Completing the quantization of Chern-Simons gauge theory with complex gauge group: a fully defined unitary TQFT, the associated two-dimensional conformal field theory, and the Hilbert space of $2+1$-dimensional gravity with positive cosmological constant

In 1991 Witten generalized the canonical quantization of Chern-Simons gauge theory from compact Lie groups to complex Lie groups $G_{\mathbb{C}}$, the case $SL(2,\mathbb{C})$ describing $2+1$-dimensional quantum gravity with Lorentz signature and positive cosmological constant, or with Euclidean signature and negative cosmological constant. Because the moduli space of flat $G_{\mathbb{C}}$-connections is non-compact and its quantization is infinite-dimensional, the 1991 construction was explicitly incomplete: the projectively flat Hitchin-Witten connection over Teichmüller space was derived heuristically with an ad hoc coefficient modification, the exotic hermitian pairing and its positivity were analyzed only in genus one, the genus-zero Knizhnik-Zamolodchikov-type equations were posited by analogy with the compact case, partition functions were computed only in genus one and special cases, and the existence of an associated two-dimensional conformal field theory was left open. Later literature genuinely resolved one facet (the anomaly coefficients and projective flatness were proven rigorously by Andersen and Gammelgaard) and advanced others (explicit genus-one mapping-class-group representations; unitary representations for all punctured surfaces from quantum Teichmüller theory, conjecturally equivalent to the complex Chern-Simons theory as the Teichmüller TQFT). The open problem is: construct complex Chern-Simons theory as a mathematically well-defined, unitary topological quantum field theory with cutting and gluing on all surfaces and 3-manifolds, incorporating abelian and non-semisimple flat connections whose contributions vanish or diverge; prove or refute the identification with the quantum Teichmüller theory; construct or exclude the associated genuine conformal field theory; and identify the physical Hilbert space of de Sitter ($\Lambda>0$) $2+1$-dimensional quantum gravity.

Background

Chern-Simons gauge theory is a three-dimensional topological quantum field theory (TQFT) - a quantum field theory whose observables are independent of the metric - with action $\frac{1}{4\pi}\int_M \mathrm{Tr}(t\,A\wedge dA+\frac{2}{3}A\wedge A\wedge A)$ for a connection $A$ on a principal bundle over a 3-manifold $M$, where the coupling constant $t$ is called the level. In canonical quantization the classical phase space is the moduli space of flat connections (gauge-equivalence classes of connections with zero curvature) on a spatial surface $\Sigma$, and quantization should assign to each $\Sigma$ a state space and to each $M$ with boundary $\Sigma$ a vector in it, compatibly with cutting and gluing of manifolds.

For compact gauge group and integer level this program is mathematically complete: Witten (1989) identified the path integral with the Jones polynomial (a celebrated invariant of knots) and with the Wess-Zumino-Witten model (a rational two-dimensional conformal field theory, CFT - rational meaning that its symmetry algebra has only finitely many unitary representations), and the geometric-quantization approach made rigorous by Hitchin (1990) gives an equivalent construction. In geometric quantization the state space consists of holomorphic sections of a power of a prequantum line bundle (the line bundle whose curvature equals the symplectic form) over the moduli space of flat connections; since the complex structure on moduli space varies with the complex structure on the surface, parametrized by Teichmüller space (the space of complex structures on $\Sigma$ modulo isotopy), the state spaces at different points must be tied together by a projectively flat connection (a connection whose curvature is a scalar multiple of the identity) - the Hitchin connection. Projective flatness guarantees that quantizations at different complex structures are identified up to phase, so the mapping class group (the group of diffeomorphisms of $\Sigma$ modulo isotopy) acts projectively on the state space.

Witten (1991) asked what survives when the compact group is replaced by a complex Lie group $G_{\mathbb{C}}$ with holomorphic level $t=k+is$ and antiholomorphic level $\bar t=k-is$, where $k$ is an integer (the level of the compact-theory limit) and $s$ is a real parameter. Non-compactness changes the theory structurally: the moduli space of flat $G_{\mathbb{C}}$-connections is generally non-compact and smooth only away from reducible connections; its geometric quantization is infinite-dimensional already in genus one (where the moduli space is essentially $G_{\mathbb{C}}$ itself); the ordinary hermitian pairing is replaced by an exotic pairing $\langle\phi,e^{2r\Delta}\psi\rangle$, with $\Delta$ a Laplace-type operator on moduli space and the constant $r$ fixed by $e^{-4kr}=-(k-is)/(k+is)$; and the CFT connection supplied for compact groups by the Wess-Zumino-Witten model is lost, leaving only candidate central charges (the pair of numbers controlling the symmetry algebra of a two-dimensional CFT) $c_L=\dim G\,(1-2h/t)$, $c_R=\dim G\,(1+2h/\bar t)$, where $h$ is the dual Coxeter number of $G$ (a constant fixed by the Lie algebra, e.g. $h=n$ for $SU(n)$), and candidate conformal blocks. Witten derived a projectively flat connection over Teichmüller space - now called the Hitchin-Witten connection, a $t$- and $\bar t$-combination of Hitchin's connection - by a formal computation whose anomaly coefficients (the curvature terms that must cancel for projective flatness) were verified only partially and with an ad hoc modification; he wrote the genus-zero Knizhnik-Zamolodchikov-type equations (the differential equations satisfied by conformal blocks on the sphere in the compact theory) only by analogy; and he analyzed the positivity of the exotic pairing only in genus one, where unitarity reduces to the complementary series - a continuous family of unitary representations of the complex group existing only in a restricted range of parameters - with positivity holding for real $s$ and failing otherwise. For $G_{\mathbb{C}}=SL(2,\mathbb{C})$ the theory should be $2+1$-dimensional quantum gravity: real $s$ corresponds to Lorentzian signature with positive cosmological constant (de Sitter space, the maximally symmetric solution of general relativity with positive cosmological constant), imaginary $s$ to Euclidean signature with negative cosmological constant; here the vierbein (the coframe field relating the metric to the gauge connection) can degenerate, the moduli space has components not seen as classical spacetimes, and Witten relied on a conjecture attributed to G. Mess relating such spacetimes to complex projective structures on the surface (an atlas of charts to the Riemann sphere whose transition maps are Möbius transformations).

Three later developments frame what remains. First, quantum Teichmüller theory - the quantization of Teichmüller space built on the quantum dilogarithm, a special function quantizing the classical dilogarithm - yields unitary projective representations of the mapping class groups of punctured surfaces on infinite-dimensional $L^2$-spaces (on $L^2(\mathbb{R}^{3g-3+n})$ for a surface of genus $g$ with $n$ punctures), exactly the unitarity the direct $G_{\mathbb{C}}$ quantization lacks; Andersen and Kashaev constructed from it a Teichmüller TQFT whose partition functions are state integrals (integrals over eigenvalue variables of products of quantum dilogarithms, associated to triangulated 3-manifolds) and argued via geometric quantization over the self-dual locally compact abelian group $\mathbb{R}\times\mathbb{Z}/N\mathbb{Z}$ (a group identified with its own Pontryagin dual, its group of continuous characters) that it is complex Chern-Simons theory for $PSL(2,\mathbb{C})$ at level $N$; the identification matches perturbation theory but is not a theorem and is expected to capture only the Teichmüller component of the moduli space. Second, the perturbative theory around a fixed irreducible flat connection is completely understood: Dimofte, Gukov, Lenells and Zagier computed the partition function to all loop orders, and state-integral models and the 3d-3d correspondence (a relation between three-dimensional supersymmetric gauge theories and the complex Chern-Simons theory of the same 3-manifold) organize much current work. Third, Witten's 2010 study of analytic continuation of Chern-Simons path integrals showed that such theories require a choice of integration cycle (a middle-dimensional steepest-descent cycle, as in Picard-Lefschetz theory) on the space of complexified fields; Mikhaylov proposed that the Teichmüller TQFT corresponds to complex Chern-Simons theory on a particular such cycle, characterized conjecturally by solutions of the Kapustin-Witten equations (a system of gauge-theoretic partial differential equations combining self-duality conditions with a Higgs field). What no construction yet delivers is the object the 1991 program pointed at: a single, fully defined quantum theory.

Problem Statement

The problem is to complete the quantization program of Witten (1991) for Chern-Simons gauge theory with complex gauge group $G_{\mathbb{C}}$ (the case $SL(2,\mathbb{C})$ being the one of gravitational interest), turning the 1991 framework into a single, fully defined quantum theory. Concretely, completing the program requires the following: (1) construct the quantum theory as a topological quantum field theory - state spaces for every closed oriented surface of arbitrary genus, with or without marked points; partition functions and wavefunctions for all compact oriented 3-manifolds; and structure maps compatible with cutting and gluing - in which all classes of flat connections (semisimple, abelian, and non-semisimple) contribute with well-defined values, resolving the known obstructions that flat connections with infinite-volume stabilizers (e.g. abelian connections on hyperbolic manifolds) force the partition function to vanish identically while continuous families of flat connections make it diverge, with no systematic regularization currently known; (2) settle the unitary structure: establish whether the exotic hermitian pairing $\langle\phi,e^{2r\Delta}\psi\rangle$ admits a positive-definite refinement beyond genus one, and prove or refute the conjectured identification of the resulting theory with the unitary quantum Teichmüller theory / Teichmüller TQFT constructions - equivalently, with complex Chern-Simons theory on the integration cycle proposed by Mikhaylov; (3) determine whether a genuine two-dimensional conformal field theory is associated with the quantization - a unitary, modular-invariant theory whose conformal blocks and central charges match Witten's candidates, with Liouville theory (the standard example of a non-rational two-dimensional CFT) and its parafermionic extensions (variants adding fields of fractional spin, between bosons and fermions) the leading candidates - and construct it rigorously if it exists, or prove that it does not; and (4) for gauge group $SL(2,\mathbb{C})$ with real $s$ (Lorentzian signature, positive cosmological constant), identify the physical Hilbert space of $2+1$-dimensional quantum gravity with the gauge-theoretic state space, resolving the treatment of degenerate vierbeins, the multiplicity of classical solutions, and the status of the conjecture attributed to G. Mess relating such spacetimes to complex projective structures on the surface (atlases of charts to the Riemann sphere whose transition maps are Möbius transformations). The anomaly-coefficient and projective-flatness question left heuristic in 1991 has been resolved by later rigorous work (Andersen-Gammelgaard 2014) and is therefore not part of the open core, except that any full construction must be compatible with it and must hold for arbitrary genus: genus-one-only constructions solve only a special case, as does any construction restricted to the Teichmüller component of the moduli space. The four parts are logically independent: establishing the identification in (2) does not by itself settle the conformal-field-theory question (3), and conversely.

References

1. E. Witten, "Quantization of Chern-Simons gauge theory with complex gauge group", Communications in Mathematical Physics 137 (1991) 29-66. DOI: 10.1007/BF02099116. https://doi.org/10.1007/BF02099116
2. E. Witten, "Quantum field theory and the Jones polynomial", Communications in Mathematical Physics 121 (1989) 351-399. DOI: 10.1007/BF01217730. https://doi.org/10.1007/BF01217730
3. N. J. Hitchin, "Flat connections and geometric quantization", Communications in Mathematical Physics 131 (1990) 347-380. DOI: 10.1007/bf02161419. https://doi.org/10.1007/bf02161419
4. E. Witten, "Analytic Continuation Of Chern-Simons Theory", preprint arXiv:1001.2933 (2010). https://arxiv.org/abs/1001.2933
5. J. E. Andersen, N. L. Gammelgaard, "The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory", preprint arXiv:1409.1035 (2014). https://arxiv.org/abs/1409.1035
6. J. E. Andersen, R. Kashaev, "A TQFT from quantum Teichmüller theory", Communications in Mathematical Physics 330 (2014) 887-934. DOI: 10.1007/s00220-014-2073-2. https://doi.org/10.1007/s00220-014-2073-2; preprint arXiv:1109.6295 (2011). https://arxiv.org/abs/1109.6295
7. J. E. Andersen, R. Kashaev, "Complex Quantum Chern-Simons", preprint arXiv:1409.1208 (2014). https://arxiv.org/abs/1409.1208
8. J. E. Andersen, R. Kashaev, "The Teichmüller TQFT", in Proceedings of the International Congress of Mathematicians (ICM 2018), World Scientific (2019) 2541-2565. DOI: 10.1142/9789813272880_0149. https://doi.org/10.1142/9789813272880_0149; preprint arXiv:1811.06853 (2018). https://arxiv.org/abs/1811.06853
9. J. E. Andersen, S. Marzioni, "The genus one complex quantum Chern-Simons representation of the mapping class group", Journal of Knot Theory and Its Ramifications 35 (2026) no. 2, published online 9 November 2023. DOI: 10.1142/S0218216523400096. https://doi.org/10.1142/S0218216523400096; preprint arXiv:1608.06872 (2016). https://arxiv.org/abs/1608.06872
10. J. E. Andersen, S. Marzioni, "Level N Teichmüller TQFT and Complex Chern-Simons Theory", preprint arXiv:1612.06986 (2016), to appear in Travaux Mathématiques. https://arxiv.org/abs/1612.06986
11. J. E. Andersen, A. Malusà, G. Rembado, "Genus-one complex quantum Chern-Simons theory", Journal of Symplectic Geometry 20 (2022) 1215-1253. DOI: 10.4310/jsg.2022.v20.n6.a1; preprint arXiv:2012.15630 (2020). https://arxiv.org/abs/2012.15630
12. J. E. Andersen, A. Malusà, "A geometric quantisation view on the AJ-conjecture for the Teichmüller TQFT", Research in the Mathematical Sciences 13 (2026), no. 1. DOI: 10.1007/s40687-025-00585-9. https://doi.org/10.1007/s40687-025-00585-9
13. J. Teschner, "An analog of a modular functor from quantized Teichmüller theory", in Handbook of Teichmüller Theory, Volume I (IRMA Lectures in Mathematics and Theoretical Physics 13), European Mathematical Society (2007) 685-760. DOI: 10.4171/029-1/17. https://doi.org/10.4171/029-1/17; preprint arXiv:math/0510174 (2005). https://arxiv.org/abs/math/0510174
14. J. Teschner, G. S. Vartanov, "Supersymmetric gauge theories, quantization of $\mathcal{M}_{\mathrm{flat}}$, and conformal field theory", Advances in Theoretical and Mathematical Physics 19 (2015) 1-135. DOI: 10.4310/atmp.2015.v19.n1.a1. https://doi.org/10.4310/atmp.2015.v19.n1.a1; preprint arXiv:1302.3778 (2013). https://arxiv.org/abs/1302.3778
15. T. Dimofte, "Perturbative and nonperturbative aspects of complex Chern-Simons Theory", preprint arXiv:1608.02961 (2016). https://arxiv.org/abs/1608.02961
16. T. Dimofte, S. Gukov, J. Lenells, D. Zagier, "Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group", Communications in Number Theory and Physics 3 (2009) 363-443. DOI: 10.4310/CNTP.2009.v3.n2.a4; preprint arXiv:0903.2472. https://arxiv.org/abs/0903.2472
17. S. Gukov, "Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial", Communications in Mathematical Physics 255 (2005) 577-627. DOI: 10.1007/s00220-005-1312-y; preprint arXiv:hep-th/0306165. https://arxiv.org/abs/hep-th/0306165
18. V. Mikhaylov, "Teichmüller TQFT vs. Chern-Simons theory", Journal of High Energy Physics 04 (2018) 085. DOI: 10.1007/JHEP04(2018)085; preprint arXiv:1710.04354. https://arxiv.org/abs/1710.04354
19. J. Cotler, K. Jensen, A. Maloney, "Low-dimensional de Sitter quantum gravity", Journal of High Energy Physics 06 (2020) 048. DOI: 10.1007/JHEP06(2020)048; preprint arXiv:1905.03780. https://arxiv.org/abs/1905.03780
20. M. Porrati, C. Yu, "Kac-Moody and Virasoro Characters from the Perturbative Chern-Simons Path Integral", Journal of High Energy Physics 05 (2019) 083. DOI: 10.1007/JHEP05(2019)083; preprint arXiv:1903.05100. https://arxiv.org/abs/1903.05100
21. E. Buffenoir, K. Noui, Ph. Roche, "Hamiltonian quantization of Chern Simons theory with SL(2,R) group", Classical and Quantum Gravity 19 (2002). DOI: 10.1088/0264-9381/19/19/313. https://doi.org/10.1088/0264-9381/19/19/313
22. S. Carlip, J. E. Nelson, "Comparative quantizations of (2+1)-dimensional gravity", Physical Review D 51 (1995) 5643. DOI: 10.1103/PhysRevD.51.5643. https://doi.org/10.1103/PhysRevD.51.5643
23. A. Nedelin, F. Nieri, M. Zabzine, "q-Virasoro modular double and 3d partition functions", preprint arXiv:1605.07029 (2016, revised 2017). https://arxiv.org/abs/1605.07029