From minimal gravity to open intersection theory

We investigated the relation between the two-dimensional minimal gravity (Lee-Yang series) with boundaries and open intersection theory. It is noted that the minimal gravity with boundaries is defined in terms of boundary cosmological constant $\mu_B$ and the open intersection theory in terms of boundary marked point generating parameter $s$. Based on the conjecture that the two different descriptions of the generating functions are related by the Laplace transform, we derive the compact expressions for the generating function of the intersection theory from that of the minimal gravity on a disk and on a cylinder.Comment: 26 page

一般化幾何学に基づく超重力理論の内部空間の新しい構成

Tohoku University綿村哲課

Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry

We consider the information metric and Berry connection in the context of noncommutative matrix geometry. We propose that these objects give a new method of characterizing the fuzzy geometry of matrices. We first give formal definitions of these geometric objects and then explicitly calculate them for the well-known matrix configurations of fuzzy S2 and fuzzy S4. We find that the information metrics are given by the usual round metrics for both examples, while the Berry connections coincide with the configurations of the Wu-Yang monopole and the Yang monopole for fuzzy S2 and fuzzy S4, respectively. Then, we demonstrate that the matrix configurations of fuzzy Sn (n=2, 4) can be understood as images of the embedding functions Sn→Rn+1 under the Berezin-Toeplitz quantization map. Based on this result, we also obtain a mapping rule for the Laplacian on fuzzy S4

Commutative geometry for non-commutative D-branes by tachyon condensation

There is a difficulty in defining the positions of the D-branes when the scalar fields on them are non-Abelian. We show that we can use tachyon condensation to determine the position or the shape of D0-branes uniquely as a commutative region in spacetime together with a non-trivial gauge flux on it, even if the scalar fields are non-Abelian. We use the idea of the so-called coherent state method developed in the field of matrix models in the context of the tachyon condensation. We investigate configurations of non-commutative D2-brane made out of D0-branes as examples. In particular, we examine a Moyal plane and a fuzzy sphere in detail, and show that whose shapes are commutative R2 and S2⁠, respectively, equipped with uniform magnetic flux on them. We study the physical meaning of this commutative geometry made out of matrices, and propose an interpretation in terms of K-homology