Gauge Fields on Torus and Partition Function of Strings

In this paper we consider the interrelation between compactified string theories on torus and gauge fields on it. We start from open string theories with background gauge fields and derive partition functions by path integral. Since the effects of background fields and compactification correlate only through string zero modes, we investigate these zero modes. From this point of view, we discuss the Wilson loop mechanism at finite temperature. For the closed string, only a few comments are mentioned.Comment: 13 pages, nofigur

The Universe as a Topological Defect in a Higher-Dimensional Einstein-Yang-Mills Theory

An interpretation is suggested that a spontaneous compactification of space-time can be regarded as a topological defect in a higher-dimensional Einstein-Yang-Mills (EYM) theory. We start with $D$-dimensional EYM theory in our present analysis. A compactification leads to a $D-2$ dimensional effective action of Abelian gauge-Higgs theory. We find a "vortex" solution in the effective theory. Our universe appears to be confined in a center of a "vortex", which has $D-4$ large dimensions. In this paper we show an example with $SU(2)$ symmetry in the original EYM theory, and the resulting solution is found to be equivalent to the "instanton-induced compactification". The cosmological implication is also mentioned.Comment: 7 pages, no figur

Double Compactification

A cosmological scenario according to which our universe experienced space-time compactifications twice in its early development is investigated through toy models. In this scenario gauge configurations on an extra space play essential roles to bring about a change of the dimensionality of the compactified space. Simple models are offered and their behaviour at finite temperature is examined. A possibility of causing inflation and problems on our scenario is argued briefly.Comment: 12 pages, no figur

Aharonov-Bohm Scattering by Vortices of Dimensionally-Reduced Yang-Mills Field

If two dimensions of six-dimensional space-time are compactified, a topological configuration of Yang-Mills gauge field appears as a cosmic string in four dimensions, whose thickness is of the same order as the size of the compact space. We consider scattering of low-energy fermions by this object.Comment: 5 pages, no figur

Condensation of Yang-Mills field at High Temperature in the Presence of Fermions

The possible condensation of the time-component of Yang-Mills field at finite temperature is discussed in the presence of Dirac fermions. We show that the condensation forms regardless of the number of fundamental and adjoint fermion species coupled to the Yang-Mills field. The effect of finite density of fermions is also investigated and it is shown that the magnitude of the condensation is also independent of the densities.Comment: 6 pages, 1 figur

Cosmic strings in compactified gauge theory

A solution of the vortex type is given in a six-dimensional $SU(2)\times U(1)$ pure gauge theory coupled to Einstein gravity in a compactified background geometry. We construct the solution of an effective abelian Higgs model in terms of dimensional reduction. The solution, however, has a peculiarity in its physically relevant quantity, a deficit angle, which is given as a function of the ratio of the gauge couplings of $SU(2)$ and $U(1)$. The size of the extra space (sphere) is shown to vary with the distance from the axis of the "string".Comment: 12 pages, 7 figure

Primitive Forms for Affine Cusp Polynomials

We determine a primitive form for a universal unfolding of an affine cusp polynomial. Moreover, we prove that the resulting Frobenius manifold is isomorphic to the one constructed from the Gromov-Witten theory for an orbifold projective line with at most three orbifold points.Comment: 57 page

A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.Comment: 15 pages. v2: minor change

A Uniqueness Theorem for Frobenius Manifolds and Gromov--Witten Theory for Orbifold Projective Lines

We prove that the Frobenius structure constructed from the Gromov-Witten theory for an orbifold projective line with at most three orbifold points is uniquely determined by the WDVV equations with certain natural initial conditions.Comment: 29 page

Vibrational properties of two-dimensional dimer packings near the jamming transition

Jammed particulate systems composed of various shapes of particles undergo the jamming transition as they are compressed or decompressed. To date, sphere packings have been extensively studied in many previous works, where isostaticity at the transition and scaling laws with the pressure of various quantities, including the contact number and the vibrational density of states, have been established. Additionally, much attention has been paid to nonspherical packings, and particularly recent work has made progress in understanding ellipsoidal packings. In the present work, we study the dimer packings in two dimensions, which have been much less understood than systems of spheres and ellipsoids. We first study the contact number of dimers near the jamming transition. It turns out that packings of dimers have "rotational rattlers", each of which still has a free rotational motion. After correcting this effect, we show that dimers become isostatic at the jamming, and the excess contact number obeys the same critical law and finite size scaling law as those of spheres. We next study the vibrational properties of dimers near the transition. We find that the vibrational density of states of dimers exhibits two characteristic plateaus that are separated by a peak. The high-frequency plateau is dominated by the translational degree of freedom, while the low-frequency plateau is dominated by the rotational degree of freedom. We establish the critical scaling laws of the characteristic frequencies of the plateaus and the peak near the transition. In addition, we present detailed characterizations of the real space displacement fields of vibrational modes in the translational and rotational plateaus.Comment: 16 pages, 16 figure