Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.Comment: Final version to be published in Commun. Math. Phys. . A new section is added and devoted to Conclusion and discussion, where, in particular, a possible relation with the generating function of the absolute Gromov-Witten invariants on CP^1 is commented. Two references are added. Typos are corrected. 32 pages. 4 figure

Subaru and Keck Observations of the Peculiar Type Ia Supernova 2006gz at Late Phases

Recently, a few peculiar Type Ia supernovae (SNe) that show exceptionally large peak luminosity have been discovered. Their luminosity requires more than 1 Msun of 56Ni ejected during the explosion, suggesting that they might have originated from super-Chandrasekhar mass white dwarfs. However, the nature of these objects is not yet well understood. In particular, no data have been taken at late phases, about one year after the explosion. We report on Subaru and Keck optical spectroscopic and photometric observations of the SN Ia 2006gz, which had been classified as being one of these "overluminous" SNe Ia. The late-time behavior is distinctly different from that of normal SNe Ia, reinforcing the argument that SN 2006gz belongs to a different subclass than normal SNe Ia. However, the peculiar features found at late times are not readily connected to a large amount of 56Ni; the SN is faint, and it lacks [Fe II] and [Fe III] emission. If the bulk of the radioactive energy escapes the SN ejecta as visual light, as is the case in normal SNe Ia, the mass of 56Ni does not exceed ~ 0.3 Msun. We discuss several possibilities to remedy the problem. With the limited observations, however, we are unable to conclusively identify which process is responsible. An interesting possibility is that the bulk of the emission might be shifted to longer wavelengths, unlike the case in other SNe Ia, which might be related to dense C-rich regions as indicated by the early-phase data. Alternatively, it might be the case that SN 2006gz, though peculiar, was actually not substantially overluminous at early times.Comment: 8 pages, 6 figures, 4 tables. Accepted for publication in The Astrophysical Journa

Optical Emission from Aspherical Supernovae and the Hypernova SN 1998bw

A fully 3D Monte Carlo scheme is applied to compute optical bolometric light curves for aspherical (jet-like) supernova explosion models. Density and abundance distributions are taken from hydrodynamic explosion models, with the energy varied as a parameter to explore the dependence. Our models show initially a very large degree ($\sim 4$ depending on model parameters) of boosting luminosity toward the polar ($z$) direction relative to the equatorial ($r$) plane, which decreases as the time of peak is approached. After the peak, the factor of the luminosity boost remains almost constant ($\sim 1.2$) until the supernova enters the nebular phase. This behavior is due mostly to the aspherical $^{56}$Ni distribution in the earlier phase and to the disk-like inner low-velocity structure in the later phase. Also the aspherical models yield an earlier peak date than the spherical models, especially if viewed from near the z-axis. Aspherical models with ejecta mass \sim 10\Msun are examined, and one with the kinetic energy of the expansion $\sim 2 \pm 0.5 \times 10^{52}$ ergs and a mass of $^{56}$Ni \sim 0.4\Msun yields a light curve in agreement with the observed light curve of SN 1998bw (the prototypical hyper-energetic supernova). The aspherical model is also at least qualitatively consistent with evolution of photospheric velocities, showing large velocities near the z-axis, and with a late-phase nebular spectrum. The viewing angle is close to the z-axis, strengthening the case for the association of SN 1998bw with the gamma ray burst GRB980425.Comment: Accepted by the Astrophysical Journal. 28 pages, 14 figure

Static and symmetric wormholes respecting energy conditions in Einstein-Gauss-Bonnet gravity

Properties of $n(\ge 5)$-dimensional static wormhole solutions are investigated in Einstein-Gauss-Bonnet gravity with or without a cosmological constant $\Lambda$. We assume that the spacetime has symmetries corresponding to the isometries of an $(n-2)$-dimensional maximally symmetric space with the sectional curvature $k=\pm 1, 0$. It is also assumed that the metric is at least $C^{2}$ and the $(n-2)$-dimensional maximally symmetric subspace is compact. Depending on the existence or absence of the general relativistic limit $\alpha \to 0$, solutions are classified into general relativistic (GR) and non-GR branches, respectively, where $\alpha$ is the Gauss-Bonnet coupling constant. We show that a wormhole throat respecting the dominant energy condition coincides with a branch surface in the GR branch, otherwise the null energy condition is violated there. In the non-GR branch, it is shown that there is no wormhole solution for $k\alpha \ge 0$. For the matter field with zero tangential pressure, it is also shown in the non-GR branch with $k\alpha<0$ and $\Lambda \le 0$ that the dominant energy condition holds at the wormhole throat if the radius of the throat satisfies some inequality. In the vacuum case, a fine-tuning of the coupling constants is shown to be necessary and the radius of a wormhole throat is fixed. Explicit wormhole solutions respecting the energy conditions in the whole spacetime are obtained in the vacuum and dust cases with $k=-1$ and $\alpha>0$.Comment: 10 pages, 2 tables; v2, typos corrected, references added; v3, interpretation of the solution for n=5 in section IV corrected; v4, a very final version to appear in Physical Review

Ginsparg-Wilson Dirac operator in the monopole backgrounds on the fuzzy 2-sphere

In the previous papers, we studied the 't Hooft-Polyakov (TP) monopole configurations in the U(2) gauge theory on the fuzzy 2-sphere,and showed that they have nonzero topological charge in the formalism based on the Ginsparg-Wilson (GW) relation. In this paper, we will show an index theorem in the TP monopole background, which is defined in the projected space, and provide a meaning of the projection operator. We also extend the index theorem to general configurations which do not satisfy the equation of motion, and show that the configuration space can be classified into the topological sectors. We further calculate the spectrum of the GW Dirac operator in the TP monopole backgrounds, and consider the index theorem in these cases.Comment: Latex2e, 37 pages, 3 figure

Discrete derivatives and symmetries of difference equations

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page