Anomalous reparametrizations and butterfly states in string field theory

Abstract

The reparametrization symmetries of Witten's vertex in ordinary or vacuum string field theories can be used to extract useful information about classical solutions of the equations of motion corresponding to D-branes. It follows, that the vacuum string field theory in general has to be regularized. For the regularization recently considered by Gaiotto et al., we show that the identities we derive, are so constraining, that among all surface states they uniquely select the simplest butterfly projector discovered numerically by those authors. The reparametrization symmetries are also used to give a simple proof that the butterfly states and their generalizations are indeed projectors.

Figures (4)

• Figure 1: Representation of the butterfly state $\langle 0 | e^{-\frac{1}{2} L_{2}}$ in two coordinate systems. The local coordinate patch around the puncture $P$ is indicated by the darker shade. In the $\hat{w}$ coordinate system the local patch is represented always as the right half disk. The rest of the surface in this case is a half disk cut into two parts along the line $[-1,0]$.
• Figure 2: Representation of the butterfly state $\langle 0 | e^{\frac{1}{4} L_{4}}$ in two coordinate systems. In the $\hat{w}$ coordinate system the local patch is represented by the right half disk, the rest of the surface on the right is cut along the line $[-1,0]$. Moreover the upper left $90^\circ$ arc is identified with the upper part of the cut and the lower left arc with the lower part of the cut.
• Figure 3: Representation of the 'wrong sign' butterfly state $\langle 0 | e^{\frac{1}{2} L_{2}}$. In the $\hat{w}$ coordinate system the upper and lower halves of the left semicircle, indicated by thick line, are identified. Note that the surface is completely regular at the midpoint.
• Figure 4: Representation of the 'wrong sign' butterfly state $\langle 0 | e^{-\frac{1}{4} L_{4}}$. In the $\hat{w}$ coordinate system the surface without the local patch is split into three independent parts, the middle part has its boundaries identified. Again the surface is perfectly regular at the midpoint.