Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 6

Volume conjecture: refined and categorified

Pages: 1669 – 1777



Hidetoshi Awata (Graduate School of Mathematics, Nagoya University, Japan)

Hiroyuki Fuji (California Institute of Technology, Pasadena, Calif., U.S.A.; Department of Physics, Graduate School of Science, Nagoya University, Japan)

Sergei Gukov (California Institute of Technology, Pasadena, Calif., U.S.A.; Max-Planck-Institut für Mathematik, Bonn, Germany)

Piotr Sułkowski (California Institute of Technology, Pasadena, Calif., U.S.A.; Institute for Theoretical Physics, University of Amsterdam, The Netherlands; Faculty of Physics, University of Warsaw, Poland)


The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the $A$-polynomial $A(x, y)$. Another “family version” of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $ħ$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a $q$-difference equation that annihilates the colored Jones polynomials and $SL(2, {\mathbb C})$ Chern– Simons partition functions. We propose refinements/categorifications of both conjectures that include an extra deformation parameter t and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined/decategorified predecessors, that correspond to $t = −1$, the new volume conjectures involve objects naturally defined on an algebraic curve $A^{ref} (x, y; t)$ obtained by a particular deformation of the A-polynomial, and its quantization $\widehat{A}^{ref} \widehat{x}, \widehat{y}; q, t)$. We compute both classical and quantum $t$-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

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