyah, Raoul Bott, section 8 of The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences

  Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

A recording of a review talk is here

• Mudumbai Narasimhan, Moduli of vector bundles on compact Riemann surfaces, talk Dec 2016, Bangalore (video)

Lecture notes include

• Jonathan Evans, Aspects of Yang-Mills theory, lecture notes,

  Narasimhan-Seshadri theorem (lecture 13 pdf, lecture 14 pdf, lecture 15 pdf, lecture 16 pdf)

A textbook providing much of the background definitions involved is

• Daniel Huybrechts, Mafred Lehn, The Geometry of the Moduli Spaces of Sheaves, 1996 (pdf)

Related discussion in the context of Hitchin connections is in

• Peter Scheinost, Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)