nLab
volume conjecture

Contents

Context

Manifolds and cobordisms

Functorial quantum field theory

Contents

Idea

The volume conjectures are a class of conjectures (slightly differing in generality and assumptions), saying that on a suitable (in particular: hyperbolic) 3-manifold XX a large N limit of SU(n)-Chern-Simons theory quantum observables (NN-colored Jones polynomials or more generally RT-invariants, TV-invariants) is equal to the volume or complex volume of XX.

For a few special cases of 3-manifolds there are explicit proofs of the volume conjecture(s). Besides this there is an abundance of numerical evidence for the volume conjectures, using computer algebra such as SnapPy (see also Zickert 07). In fact experimentation with these numerics is what has been driving the formulation of further variants of the volume conjecture.

Hence experimental mathematics strongly suggests that the volume conjectures are true. But a conceptual explanation (let alone proof) in terms of quantum field theory has remained open (Witten 14, bottom of p. 4). But an explanation in terms of wrapped M5-branes (3d-3d correspondence) is claimed in Gang-Kim-Lee 14, 3.2, see also Gang-Kim 18 (21).

For SU(2)SU(2) on knot complements

The original volume conjecture (also “Kashaev’s conjecture”, due to Kashaev 95, and understood in terms of the NN-colored Jones polynomial by Murakami-Murakami 01) states that the large N limit of the NN-colored Jones polynomial (for gauge group SU(2)) of a knot KK gives the simplicial volume of its complement in the 3-sphere (for hyperbolic knots this is the volume of the complementary hyperbolic 3-manifold)

(1)lim N(2πlogN|V N(K;q=e 2πiN)|)=vol(K). lim_{N \to \infty} \left( \frac{ 2 \pi log } {N} \left\vert V_N(K; q = e^{\frac{2 \pi i}{N}}) \right\vert \right) \;=\; vol(K).

Here V N(K;q)V_N(K; q) is the ratio of the values of the NN-colored Jones polynomial of KK and of the unknot

V N(K;q)=J N(K;q)J N(;q). V_N(K; q) = \frac{J_N(K; q)}{J_N(\bigcirc; q)}.

The simplicial volume of a knot complement can be found via its unique torus decomposition into hyperbolic pieces and Seifert fibered pieces by a system of tori. The simplicial volume is then the sum of the hyperbolic volumes of the hyperbolic pieces of the decomposition.

If one omits the absolute value in (1) then the volume conjecture instead involves the complex volume (MMOTY 02, Conjecture 1.2).

For SU(2)SU(2) on general 3-manifolds

More generally, volume conjectures state convergence of the Turaev-Viro invariants or Reshetikhin-Turaev invariants on general hyperbolic 3-manifolds to the volume or complex volume, respectively.

See (Chen-Yang 15)

For SU(n)SU(n)

Generalization from gauge group SU(2) to SU(n): Chen-Liu-Zhu 15

References

Original articles include

Generalization to Reshetikhin-Turaev construction on closed manifold, to the Turaev-Viro construction on manifolds with boundary, and to more general roots of unity than considered before is in

Generalization to SU(n):

Review includes

See also

Speculative discussion in terms of quantum field theory or string theory includes

A conceptual explanation of the volume conjecture via analytically continued Chern-Simons theory was proposed in

(but it seems that as a sketch or strategy for a rigorous proof, it didn’t catch on).

In terms of M5-branes wrapped on the hyperbolic 3-manifold (3d-3d correspondence):

  • Dongmin Gang, Nakwoo Kim, Sangmin Lee, Holography of Wrapped M5-branes and Chern-Simons theory, j.physletb.2014.04.051 (arXiv:1401.3595)

  • Dongmin Gang, Nakwoo Kim, Sangmin Lee, Section 3.2 of Holography of 3d-3d correspondence at Large N, JHEP04(2015) 091 (arXiv:1409.6206)

  • Dongmin Gang, Nakwoo Kim, aound (21) of Large NN twisted partition functions in 3d-3d correspondence and Holography, Phys. Rev. D 99, 021901 (2019) (arXiv:1808.02797)

Last revised on May 22, 2019 at 15:56:25. See the history of this page for a list of all contributions to it.