Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of C N ⊗ C N , N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 → 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N = 1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds W endowed with a flat principal P SL(2, C)-bundle ρ, and a fixed non empty link L if N > 1, and for (possibly "marked") cusped hyperbolic 3-manifolds M . When N = 1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N > 1, the invariants for M are new; those for triples (W, L, ρ) coincide with the quantum hyperbolic invariants defined in [3], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N = 1 and N > 1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N → ∞.

Quantum hyperbolic geometry

“…We will analyze the relations between H N and H N partition functions. When N D 1, we know a geometric interpretation of the invariants H 1 .W; L F ; / only when extends to the whole of W : by using Theorem 6.7 of the present paper we can reach the setup of [24] (see also [4]) and, as mentioned above, identify H 1 .W; L F ; / with the Cheeger-Chern-Simons invariant of the pair .W; /, the link being immaterial. If does not extend, we get a relative version, the meaning of which deserves careful attention.…”

“…; b; w; f; c/, and matrix dilogarithms R N ..b; w; f; c// 2 Aut.‫ރ‬ N˝‫ރ‬N / defined for every odd positive integer N , which were introduced in [4]. In fact we consider as an abstract oriented simplex equipped with an additional decoration.…”

“…The key geometric object on y ‫ރ‬ is the smooth tangent 2-vector 2 ƒ 2 T y ‫ރ‬ given by (see Choi [12]) (2)(3)(4)(5) .…”

“…Consider the connected component y ‫ރ‬ 0;0 of y ‫ރ‬ with even-valued flattenings, ie the Riemann surface of the map ' 0;0 in (2-4). We have an embedding (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) y ‫ރ‬ 0;0 ! y X .uI p; q/ 7 !…”

“…In [4] we defined also quantum hyperbolic invariants H N .M / for cusped manifolds M . Although these invariants are not immediately QHFT partition functions, we will show how they can be obtained in terms of these last.…”