Trivalent graphs, volume conjectures and character varieties

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the fundamental group of the complement of a trivalent graph with $E$ edges in $S^3$ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus $g=E/3+1$. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel-Nielsen coordinates. Moreover, the $q$-holonomic difference equation of the quantum invariants provides the quantization of the character variety.Comment: 11 pages, 2 figure

Multiplicity-free quantum 6j-symbols for U_q(sl_N)

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.Comment: 8 pages; v2 typos corrected; v3 minor corrections and reference adde

Colored HOMFLY polynomials from Chern-Simons theory

We elaborate the Chern-Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for U_q(sl_N), we present explicit evaluations of the HOMFLY invariants colored by symmetric representations for a variety of knots, two-component links and three-component links.Comment: 40 pages, 23 figures, a Mathematica notebook linked on the right as an ancillary file; v2 typos corrected; v3 corrections in section 4.2 and cosmetic changes; v4 corrections in two-component link

Supersymmetric Galilean conformal blocks

We set up the bootstrap procedure for supersymmetric Galilean Conformal (SGC) field theories in two dimensions by constructing the SGC blocks in the $\mathcal{N}=1$ and two possible $\mathcal{N} =2$ extensions of the Galilean conformal algebra. In all analyzed cases, we present the bootstrap equations by crossing symmetry of the four point function. In addition, we compute the global SGC blocks analytically by solving the differential equations obtained by acting with the Casimirs of the global subalgebras inside the four point function. These global blocks agree with the general SGC blocks in the limit of large central charge. We comment on possible applications to supersymmetric BMS$_3$ invariant field theories and flat holography.Comment: 43 pages, v2: references added and typos fixed, v3: refs added, minor errors in the expression for the despotic blocks fixed. Matches published versio