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Register a new userConnes distance function on fuzzy sphere and the connection between geometry and statistics
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An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the $su(2)$ algebra. This has been computed for both the discrete, as well as for the Perelemovs $SU(2)$ coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by $ninmathbb{Z}/2$.
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We revise and extend the algorithm provided in [1] to compute the finite Connes distance between normal states. The original formula in [1] contains an error and actually only provides a lower bound. The correct expression, which we provide here, involves the computation of the infimum of an expression which involves the transverse component of the algebra element in addition to the longitudinal component of [1]. This renders the formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a non-trivial task, but under rather generic conditions it turns out that the Connes distance is proportional to the trace norm of the difference in the density matrices, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [2] in our Hilbert-Schmidt operatorial formulation. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal non-commutativity ($n = 1/2$), the finite distance is shown to coincide exactly with the above mentioned lower bound, with the transverse component playing no role. This, however starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our new and modified algorithm.
The Brezin-Gross-Witten (BGW) model is one of the basic examples in the class of non-eigenvalue unitary matrix models. The generalized BGW tau-function $tau_N$ was constructed from a one parametric deformation of the original BGW model using the generalized Kontsevich model representation. It is a tau-function of the KdV hierarchy for any value of $Ninmathbb C$, where the case $N=0$ reduces to the original BGW tau-function. In this paper, we present a bosonic representation of $tau_N$ in terms of the $W_{1+infty}$ operators that preserves the KP integrability. This allows us to establish a connection between the (generalized) BGW and Kontsevich-Witten tau-functions using $GL(infty)$ operators, both considered as the basic building blocks in the theory of matrix models and partition functions.
We present a detailed calculation of the distance-dependent two-point function for quadrangulations with no multiple edges. Various discrete observables measuring this two-point function are computed and analyzed in the limit of large maps. For large distances and in the scaling regime, we recover the same universal scaling function as for general quadrangulations. We then explore the geometry of minimal neck baby universes (minbus), which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges, the mother universe. We give a number of distance-dependent characterizations of minbus, such as the two-point function inside a minbu or the law for the distance from a random point to the mother universe.
We construct various systems of coherent states (SCS) on the $O(D)$-equivariant fuzzy spheres $S^d_Lambda$ ($d=1,2$, $D=d!+!1$) constructed in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451] and study their localizations in configuration space as well as angular momentum space. These localizations are best expressed through the $O(D)$-invariant square space and angular momentum uncertainties $(Deltaboldsymbol{x})^2,(Deltaboldsymbol{L})^2$ in the ambient Euclidean space $mathbb{R}^D$. We also determine general bounds (e.g. uncertainty relations from commutation relations) for $(Deltaboldsymbol{x})^2,(Deltaboldsymbol{L})^2$, and partly investigate which SCS may saturate these bounds. In particular, we determine $O(D)$-equivariant systems of optimally localized coherent states, which are the closest quantum states to the classical states (i.e. points) of $S^d$. We compare the results with their analogs on commutative $S^d$. We also show that on $S^2_Lambda$ our optimally localized states are better localized than those on the Madore-Hoppe fuzzy sphere with the same cutoff $Lambda$.
The short-distance expansion of the tau function of the radial sine-Gordon/Painleve III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.
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