complex volume



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Differential cohomology



The Cheeger-Simons classes are complexified secondary invariants.

Under identifying the fundamental class of a hyperbolic 3-manifold XX as an element in the Bloch group, the corresponding degree-3 Cheegers-Simons invariant is the complex volume of the 3-manifold, namely the linear combination

CS+ivol CS + i vol

of its Chern-Simons invariant and its volume (e.g. Neumann 11, section 2.3, Garoufalidis-Thurston-Zickert 11).

This appears as the action in analytically continued Chern-Simons theory.

(It is, incidentally, also the contribution of a corresponding membrane instanton wrapping a hyperbolic 3-cycle.)


Volume conjecture

The volume conjecture for the Reshetikhin-Turaev construction states that in the classical limit it converges to the complex volume (MMOTY 02, Conjedtcure 1.2, see also Chen-Yang 15)


  • Jeff Cheeger, Jim Simons, Differential characters and geometric invariants, in Geometry and Topology, Proceedings of the Special Year, University of Maryland 1983-84, eds. J. Alexander and J. Harer, Lecture Notes in Math. 1167, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 50–80.

  • Johan Dupont, Richard Hain, Steven Zucker, Regulators and characteristic classes of flat bundles (arXiv:alg-geom/9202023)

  • Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)

  • Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541, Amer. Math. Soc. 2011, 233–246 (arXiv:1108.0062)

  • Stavros Garoufalidis, Dylan Thurston, Christian Zickert, The complex volume of SL(n,)SL(n,\mathbb{C})-representations of 3-manifolds (arXiv:1111.2828, Euclid)

The volume conjecture (Kashaev’s conjecture) for complex volume is due to

see also

Relation to analytic torsion is discussed in

  • Varghese Mathai, section 6 of L 2L^2-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386

  • John Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 no 2 (1992) (pdf)

Last revised on May 22, 2019 at 05:07:34. See the history of this page for a list of all contributions to it.