Stringy profiles of gauge field tadpoles near orbifold singularities: I. heterotic string calculations

TL;DR

This paper calculates gauge field tadpole profiles in heterotic string orbifolds, revealing Gaussian distributions influenced by string excitations, with implications for understanding localized states near orbifold singularities.

Contribution

It provides a detailed string-theoretic computation of gauge tadpole profiles, including effects of massive and tachyonic states, and clarifies the role of propagator normalization constants.

Findings

Profiles are Gaussian distributions over the orbifold, shaped by string propagators.

Massive and tachyonic states contribute to the profiles, with tachyons canceling out after integration.

Normal ordering constant bounds are necessary for the Gaussian profiles to be well-defined.

Abstract

Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profiles. These propagators are determined up to a single normal ordering constant which must be bounded below to allow the existence of the coordinate space representation of these Gaussians. Apart from the expected massless states, massive and even tachyonic string excitations contribute to the profiles in some anomalous U(1) models. However, when a tadpole is integrated over the internal dimensions, these tachyonic contributions cancel in a non-trivial manner.