In this paper, we prove the mirror symmetry conjecture between the Saito–Givental theory
of exceptional unimodular singularities on the Landau–Ginzburg B-side and the Fan–Jarvis–
Ruan–Witten theory of their mirror partners on the Landau–Ginzburg A-side. On the B-side, we
develop a perturbative method to compute the genus-0 correlation functions associated to the primitive
forms. This is applied to the exceptional unimodular singularities, and we show that the numerical
invariants match the orbifold-Grothendieck–Riemann–Roch and WDVV calculations in FJRW
theory on the A-side. The coincidence of the full data at all genera is established by reconstruction
techniques. Our result establishes the first examples of LG-LG mirror symmetry for weighted
homogeneous polynomials of central charge greater than one (i.e. which contain negative degree
We present a 3D topological picture-language for quantum information.
Our approach combines charged excitations carried by
strings, with topological properties that arise from embedding the
strings in the interior of a 3D manifold with boundary. A quon is
a composite that acts as a particle. Specifically, a quon is a hemisphere
containing a neutral pair of open strings with opposite
charge. We interpret multiquons and their transformations in a
natural way. We obtain a type of relation, a string–genus “joint
relation,” involving both a string and the 3D manifold. We use
the joint relation to obtain a topological interpretation of the C*
Hopf algebra relations, which are widely used in tensor networks.
We obtain a 3D representation of the controlled NOT (CNOT) gate
that is considerably simpler than earlier work, and a 3D topological
protocol for teleportation.
We find the shock wave solutions in a class of cosmological backgrounds with a null singularity, each of these backgrounds admits a matrix description. A shock wave solution breaks all supersymmetry meanwhile indicates that the interaction between two static D0-branes cancel, thus provides basic evidence for the matrix description. The probe action of a D0-brane in the background of another suggests that the usual perturbative expansion of matrix model breaks down.
We compute the one-loop effective action of two D0-branes in the matrix model for a cosmological background, and find vanishing static potential. However, there is a non-vanishing v2 term not predicted in a supergravity calculation. This term is complex and signals an instability of the two D0-brane system, it may also indicate that the matrix model is incorrect.
We discuss a few examples in 2+1 dimensions and 1+1 dimensions supporting a recent conjecture concerning the relation between the Planck scale and the coupling strength of a non-gravitional interaction, unlike those examples in 3+1 dimensions, we do not have to resort to exotic physics such as small black holes. However, the result concerning these low dimensional examples is a direct consequence of the 3+1 dimensional conjecture.