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Orientifolds and twisted boundary conditions

Arjan Keurentjes

Abstract

It is argued that the T-dual of a crosscap is a combination of an O+ and an O- orientifold plane. Various theories with crosscaps and D-branes are interpreted as gauge-theories on tori obeying twisted boundary conditions. Their duals live on orientifolds where the various orientifold planes are of different types. We derive how to read off the holonomies from the positions of D-branes in the orientifold background. As an application we reconstruct some results from a paper by Borel, Friedman and Morgan for gauge theories with classical groups, compactified on a 2-- or 3--torus with twisted boundary conditions.

Orientifolds and twisted boundary conditions

Abstract

It is argued that the T-dual of a crosscap is a combination of an O+ and an O- orientifold plane. Various theories with crosscaps and D-branes are interpreted as gauge-theories on tori obeying twisted boundary conditions. Their duals live on orientifolds where the various orientifold planes are of different types. We derive how to read off the holonomies from the positions of D-branes in the orientifold background. As an application we reconstruct some results from a paper by Borel, Friedman and Morgan for gauge theories with classical groups, compactified on a 2-- or 3--torus with twisted boundary conditions.

Paper Structure

This paper contains 16 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The T-dual of a cross-cap
  3. Twisted boundary conditions on the 2--torus
  4. Twist in unitary groups
  5. Twist in symplectic groups on the 2--torus
  6. Twist in orthogonal groups on the 2--torus
  7. No spin structure
  8. No vector structure
  9. Compactifications on a 3--torus
  10. Commuting triples for orthogonal and symplectic groups
  11. $c$-triples for symplectic groups
  12. $c$-triples for orthogonal groups
  13. No spin structure
  14. No vector structure
  15. Spin nor vector structure
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The Klein-bottle: a double cover of the Klein-bottle, arrows indicating the direction of identifications (middle); an attempt to draw the standard representation of the Klein-bottle, obtained by taking the lower half of the double cover as fundamental domain (right); the cylinder with two cross-caps, obtained by taking the left half of the fundamental domain (left). We also drawn an example of a brane in all three pictures (depicted twice on the double cover) as it is positioned after the first T-duality
  • Figure 2: The Möbius strip: a double cover of the Möbius strip, arrows indicating the direction of identifications, fat lines the edges (middle); the standard representation of the Möbius strip, obtained by taking the lower half of the double cover as fundamental domain (right); the cylinder with one cross-cap, obtained by taking the left half as fundamental domain (left)