DLCQ strings and branched covers of torii

Abstract

In this lecture I will review some results about the discrete light-cone quantization (DLCQ) of strings and some connections of the results with matrix string theory. I will review arguments which show that, in the path integral representation of the thermal free energy of a string, the compactifications which are necessary to obtain discrete light-cone quantization constrains the integral over all Riemann surfaces of a given genus to the set of those Riemann surfaces which are branched covers of a particular torus. I then review an explicit check of this result at genus 1. I discuss the intriguing suggestion that these branched covers of a torus are related to those which are found in a certain limit of the matrix string model. PreambleIn this lecture I will review some aspects of the discrete light-cone quantization (DLCQ) of strings. This is a summary of work which has appeared in a series of publications about this subject, its relationship with the Matrix model of M-theory and related issues, [1]- [5].One of the essential points will be that, when the light-cone is compactified, the degrees of freedom of the string are thinned somewhat. We will see this by examining the thermodynamic partition function. The thinning of degrees of freedom can be quantified geometrically and summarized in a simple statement about string worldsheets: The set of string worldsheets in DLCQ is the set of those Riemann surfaces which are branched covers of a particular torus, rather than the bigger set of all Riemann surfaces. This means that the string fluctuates less. A further lesson is that these branched covers appear naturally in certain models of random matrices [6]- [12]. The example that we are particularly interested in here is matrix string theory where the limit of weak string * This work is supported in part by NSERC of Canada. coupling of the model is related to string degrees of freedom living on branched covers.After discussing the general result, we will illustrate by doing three different computations of the same quantity: the thermodynamic partition function of discrete light-cone quantized strings. The first computation uses the path Polyakov path integral. The second computation takes the operator quantization of the free string and solves for the spectrum and then constructs the thermodynamic partition function from that spectrum. The third computation begins with a certain limit of the matrix model of M-theory. In this limit the model reduces to a statistical theory of eigenvalues of the matrices which live on branched covers of a torus. The partition function should be compared with that of free type II strings. In all three cases, we obtain an identical result.An essential tool that we use is the thermodynamic partition function. In our case, we simply regard it as a generating function for the energy spectrum of free string theory. Of course, a certain amount of caution must always be used when discussing string theory at finite temperature. Closed string theory is a theory of quantum

Matrix Model Thermodynamics

“…That is a strong coupling limit which kills the off-diagonal degrees of freedom of the matrices. The remaining, diagonal degrees of freedom are N in number and it can be shown explicitly [23,24,25] that the free energy is negative and is proportional to N . Of course, this is just the correct behavior for a string theory, there will be a Hagedorn temperature where the large N terms in the sum in (9) go from being exponentially suppressed to growing exponentially.…”

String vertex operators and cosmic strings

“…We learn that the underlying spacetime manifold is compactified in a light-like spacetime direction, that is we are considering the DLCQ [163] of string theory. Lightlike compactifications show up in the connection of M(atrix) models to string theories: DLCQ of M-theory has been conjectured [164] to be equivalent to U(N) super Yang-Mills at finite N. (See for example, [163][164][165][166] and also [167][168][169].) Although lightlike compactifications are in general rather non-trivial [166], various properties of a vertex operator in a lightlike compactified spacetime can be extracted rather straightforwardly as we show next.…”

Black Holes and Thermodynamic States of Matrix Models

“…That is a strong coupling limit which kills the off-diagonal degrees of freedom of the matrices. The remaining, diagonal degrees of freedom are N in number and it can be shown explicitly [23][24][25] that the free energy is negative and is proportional to N . Of course, this is just the correct behavior for a string theory, there will be a Hagedorn temperature where the large N terms in the sum in (27) go from being exponentially suppressed to growing exponentially.…”