Anomaly Cancellations in the Type I D9-anti-D9 System and the USp(32) String Theory

We check some consistency conditions for the D9-anti-D9 system in type I string theory. The gravitational anomaly and gauge anomaly for SO(n) x SO(m) gauge symmetry are shown to be cancelled when n-m=32. In addition, we find that a string theory with USp(n) x USp(m) gauge symmetry also satisfies the anomaly cancellation conditions. After tachyon condensation, the theory reduces to a tachyon-free USp(32) string theory, though there is no spacetime supersymmetry.Comment: 17 pages + 10 eps figures, LaTeX; minor corrections, reference added, version to appear in Prog. Theor. Phy

Geometric transition in Non-perturbative Topological string

We study a geometric transition in non-perturbative topological string. We consider two cases. One is the geometric transition from the closed topological string on the local $\mathcal{B}_{3}$ to the closed topological string on the resolved conifold. The other is the geometric transition from the closed topological string on the local $\mathcal{B}_{3}$ to the open topological string on the resolved conifold with a toric A-brane. We find that, in both cases, the geometric transition can be applied for the non-perturbative topological string. We also find the corrections of the value of K\"ahler parameters at which the geometric transition occurs.Comment: 21 pages, 5 figures, v2: typos are corrected, reference is added, published in Physical Review

Comments on Duality in MQCD

We clarify some ambiguous points in a derivation of duality via brane exchange using M-theory language, and propose a ``proof'' of duality in MQCD. Actually, duality in MQCD is rather trivial and does not need a complicated proof. The problem is how to interpret it in field theory language. We examine BPS states in N=2 theory and find the particle correspondence under duality. In the process, we also find some exotic particles in N=2 MQCD, and we observe an interesting phenomenon in type IIA string theory, namely, that fundamental strings are converted into D2-branes via the exchange of two NS5-branes. We also discuss how we should understand Seiberg's N=1 duality from exact duality in MQCD.Comment: 29 pages + 20 uuencoded eps figures, LaTeX with PTPTeXsty, typo corrected, version to appear in Prog. Theor. Phy

The sharp energy-capacity inequality on convex symplectic manifolds

In symplectic geometry, symplectic invariants are useful tools in studying symplectic phenomena. Hofer-Zehnder capacity and displacement energy are important symplectic invariants. Usher proved the so-called sharp energy-capacity inequality between Hofer-Zehnder capacity and the displacement energy for closed symplectic manifolds. In this paper, we extend the sharp energy-capacity inequality to convex symplectic manifolds

Rotationally symmetric tilings with convex pentagons belonging to both the Type 1 and Type 7

Rotationally symmetric tilings by a convex pentagonal tile belonging to both the Type 1 and Type 7 families are introduced. Among them are spiral tilings with two- and four-fold rotational symmetry. Those rotationally symmetric tilings are connected edge-to-edge and have no axis of reflection symmetry.Comment: 13 pages, 16 figures. arXiv admin note: text overlap with arXiv:2005.08470, arXiv:2005.1270

Convex pentagons and convex hexagons that can form rotationally symmetric tilings

In this study, the properties of convex hexagons that can generate rotationally symmetric edge-to-edge tilings are discussed. Since the convex hexagons are equilateral convex parallelohexagons, convex pentagons generated by bisecting the hexagons can form rotationally symmetric non-edge-to-edge tilings. In addition, under certain circumstances, tiling-like patterns with an equilateral convex polygonal hole at the center can be formed using these convex hexagons or pentagons.Comment: 23 pages, 28 figures. arXiv admin note: text overlap with arXiv:2005.0847