Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

By Sergei Gukov. 2003

334 citations recorded by the source corpus.

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,ℂ) partition function and the colored Jones polynomial.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/debec40e9e09ee932a3f238d1cff08bbf39c050c

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In Geometry and Topology

1. Sergei Gukov
2. Tudor Dimofte
3. C. Vafa
4. E. Witten
5. D. Gaiotto
6. Amer Iqbal
7. R. Dijkgraaf
8. Abhijit Gadde

In Nuclear and High Energy Physics

1. C. Vafa
2. Sergei Gukov
3. Tudor Dimofte
4. E. Witten
5. Marcos Mariño
6. D. Gaiotto
7. R. Dijkgraaf
8. N. Nekrasov

Citations in the corpus

• 3-Manifolds and 3d Indices (2011, background): “quot;) edge. Otherwise, it has been proven that all hyperbolic knot complements have non-univalent triangulations [58], and the same seems to hold even for non-hyperbolic knots [56]. { 27 { (see e.g. [28]): P @M = C C Z 2 = f(‘;m)g (‘;m) ˘(‘ 1;m 1) : (4.14) Finally, we take the canonical position ”
• Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function (2006, background): “…is conjectured to dominate the asymptotics of the invariant J K N; e 2πi/N in the large-N limit N → ∞, lim N→∞ 2 π N log J K N; e 2πi/N = Vol(S 3 \ K) (1.1) This " volume conjecture " is generalized for other values (near the N-th root of unity) of the Ncolored Jones polynomial [18] (see also Refs.”
• A-polynomial, B-model, and quantization (2011, methodology): “,C) Chern-Simons theory with a Wilson loop, the polynomial A(x,y) is a topological invariant called the A-polynomial and plays a role similar to that of the Seiberg-Witten curve in N = 2 gauge theory [3]. The parameter ~is the coupling constant of Chern-Simons theory. 3. Matrix models: In matrix mod”
• 6j–symbols, hyperbolic structures and the volume conjecture (2006, methodology): “In [ 10 ], S. Gukov proposed the following generalization of the Volume Conjecture for hyperbolic knots in S3:”
• On dualizability of braided tensor categories (2018, background): “As such, it should provide a foundational framework for: quantum A-polynomials [Dim13, Gar04, Guk05, GS12], DAHAJones polynomials and skein modules of knots complements [Che13, BS16, Sam17], and refined Chern–Simons invariants [AS15], among others, each of which yet lacks a topologically invariant f”
• Fivebranes and 4-manifolds (2013, background): “slightly more interesting theory φai ◦T[S1 ×D2], whose space of supersymetric vacua (2.12) is simply an SL(2,Z) transform of (2.44): Vφ ai◦T[S1×D2] = ˆ (xi,yi) ∈ TC×TC W ixa i yi= 1 ˙ (2.45) See e.g. [42] for more details on Dehn surgery in the context of complex Chern-Simons theory. The space of va”
• Chern-Simons theory, analytic continuation and arithmetic (2007, background): “p τK,G(z) = L p K,G(−z). Unlike the case of Conjecture 2, this is an easy corollary of its very definition. Remark 1.3. Chern-Simons theory with complex gauge group was studied extensively by Gukov in [Gu]. It is an interesting problem to compare forthcoming work of Gukov-Zagier on modularity propert”
• The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces (2004, methodology): “ith E the figure-eight knot. Then it turns out that it determines the potential function introduced by W. Neumann and D. Zagier [16] to study the volumes of Dehn surgered spaces along knots. (See also [1].) The we apply T. Yoshida’s theorem [24] to show that the limit of log JN E;exp(2πr √ −1/N) /N”
• Exact results for perturbative Chern-Simons theory with complex gauge group. (2009, background): “Every such M can be represented as a quotient M = H3/Γ (2.6) of the hyperbolic space H3 by a discrete, torsion-free subgroup Γ ⊂ PSL(2,C), which is a holonomy representation of the fundamental group π1(M) into Isom +(H3) = PSL(2,C).”
• Quantum foam and topological strings (2003, background): “space of connections is an enlargement of the space of allowed metrics [11] (see [ 17 ] for”
• Gauge Theories Labelled by Three-Manifolds (2011, background): “nections on @Mg (gauge equivalence): (2.1) 1The equivalence between at connections and hyperbolic geometry results from the fact that the isometry group of hyperbolic three-space is (P)SL(2;C), cf. [6, 7, 8, 9]. Almost all at connections can be realized as (possibly degenerate) hyperbolic metrics; ”
• Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants (2014, neutral): “ C. BLANCHET, F. COSTANTINO, N. GEER, AND B. PATUREAU-MIRAND Lemma A.5 (General hopf links). 8i;j2f0;1;:::;r 2gand 8;2C, one has V d;V f = ( r1) 1r ( ) q Id V S i;S j = ( 1) i f(i+ 1)(j+ 1)g j+ 1g (35) Id V j S i;V = f(i+ 1)g fg Id V d P i;V = ( 1) r 1r q (r 1 i) + q (r 1 i) ( ) (36) Id V S i;P j = ”
• Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “ere A M(‘;m) is the so-called A-polynomial of M[3]. This condition cuts out a Lagrangian submanifold L M = fA M(‘;m) = 0g ˆ P T2 = f(‘;m)gˇC C ; (1.4) or a semi-classical state in Chern-Simons theory [4]. We would like to quantize the A-polynomial, promoting it to an operator A^ M(^‘; m^;q) that ann”