By Recknagel A., Schomerus V.

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**Example text**

We assume that V comes equipped with a constant antisymmetric d × d matrix Θij , which deﬁnes a Poisson bracket for functions on V . More concretely, when evaluated on the coordinate functions xj , j = 1, . . , d, the Poisson structure reads, { xj , xl } = Θjl . 50) Quantisation means to associate a self-adjoint operator x ˆj : H → H on some state space H to each coordinate function such that the commutators are [x ˆj , x ˆl ] = i Θjl . 51) More generally, one would like to associate a self-adjoint operator Q(f ) to any complex-valued function f on V such the commutator [Q(f1 ), Q(f2 )] is approximated by the Poisson bracket {f1 , f2 } in a sense that we shall make more precise below.

The power of the leading singularity is determined by the conformal weights of our boundary ﬁelds. One may recall that the phase factor involving the crossed product k × k also appears when a linear space with constant Poisson bracket is quantised, and we shall brieﬂy review the relation before concluding this section. The Weyl product Let us digress for a moment and recall a few facts about the quantisation of a d-dimensional linear space V . We assume that V comes equipped with a constant antisymmetric d × d matrix Θij , which deﬁnes a Poisson bracket for functions on V .

55). The fermionic modes ψnμ and ψrμ obey anticommutation relations of the form ν } = g μν δm+n,0 (R) , {ψnμ , ψm {ψrμ , ψsν } = g μν δr+s,0 (NS) . Note that, in contrast to the boundary theory, the left- and right-moving fermion z ) have independent (anti-commuting) modes if the theory is ﬁelds ψ(z) and ψ(¯ deﬁned on the full complex plane; those bulk ﬁeld modes will appear when we work with boundary states below. Let us now construct the state space of the boundary theory. This is particularly simple for the Neveu–Schwarz sector, which has a unique ground state |0 such that ψrμ |0 = 0 for r > 0 .