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Consistency Conditions for Orientifolds and D-Manifolds

Eric G. Gimon, Joseph Polchinski

TL;DR

The paper establishes two key consistency conditions for orientifolds and D-manifolds: algebraic closure of the Chan-Paton action under the orientifold group and cancellation of one-loop divergences. It shows that D5-branes in Type I must carry a symplectic projection, and derives a rich set of Type I on K3 orbifold theories with unitary and symplectic gauge factors via a detailed tadpole analysis. A unique consistent Chan-Paton action is found by solving the algebra and divergence constraints, yielding a 6D N=1 spectrum with anomaly cancellation for the resulting gauge groups such as $U(16) imesigl( ext{sum of }U(m_I)igr) imes USp(n'_J)$; the work also notes the non-realization of a spin-connection-embedded orbifold as a free CFT. The discussion connects these results to string dualities and the broader landscape of orientifold backgrounds, highlighting the role of 59 sectors and potential 59-background interpretations.

Abstract

We study superstrings with orientifold projections and with generalized open string boundary conditions (D-branes). We find two types of consistency condition, one related to the algebra of Chan-Paton factors and the other to cancellation of divergences. One consequence is that the Dirichlet 5-branes of the Type I theory carry a symplectic gauge group, as required by string duality. As another application we study the Type I theory on a $K3$ $Z_2$ orbifold, finding a family of consistent theories with various unitary and symplectic subgroups of $U(16) \times U(16)$. We argue that the $K3$ orbifold with spin connection embedded in gauge connection corresponds to an interacting conformal field theory in the Type I theory.

Consistency Conditions for Orientifolds and D-Manifolds

TL;DR

The paper establishes two key consistency conditions for orientifolds and D-manifolds: algebraic closure of the Chan-Paton action under the orientifold group and cancellation of one-loop divergences. It shows that D5-branes in Type I must carry a symplectic projection, and derives a rich set of Type I on K3 orbifold theories with unitary and symplectic gauge factors via a detailed tadpole analysis. A unique consistent Chan-Paton action is found by solving the algebra and divergence constraints, yielding a 6D N=1 spectrum with anomaly cancellation for the resulting gauge groups such as ; the work also notes the non-realization of a spin-connection-embedded orbifold as a free CFT. The discussion connects these results to string dualities and the broader landscape of orientifold backgrounds, highlighting the role of 59 sectors and potential 59-background interpretations.

Abstract

We study superstrings with orientifold projections and with generalized open string boundary conditions (D-branes). We find two types of consistency condition, one related to the algebra of Chan-Paton factors and the other to cancellation of divergences. One consequence is that the Dirichlet 5-branes of the Type I theory carry a symplectic gauge group, as required by string duality. As another application we study the Type I theory on a orbifold, finding a family of consistent theories with various unitary and symplectic subgroups of . We argue that the orbifold with spin connection embedded in gauge connection corresponds to an interacting conformal field theory in the Type I theory.

Paper Structure

This paper contains 8 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Orientifolds and Chan-Paton Factors
  3. Tadpoles
  4. General Framework
  5. Type I Theory on a $K3$ Orbifold
  6. Tadpole Calculation
  7. Solutions
  8. Discussion

Figures (2)

  • Figure 1: Riemann surfaces described by eq. (\ref{['cond1']}). a) Klein bottle. b) Möbius strip. c) Cylinder.
  • Figure 2: Tadpoles in the $g$-twisted sector. a) Crosscap: fields at opposite points differ by an $\Omega h$ transformation, where $g = (\Omega h)^2$. b) Boundary in state $i$. The manifold $M_i$ must be fixed under $g$.