nLab complex volume

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Differential cohomology

Contents

Idea

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.)

Properties

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)

References

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 June 4, 2020 at 13:56:52. See the history of this page for a list of all contributions to it.