Anomalies and Nonsupersymmetric D-Branes

TL;DR

This work analyzes D-branes through worldsheet anomalies, showing that boundary-condition flips to Dirichlet on bosons necessitate paired flips on fermions, which induces a mod $2$ anomaly in Type II and a mod $8$ anomaly in Type I. The authors identify three manifestations of the same Z2 anomaly in Type II (boundary Majorana modes, bulk spin-structure signs, and boundary fermions), and demonstrate how anomaly cancellation dictates the existence and properties of supersymmetric and nonsupersymmetric D-branes, including their tensions, vertex operators, and gauge content. In Type I theories, a comprehensive mod 8 analysis of time-reversal and orientifold symmetries determines the allowed boundary degrees of freedom and leads to explicit brane constructions with orthogonal or symplectic Chan-Paton factors, as well as realizations with boundary Majorana modes. Collectively, the results connect anomaly cancellation to brane stability, tachyon content, and reducibility, providing a unifying framework for understanding nonsupersymmetric D-branes and their duality relations to heterotic strings and orientifold projections.

Abstract

We revisit some aspects of D-brane theory from the point of view of anomalies. When the boundary condition on a worldsheet boson is flipped from Neumann to Dirichlet, worldsheet supersymmetry requires also reversing the sign of the boundary condition of the corresponding worldsheet fermion. This induces an anomaly which is a mod 2 anomaly in Type II superstring theory and a mod 8 anomaly in Type I superstring theory. The same anomaly also receives contributions from a sign in the sum over bulk spin structures (in Type IIA superstring theory), Chan-Paton factors of symplectic type (in Type I superstring theory), and Majorana fermions that propagate only on the worldsheet boundary. The need to cancel the anomaly accounts for many properties of supersymmetric and especially nonsupersymmetric D-branes in Type I and Type II superstring theory.

Figures (5)

• Figure 1: A Riemann surface with boundary.
• Figure 2: An annulus $S^1\times I$. If two Majorana fermions satisfy the same boundary condition on one boundary of the annulus but opposite boundary conditions on the other boundary, then precisely one of the two fermions has a zero-mode along $I$. The theory then reduces along $S^1$ to the anomalous theory of a single Majorana fermion in one spacetime dimension.
• Figure 3: Opposite sides are identified to make a Riemann surface of genus 1. Spin structures are conveniently labeled as $\pm \pm$, where the first (second) sign indicates whether a fermion is periodic or antiperiodic in the horizontal (vertical) direction.
• Figure 4: On a worldsheet boundary labeled by a "wrong parity" Dp-brane, the number of vertex operators insertions of the type $\lambda V$ (in other words, insertions of vertex operators that are GSO-odd if one does not take the boundary fermion $\lambda$ explicitly into account) must be even (odd) in the case of a boundary of NS (Ramond) type, in order to avoid vanishing of the $\lambda$ path integral. The NS case is depicted here.
• Figure 5: (a) The orientifolding symmetry is a reflection ${\sf R}$ that exchanges the two ends of an open string. (b) Introducing a vertex operator $V$ that creates the open string state in question, ${\sf R}$ leaves fixed the point $p$ at which $V$ is inserted and reverses the boundary orientation near $p$. (c) The reflection ${\sf R}$ reverses the order with which vertex operators $V_1$, $V_2$ are inserted on the boundary.