Open String on Symmetric Product

TL;DR

This paper studies the properties of open strings on symmetric products within the framework of discrete lightcone quantization, classifying boundary states, analyzing amplitudes, and deriving gauge groups.

Contribution

It classifies boundary states and solutions for open strings on symmetric products, connecting short and long string topologies, and derives the gauge group in DLCQ.

Findings

Classified boundary and cross-cap states for short strings.

Calculated orbifold amplitudes from boundary state and Hilbert space perspectives.

Derived the gauge group SO(2^{13}) from tadpole cancellation.

Abstract

We develop some basic properties of the open string on the symmetric product which is supposed to describe the open string field theory in discrete lightcone quantization (DLCQ). After preparing the consistency conditions of the twisted boundary conditions for Annulus/M\"obius/Klein Bottle amplitudes in generic non-abelian orbifold, we classify the most general solutions of the constraints when the discrete group is $S_N$. We calculate the corresponding orbifold amplitudes from two viewpoints -- from the boundary state formalism and from the trace over the open string Hilbert space. It is shown that the topology of the world sheet for the short string and that of the long string in general do not coincide. For example the annulus sector for the short string contains all the sectors (torus, annulus, Klein bottle, M\"obius strip) of the long strings. The boundary/cross-cap states of the short strings are classified into three categories in terms of the long string, the ordinary boundary and the cross-cap states, and the ``joint'' state which describes the connection of two short strings. We show that the sum of the all possible boundary conditions is equal to the exponential of the sum of the irreducible amplitude -- one body amplitude of long open (closed) strings. This is typical structure of DLCQ partition function. We examined that the tadpole cancellation condition in our language and derived the well-known gauge group $SO(2^{13})$.