Perturbative and nonperturbative aspects of complex Chern-Simons Theory
Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON N2J 2Y5, Canada
(on leave) Department of Mathematics, University of California, Davis, CA 95616, USA
We present an elementary review of some aspects of Chern-Simons theory with complex gauge group . We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d-3d correspondence. The 3d-3d correspondence relates partition functions (and other aspects) of complex Chern-Simons theory on a 3-manifold to supersymmetric partition functions (and other observables) in an associated 3d theory . Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d-3d correspondence.
This is a contribution to the review volume “Localization techniques in quantum field theories” (eds. V. Pestun and M. Zabzine) which contains 17 Chapters available at 
- 1 Introduction
- 2 Warmup: quantization of
- 3 Complex Chern-Simons theory as a TQFT?
- 4 Three connections to three-manifold topology
- 5 The Quantum Modularity Conjecture
Chern-Simons theory with complex gauge group came to prominence in the late 80’s, partly as a tool for understanding three-dimensional gravity with a negative cosmological constant [2, 3, 4, 5]. Many early developments were due to Witten. Since then, it has found a multitude of applications and deep connections with many parts of theoretical physics and mathematics. A highly incomplete list includes:
Chern-Simons theory with gauge group can naturally be embedded in string/M-theory, opening up many powerful perspectives and techniques for analyzing the former. As a notable example, the compactification of M5 branes on the product of an ellipsoidally deformed lens space and a three-manifold (with a topological twist along ) leads equivalently to Chern-Simons theory at level on [8, 9, 10] or an supersymmetric theory on the lens space [11, 12, 13, 14]. This duality, known as the 3d-3d correspondence, fits into a series of dualities involving the compactification of five-branes on various -dimensional manifolds , including the AGT correspondence  and the duality of Gukov-Gadde-Putrov relating Vafa-Witten partition functions on and elliptic genera .
There is a multitude of applications to three-dimensional geometry and topology. Fundamentally, partition functions of complex Chern-Simons theory on three-manifolds provide new topological invariants, generalizing the famous invariants (including knot polynomials) associated with compact Chern-Simons theory [17, 18]. As yet, a systematic computation of the complex invariants only exists for certain classes of manifolds (e.g. hyperbolic ones [19, 20, 21, 22]), though new tools to attack the general case are under development [23, 24].
The perturbative expansion of Chern-Simons theory on a three-manifold encodes various topological invariants of , such as its hyperbolic volume and twisted analytic torsion. In the case that is a knot complement in , it was conjectured by Gukov  that this expansion agrees with a (highly nontrivial) asymptotic limit of colored Jones polynomials of , providing physical motivation for a mathematical statement known as the Volume Conjecture [26, 27]. The relation between complex Chern-Simons theory and knot polynomials is essentially a result of analytic continuation, albeit a subtle one .
The perturbative expansion of Chern-Simons theory on knot complements has been successfully reproduced [29, 30, 31] using the topological recursion of Eynard-Orantin , a far-reaching formalism for the quantization of spectral curves.
The study of five-brane systems related to complex Chern-Simons theory recently led to a vast generalization of the Volume Conjecture, involving asymptotic limits of colored HOMFLY polynomials and their categorification .
There are several hints that complex Chern-Simons theory has quasi-modular properties, cf. [12, 34], though a complete physical characterization of these properties is still missing. The asymptotic expansions of Chern-Simons theory around various singular points in the space of coupling constants (levels) , related by an action of the modular group, provide evidence for the Quantum Modularity Conjecture of Zagier .
There are close connections between complex Chern-Simons theory and the mathematical theory of cluster algebras, cf. [37, 38]. Cluster algebras play an essential role in the (local) description and quantization of phase spaces that complex Chern-Simons theory attaches to two-dimensional boundaries, cf. [39, 40, 41], and Chern-Simons theory on three-manifolds is associated with cluster-algebra morphisms.
In this short review, we will only be able to touch upon a few of these topics and connections. We will actually begin in Section 2 with some basic concepts in complex Chern-Simons theory, including the definition of the Hilbert spaces assigned to two-dimensional oriented manifolds. One of the most prominent distinctions between Chern-Simons theory with complex and compact gauge groups is that, in the complex case, these Hilbert spaces are infinite-dimensional. As an illustrative example, we will outline the simple quantization of the torus Hilbert space for gauge group , and its dependence on the coupling constants or “levels” of the theory. We also review some features of the refined or equivariant quantization of Hilbert spaces recently developed by Gukov and Pei .
This prepares us in Section 3 to discuss one of the most fundamental open problems in complex Chern-Simons theory: defining the theory as a full TQFT. In essence, this means being able to assign Hilbert spaces to any oriented surface and wavefunctions to any oriented three-manifold, in such a way that the standard cutting-and-gluing axioms of Atiyah and Segal are obeyed . As we shall review, the difficulty with cutting and gluing in complex Chern-Simons theory stems from the infinite-dimensional nature of Hilbert spaces, and the fact that, naively, wavefunctions often vanish or diverge. Some very promising routes to overcoming these difficulties are suggested by embedding complex Chern-Simons theory in string/M-theory, and using additional symmetries to regulate zeroes or infinities [44, 23, 24]. Interestingly, these symmetries are related to categorification of Chern-Simons theory.
In the second half of this review, we then discuss a few relations between complex Chern-Simons theory and the topology and geometry of three-manifolds. In each case, we view these relations in light of string/M-theory and the 3d-3d correspondence. In Section 4, we will discuss 1) asymptotic expansions of Chern-Simons partition functions, their relation to hyperbolic volumes, and their behavior at large vis à vis holography of five-brane systems; 2) state-sum/integral models for Chern-Simons theory, their relation to positive angle structures on ideal triangulations of three-manifolds, and the corresponding implication for positivity of operator dimensions in theories built from ideal triangulations; 3) the interpretation of the partition function at level as counting surfaces in a three-manifold , BPS operators in , and BPS M2 branes ending on wrapped M5 branes in M-theory (related to recent mathematical work ). In Section 5, we will state some of the observations and conjectures about “quantum” modularity in Chern-Simons theory.
We emphasize that the 3d-3d correspondence provides the main link between complex Chern-Simons theory and localization methods in supersymmetric gauge theories, which are the focus of this collection of articles. Particularly relevant are the reviews of T. Dumitrescu (Contribution ) and B. Willett (Contribution ). The basic idea is that whenever can be explicitly described as (say) a gauge theory, its partition function on spaces such as or is readily computed by supersymmetric localization. This has led to new formulations and refinements of Chern-Simons partition functions on , which in turn produce invariants of 3-manifolds,
Unfortunately, we will not say very much about connections of complex Chern-Simons theory to cluster algebras, topological recursion, categorification, gravity, or many other fascinating topics. We hope that some of the references above will guide readers interested in these subjects.
2 Warmup: quantization of
To get a feel for the structure of complex Chern-Simons theory, we begin with a (seemingly) elementary exercise: the quantization of the phase space that Chern-Simons theory attaches to a two-torus.
First, some generalities. As discussed in , the action of complex Chern-Simons theory on a Euclidean three-manifold takes the form
where is the usual Chern-Simons functional. Here is a connection on an bundle over , and is its complex conjugate. The group contains as its maximal compact subgroup (in fact, as a complex manifold, ), and on a compact 3-manifold there can be large gauge transformations that wrap nontrivially around the compact . The path-integral integration measure is invariant under large gauge transformations on a closed so long as . On the other hand, the coupling constant is unconstrained.
For a unitary theory — which in Euclidean space means that partition functions are conjugated under orientation reversal — the action must be real, which forces to be real. We will assume this to be true in the quantization below, though eventually in partition functions we will find that we can analytically continue in a straightforward manner. There also exist exotic unitarity structures with imaginary , which will lead to slightly different Hilbert spaces.
As an aside, in the relation to 3d Euclidean gravity with a negative cosmological constant, one identifies the Hermitian and anti-Hermitian parts of as a vielbein and a spin connection . The part of the action proportional to becomes the usual Einstein-Hilbert action, while the part proportional to is a gravitational Chern-Simons term . The classical solutions of Chern-Simons theory on a three-maniofld are flat connections, which become identified with (possibly degenerate) metrics of constant negative curvature, i.e. hyperbolic metrics, in 3d gravity.
Geometric quantization of complex Chern-Simons theory on a general surface was first discussed in  and recently revisited in , using a holomorphic polarization. A more modern perspective on quantization, based on the topological A-model, appears in , following  (see also ). In the case of , we can take a more pedestrian approach, following [22, 21].
The Hilbert space that Chern-Simons theory assigns to any surface is a quantization of the classical phase space
This is the space of complex flat connections on (modulo gauge transformations), or equivalently, the space of representations of the fundamental group of in . The space is a finite-dimensional complex symplectic variety, possibly singular, equipped with the Atiyah-Bott holomorphic symplectic form
Of course, the actual symplectic form we use for quantization should be real; the Chern-Simons action (2.1) tells us to take
Famously, the space is hyperkähler. It admits an entire of complex structures, one of which is singled out in the description (2.2) as a space of complex flat connections. The other complex structures can be made manifest by rewriting as Hitchin’s moduli space associated to the compact group . Similarly, the space admits a of real symplectic forms, spanned by the hyperkähler triplet , where and above. The third form is a Kähler form in our chosen complex structure. Notably, represents a nontrivial integral cohomology class in , while are cohomologically trivial. This provides another explanation for the quantization of the level : in geometric quantization, one requires to be the first Chern class of a line bundle, which can only happen if it has integral periods, whence , but is unconstrained.
Now let us specialize to . Flat connections on a torus are determined by the holonomies along the and cycles, up to conjugation. Since the fundamental group is abelian, the holonomies commute and can be simultaneously diagonalized.111More precisely: the holonomies can simultaneously be put in Jordan normal form. Letting denote their eigenvalues, we find
The action here is just that of the Weyl group, acting as residual gauge transformations. The holomorphic symplectic form is
reflecting the nontrivial intersection of A and B cycles on , whence
As anticipated, the period of on the compact cycle in is equal to , and is properly quantized.
In order to diagonalize the real symplectic form , we define to be the complex number with and
and make a change of variables222The new variables absorb some powers of the coupling constants , which obfuscates the effect of the classical limit , e.g. in (2.11). It is important to keep in mind that the natural functions on the phase space are still . from to :
Notice that if is real then , so and are complex conjugates, as written. Then the symplectic form collapses to
We proceed to quantize the space as if it were just , and restore Weyl invariance later. The functions and can simply be quantized to operators with canonical commutation relations
Of course, since is periodic the spectrum of is quantized, and vice versa; altogether, both the eigenvalues of both and must belong to . The well-defined operators are actually quantizations of the -valued functions in (2.9), with , etc. For these we find
Thus, abstractly, we see that the quantized operator algebra consists of two independent Weyl algebras (or “quantum torus” algebras), one in and one in .
There are many equivalent ways to represent the operator algebra on a Hilbert space . The simplest is to take
to consist of functions of a real variable and an integer . Equivalently, we may take functions of and . Formally, in geometric quantization, this corresponds to choosing a particular “real” polarization — taking sections of a line bundle with that are covariantly constant with respect to and . The operators act on as multiplication by , while are shifts
To restore Weyl-invariance, we restrict to the -invariant part of the Hilbert space, i.e. functions that are even
Correspondingly, we should restrict to a subalgebra of the operator algebra that is invariant under . This subalgebra is generated by operators , , and (and their conjugates), which obey
When , complex Chern-Simons theory still make sense (as long as ), but the above quantization procedure requires a slight modification . The change of variables (2.9) does not make sense, and is not necessary, since is already diagonalized. Indeed, is the canonical symplectic form on when viewed as the cotangent bundle . We expect quantization to produce . To see it, simply write . The symplectic form becomes
These canonically-conjugate functions are quantized to operators with . Since are periodic with period , the eigenvalues of must be integers. We can represent the operator algebra (say) on functions of two integers, such that
where now (2.13) reduces to
With these new definitions of and , the operators satisfy the standard quantum-torus relations (2.12). Alternatively, and equivalently, we may take the Hilbert space to contain functions of an integer and a phase , with
Again, we impose Weyl-invariance at the end by restricting to even functions or .
2.1 Equivariant quantization
There are several things to notice about (2.14). Perhaps the most salient is that, unlike in the case of Chern-Simons theory with compact gauge group [17, 53], the Hilbert space is infinite-dimensional. This is no surprise, since it comes from quantizing a noncompact phase space. One may also recognize the factor as being related to the standard Hilbert space for theory at (bare) level . Indeed, if we ignore and consider functions of an integer , such that as in (2.16), we find exactly independent values that determine a state in the finite-dimensional Hilbert space of Chern-Simons.
What is less obvious, in particular for general , is that the infinite-dimensional Hilbert space of complex Chern-Simons theory admits an additional symmetry, introduced in . The graded components of (i.e. the subspaces of fixed charge) turn out to be finite-dimensional, and in particular the subspace of zero charge is just the familiar Hilbert space.
The extra symmetry comes from viewing as the Hitchin moduli space. There is a canonical metric isometry of the Hitchin moduli space that rotates the of complex structures about an axis. In particular, it rotates and into each other. This is an isometry of our quantization problem at least when , since it preserves , and leads to the desired symmetry of . (Since the Hilbert space, abstractly, does not depend on , one might then hope to endow even spaces at with the symmetry.)
In the case , it is easy to describe the symmetry: when we view as the cotangent bundle of the space of flat connections, simply rotates the cotangent fibers. The -invariant subspace of simply consists of the functions that are independent of — i.e. the Hilbert space we found above. The full action is trickier to describe in the polarization we are using. Roughly, one observes that at and the variables can be written as
where and . Then, on functions of that are analytic in , the symmetry just acts as rotations . The subspaces of fixed weight contain monomials in . After imposing Weyl invariance, the graded dimension of the Hilbert space becomes
2.2 Holomorphic polarizations and CFT
Often in geometric quantization of Chern-Simons theory, one uses a holomorphic polarization instead of the real polarization above. In Chern-Simons theory with compact gauge group (cf. ), this means to choose a complex structure ‘’ on a surface , to write the connection one-form as in local complex coordinates, and, when quantizing, to define the Hilbert space to consist of sections of the line bundle that are covariantly constant with respect to .
Naively, it may appear that choosing such a complex polarization needlessly complicates the problem. However, a complex polarization has three great advantages. First, it allows one to ask analyze the Hilbert space varies with the choice of complex structure. Locally the variation is trivial, expressed formally by saying that the bundle of Hilbert spaces over the space of complex structures (i.e. over the Teichmüller space of ) has a projectively flat connection. However, globally, one derives an action of the mapping class group of on the Chern-Simons Hilbert space. Second, and related to this idea, a holomorphic polarization allows one to identify the Hilbert space of Chern-Simons theory with the space of conformal blocks in a particular boundary CFT. In the case of compact Chern-Simons theory, the boundary CFT is a famously WZW model . The projectively flat connection on Teichmüller space is the Knizhnik-Zamolodchikov connection of the CFT. Finally, for generic surfaces a real polarization as above simply isn’t available! Thus, using a holomorphic polarization is the only way to go.
In the case of complex Chern-Simons theory, there are actually multiple choices of complex polarizations. If we write a complex connection and its conjugate as , , then we can ask that sections of be covariantly constant with respect to
The first choice was analyzed by Witten , and leads to a boundary CFT whose conformal blocks must contain contributions from both chiral and anti-chiral sectors. This polarization plays a central role in the relation between Chern-Simons theory and quantum Hall systems [42, 54]. The second choice is related to Liouville theory coupled to parafermions [55, 56].
3 Complex Chern-Simons theory as a TQFT?
Now, having seen very explicitly that complex Chern-Simons Hilbert spaces are infinite-dimensional (and exactly how they’re infinite-dimensional), let us think a bit about the properties of partition functions.
For a closed three-manifold , it is expected that the complex Chern-Simons partition function takes the form
The sum here is over flat complex connections on , which are the critical points of the Chern-Simons path integral; and , are holomorphic and antiholomorphic contributions to the path integral from quantum fluctuations around the critical point, with defined by (2.13). The prefactor is the volume of the stabilizer of , i.e. the volume of the subgroup of the gauge group that preserves a particular flat connection.
One certainly expects such a formula to be valid perturbatively, due to standard properties of path integrals in quantum field theory. (The perturbative version of (3.1) formed the basis for the physical explanation of the Volume Conjecture in .) It was argued in , however, that the formula is actually valid non-perturbatively as well. Roughly, one should think of (3.1) as expanding the integration cycle in the Chern-Simons path integral into a sum of integration cycles defined by gradient flow off of each critical point with respect to the Chern-Simons action. The can further be written as products333In general, there could be a nontrivial matrix connecting holomorphic and anti-holomorphic sectors. However, when one considers unitary Chern-Simons theory with an integration contour along which is honestly the conjugate of , the matrix is just the identity. of cycles in the space of (holomorphic)(anti-holomorphic) connections, leading to the factorization . 444The 3d-3d correspondence relates holomorphic-antiholomorphic factorization in complex Chern-Simons theory to a rather nontrivial statement about lens-space partition functions of 3d theories .
For a three-manifold with boundary , the same type of formula holds after properly accounting for boundary conditions. In particular, the sum is over flat connections with a fixed behavior at , and each summand becomes a wavefunction in . For example if is a torus, we would fix A-cycle holonomy eigenvalues of a flat connection as in Section 2, and the individual wavefunctions would have the factorized form .
What can we learn from (3.1)? In the best-case scenario, there is a finite number of flat connections on , and all the flat connections have finite-volume stabilizers. This would lead to a finite, well-defined (in principle) . In contrast:
If a flat connection is isolated but its stabilizer has infinite volume, its contribution to (3.1) vanishes.
If flat connections come in a continuous family on which the Chern-Simons action is constant, then the contribution to (3.1) can be infinite.
Unfortunately, the best-case scenario never holds, and both of these potentially bad situations can arise. We consider some examples.
If is hyperbolic (meaning that it admits a hyperbolic metric), it is expected that there are a finite number of flat connections on . Intuitively, for hyperbolic the fundamental group is sufficiently complicated that the representations are isolated.555On hyperbolic , one particular flat connection — the one corresponding to the global hyperbolic metric — is well known to be isolated [58, 59]. Computational experiments suggest that in fact all flat connections are isolated, but there exists no general proof of this statement. Then the partition function is finite. As we discuss in Section 4.2, the partition function can actually be computed. However, there is always at least one flat connection whose holonomies belong to the maximal torus . The stabilizer of contains constant -valued gauge transformations; since has infinite volume, does not contribute at all to the partition function. This becomes hugely problematic when trying to formulate complex Chern-Simons as a TQFT, as all flat connections must be accounted for during cutting and gluing .
There are some simple manifolds whose fundamental group is abelian. For example, if is a lens space , the fundamental group is . In this case, every single flat connection has holonomy in the maximal torus of the gauge group, the volume of the stabilizer is always infinite, and vanishes identically.
In the opposite extreme are manifolds on which the flat connections are not isolated. In this case, we expect that diverges. For example, consider . Then (where is the maximal torus of ). The partition function is
Both the zeroes and infinities appearing here obstruct the definition of consistent cutting and gluing rules needed to make Chern-Simons theory a TQFT. The zeroes and infinities have to be regularized. While no systematic approach to regularization has been formulated so far, there are exist several promising and exciting proposals. Almost all of them are motivated by string/M-theory and the 3d-3d correspondence.
3.1 Symmetries, regularizations, and the 3d-3d correspondence
In principle, the 3d-3d correspondence itself may suffice to resolve the difficulties with cutting and gluing in complex Chern-Simons theory.666We will not review the 3d-3d correspondence here. For recent reviews and discussions, especially in the context of Chern-Simons theory and TQFT, see [60, 44, 23, 24] as well as the related . The correspondence assigns to a closed 3-manifold and a Lie algebra of type (and a bit of extra discrete data) a three-dimensional field theory with the property that that
where . When , the theory is the effective low-energy worldvolume theory of M5 branes compactified on ; the branes wrap in the M-theory geometry . Similarly, the correspondence assigns to a three-manifold with boundary a boundary condition for the four-dimensional theory of class [62, 63].
Typically is an superconformal theory, though both supersymmetry and conformal invariance might be broken. Unfortunately, it is not completely understood what conditions on guarantee superconformal theories. We assume for the present heuristic argument that we do have superconformal theories.
The basic idea, then, would be to replace Chern-Simons theory on with , which is a much more powerful object. Even when the Chern-Simons partition functions (equivalently, lens-space partition functions of ) are ill-defined, the theory itself should still make sense. Moreover, the theories obey cutting and gluing rules. Gluing corresponds to “sandwiching” the four-dimensional theory between boundary conditions and , and colliding the boundaries together to produce a new effective theory . In this way, we reproduce the structure of a three-dimensional TQFT. If we should ever want to recover Chern-Simons partition functions, we just place the superconformal theories on a lens space .
There are two practical difficulties with this proposal that will hopefully be overcome soon. First, the full theories are not actually known for most manifolds, for any nonabelian . A construction using ideal triangulations was outlined in [14, 38] (also [64, 65]); however, that construction produces subsectors of the full theories that are missing some branches of vacua, the same way partition functions of Chern-Simons theory on hyperbolic manifolds are “missing” abelian flat connections. Examples of complete theories for a handful of manifolds (including a hyperbolic one) were postulated in , and theories for lens spaces were studied in [44, 23, 24].
The second difficulty, or potential shortcoming, is that zeroes and infinities still remain in the actual Chern-Simons partition functions. Here, however, another solution presents itself: the theories often have extra symmetries; and parameters associated to these symmetries (twisted masses or fugacities) can be used to refine the squashed-lens-space partition functions of — thus literally regularizing the Chern-Simons zeroes and infinities.
We already met one such symmetry in Section 2.1: the that gave an equivariant quantization of the Hilbert space . Via the 3d-3d correspondence, this Hilbert space is mapped to the BPS Hilbert space of the 4d class- theory on . This 4d theory has an additional R-symmetry that commutes with the supercharge used to define the “BPS” Hilbert space, and provides the grading.
Similarly, the three-dimensional theory (obtained by compactifying on a circle) has rather than supersymmetry. The larger R-symmetry group of the theory contains , and including its twisted mass in partition functions leads to finite answers that encapsulate the graded dimension (2.23).
Theories for Seifert-fibered three-manifolds should also retain this symmetry. In this case, it can ultimately be traced back to an exceptional isometry777This very same symmetry played a central role in defining refined (compact) Chern-Simons theory on Seifert-fibered manifolds . of the -theory geometry . An simple example of such a manifold is a lens space , whose refined partition functions were analyzed in , and put precisely into the factorized form (3.1) — with the factors now regularized.
When is generic (e.g. hyperbolic) this exceptional symmetry is, unfortunately, absent. It was nevertheless proposed in  that there exists yet another symmetry in any theory , related to the standard R-symmetry of three-dimensional theories — as well as to categorification of colored knot polynomials. This was used to regularize Chern-Simons partition functions for the trefoil and figure-eight knot complements (Seifert-fibered and hyperbolic manifolds, respectively), producing sums of the form (3.1) that included all flat connections, even abelian ones.
4 Three connections to three-manifold topology
As discussed in Section 3, the partition function of Chern-Simons theory with gauge group will take a finite, well-defined value on manifolds that only admit finitely many flat connections. For and possibly , hyperbolic manifolds are expected to be of this type. (Indeed, there exist systematic computations of partition functions for hyperbolic manifolds with boundary of genus .) One may then try to relate properties of the partition function with the topology of . We proceed to outline some of the more striking relations. In each case, M-theory and/or the 3d-3d correspondence provides valuable insight.
4.1 Hyperbolic volumes, twisted torsion, and large
The most fundamental relation between complex Chern-Simons theory and hyperbolic geometry has been understood for a long time, and concerns the semi-classical asymptotic expansion of partition functions  (see also [25, 67]). It is easiest to formulate it first in terms of the “holomorphic blocks” appearing in (3.1), labelled by flat connections :
where is the classical Chern-Simons action evaluated on a particular flat connection and is the analytic Ray-Singer torsion twisted by the flat connection . This is the standard result expected from Chern-Simons perturbation theory . In the presence of a boundary, the classical action and torsion on the RHS depend on the choice of boundary conditions (boundary holonomies) for . From (3.1), it then follows that if both and
The formula is particularly meaningful if is analytically continued to (say) positive imaginary values. Then each term in (4.2) shows exponential growth or decay at large , controlled by .
Now suppose that is hyperbolic, meaning that it admits a hyperbolic metric. The hyperbolic metric is unique (given suitable boundary conditions) and is a topological invariant of [58, 59]. Moreover, the vielbein and spin connection of the hyperbolic metric can be rewritten as a flat connection , with the property that is the hyperbolic volume of . The real part is known as the Chern-Simons invariant of the hyperbolic structure, and provides a natural complexification of the hyperbolic volume [70, 71]. It is also useful to note that the connection necessarily has a trivial stabilizer – the connection is fundamentally non-abelian.
Therefore, if the gauge group is and is hyperbolic, the sum (4.2) contains a term that is controlled by the hyperbolic volume of . Typically, is larger than the “volume” of any other flat connection, and the entire sum (4.2) is dominated by the hyperbolic volume. (It is expected that always dominates, but no general result of this type has been proven.)
It is often useful to strip off the holomorphic part of the asymptotic expansion. This can be done by taking a singular limit: we fix , analytically continue to imaginary values, and send . This has the effect of sending but , which trivializes the anti-holomorphic blocks, . In this singular limit, we expect
This sort of limit played a major role in early analyses of partition functions for complex Chern-Simons theory [72, 49], though it was not realized at the time that the partition functions (derived from quantum Teichmüller theory) being analyzed had fixed level .
The fact that the perturbative expansion of complex Chern-Simons theory (or individual holomorphic blocks, as in (4.1)) contains geometric invariants of played a major role in providing a physical justification for the Volume Conjecture, and generalizing it. The original Volume Conjecture [26, 27] claims that a particular double-scaling limit of colored Jones polynomials of a knot leads to exponential growth, controlled by the hyperbolic volume of the knot complement . This was justified in  by embedding Chern-Simons theory (which computes colored Jones polynomials) into a holomorphic sector of Chern-Simons theory, and arguing that the asymptotic expansions of and (holomorphic) theories should coincide. The argument immediately led to generalizations, involving higher-order terms in the asymptotic expansion and a dependence on boundary conditions, which have been carefully checked in many computations, cf. [67, 49, 73].
It is also interesting to consider “large-” limits in complex Chern-Simons theory, taking the gauge group to be and sending . Physically, such limits are most conveniently studied by realizing Chern-Simons theory on a stack of M5 branes (Section 3.1), and using AdS/CFT or large- duality. A study of the five-brane system [74, 75, 76] predicts that the leading asymptotic growth of the partition function as in (4.2) is scales as at large . This is not surprising: the hyperbolic flat connection can be embedded into by using the -dimensional representation , and the Chern-Simons functional evaluates to on , cf. . As long as (4.2) is dominated by the flat connection for any , one quickly recovers the scaling prediction. Much more non-trivially, the M-theory analysis predicts that the logarithm of the torsion will grow as [78, 79] at large as well. This latter result was recently proved by Porti and Menal-Ferrer .
4.2 State-integral models and angle structures
When is an oriented hyperbolic knot or link complement (or, more generally, an oriented hyperbolic manifold with non-empty boundary of genus ), there exists a systematic construction of Chern-Simons partition functions for all levels . The full definition of these partition functions appears in [81, 82] for , and [22, 21] for , respectively. It is a culmination of much previous work, including [72, 49, 83, 19, 84] for ;  for (and indirectly [85, 86, 87]); and (indirectly, via 3d-3d correspondence) [88, 89] for . The definition extends to using techniques of .
The construction of these partition functions uses a topological ideal triangulation of the three-manifold . This is a tiling of by truncated tetrahedra , as in Figure 4.1, such that 1) various pairs of large hexagonal faces are glued together; but 2) the small triangles at the truncated vertices are left untouched, and become part of the boundary .
One then proceeds in standard TQFT fashion. For gauge group (or more precisely, ), the boundary of each tetrahedron is assigned a Hilbert space . The Hilbert space comes from quantizing an open subset of the space of flat connections on , viewed as a four-punctured sphere; in this case , and the quantization of Section 2 applies in a straightforward manner, with no Weyl quotient. Each tetrahedron is assigned a canonical partition function , which has the form of a “quantum dilogarithm” function; for , this is
The partition function is then obtained by taking a product , and integrating out all pairs of variables associated to the interior of , leaving behind some variables associated to the boundary. It is slightly tricky to describe this operation in precise terms, because the tetrahedron Hilbert space does not easily factorize into contributions from the tetrahedron’s four large faces.888The Hilbert space can be factorized by introducing some redundant degrees of freedom , in a manner directly analogous to Kashaev’s quantization of Teichmüller space . The right prescription comes from viewing gluing in TQFT somewhat more globally, as a quantum symplectic reduction. It turns out that the combinatorics of ideal triangulations have some fundamental symplectic properties, first discovered by Neumann and Zagier [90, 91], that allow the quantum symplectic reduction to be defined.
As an example, take to be a knot complement. In this case, there is a canonical “A-cycle” on the boundary torus , defined by a small loop linking the knot in (called the meridian cycle); and there is a canonical boundary condition for connections, defined by requiring their holonomy along the meridian cycle to have trivial eigenvalues. (Crucially, this does not require the actual holonomy to be trivial; in there are parabolic matrices with trivial eigenvalues.) In the notation of Section 2, the boundary condition sets , or restricts functions to their values at . Suppose that the knot complement is glued from tetrahedra . The combinatorics of the triangulation define a “Neumann-Zagier datum”
consisting of two integer matrices (that encode adjacency relations for edges of the tetrahedra and satisfy the symplectic property ) and a vector of integers. A precise definition is given in . Then the partition function with the canonical boundary condition takes the concise form 
with . The prefactor depends on an integer solution to .
There are several things to note about this partition function:
It approximately takes the form of a “state sum” or “state integral,” with the partition function (4.4) assigned to every tetrahedron building-block. The “state variables” and are summed/integrated over. The number of sums/integrals is the same as the number of tetrahedra which is also the same as the number of internal edges in the triangulation.
Perhaps most interestingly, the precise definition of the integration contour in (4.6) and the convergence of the integral depends crucially on the existence of a positive angle structure on the triangulation being used. This is ultimately what restricts the computation to a particular class of 3-manifolds that includes all hyperbolic ones. It also beautifully makes contact with the three-dimensional superconformal theory defined combinatorially in  using an ideal triangulation — the definition of does not make sense (cannot produce a superconformal theory) unless a positive angle structure exists. We proceed to explain this idea momentarily.
Via the 3d-3d correspondence, the integral (4.6) has a dual interpretation as a partition function of on the lens space . This partition function is computed by localization methods, as in Section 4.1 of B. Willett’s article, Contribution . Each is the contribution of a chiral multiplet to the partition function, and the additional prefactor comes from background Chern-Simons terms involving flavor and R-symmetries.
Changes of ideal triangulations are generated by 2–3 Pachner moves, which replace a pair of tetrahedra glued along a common face by a triplet glued along three common faces and a central edge. The state-integral (4.6) is invariant under 2–3 moves that preserve the positive or non-negative angle structure (as appropriate), due to a 5-term integral identity for the quantum dilogarithm (4.4). In the case , this identity was first discovered by Faddeev  (see also ).
An angle structure on a topological ideal triangulation is an assignment of real parameters (“angles”) to the six long edges of each tetrahedron, in such a way that 1) angles on opposite edges are equal (leaving three angles per tetrahedron ); 2) the sum of angles around any tetrahedron vertex equals (thus ); and 3) the sum of angles around every internal edge in the triangulation equals . In terms of the Neumann-Zagier gluing datum (4.5) for a knot complement, the last condition translates to999Technically, only out of the components of (4.7) correspond to internal edges; the last component of (4.7) relates to a boundary condition at , and can be removed from the angle-structure analysis. We are bypassing such subtleties in this heuristic discussion.
The idea of an angle structure is motivated by hyperbolic geometry. In an ideal triangulation by hyperbolic tetrahedra, the dihedral angles precisely obey the three conditions above .
A positive (respectively, non-negative) angle structure is one where all angles additionally obey (). An oriented hyperbolic three-manifold with genus-one boundary always admits an triangulation with non-negative dihedral angles, hence there a triangulation with a non-negative angle structure . It is conjectured and strongly believed that the same holds for strictly positive angle structures. (As evidence for this, the default triangulations produced by the computer program SnapPy  for the first few thousand hyperbolic knot complements all admit positive angle structures.)
It was shown in [81, 82] that analogue of the partition function (4.6) at (cf. Section 4.3) is well-defined when the triangulation admits a non-negative angle structure. Similarly, it was shown in  for and [22, 21] for that the partition function (4.6) is well defined if the triangulation admits a positive angle structure. In each case, the angle structure specifies a canonical convergent integration contour.