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June 24, 2003

Gukov on Knots

One of the most celebrated connections between math and physics is the relation between knot theory and 3D Chern-Simons theory. If you consider a Chern-Simons gauge theory for a compact gauge group, GG, the natural observables are the expectation-values of Wilson loops in unitary representations, R iR_i of GG. These compute the Jones polynomial (for G=SU(2)G=SU(2)) and its generalizations.

3D gravity can also be cast in the form of a Chern-Simons theory,

Chern-Simons gauge groups for 3D gravity
Λ>0\Lambda\gt 0SU(2)×SU(2)SU(2)\times SU(2)SL(2,)SL(2,ℂ)
Λ=0\Lambda = 0ISO(3)ISO(3)ISO(2,1)ISO(2,1)
Λ<0\Lambda\lt 0SL(2,)SL(2,&complexes;)SL(2,)×SL(2,)SL(2,&reals;)\times SL(2,&reals;)

At least, with Euclidean signature, you might ask whether there’s some relation between 3D quantum gravity and knot theory. Recently, Gukov has proposed that, with negative cosmological constant, 3D quantum gravity is related to the A-polynomial of knot-theory

Consider the complement of a knot, γ\gamma, in S 3S^3. A theorem of Thurston states that “most” knots, γ\gamma, are “hyperbolic”: S 3γS^3\setminus\gamma can be endowed with a hyperbolic metric. The exceptions are torus knots (knots which can be inscribed on the surface of a torus) and satellite knots (a torus knot, where the torus, as embedded in S 3S^3, is itself knotted – with some restrictions to exclude trivial cases).

Classical solutions to SL(2,)SL(2,&complexes;) Chern-Simons theory

(1)S=i(k+1/4G N)8πTr𝒜d𝒜+23𝒜𝒜𝒜i(k1/4G N)8πTr𝒜¯d𝒜¯+23𝒜¯𝒜¯𝒜¯ S= -{\frac{i(k+1/4G_N)}{8\pi}\int Tr\mathcal{A}\wedge d\mathcal{A} +\frac{2}{3} \mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A}} -{\frac{i(k-1/4G_N)}{8\pi}\int Tr \overline{\mathcal{A}}\wedge d\overline{\mathcal{A}} +\frac{2}{3} \overline{\mathcal{A}}\wedge\overline{\mathcal{A}}\wedge\overline{\mathcal{A}}}

are flat SL(2,)SL(2,&complexes;) connections which translates into a hyperbolic metric and its associated Levi-Cevita connection, when we identify 𝒜=ω+ie\mathcal{A}=\omega +i e. Under a reversal of orientation of MM, eee\to -e, and hence 𝒜𝒜¯\mathcal{A}\to \overline{\mathcal{A}}. Thus the Euclidean action gets complex-conjugated under orientation reversal, as it should.

The boundary of S 3γS^3\setminus\gamma is Σ=T 2\Sigma=T^2, and a flat SL(2,)SL(2,&complexes;) connection is specified by a homomorphism from π 1(Σ)=\pi_1(\Sigma)=&integers;\oplus&integers; to SL(2,)SL(2,&complexes;), up to conjugation. That is, it’s specified by a pair of elements,

(2)g a=(x * 0 1/x),g b=(y * 0 1/y)SL(2,) g_a = \left(\array{x&*\\ 0&1/x}\right), g_b=\left(\array{y&*\\ 0&1/y}\right)\in SL(2,&complexes;)

(You can neglect the entries marked “*”: demanding that g ag_a and g bg_b commute generically constrains one linear combination and by a conjugation, we can rescale the other.) Not every pair, (x,y) *× *(x,y)\in &complexes;^*\times &complexes;^*, however, can be the boundary value of a flat SL(2,)SL(2,&complexes;) connection. The constraint that the connection does extend to S 3γS^3\setminus\gamma can be expressed as a polynomial equation [D. Cooper, M. Culler, H. Gillet, D.D. Long, P.B. Shalen, “Plane curves associated to character varieties of 3-manifolds,” Invent. Math. 118 (1994) 47.]

(3)A(x,y)=0 A(x,y)=0

which is the A-polynomial mentioned above.

The algebraic curve L={A(x,y)=0,(x,y) *× *}L=\{A(x,y)=0, (x,y)\in &complexes;^*\times &complexes;^*\} has several components. In particular, there’s alway a factor of x1x-1, corresponding to the trivial representation. Quantizing the system can be done by breaking up the path integral into an integration over the fields on S 3N(γ)S^3\setminus N(\gamma), where N(γ)N(\gamma) is a tubular neighbourhood of the knot, producing a state on the boundary T 2T^2. This then gets paired with the state associated to the solid torus containing the Wilson line.

Gukov does the semiclassical (WKB) computation of the boundary state, and ties in the result with a number of conjectures in knot theory. For completely independent reasons, I’ve been interested in SL(2,)SL(2,&complexes;) Chern-Simons theory, so this was a very nice paper to see.

Posted by distler at June 24, 2003 1:30 PM

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