Chern-Simons theory as a quantum TQFT, the Jones polynomial, and Reshetikhin-Turaev

Intuition Beginner

Imagine a knot tied in a loop of string, then sealed so the ends never come apart. How can you tell two such knots apart without untying them? You need a number, or a formula, that you can compute from a picture of the knot and that never changes when you wiggle the string. Such a quantity is called a knot invariant. Finding good ones is hard, because a single knot has infinitely many pictures.

Chern-Simons theory is a kind of physics built on a three-dimensional space. Its remarkable feature is that it has no local action at all: it does not care about distances, only about how loops are linked and knotted. The only things you can measure in it are loops of "charge" threaded through the space, and the answer the theory returns for a loop is a knot invariant.

The surprise discovered in 1989 is that for one natural choice of the theory, the number it assigns to a knot is exactly the Jones polynomial, a knot invariant found four years earlier by completely different means. So a quantum field theory turned out to be a machine that distinguishes knots, and the same machine assigns a number to the whole three-dimensional space as well.

Visual Beginner

The picture shows a three-dimensional region with two knotted, linked loops inside it. In Chern-Simons theory each loop is a "Wilson line" carrying a label, like a color. The theory hands back a single number built from how the loops cross and link. Slide the loops around without cutting them and the number stays put. Replace a crossing where one strand passes over another, and the number changes in a controlled way: that controlled change is the skein rule, and it is enough to compute the Jones polynomial of any knot from its picture.

Worked example Beginner

Take the simplest interesting link: two plain circles that pass through each other once, like two links of a chain. This is called the Hopf link. In the abelian version of the theory, where the charge is just an ordinary number, the theory assigns to two loops a phase built from how many times one loop winds through the other. For the Hopf link that winding number is 1.

With level $k=4$ and unit charges, the abelian theory returns the value $\exp (2\times 3.14159\times 1/4)=\exp (1.5708)$, read as a point on the unit circle a quarter of the way around. Two unlinked circles wind through each other 0 times, so they return $\exp (0)=1$.

The two answers differ. That difference, between linked and unlinked, is the whole point: the number sees the linking and reports it. What this tells us is that even the simplest version of the theory already measures a topological fact about the loops, and the richer non-abelian version measures genuine knotting.

Check your understanding Beginner

Formal definition Intermediate+

Fix a compact, simple, simply connected Lie group $G$ with Lie algebra $\mathfrak{g}$, and a closed oriented three-manifold $M$. The classical Chern-Simons functional 03.07.17 of a connection $A$ on the (necessarily globally split) principal $G$-bundle over $M$ is $$ S_k(A) = \frac{k}{4\pi} \int_M \mathrm{tr}!\left( A \wedge dA + \tfrac{2}{3} A \wedge A \wedge A \right), $$ where the trace is the basic invariant form normalized so that $S_k$ is well defined modulo $2\pi k\mathbb{Z}$ under large gauge transformations, forcing the level $k$ to be an integer ^{[tong Gauge Theory lectures, Chern-Simons term]}.

The quantum Chern-Simons theory at level $k$ is the assignment, to each $M$, of the partition function $$ Z_k(M) = \int e^{i S_k(A)} , \mathcal D A, $$ a heuristic path integral over the space of connections modulo gauge. The content of Witten's construction is that this assignment, together with its values on surfaces, satisfies the Atiyah-Segal axioms 03.16.01: it is a symmetric monoidal functor $Z_k$ from the bordism category of oriented surfaces and three-dimensional bordisms to finite-dimensional complex vector spaces. To a closed oriented surface $\Sigma$ it assigns the state space $Z_k(\Sigma )$, a finite-dimensional Hilbert space; to a bordism $M:{\Sigma }_0\to {\Sigma }_1$ it assigns a linear map; to a closed $M$ the partition-function number $Z_k(M)\in \mathbb{C}$ ^[Witten].

The state space is identified geometrically. Canonical quantization of the theory on $\Sigma \times \mathbb{R}$ has phase space the moduli space ${\mathcal{M}}_{\Sigma }$ of flat $G$-connections on $\Sigma$, a finite-dimensional symplectic manifold whose symplectic form has cohomology class $k$ times a generator. Geometric quantization at level $k$ then yields $$ Z_k(\Sigma) ;\cong; H^0!\left(\mathcal M_\Sigma, \mathcal L^{\otimes k}\right), $$ the space of holomorphic sections of the $k$-th power of the prequantum line bundle, once a complex structure on $\Sigma$ is chosen. This space is canonically the space of conformal blocks of the Wess-Zumino-Witten model 03.10.03 on $\Sigma$ at level $k$. Its dimension is computed by the Verlinde formula ^[Verlinde].

The observables are framed Wilson loops. For an oriented framed knot $K\subset M$ and an irreducible representation $R$ of $G$, set $W_R(K)={\mathrm{tr}}_R{\mathrm{Hol}}_K(A)$, the trace in $R$ of the holonomy around $K$. The link invariants of the theory are the normalized expectation values $$ \big\langle W_{R_1}(K_1) \cdots W_{R_n}(K_n) \big\rangle = \frac{1}{Z_k(M)} \int \Big( \textstyle\prod_i W_{R_i}(K_i) \Big) , e^{i S_k(A)} , \mathcal D A . $$ A framing of each component is part of the datum: the holonomy of a loop against itself is ill defined without a choice of normal pushoff, and the invariant depends on this choice in a controlled way.

Key theorem with proof Intermediate+

The signature computation is the skein relation for $SU(2)$ Wilson loops in the fundamental representation, from which the Jones polynomial follows.

Theorem (Witten). Let $G=SU(2)$ at level $k$, with each component colored by the two-dimensional representation. Set $q=\exp (2\pi i/(k+2))$. For three framed links $L_{+}$, $L_{-}$, $L_0$ that are identical except inside a small ball, where they show an overcrossing, an undercrossing, and the oriented smoothing respectively, the expectation values satisfy $$ q^{1/2}, \big\langle W(L_+) \big\rangle - q^{-1/2}, \big\langle W(L_-) \big\rangle = \big(q^{1/4} - q^{-1/4}\big), \big\langle W(L_0) \big\rangle . $$ This is the defining skein relation of the Jones polynomial $V_L(q)$ ^[Jones], so the normalized $SU(2{)}_k$ Wilson-loop invariant equals $V_L$ evaluated at this root of unity.

Proof. Work in a solid ball $B$ containing the relevant crossing, with the rest of the link fixed outside. The path integral over $B$ with two marked points on each of the two boundary disks, colored by the fundamental representation $V$, produces a vector in the state space $Z_k(\Sigma )$ for $\Sigma$ the four-punctured sphere bounding $B$. That state space is the space of conformal blocks of the WZW model on the four-punctured sphere with four fundamental insertions, identified above with $H^0({\mathcal{M}}_{\Sigma },{\mathcal{L}}^{\otimes k})$.

For $SU(2{)}_k$ with four fundamental insertions, this space is two-dimensional: the tensor product $V\otimes V$ decomposes into the spin-0 and spin-1 channels, and exactly these two channels survive the level-$k$ fusion truncation for $k\ge 2$. Any three states obtained by inserting an overcrossing, an undercrossing, or the planar smoothing into the same ball therefore lie in this two-dimensional space, so they satisfy one linear relation.

The coefficients of that relation are fixed by the monodromy of the conformal blocks. The braiding that exchanges two fundamental insertions acts on the two channels with eigenvalues ${\lambda }_0$ and ${\lambda }_1$ governed by the conformal weights $h_j$ of the spin-$j$ primary fields, namely ${\lambda }_j=\pm \exp (i\pi (h_{2j}-2h_1))$ up to a common framing phase. With $h_j=j(j+1)/(k+2)$, the eigenvalues are $q^{1/4}$ and $-q^{-3/4}$ after normalization, where $q=\exp (2\pi i/(k+2))$. A crossing is this braiding, its inverse is the opposite crossing, and the identity (the smoothing) is the third operator; the characteristic polynomial $(x-{\lambda }_0)(x-{\lambda }_1)=0$ of the braiding, rewritten in terms of $L_{+},L_{-},L_0$, is precisely the displayed skein relation. Comparing with Jones's original relation identifies the normalized invariant with $V_L(q)$. $\square$

Bridge. This skein relation builds toward the entire combinatorial theory of quantum invariants, and the two-dimensionality of the four-point block appears again in the Reshetikhin-Turaev construction as the rank of a fusion space. The foundational reason the coefficients are roots of unity is that the conformal weights $h_j=j(j+1)/(k+2)$ carry the level in their denominator, so this is exactly the same shift by the dual Coxeter number that fixed the central charge of the WZW model in 03.10.03; the braiding eigenvalues generalise the abelian linking phase of the Beginner example, and putting these together shows the bridge is fusion: the Hilbert space the path integral builds over a punctured surface is the representation-theoretic fusion space, and the knot polynomial is its braid-group monodromy.

Exercises Intermediate+

Advanced results Master

The combinatorial face of the theory removes the path integral entirely. The input is a modular tensor category $\mathcal{C}$: a semisimple ribbon braided monoidal category with finitely many isomorphism classes of simple objects $\{X_0=1,X_1,\dots ,X_m\}$, whose $S$-matrix, recording the trace of the double braiding $X_i\otimes X_j\to X_j\otimes X_i\to X_i\otimes X_j$, is invertible. For the present theory $\mathcal{C}$ is the category of integrable level-$k$ representations of the affine algebra $\widehat{\mathfrak{su}(2)}$, equivalently the representation category of the quantum group $U_q({\mathfrak{sl}}_2)$ at $q=\exp (i\pi /(k+2))$, with the unstable representations quotiented out by the tilting (negligible) ideal. The fusion product is the Verlinde fusion, and the twist ${\theta }_{X_j}=\exp (2\pi i{h}_j)$ records the conformal weight.

From $\mathcal{C}$ one defines the Reshetikhin-Turaev invariant ^{[Reshetikhin Turaev]}. A closed oriented three-manifold $M$ is presented by surgery on a framed link $L=L_1\cup \cdots \cup L_n\subset S^3$. Color each component by the formal sum $\omega ={\sum }_j({\dim }_q{X}_j)X_j$, the Kirby color, where ${\dim }_q{X}_j=S_{0j}/S_{00}$ is the quantum dimension. Let $J(L;\omega )$ be the colored Jones evaluation of $L$ with this coloring, computed by the ribbon functor that sends a tangle diagram to a morphism in $\mathcal{C}$. Writing $\sigma (L)$ for the signature of the linking matrix and ${\Delta }_{\pm }={\sum }_j{\theta }_{X_j}^{\pm 1}({\dim }_q{X}_j{)}^2$ for the Gauss sums, the invariant is $$ \tau(M) = \frac{J(L; \omega)}{\Delta_+^{,b_+} , \Delta_-^{,b_-}}, \qquad b_\pm = \tfrac{1}{2}\big(n \pm \sigma(L)\big), $$ normalized so that $\tau (S^3)=1$. Invariance under handle slides follows from a sliding property of the Kirby color, and invariance under stabilization is exactly what the ${\Delta }_{\pm }$ denominators correct, so $\tau (M)$ depends only on $M$.

The two constructions agree where both apply: ${\tau }_{SU(2{)}_k}(M)$ reproduces Witten's $Z_k(M)$, and the colored link evaluations reproduce the Wilson-loop expectation values, with the Jones polynomial recovered for the unknotted ambient $S^3$. The state space $Z_k(\Sigma )$ reappears as the space of morphisms in $\mathcal{C}$ associated to a decomposition of $\Sigma$ into pairs of pants, and the Verlinde formula $$ \dim Z_k(\Sigma_{g,n}^{j_1 \ldots j_n}) = \sum_{a} (S_{0a})^{2-2g} \prod_{r=1}^n \frac{S_{j_r a}}{S_{0 a}} $$ expresses this dimension purely through the modular $S$-matrix.

The theory carries a framing anomaly. The partition function is not quite a number attached to $M$ alone but to $M$ equipped with a $2$-framing, equivalently a chosen splitting of $2p_1$ of the tangent bundle. Changing the framing by one unit multiplies $Z_k(M)$ by $\exp (2\pi ic/24)$ with $c=k\dim G/(k+h^{\vee })$ the central charge of the associated WZW model. In the language of 03.16.05 the anomaly is the partition function of an invertible field theory in one dimension higher, evaluated against the framing data; the same phase governs the framing dependence of Wilson loops, contributing ${\theta }_{X_j}$ per full twist. The abelian $G=U(1)$ specialization strips this to its simplest core: there the Wilson-loop expectation is $\exp (\frac{2\pi i}{k}{\sum }_{a<b}{n}_a{n}_b\mathrm{lk}(K_a,K_b))$, a pure linking-number phase, and the framing dependence reduces to the self-linking term of the Beginner example.

Synthesis. The central insight is that one integer, the level $k$, organises three faces of the same theory, and tracing it is the foundational reason the constructions cohere. This is exactly the data of the WZW model of 03.10.03 read in a third register: the level was the curvature period of a gerbe there, the central term of an affine algebra, and now the deformation parameter of the quantum group whose representation category is the modular tensor category. The bridge is conformal blocks: the geometric quantization that builds $Z_k(\Sigma )$ generalises the holomorphic sections counted by the Verlinde formula, and the Reshetikhin-Turaev surgery formula is dual to the path integral, computing the same $Z_k(M)$ from a combinatorial presentation. Putting these together, the framing anomaly is exactly the obstruction to having a number rather than a number-up-to-a-root-of-unity, and the abelian linking phase is the foundational shadow of the whole structure, the case where fusion is the addition of charges and the Jones polynomial collapses to a Gauss linking number.

Full proof set Master

Proposition. Let $\mathcal{C}$ be a modular tensor category with $S$-matrix $S$ and twists $\theta$, and let $\tau$ be the Reshetikhin-Turaev invariant defined above. Then $\tau (M)$ is independent of the surgery link $L$ presenting $M$, so it is an invariant of the closed oriented three-manifold $M$ with its $2$-framing.

Proof. By Kirby's theorem two framed links in $S^3$ present the same closed oriented three-manifold by surgery if and only if they are related by a finite sequence of handle slides and stabilizations. It suffices to check that the ratio defining $\tau$ is unchanged by each move.

Consider first a handle slide of component $L_i$ over component $L_j$. The colored evaluation $J(-;\omega )$ uses the Kirby color $\omega ={\sum }_a({\dim }_q{X}_a)X_a$ on every component. The defining property of $\omega$ in a modular tensor category is the encircling identity: an $\omega$-colored meridian around a strand colored by any simple object $X$ acts as multiplication by the scalar that projects onto the unit sector, so sliding a strand through the $\omega$-colored component leaves the morphism in $\mathcal{C}$ unchanged. Concretely, ${\sum }_a({\dim }_q{X}_a)(\text{double braiding of }{X}_a\text{ with }X)={\delta }_{X,1}{\Delta }_{+}{\Delta }_{-}^{}/|\cdot |$ on simple $X$, and the normalization by the Gauss sums is independent of which components participate. The linking matrix signature $\sigma (L)$ and the count $n$ are unchanged by a handle slide because the move preserves the diffeomorphism type of the four-manifold the link bounds, hence its intersection form. Therefore both $J(L;\omega )$ and the denominator are unchanged, and the ratio is preserved.

Now consider a stabilization that adjoins a disjoint unknot $U$ with framing $\pm 1$. Coloring $U$ by $\omega$ and evaluating, the disjoint unknot of framing $\pm 1$ contributes the factor ${\sum }_a{\theta }_{X_a}^{\pm 1}({\dim }_q{X}_a{)}^2={\Delta }_{\pm }$ to $J(L\bigsqcup U;\omega )={\Delta }_{\pm }\cdot J(L;\omega )$, because the framed unknot evaluates to its quantum dimension weighted by the twist, summed over the Kirby color. The move changes $n$ to $n+1$ and shifts $\sigma (L)$ by $\pm 1$, hence changes exactly one of $b_{+},b_{-}$ by one, multiplying the denominator ${\Delta }_{+}^{b_{+}}{\Delta }_{-}^{b_{-}}$ by the same ${\Delta }_{\pm }$. The new numerator and new denominator therefore acquire a common factor ${\Delta }_{\pm }$, which cancels, and $\tau$ is unchanged. Since the two moves generate all surgery equivalences, $\tau (M)$ depends only on $M$ with its $2$-framing; the residual framing dependence is the central-charge phase recorded in the Advanced results. $\square$

The geometric counterpart, that $Z_k(\Sigma )=H^0({\mathcal{M}}_{\Sigma },{\mathcal{L}}^{\otimes k})$ has the Verlinde dimension, is stated without a self-contained proof here; the identification of conformal blocks with these sections is the Verlinde conjecture, proved for $\widehat{\mathfrak{su}(2)}$ via factorization by Tsuchiya-Ueno-Yamada and in general by Faltings and by Beauville-Laszlo, cited in the bibliography.

Connections Master

The classical input is the Chern-Simons functional of 03.07.17: that unit constructs $S_k(A)$ as a circle-valued function on the space of connections whose critical points are flat connections, and the present unit quantizes it, turning the same functional into the phase of a path integral and the critical loci into the labels of the resulting Hilbert spaces.

The state spaces are the conformal blocks of the Wess-Zumino-Witten model of 03.10.03: the level $k$ that controlled the central charge and the affine symmetry there is the level that controls the dimension of $Z_k(\Sigma )$ here, and the bulk-boundary relation between three-dimensional Chern-Simons theory and two-dimensional WZW theory is the geometric form of that shared integer.

The categorical-axiomatic frame is 03.16.01: the partition function and state spaces of this unit are an instance of the Atiyah-Segal functor, and Chern-Simons theory is the canonical example of a TQFT whose state spaces are nonzero and finite-dimensional yet not built from a finite set, in contrast to the finite-gauge-theory examples that open that chapter.

Laterally, the framing dependence connects to 03.16.05: the central-charge phase that obstructs $Z_k(M)$ from being a plain number is the partition function of an invertible field theory in one higher dimension, so the framing anomaly here is a concrete instance of the anomaly-as-invertible-theory reframing developed there.

Historical & philosophical context Master

Vaughan Jones discovered his polynomial invariant in 1984 while studying subfactors of von Neumann algebras and the braid-group representations arising from the Temperley-Lieb algebra, publishing it in 1985 ^[Jones]. The invariant was unexpected, sharper than the Alexander polynomial, and its origin in operator algebras gave no hint of why it should be a knot invariant at all. Edward Witten supplied the conceptual source in 1989, constructing a three-dimensional gauge theory whose Wilson-loop observables reproduce the Jones polynomial and simultaneously assign invariants to three-manifolds, work for which he received the Fields Medal in 1990 ^[Witten]. The path integral was heuristic, and the demand for rigor was met by Nicolai Reshetikhin and Vladimir Turaev in 1991, who built the three-manifold invariants combinatorially from the representation theory of quantum groups at roots of unity and the Kirby calculus of surgery ^{[Reshetikhin Turaev]}. The dimension count of the state spaces had been anticipated by Erik Verlinde in 1988, whose fusion-rule formula expresses the conformal-block dimensions through the modular $S$-matrix ^[Verlinde]. The convergence of subfactor theory, gauge theory, two-dimensional conformal field theory, and quantum-group representation theory on a single family of invariants fixed the modern shape of quantum topology.

Bibliography Master

@article{Witten1989Jones,
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}

@article{ReshetikhinTuraev1991,
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}

@article{Jones1985,
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}

@article{TUY1989,
  author  = {Tsuchiya, Akihiro and Ueno, Kenji and Yamada, Yasuhiko},
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@book{BakalovKirillov2001,
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