Volume Conjecture: Refined and Categorified

TL;DR

This work develops a refined and categorified version of the volume conjecture by introducing a deformation parameter $t$ alongside the quantum parameter $q$ and by formulating refined algebraic curves $A^{ref}(x,y;t)=0$ and their quantizations $\widehat{A}^{ref}(\widehat{x},\widehat{y};q,t)$. It establishes refined quantum and generalized volume conjectures, connecting colored knot invariants to triply-graded knot homologies and refined BPS invariants, and derives explicit refined A-polynomials and quantum curves for key knots, including the unknot, trefoil, and $(2,2p+1)$ torus knots. The paper develops the refined braid operators, gamma factors, and Macdonald-polynomial machinery necessary to compute refined colored superpolynomials, and demonstrates how these objects satisfy recursion relations that generalize the AJ-conjecture. A saddle-point analysis confirms that the leading refined action $S_0(u,t)=\int v\,du$ computed from the refined curves matches asymptotics of the refined invariants, and quantizability constraints enforce that $t$ be a root of unity, tying in with temperedness in algebraic K-theory. Beyond knot theory, the work connects refined mirror curves to open refined BPS amplitudes in toric Calabi–Yau geometries, interpreting refinement as a twisted mass parameter and highlighting the role of refined topological strings and Macdonald polynomials in producing open-closed BPS data. These results offer a cohesive framework linking refined A-polynomials, quantum curves, knot homologies, and refined BPS invariants, with implications for algebraic K-theory, mirror symmetry, and geometric engineering of supersymmetric gauge theories.

Abstract

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 $\hbar$; 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 $SL(2,\C)$ 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 $\hat A^{ref} (\hat x, \hat y; q, t)$. We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

Figures (24)

• Figure 1: Deformation and quantization of the $A$-polynomial. The horizontal arrows describe a deformation / refinement, such that the unrefined case corresponds to $t=-1$. The vertical arrows represent quantization, i.e. a lift of classical polynomials $A(x,y)$ and $A(x,y;t)$ to quantum operators.
• Figure 2: Categorification of quantum knot invariants. To help the reader navigate through this picture we suggest to keep track of the variables $a$, $q$, $t$, as well as the rank of $sl(N)$. Note, the polynomial (resp. homological) knot invariants which have $a$-dependence (resp. $a$-grading) are not labeled by $sl(N)$.
• Figure 3: Slicing and braiding.
• Figure 4: Conformal block $\phi_{Q^{\prime}}(R_1,R_2,\overline{R}_3,\overline{R}_4)$ for the four point function.
• Figure 5: Slicing along $\Sigma_2 \simeq {\bf S}^2$.