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The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a boundary term, for example the extrinsic curvature in two dimensional case.

I am wondering if there is an analogous topological invariant for a gauge theory (or principal bundle) on manifolds with boundary, and if so, what is its boundary term. I am particularly interested in the case of the two dimensional base manifold.

Note added: I am from a physics background and pretty new to this kind of subject. It would be especially great if one can also point to physics-oriented reference on this.

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    $\begingroup$ Atiyah Patodi Singer. For example, see Melrose's book. $\endgroup$
    – AHusain
    Nov 7, 2016 at 22:08
  • $\begingroup$ Fantastic! Thank you. I will look into it. Can anyone explain what the topological invariant looks like in terms of local gauge fields/field strength, say for U(1) principal bundle over a two dimensional manifold with boundary? In particular, what is the boundary term looks like? It would be very helpful to know this in advance before I decide whether to really learn the APS theorem for my (physics) research. $\endgroup$
    – BK736
    Nov 8, 2016 at 0:14
  • $\begingroup$ I would highly recommend taking a look at Nakahara's book "Geometry topology and physics". This book is aimed at people with physics background. Chapter 11 describes characteristic classes obtained via Chern-Weil (which expresses them as invariant polynomials of the field strength). Chapter 12 describes both the AS index theorem and APS index theorem in terms of these invariants. $\endgroup$ Dec 9, 2016 at 15:10

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