IIA/B, Wound and Wrapped

Abstract

We examine the T-duality relation between 1+1 NCOS and the DLCQ limit of type IIA string theory. We show that, as long as there is a compact dimension, one can meaningfully define an `NCOS' limit of IIB/A string theory even in the absence of D-branes (and even if there is no B-field). This yields a theory of closed strings with strictly positive winding, which is T-dual to DLCQ IIA/B without any D-branes. We call this the Type IIB/A Wound String Theory. The existence of decoupled sectors can be seen directly from the energy spectrum, and mirrors that of the DLCQ theory. It becomes clear then that all of the different p+1 NCOS theories are simply different states of this single Wound IIA/B theory which contain D-branes. We study some of the properties of this theory. In particular, we show that upon toroidal compactification, Wound string theory is U-dual to various Wrapped Brane theories which contain OM theory and the ODp theories as special states.

Figures (4)

• Figure 1: The $1+1$ NCOS/DLCQ IIA duality web. Seiberg's derivation of a non-perturbative Matrix formulation for DLCQ IIA (DLCQ M-theory on a transverse ${\mathbf{S}}^{1}$) proceeds along the top (bottom) line and then down to arrive at $1+1$ SYM. $\beta_{1}$ denotes a boost along $x^{1}$.
• Figure 2: A closed string vertex encircling another, and the analogous operation for open string vertices, which inverts their order. In the latter case there is an extra phase due to the fact that $\langle X_\mu(z) \tilde{X}_\nu(\bar{w}) \rangle\neq 0$ .
• Figure 3: A portion of the duality web for Type II Wound/DLCQ theories on a transverse $\mathbf{T}^{p-1}$ for $p=1,3,5$, including the images of $K$ Wound D$p$-branes and $N$ units of F-string winding in the various descriptions. As explained in Section \ref{['ncossec']}, $N$ must be strictly positive, but $K$ is arbitrary. The Wound IIB setup in the upper-left corner of the diagram has hitherto been known as $(p+1)$-dimensional NCOS theory. Non-vanishing-size compactifications are not mentioned. $\beta_{1}$ denotes a boost along $x^{1}$ together with a change of units. $P_{i}$ stands for Kaluza-Klein units of momentum along $x^{i}$. The notation Xq:$i_{1}\cdot\cdot i_{q-1}$ indicates an X$p$-brane wrapping the $i_{1},\ldots,i_{q-1}$ cycles of the torus. Triple $K$-entries apply respectively to the cases $p=1,3,5$. Seiberg's derivation of a non-perturbative (Matrix) formulation for DLCQ IIA on a transverse $\mathbf{T}^{p-1}$ (DLCQ M on a transverse $\mathbf{T}^{p}$) follows the horizontal arrows in the top (middle) line of the diagram, and then proceeds down and diagonally to arrive at the $\widehat{\hbox{IIB}}$ theory. See text for further discussion.
• Figure 4: Same as Fig. 2, for $p=2,4$. The two figures are related through transverse T-duality. Again, compact directions are not mentioned unless they have vanishing size in the relevant metric. Double entries for the object with multiplicity $K$ apply respectively to the $p=2,4$ cases. The dotted arrow connecting M-theory to $\widehat{\hbox{IIA}}$ applies only in the $p=2$ case. The Wound IIA setup in the upper-left corner has hitherto been known as $p+1$ NCOS theory. Seiberg's derivation of a non-perturbative (Matrix) description of DLCQ M-theory on $\mathbf{T}^{p}$ proceeds along the bottom line of the figure.