相关论文: Quantum Holonomies from Spectral Networks and Fram…

We consider the quantum UV-IR map for line defects in class $S$ theories of $\mathfrak{gl}(3)$-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric…

A quantum holonomy reflects the curvature of some underlying structure of quantum mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace…

Quantum invariants like the colored Jones polynomial are algebraic in nature but are conjectured to detect important information about the geometry of links. In this thesis we explore these connections using an enhanced version of the RT…

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket…

Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum…

We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of…

In the context of (2+1)--dimensional quantum gravity with negative cosmological constant and topology R x T^2, constant matrix--valued connections generate a q--deformed representation of the fundamental group, and signed area phases relate…

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and…

We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the…

We present a new formulation of quantum holonomy theory, which is a candidate for a non-perturbative and background independent theory of quantum gravity coupled to matter and gauge degrees of freedom. The new formulation is based on a…

In the context of (2+1)--dimensional gravity, we use holonomies of constant connections which generate a $q$--deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to…

I present a simple graphical method to find the BPS spectra of $A_1$ theories of class S. BPS graphs provide a bridge between spectral networks and BPS quivers, the two main frameworks for the study of BPS states. Here I show how to…

We consider quantum holonomy of some three-dimensional general covariant non-Abelian field theory in Landau gauge and confirm a previous result partially proven. We show that quantum holonomy retains metric independence after explicit gauge…

BPS spectrum with finite number of states are found for higher rank four dimensional N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on a Riemann surface with various kinds of defects. The wall crossing formula is…

Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on…

We construct $ 1/2 $ BPS 1-instanton states in the Hilbert space of $ 5d $ gauge theories. The states are local with the scale size $ \rho=0 $. With the degeneracy from the gauge orientation taken into account, the BPS spectrum consists of…

The holonomy group $G$ of a pseudo-quaternionic-K\"ahlerian manifold of signature $(4r,4s)$ with non-zero scalar curvature is contained in $\Sp(1)\cdot\Sp(r,s)$ and it contains $\Sp(1)$. It is proved that either $G$ is irreducible, or $s=r$…

We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and…

The manifestation of entanglement within geometric phase is elucidated for spatially-structured bi-photons. Entanglement parameters are shown to influence holonomy in two distinct ways: through statistical superpositions of separable…

We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. Our construction is based on SU(2) Chern-Simons topological quantum field…