|DATE||September 17 (Thu), 2020|
|TITLE||Complex volume potential function of 3-manifold and PSL(2,C)-character variety|
|ABSTRACT||For a $PSL(2,\mathbb C)$)-representation $\rho$ of a 3-manifold, we can define hyperbolic volume of $\rho$. In the study of the volume conjecture: the asymptotic growth rate of colored Jones polynomials of a hyperbolic knot $K$ gives hyperbolic volume of $K$, we usually obtain a complex volume potential function $V(z_1,\ldots,z_N)$ as an optimistic limit from the quantum invariant. We review an interesting phenomenon in this situation that the derivative of $V$ is related to $PSL(2,\mathbb C)$-character variety as an affine algebraic set. In the previous talk by Dr. Seokbeom Yoon, he reported an observation that the Jacobian of the defining equations for character variety is related to the adjoint Reidemeister torsion. So, the Hessian of volume potential function $V$ also has a similar relation, which can be seen as an origin of 1-loop conjecture.
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