In its simplest form, a \((d+1)\)-dimensional topological quantum field theory (TQFT) associates a vector space to \(d\)-dimensional manifolds and a vector to a \((d+1)\)-dimensional manifold with boundary (in the vector space associated to the boundary) satisfying Atiyah’s axioms [M. Atiyah, Publ. Math., Inst. Hautes Étud. Sci. 68, 175–186 (1988; Zbl 0692.53053)]; it defines a (modular) functor from the category whose objects are \(d\)-dimensional manifolds (perhaps with extra structure) and morphisms are cobordisms, to the category of vector spaces.
It is by now well-established that the three main ways of defining the Witten-Reshetikhin-Turaev (WRT) 2+1-dimensional TQFT are equivalent for \(SU(n)\), namely via the representation theory of quantum groups (or equivalent combinatorics, initially for \(U_qsl_2\), cf. N. Yu. Reshetikhin and V. G. Turaev [Commun. Math. Phys. 127, No. 1, 1–26 (1990; Zbl 0768.57003); Invent. Math. 103, No. 3, 547–597 (1991; Zbl 0725.57007)]), via conformal field theory (conformal blocks of the Wess-Zumino-Witten model, A. Tsuchiya, K. Ueno and Y. Yamada [Adv. Stud. Pure Math. 19, 459–566 (1989; Zbl 0696.17010)], J. E. Andersen and K. Ueno [Invent. Math. 201, No. 2, 519–559 (2015; Zbl 1328.57030)]) and via Witten-Chern-Simons theory, by which we mean gauge theoretic constructions (geometric quantisation, N. J. Hitchin [Commun. Math. Phys. 131, No. 2, 347–380 (1990; Zbl 0718.53021)], S. Axelrod et al. [J. Differ. Geom. 33, No. 3, 787–902 (1991; Zbl 0697.53061)] making rigorous Witten’s path integral with Chern-Simons action [E. Witten, Commun. Math. Phys. 121, No. 3, 351–399 (1989; Zbl 0667.57005)]. The theory is defined from the data of an integer \(k\) (appearing in CFT as the level or as a multiplicative parameter in the phase defined by the path-integral action) and a simple Lie group \(G\) (here \(SU(n)\)), while the root of unity \(q\) in the quantum group picture is related by \(q=e^{\frac{2\pi{}i}{k+c^\vee_G}}\); here \(c^\vee_G\) is the dual Coxeter number of \(G\) (here \(n\)). The identification of the presentations was proved in [Y. Laszlo, J. Differ. Geom. 49, No. 3, 547–576 (1998; Zbl 0987.14027)] along with a series of papers of J. E. Andersen and K. Ueno [J. Knot Theory Ramifications 16, No. 2, 127–202 (2007; Zbl 1123.81041); Quantum Topol. 3, No. 3–4, 255–291 (2012; Zbl 1263.57025)].
The theory ‘evaluates’ to give a (finite-dimensional) Hilbert space \(Z(\Sigma)\) associated to a 2-dimensional surface \(\Sigma\) with marked points (and the extra data of an element of the centre \(Z(G)\) at each marked point) and a vector \(Z(M)\) in this space, associated to a 3-dimensional manifold \(M\) (with boundary \(\Sigma\)) containing links (with boundary at those marked points) or (in the case of empty boundary) a scalar invariant associated with a link inside a (compact connected oriented) 3-manifold. This evaluation is the Jones polynomial of links, for links in \(S^3\), and the Witten-Reshetikhin-Turaev invariant of 3-manifolds, for the pair of an empty link in a closed 3-manifold (for \(g=sl_2\)).
The formulation of the WRT TQFT via the Witten-Chern-Simons path integral \(\int_{\mathcal A}e^{\frac{ik}{4\pi}CS(A)}DA\) where the integral is over \(G\)-connections \(A\) over the manifold (up to gauge equivalence) and \(CS(A)\) is the Chern-Simons invariant of \(A\), leads to the expectation of the existence of an asymptotic expansion in \(k^{-1}\) for large \(k\), known as the asymptotic expansion conjecture. Various results are known.
The current paper deals with the case of \(G=SU(n)\) with \(\Sigma\) being a once-punctured torus of genus at least two. Let \(\mathcal{M}\) denote the moduli space of flat \(SU(N)\) connections on \(\Sigma\) (with fixed holonomy around the puncture); by the Narasimhan-Seshadri theorem [M. S. Narasimhan and C. S. Seshadri, Math. Ann. 155, 69–80 (1964; Zbl 0122.16701)], this is a compact, simply connected, symplectic manifold (denote the dimension as \(2n_0\)) with Kähler structure induced by a choice of complex structure \(\sigma\) on \(\Sigma\). The Hilbert space \(Z(\Sigma)\) is identified as \(H^0(\mathcal{M}_\sigma,\mathcal{L}^{\otimes{}k})\) where \(\mathcal{L}\) is the Chern-Simons line bundle; it is the fibre of the Verlinde bundle over the point \(\sigma\) in Teichmüller space, on which the Hitchin connection provides a (projectively) flat connection.
An element \(\phi\) of the mapping class group \(\Gamma\) of \(\Sigma\) determines a mapping torus by identifying the boundary components of \(\Sigma\times[0,1]\) via \(\phi\). Also \(\phi\) acts on \(\mathcal{M}\) and this lifts to an action \(Z^{(k)}_{CS}(\phi)\) on the Verlinde bundle whose trace is, by the axioms of TQFT, the WRT invariant of the mapping torus. The paper under review uses a delicate analysis of general oscillatory integrals to provide an asymptotic expansion of this trace in more general cases than was previously accessible. Set \(r=k+n\) and let \(\mathcal{M}^\phi\subset\mathcal{M}\) denote the fixed point set of \(\phi\).
(1) For any \(N\in\mathbb{N}\), \[tr(Z^{(k)}_{CS}(\phi))=r^{n_0}\sum_{n=0}^Nr^{-n}\int_{\mathcal{M}}e^{rP_\phi}\Omega_n+O(k^{n_0-N-1})\] where \(P_\phi:\mathcal{M}\to\mathbb{C}/2\pi{}i\mathbb{Z}\) is a smooth function, real analytic near \(\mathcal{M}_\phi\) with strictly negative real part away from \(\mathcal{M}^\phi\), while \(\Omega_m\) (\(m\geq0\)) are top-dimensional smooth forms on \(\mathcal{M}\). Every point in \(\mathcal{M}^\phi\) is a stationary point of \(P_\phi\), with corresponding stationary value \(2\pi{}i\) times the value of the Chern-Simons action.
This holds without restriction on the mapping class \(\phi\). Using a stationary phase approximation (Theorem 1.4) for certain general oscillatory integrals of the form \(\int_{\mathbf{R}^n}e^{kf(x)}\phi(x)dx\) based on work of Hörmander and Malgrange, they then deduce the following results.
(2a) If \(\mathcal{M}^\phi\) is cut out transversely (that is, any connected component \(Y\) of \(\mathcal{M}^\phi\) is smooth and satisfies \(TY=Ker(d\phi-I)|_Y\)) then \[tr(Z^{(k)}_{CS}(\phi))\sim\sum_je^{2\pi{}ir\theta_j}r^{m_j}\sum_{\alpha=0}^\infty{}r^{-\alpha/2}\int_{\mathcal{M}^\phi}\Sigma_\alpha^j\] where \(\{\theta_j\}\) are the critical Chern-Simons values and correspondingly \(2m_j=\max_z\left(\dim(\ker(d\phi_z-I))\right)\) amongst \(z\in\mathcal{M}^\phi\) for which \(P_\phi(z)=2\pi{}i\theta_j\); and \(\Omega_\alpha^j\) are suitable differential forms on \(\mathcal{M}^\phi\).
(2b) Under the further assumption on \(\phi\), that for every \(z\in\mathcal{M}^\phi\) either (a) \(z\) is a smooth point with \(T_z\mathcal{M}^\phi=\ker(d\phi_z-I)\) or (b) \(\dim\ker(d\phi_z-I)\leq1\) or (c) \(z\) is an isolated stationary point of the germ of the holomorphic extension of \(P_\phi\) to a neighbourhood of \(z\) in \(\mathcal{M}\), there is an asymptotic expansion \[tr(Z^{(k)}_{CS}(\phi)\sim\sum_je^{2\pi{}ir\theta_j}r^{n_j} \sum_{\alpha\in{}A_j,\beta\in{}B_j}c_{\alpha,\beta}r^\alpha(\ln{r})^\beta\] where \(n_j\) are non-negative rationals, each \(B_j\) is a finite set of natural numbers and each \(A_j\) is a union of finitely many (one-sided infinite) arithmetic progressions of non-positive rational numbers.
(3) Without the assumption of being cut out transversally, they show for the case of the moduli space \(\mathcal{M}_l\) of flat SU(2) connections on a once-punctured (genus 1) torus whose holonomy around the puncture has trace \(l\in(-2,2)\), that the same form of asymptotic expansion as in (2b) applies under the conditions of (2b) alone. They also explicitly construct an example of \(\phi\in\Gamma_1^1\) for which \(\mathcal{M}_{-1/4}^\phi\) is not transversally cut out, but the conditions here do still hold.