BMS Particles in Three Dimensions by Blagoje Oblak

Author:Blagoje Oblak
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Up to transformations, the solution (7.73)-(7.74) is the unique solution of Hill’s equation with non-degenerate parabolic monodromy

(7.78)

and winding number n.

7.2.7 Summary: A Map of Virasoro Orbits

The above analysis exhausts all coadjoint orbits of the Virasoro group. Since these orbits will play a key role in the remainder of this thesis, we now briefly summarize the salient features of the classification.

The schematic drawings of Figs. 7.1 and 7.2 represent Virasoro orbits. The only orbits which are not accounted for by these pictures are massless ones; in order to include them we use the same trick as in Fig. 4.​3b, where massless orbits are represented by two dots near the origin (one with positive energy, the other with negative energy). We will use the same notation here, except that such a pair of massless orbits occurs for all positive integers . With this convention, Fig. 7.2 turns into the complete map of Virasoro coadjoint orbits displayed in Fig. 7.3.

Fig. 7.3The map of Virasoro coadjoint orbits at positive central charge. Note the similarity with Fig. 4.​3b. Roughly speaking, the map consists of an infinity of copies of Poincaré momentum orbits glued together and labelled by the winding number n. Locally (near a node n), the two pictures look identical. This is not surprising given that Poincaré momentum orbits in three dimensions coincide with coadjoint orbits, which in turn are classified similarly to the conjugacy classes of that were instrumental for Virasoro coadjoint orbits. This hints that there exists a relation between Virasoro and Poincaré symmetry; we shall see in part III that this relation is embodied by the BMS group

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