nLab quantization of D=3 Chern-Simons theory

Idea

This entry discusses the full (non-perturbative) quantization of the prequantum data of standard 3d Chern-Simons theory (induced from a suitable Lie group and invariant polynomial/second Chern class action functional) to a 3d TQFT.

(For the perturbative quantization of Chern-Simons theory see there).

Existing literature knows three sectors of this problem, which overlap but do not coincide

1. path integral quantization. This may be made precise sense of in perturbation theory where it involves lots of interesting structure such as analytic torsion (Witten 89). However, being just perturbation theory it is just an approximation to the full answer.

2. geometric quantization yields the full (non-perturbative quantization) in codimension 1, but does not say anything about codimension 0.

3. The Reshetikhin-Turaev construction produces a 3d TQFT from algebraic data that is naturally associated with the prequantum data defining Chern-Simons theory (such as the category of positive energy representations of the loop group of the given gauge group $G$, or else of a quantum group Sawin 06), but it is not a priori clear that this 3d quantum field theory is genuinely the result of quantizing the Chern-Simons action functional.

The known relation between the second and the third point here is the following:

That the complex-geometric modular functor obtained from geometric quantization of Chern-Simons theory as in (Axelrod-Pietra-Witten 91, Hitchin 90) coincides with that of conformal blocks of the WZW model was shown in (Laszlo 98,see also Andersen 11, Andersen 12). That this in turn indeed satisfies the required sewing law (and hence really is a modular functor in the strong sense) was shown in (Tsuchiya-Ueno-Yamada). By deprojectivization these constructions yield a topological modular functor of the form also obtained from the Reshetikhin-Turaev construction.

These (topological) modular functors are fixed by their genus-0 data (Andersen-Ueno 06) which is equivalently the datum of a (weakly) modular tensor category. Hence for matching geometric quantization of 3d Chern-Simons theory to the Reshetikhin-Turaev construction one has to match the modular tensor categories obtained from the conformal blocks of the WZW model in genus-0 to that associated with the coresponding quantum groups. This works (Ostrik 14).

Something more

random notes, needs to be brought into shape

Space of Chern-Simons quantum states

Chern-Simons action functional

$$\exp(\tfrac{i}{\hbar}S_{CS}) \colon \mathbf{Fields}_{CS} \longrightarrow \mathbf{B}^{n+1} U(1)_{conn}$$

given a closed manifold $\Sigma_n$ then choice of complex structure $\mathbf{\Sigma}_n$ on $\Sigma_n$ is supposed to naturally induce a complex structure on the space of (on-shell) fields over $\Sigma_n$

$$\mathbf{Fields}_{CS}(\mathbf{\Sigma}_n)$$

Moreover, transgression of $\exp(\tfrac{i}{\hbar}S)$ to $\mathbf{Fields}_{CS}(\Sigma)$ is supposed to yield a holomorphic line bundle with connection with respect to that complex structure

$$\exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n)) \colon \mathbf{Fields}_{CS}(\mathbf{\Sigma}_n) \longrightarrow \mathbf{B}U(1)_{conn} \,.$$

This is the prequantum line bundle of the Chern-Simons theory, already equipped with a Kähler polarization.

Accordingly, the geometric quantization of the CS action functional assigns to $\Sigma_n$ the Hilbert space $\mathcal{H}_{\mathbf{\Sigma}}$ of holomorphic sections of $\exp(\tfrac{i}{\hbar}S(\mathbf{\Sigma}_n))$.

As the complex structure $\mathbf{\Sigma}_n$ on $\Sigma_n$ varies over the moduli stack of complex structures $\mathcal{M}_{\Sigma}$, these vector spaces $\mathcal{H}_{\mathbf{\Sigma}_n}$ form a vector bundle with projective flat connection (the Hitchin connection) on the moduli stack

$$\array{ \mathcal{H} \\ \downarrow \\ \mathcal{M}_{\Sigma} }$$

The assignment

$$\Sigma_n \mapsto \mathcal{H}_{\Sigma}$$

natural in diffeomorphisms of $\Sigma$ is called the modular functor, this we focus on more below

One such section $\Psi$ is to be singled out. For instance if $\exp(\tfrac{i}{\hbar}S(\Sigma_n))$ is a theta characteristic then there is up to scale a unique holomorphic section. This singling-out is formalized by the FRS-formalism. See there for more.

Space of WZW-model pre-correlators

Under the AdS3-CFT2 and CS-WZW correspondence the states of Chern-Simons theory also correspond to partition functions of the gauged WZW model and hence to generating functions for correlation functions of the actual WZW model.

The fields $\mathbf{Fields}(\Sigma_n)$ may also be thought of as the sources of a (higher) gauged WZW model on $\Sigma$.

The holographic principle says that the quantum state/wavefunction

$$\Psi \in \Gamma(\mathbf{Fields}_{CS}(\mathbf{\Sigma}_n), L)$$

is also the generating function for the correlators of the WZW model on $\Sigma_n$, meaning that its functional derivatives

$$\frac{\delta}{\delta A^{a_1}(z_1)} \cdots \frac{\delta}{\delta A^{a_n}(z_n)} \Psi = \left\langle J^{a_1}(z_1) \cdots J^{a_n}(z_n) \right\rangle$$

with respect to the Chern-Simons-fields, hence the WZW sources, are the n-point functions of the WZW model for current algebra insertions, as indicated

(e.g. Gawędzki 99 (4.23), 5.1)

Modular functor and equivariant elliptic cohomology

Discussion of equivariant elliptic cohomology (see there at interpretation in QFT) shows that the construction of the modular functor refines from equipping $\Sigma$ with complex structure to equipping it with arithmetic structure. Hence for $\Sigma$ a torus it refines from structures of elliptic curves over the complex numbers to general arithmetic elliptic curves (over the integers) and in fact to derived elliptic curves (over the sphere spectrum). Hence eventually the theory of geometric quantization needs to be refined to admit polarizations in arithmetic geometry. See at differential cohesion and idelic structure for more on this.

Related concepts

• Donaldson-Uhlenbeck-Yau theorem

• Narasimhan-Seshadri theorem

• Harder-Narasimhan theorem

• equivariant elliptic cohomology

References

Non-perturbative geometric quantization

Discussion of non-perturbative geometric quantization of Chern-Simons theory:

Basics are recalled for instance in

• Fernando Falceto, Krzysztof Gawedzki, Chern-Simons States at Genus One, Commun.Math.Phys. 159 (1994) 549-580 (arXiv:hep-th/9211003)

• Krzysztof Gawedzki, Conformal field theory: a case study, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) Conformal Field Theory – New Non-perturbative Methods In String And Field Theory, CRC Press (2000) [arXiv:hep-th/9904145, doi:10.1201/9780429502873]

• Yasuhiro Abe, Application of abelian holonomy formalism to the elementary theory of numbers (arXiv:1005.4299)

The geometric quantization of 3d CS theory in codimension 1 is due to

• Scott Axelrod, S. Della Pietra, Edward Witten, Geometric quantization of Chern-Simons gauge theory, Jour. Diff. Geom. 33 (1991), 787-902. (EUCLID)

• Nigel Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)

(see also at Hitchin connection).

The 3d TQFT candidate for quantum CS theory in the form of the Reshetikhin-Turaev construction and the corresponding modular tensor category data is discussed in

• Nicolai Reshetikhin, Vladimir Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), no. 3, 547–597. (pdf)

• B. Bakalov & Alexandre Kirillov, Lectures on tensor categories and modular functors AMS, University Lecture Series, (2000) (web).

• Stephen Sawin, Quantum groups at roots of unity and modularity J. Knot Theory Ramifications 15 (2006), no. 10, 1245–1277 (arXiv:0308281)

• Victor Ostrik, MO comment August 2014

The relation between the modular functor obtained from the conformal blocks of the WZW model and from geometric quantization of CS theory is discussed in

• Yves Laszlo, Hitchin’s and WZW connection are the same, J. Differential Geom. 49 (1998), no. 3, 547–576 (pdf)

• Tsuchiya; K Ueno; Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics, 459–566.

• Jørgen Andersen, K. Ueno, Modular functors are determined by their genus zero data, Journal of Quantum Topology (arXiv:math/0611087)

Discussion specific to special unitary gauge group is in

• Jørgen Andersen, K. Ueno, Abelian Conformal Field theories and Determinant Bundles, International Journal of Mathematics, 18 919 - 993 (2007).

• Jørgen Andersen, K. Ueno, Geometric Construction of Modular Functors from Conformal Field Theory, Journal of Knot theory and its Ramifications, 16 127 – 202, (2007).

• Jørgen Andersen, K. Ueno, Construction of the Reshetikhin-Turaev TQFT via Conformal Field Theory (arXiv:1110.5027)

• Jørgen Andersen, A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications (arXiv:1206.2785)

For discussion of the state of the proof see also

• MO, Why hasn’t anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

and in particular this reply by Andre Henriques.

Detailed review in the case of abelian Chern-Simons theory includes

• Spencer D. Stirling, Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories (arXiv:http://arxiv.org/abs/0807.2857v1)

Discussion in terms of Weyl quantization of Wilson lines and details on the role of theta functions is in

Discussion of quantization of Chern-Simons theory in terms of Weyl quantization and skein relations is in

• Jørgen Andersen, Deformation quantization and geometric quantization of abelian moduli spaces, Commun. Math. Phys., 255 (2005), 727–745

• Razvan Gelca, Alejandro Uribe, The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same, Commun.Math.Phys. 233 (2003) 493-512 (arXiv:math-ph/0201059)

• Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)

• Razvan Gelca, Alejandro Uribe, Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$ (arXiv:1007.2010)

Another approach is

• Daniel Freed, Mike Hopkins, Constantin Teleman, Jacob Lurie, TQFT from compact Lie groups – 3d Chern-Simons as a fully extended TQFT.

Perturbative quantum Chern-Simons theory

Discussion of perturbative quantization of Chern-Simons theory (via Kontsevich integrals/knot graph cohomology on Jacobi diagrams, regarding Feynman amplitudes as differential forms on configuration spaces of points and yielding universalVassiliev invariants):

• Dror Bar-Natan, Perturbative aspects of the Chern-Simons topological quantum field theory, thesis 1991 (spire:323500, proquest:303979053, BarNatanPerturbativeCS91.pdf)

• Scott Axelrod, Isadore Singer, Chern-Simons Perturbation Theory, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; (arXiv:hep-th/9110056)

• Scott Axelrod, Isadore Singer, Chern–Simons Perturbation Theory II, J. Diff. Geom. 39 (1994) 173-213 (arXiv:hep-th/9304087)

• Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)

• Maxim Kontsevich, Feynman diagrams and low-dimensional topology, in First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121, Birkhäuser, Basel, 1994. (pdf)

• Dror Bar-Natan, Perturbative Chern-Simons theory, Journal of Knot Theory and Its RamificationsVol. 04, No. 04, pp. 503-547 (1995) (doi:10.1142/S0218216595000247)

• Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 $[$arXiv:q-alg/9603010, doi:10.1007/s002200050136$]$

• Pascal Lambrechts, Ismar Volić, sections 6 and 7 of Formality of the little N-disks operad, Memoirs of the American Mathematical Society ; no. 1079, 2014 (arXiv:0808.0457, doi:10.1090/memo/1079)

Review:

• Robbert Dijkgraaf, Perturbative topological field theory, In: Trieste 1993, Proceedings, String theory, gauge theory and quantum gravity ‘93 189-227 (spire:399223, pdf)

• Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 4 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv/1103.5628, doi:10.1017/CBO9781139107846)

See also at correlator as differential form on configuration space of points and see at graph complex as a model for the spaces of knots.

The “Wheels theorem”, saying that the perturbative Chern-Simons Wilson loop observable of the unknot is, as a universal Vassiliev invariant, a series of wheel-shaped Jacobi diagrams with coefficients the modified Bernoulli numbers, is due to

• Dror Bar-Natan, Thang T Q Le, Dylan Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. Volume 7, Number 1 (2003), 1-31 (euclid.gt/1513883092)

following

• Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, Dylan Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot (q-alg/9703025)