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Chern–Simons theory and cohomological invariants of representation varieties

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Abstract

We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space G/H is constant under deformation of the G/H-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter 3-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of G-invariant forms on G/H is generated by “Chern–Weil forms” and “Chern–Simons forms”.

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Notes

  1. While the complexification is perfectly well-defined at the level of Lie algebras, there is something arbitrary in our definition of the complexification of a Lie group. For instance, the complexification of \(\textrm{SL}(k,\mathbb {R})\) is the adjoint group \(\textrm{PSL}(k,\mathbb {C})\). We did not settle with a more algebraic definition because we want to consider connected simple real Lie groups such as \(\textrm{SO}_\circ (p,1)\), which are not necessarily algebraic.

  2. Note that both connections are indeed bi-invariant because the right-translate of a left-invariant vector field is another left-invariant vector field.

  3. Note that this does not require \(\textrm{Hom}(\pi _1(M),G)\) to be smooth, since we do not assume that these smooth arcs are immersed.

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Acknowledgements

This paper is very much indebted to François Labourie, who explained to me the nuts and bolts of Chern–Simons theory and how it could be used to prove volume rigidity of locally homogeneous manifolds. I thank him more generally for the inspiration that his work has been for my research. I am also thankful to Michelle Bucher and Marc Burger for their interest in this work and for pointing out several references. Finally, I thank the anonymous referee for his valuable comments.

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  1. ÉNS – PSL, CNRS, 45 rue d’Ulm, 75005, Paris, France

    Nicolas Tholozan

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  1. Nicolas Tholozan
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Correspondence to Nicolas Tholozan.

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To François Labourie, for his 60th + $$\varepsilon $$ ε birthday

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Tholozan, N. Chern–Simons theory and cohomological invariants of representation varieties. Geom Dedicata 218, 82 (2024). https://doi.org/10.1007/s10711-024-00926-y

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