Moving NS Punctures on Super Spheres

One of the subtleties that has made superstring perturbation theory intricate at high string loop order is the fact that, as shown by Donagi and Witten, supermoduli space is not holomorphically projected, nor is it holomorphically split. In recent years, Sen introduced the notion of vertical integration in moduli space (further refined by Sen and Witten). This enables one to use the traditional (only locally-defined) gauge fixing for the worldsheet gravitino in local patches, allowing one to formulate the theory on the moduli space of ordinary Riemann surfaces, and then prescribes certain correction terms to account for the incorrect gauge fixing to restore BRST invariance. This approach makes use of the fact that there is no obstruction to a smooth splitting of supermoduli space. It may, however, not necessarily be the most convenient or natural solution to the problem. There may be situations where one would like to have a well-defined path integral at arbitrary string loop order from the outset. In this paper I initiate an alternative approach that implements the fact that a smooth gauge slice for supermoduli space always exists. As a warmup, I focus specifically on super Riemann surfaces with the topology of a 2-sphere in heterotic string theory, incorporating the corresponding super curvature locally, and introduce a new well-defined smooth gauge fixing that leads to a globally defined path integral measure that translates fixed ($-1$) picture vertex operators (or handle operators) (that may or may not be offshell) to integrated (0) picture. I also provide some comments on the extension to arbitrary super Riemann surfaces.Comment: 39 pages, no figure

Vertex operators for cosmic strings

Superstring theory posits that as complicated as nature may seem to the naive observer, the variety of observed phenomena may be explained by postulating that at the fundamental scale, matter is composed of lines of energy, namely strings. These oscillating lines would be elementary and would hence have no substructure. They are expected to be incredibly tiny, their line-like structure would become noticeable at scales close to the string scale (which may lie anywhere from the TeV scale all the way up to the Planck scale) and would appear to be point-like to the macroscopic observer. Internal consistency then also requires the presence of higher dimensional objects, namely D-branes, all of which conspire and combine in such a way so as to give rise to the observable Universe. Advances in cosmology suggest the early universe was much hotter and denser than is the Universe at present, that the Universe has expanded and continues to expand (exponentially in fact) at present. This in turn has led a number of theorists to point out the remarkable possibility that some of these strings or D-branes were also stretched with the expansion. The resulting macroscopic strings, the so-called cosmic strings, would potentially stretch across the entire Universe. Cosmic strings make their presence manifest by oscillating, scattering off other structures, by decaying, producing gravitational waves and so on, and this in turn hints at the available handles that may be used to observe them. Before we can hope to observe cosmic strings however, the first step is then clearly to understand these properties which determine their evolution. A number of approximate (classical) descriptions of cosmic strings have been constructed to date, but approximations break down, especially when potentially interesting things happen (e.g. close to cusps, i.e. points on the string that reach the speed of light) and can obscure the physics. Thankfully, one can go beyond these approximations: all properties of cosmic strings can be concisely and accurately contained or encoded in a single object, the so-called fundamental cosmic string vertex operator. In the present thesis I construct precisely this, covariant vertex operators for general cosmic strings and this is the first such construction. Cosmic strings, being macroscopic, are likely to exhibit classical behaviour in which case they would most accurately be described by a string theory analogue of the well known harmonic oscillator coherent states. By minimally extending the standard definition of coherent states, so as to include the string theory requirements, I go on to construct both open and closed covariant coherent state vertex operators. The naive construction of the latter requires the existence of a lightlike compactification of spacetime. When the lightlike winding states in the underlying Hilbert space are projected out, the resulting vertex operators have a classical interpretation and can consistently propagate in noncompact spacetime. Using the DDF map I identify explicitly the corresponding general lightcone gauge classical solutions around which the exact macroscopic quantum states are fluctuating. We go on to show that both the covariant gauge coherent vertex operators, the corresponding lightcone gauge coherent states and the classical solutions all share the same mass and angular momenta, which leads us to conjecture that the covariant and lightcone gauge states are different manifestations of the same state and share identical interactions. Apart from the coherent state vertices I also present a complete set of covariant mass eigenstate vertex operators and these may also be relevant in cosmic string evolution. Finally, I also present the first amplitude computation with the coherent states, the graviton emission amplitude (including the effects of gravitational backreaction) for a simple class of cosmic string loops. As a byproduct of the above, I find that the fundamental building blocks of arbitrarily massive covariant string states are given by elementary Schur polynomials (equivalently complete Bell polynomials). This construction enables one to address the aforementioned questions concerning the properties of cosmic strings, their cosmological signatures, and may lead to the first observations of such objects in the sky. This in turn would be a remarkable way of verifying Superstring theory as the framework underlying the structure of our Universe

Handle Operators in String Theory

We derive how to incorporate topological features of Riemann surfaces in string amplitudes by insertions of bi-local operators called handle operators. The resulting formalism is exact and globally well-defined in moduli space. After a detailed and pedagogical discussion of Riemann surfaces, complex structure deformations, global vs local aspects, boundary terms, an explicit choice of gluing-compatible and global (modulo U(1)) coordinates (termed `holomorphic normal coordinates'), finite changes in normal ordering, and factorisation of the path integral measure, we construct these handle operators explicitly. Adopting an offshell local coherent vertex operator basis for the latter, and gauge fixing invariance under Weyl transformations using holomorphic normal coordinates (developed by Polchinski), is particularly efficient. All loop amplitudes are gauge-invariant (BRST-exact terms decouple up to boundary terms in moduli space), and reparametrisation invariance is manifest, for arbitrary worldsheet curvature and topology (subject to the Euler number constraint). We provide a number of complementary viewpoints and consistency checks (including one-loop modular invariance, we compute all one- and two-point sphere amplitudes, glue two three-point sphere amplitudes to reproduce the exact four-point sphere amplitude, etc.).Comment: 324 pages, 34 figure

Complete normal ordering 1:Foundations

We introduce a new prescription for quantising scalar field theories perturbatively around a true minimum of the full quantum effective action, which is to `complete normal order' the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all `cephalopod' Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of `complete normal ordering' (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting `trick' we extend this result to theories with derivative interactions, such as those appearing as non-linear sigma-models in the world-sheet formulation of string theory. We focus here on theories with trivial vacua, generalising the discussion to non-trivial vacua in a follow-up paper.Comment: 40 pages; a significantly condensed version compared to v1; matches published article. Refer to v1 for a pedagogical expositio

Duality and decay of macroscopic F-strings

We study the decay of fundamental string loops of arbitrary size L/min(n,m)≫√α′, labeled by (n, m; λn, ¯λm), where n, m correspond to left- and right-mover harmonics and λn, ¯λm to polarization tensors, and find that a description in terms of the recent coherent vertex operator construction of Hindmarsh and Skliros is computationally very efficient. We primarily show that the decay rates and mass shifts of vertex operators (n, m; λn, ¯λm) and their “duals” (n, m; λn, ¯λ∗m) are equal to leading order in the string coupling, implying, for instance, that decay rates of epicycloids equal those of hypocycloids. We then compute the power and decay rates associated with massless IR radiation for the trajectory (1, 1; λ1, ¯λ1), and find that it is precisely reproduced by the low energy effective theory of Dabholkar and Harvey. Guided by this correspondence, we conjecture the result for arbitrary trajectories (n, m; λn, ¯λm) and discover a curious relation between gravitational and axion plus dilaton radiation. It is now possible to start exploring string evolution in regimes where a low energy effective description is less useful, such as in the vicinity of cusps

Large Radius Hagedorn Regime in String Gas Cosmology

We calculate the equation of state of a gas of strings at high density in a large toroidal universe, and use it to determine the cosmological evolution of background metric and dilaton fields in the entire large radius Hagedorn regime, (ln S)^{1/d} << R << S^{1/d} (with S the total entropy). The pressure in this regime is not vanishing but of O(1), while the equation of state is proportional to volume, which makes our solutions significantly different from previously published approximate solutions. For example, we are able to calculate the duration of the high-density "Hagedorn" phase, which increases exponentially with increasing entropy, S. We go on to discuss the difficulties of the scenario, quantifying the problems of establishing thermal equilibrium and producing a large but not too weakly-coupled universe.Comment: 12 pages, 4 figures, more details presented in string thermodynamics section, to be published in Physical Review

Covariant Closed String Coherent States

We give the first construction of covariant coherent closed string states, which may be identified with fundamental cosmic strings. We outline the requirements for a string state to describe a cosmic string, and using DDF operators provide an explicit and simple map that relates three different descriptions: classical strings, lightcone gauge quantum states and covariant vertex operators. The naive construction leads to covariant vertex operators whose existence requires a lightlike compactification of spacetime. When the lightlike compactified states in the underlying Hilbert space are projected out the resulting coherent states have a classical interpretation and are in one-to-one correspondence with arbitrary classical closed string loops.Comment: 4 page