Let $\Sigma$ be a finite type surface, and $G$ complex algebraic simple Lie group with algebra $\mathfrak{g}$. The quantum moduli of $(\Sigma,G)$ is quantization the ring functions $X_G(\Sigma)$, variety $G$-characters $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus Buffenoir-Roche in mid '90s. It can realized as invariant subalgebra so-called graph algebras, which are $U_q(\mathfrak{g...

For a Seifert fibered homology sphere X $X$ , we show that the q $q$ -series invariant Z ̂ 0 ( ; ) $\hat{\operatorname{Z}}_0(X;q)$ introduced by Gukov–Pei–Putrov–Vafa, is resummation of Ohtsuki series $\operatorname{Z}_0(X)$ . We for every even k ∈ N $k \in \mathbb {N}$ there exists full asymptotic expansion $ \hat{\operatorname{Z}}_0(X;q)$ tending to e 2 π i / $e^{2\pi i/k}$ and in particular l...

The results here are based on formulae in the two appendices in R. Schack, C. M. Caves, and G. M. D’Ariano, “Hypersensitivity to perburbation in the quantum kicked top,” Phys. Rev. E 50, 972–987 (1994). The relevant equations in the two appendices are (A18) and (B5)–(B6). A preliminary version of the analysis here can be found at the end of Sec. V of R. Schack and C. M. Caves, “Information-theo...

2 on their strength stemming from the maximum of possible torsion within all norm-composing parallelizable manifolds. A profound interplay is thus uncovered between the existence and strength of quantum correlations and the parallelizability of the spheres S, S, S, and S necessitated by the four real division algebras. In particular, parallelization within a unit 3-sphere is shown to be respons...

We discuss the relations between (topological) quantum field theories in 4 dimensions and the theory of 2-knots (embedded 2-spheres in a 4-manifold). The so-called BF theories allow the construction of quantum operators whose trace can be considered as the higher-dimensional generalization of Wilson lines for knots in 3-dimensions. First-order perturbative calculations lead to higher dimensiona...

We discuss the question of what quantummethods (J-holomorphic curves and quantum homology) can tell us about the symplectomorphism group and its compact subgroups. After describing the rather complete information we now have about the case of the product of two 2-spheres, we describe some recent results of McDuff– Tolman concerning the symplectomorphism group of toric manifolds. This leads to a...

We developed a novel method to fabricate nanocomposite monodisperse SiO2 spheres (approximately 100 nm) containing homogeneously dispersed Ag quantum dots (approximately 2 to 5 nm). The inclusion morphology is controlled through the timing of the photochemical reduction of silver ions during hydrolysis of tetraethoxysilane in a microemulsion. Depending on the timing, Ag quantum dots can be dire...

We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal d-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space S(Y ), with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories HLQFT and c...

We show that the braided tensor product algebra A1⊗A2 of two module algebras A1,A2 of a quasitriangular Hopf algebraH is equal to the ordinary tensor product algebra of A1 with a subalgebra of A1⊗A2 isomorphic to A2, provided there exists a realization of H within A1. In other words, under this assumption we construct a transformation of generators which ‘decouples’ A1,A2 (i.e. makes them commu...