Gauge Invariance and Equations of Motion for Closed String Modes

We continue earlier discussions on the exact renormalization group and loop variables on the string world sheet for closed and open string backgrounds. The world sheet action with a UV regulator is written in a generally background covariant way by introducing a background metric. It is shown that the renormalization group gives background covariant equations of motion - this is the gauge invariance of the graviton. Interaction is written in terms of gauge invariant and generally covariant field strength tensors. The basic idea is to work in Riemann normal coordinates and covariantize the final equation. It turns out that the equations for massive modes are gauge invariant only if the space time curvature of the (arbitrary) background is zero. The exact RG equations give quadratic equations of motion for all the modes {\em including} the physical graviton. The level $(2,\bar 2)$ massive field equations are used to illustrate the techniques. At this level there are mixed symmetry tensors. Gauge invariant interacting equations can be written down. In flat space an action can also be written for the free theory.Comment: 41 page

Scattering Equations, Twistor-string Formulas and Double-soft Limits in Four Dimensions

We study scattering equations and formulas for tree amplitudes of various theories in four dimensions, in terms of spinor helicity variables and on-shell superspace for supersymmetric theories. As originally obtained in Witten's twistor string theory and other twistor-string models, the equations can take either polynomial or rational forms, and we clarify the simple relation between them. We present new, four-dimensional formulas for all tree amplitudes in the non-linear sigma model, a special Galileon theory and the maximally supersymmetric completion of the Dirac-Born-Infeld theory. Furthermore, we apply the formulas to study various double-soft theorems in these theories, including the emissions of a pair of soft photons, fermions and scalars for super-amplitudes in super-DBI theory.Comment: 22 pages, 2 tables; v2: ref added, minor typos fixe

Massless spinning particle and null-string on AdSdAdS_d: projective-space approach

Massless spinning particle and tensionless string models on $AdS_d$ background in the projective-space realization are proposed as constrained Hamiltonian systems. Various forms of particle and string Lagrangians are derived and classical mechanics is studied including the Lax-type representation of the equations of motion. After that transition to the quantum theory is discussed. Analysis of potential anomalies in the tensionless string model necessitates introduction of ghosts and BRST charge. It is shown that quantum BRST charge is nilpotent for any $d$ if coordinate-momentum ordering for the phase-space bosonic variables, Weyl ordering for the fermions and $cb$ ($\gamma\beta$) ordering for ghosts is chosen, while conformal reparametrizations and space-time dilatations turn out to be anomalous for the ordering in terms of positive and negative Fourier modes of the phase-space variables and ghosts.Comment: 23 pages, v4: In the title and abstract added adjective 'Massless'. In subsection 2.2 the material has been rearranged to streamline the presentation and also a comment added to explain the notation. Minor improvements of the text and further typos corrected. Version accepted for publicatio

Comparison of the oscillatory behaviors of a gravitating Nambu-Goto string with a test string

Comparison of the oscillatory behavior of a gravitating infinite Nambu-Goto string and a test string is investigated using the general relativistic gauge invariant perturbation technique with two infinitesimal parameters on a flat spacetime background. Due to the existence of the pp-wave exact solution, we see that the conclusion that the dynamical degree of freedom of an infinite Nambu-Goto string is completely determined by that of gravitational waves, which was reached in our previous works [K. Nakamura, A. Ishibashi and H. Ishihara, Phys. Rev. D{\bf 62} (2002), 101502(R); K. Nakamura and H. Ishihara, Phys. Rev. D{\bf 63} (2001), 127501.], do not contradict to the dynamics of a test string. We also briefly discuss the implication of this result.Comment: 32 pages, 1 figure, PTPTeX ver.0.8 (LateX2e), Accepted for publication to Progress of Theoretical Physic

Non-supersymmetric Black Holes and Topological Strings

We study non-supersymmetric, extremal 4 dimensional black holes which arise upon compactification of type II superstrings on Calabi-Yau threefolds. We propose a generalization of the OSV conjecture for higher derivative corrections to the non-supersymmetric black hole entropy, in terms of the one parameter refinement of topological string introduced by Nekrasov. We also study the attractor mechanism for non-supersymmetric black holes and show how the inverse problem of fixing charges in terms of the attractor value of CY moduli can be explicitly solved.Comment: 47 pages, harvmac. v2: footnote(4) expanded, references adde

Polynomial Time Algorithms for Multi-Type Branching Processes and Stochastic Context-Free Grammars

We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in time polynomial in both the encoding size of the system of equations and in log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We use this result to resolve several open problems regarding the computational complexity of computing key quantities associated with some classic and heavily studied stochastic processes, including multi-type branching processes and stochastic context-free grammars

Toda Lattice Hierarchy and Generalized String Equations

String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33), (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references are update