The Quantum Structure of Space and Time (293 pages)

154 The Quantum Structure of Space and Time

n n n n n A n

Ql Q2 Q3 Q4 Q5 a6 Q7 a8 Q9

Fig. 4.1 Dynkjn diagram of elo.

However, it is known that, generically, they grow exponentially as a2 4 -00, like

the number of massive string states. In order to get a handle at least on low-lying

generators, one analyzes el0 w.r.t. certain finite-dimensional regular subalgebras

by means of a level decomposition: pick one special node ao, and write a given

root as Q = Cjrnjaj +~QO with aj E subalgebra, and where t is the ‘level’. For
instance, decomposing w.r.t. the subalgebra Ag = 5110 (i.e. QO = ale) we obtain
the following table for t 5 3:

A9 module Tensor

[000000 1001

l0100000011 EUl ... us la9

[OOO 100000] Eal ... a6

These tensors correspond to the bosonic fields of D = 11 supergravity and their

‘magnetic’ duals. Similar low level decompositions of el0 w.r.t. its other distin-

guished rank-9 subalgebras Dg -- sa(9,9) and As @ A1 = 5Ig @ 512 yield the correct

bosonic multiplets, again with ‘magnetic’ duals, of (massive) type IIA and type

IIB supergravity, respectively. Furthermore, for the Dg decomposition, one finds

that the (Neveu-Schwarz)2 states (at even levels) and the (Ramond)2 states (at

odd levels), respectively, belong to tensorial and spinorial representations of the T-

duality group SO(9,9), and that the truncation to even levels contains the rank-10

hyperbolic Kac-Moody algebra DElo, corresponding to type-I supergravity, as a

subalgebra.

Dynamics (cf. [2, 3, 51): The equations of motion are derived from the following

(essentially unique) ‘geodesic’ a-model over Elo/K(Elo):

where P := PV-’ -w(PV-~) is the ‘velocity’, (.I.) is the standard invariant bilinear

form, and n(t) a one-dimensional ‘lapse’ needed to ensure (time) reparametrisation

invariance. When truncated to levels 5 3, the corresponding equations of motion

coincide with the appropriately truncated bosonic supergravity equations of motion,

where only first order spatial gradients are retained [2]. Analogous results hold for