Local Fields in Boundary Conformal QFT

Abstract

Dedicated to Detlev Buchholz on the occasion of his 60th birthdayConformal quantum field theory on the half-space x > 0 of Minkowski space-time ("boundary CFT") is analyzed from an algebraic point of view, clarifying in particular the algebraic structure of local algebras and the bi-localized charge structure of local fields. The field content and the admissible boundary conditions are characterized in terms of a non-local chiral field algebra.It follows that the components T 10 = T 01 , T 11 = T 00 of the stress-energy tensor are of the formi.e. bi-local expressions in terms of the chiral field T (cf. Fig. 1). 909 Rev. Math. Phys. 2004.16:909-960. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. Rev. Math. Phys. 2004.16:909-960. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. Local Fields in Boundary Conformal QFT 911has additional contributions at t 1 +x 1 = t 2 −x 2 and at t 1 −x 1 = t 2 +x 2 . But within a wedge region M + ⊃ W : x > |t| (⇔ t − x < 0 < t + x), the latter contributions are ineffective. The same holds for any time translate of W . A slightly stronger version of this algebraic indistinguishability is the following. It has been shown that the chiral stress-energy tensor satisfies the split property: namely for every pair of intervals J < I which do not touch (thus allowing to smooth out the UV singularities), there exists a state ϕ in the vacuum Hilbert space H 0 of T (depending on I and J; in particular not the vacuum state) which has no correlation between T (u 1 ) and T (u 2 ) when u 1 ∈ I and u 2 ∈ J. In other words, ϕ factorizes on products of T (u i ) with u i ∈ I ∪ J according to( 1.6) This implies that, for every double-cone O not touching the boundary (hence t − x and t + x belong to non-touching intervals as before), there is a state ϕ such that products of T µν (t, x) given by (1.2) in the boundary CFT with (t, x) ∈ O have the same expectation values in the state ϕ as the same products of T µν (t, x) given by (1.5) in the 2D Minkowski space CFT have in the state ϕ ⊗ ϕ. This property exhibits the local "decoupling" of left and right chiral components. Exactly as the split property fails when the intervals I and J touch, the decoupling of left-and right-movers breaks down at the boundary in BCFT. We shall assume the split property for all chiral fields of a boundary CFT. This property is known to be related to phase space properties of the CFT (existence of Tr exp(−βL 0 )) [9, 1], and it has been established for large classes of chiral models ([45] and references therein).Our aim in the present article is to understand the structure of local fields in boundary CFT which do not decompose in the manner of (1.2) or (1.4). These nonchiral fields have to satisfy local commutativity with the chiral fields and with each other, and transform covariantly under the conformal (Möbius) group generated by the chiral stress-energy tensor T . The crucial observation will be that non-chiral...

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

“…is irreducible with finite index [18], and hence has a unique conditional expectation µ : The structure of B + (O 0 ) has been described in [18]. By complete rationality, A has finitely many irreducible superselection sectors [14].…”

“…We compute the limit and the 2D vacuum expectation value separately. Using the decomposition of products ψ 1 ψ 2 into finite sums of operators of the form ι(T 1 T 2 ) · ψ [18], where T i are intertwiners between DHR endomorphisms of A, we see that the product b 1 · · · b n is a finite sum of operators of the same form ι(a 1 a 2 ) · ψ.…”

“…In [18], the authors have formulated boundary conformal field theory (BCFT) in real time (Lorentzian signature) in the algebraic framework of quantum field theory. BCFT is a local Möbius covariant QFT B + on the two-dimensional Minkowski halfspace M + (given by x > 0), which contains a (given) local chiral subtheory A, e.g., the stressenergy tensor.…”

“…1). Although the chiral net B is not necessarily local, The main result in [18] is that every BCFT is contained in a maximal (Haag dual) BCFT of the form (1.2).…”

“…One might argue that such a "reduction of degrees of freedom" is a characteristic feature of QFT with a boundary. But this point of view cannot be maintained, because on the other hand, it was shown in [18] that the resulting BCFT B + is locally equivalent to another CFT B 2D on the full two-dimensional (2D) Minkowski spacetime, which has all the degrees of freedom of a 2D QFT, and in particular contains a full 2D stress-energy tensor (two commuting copies of the Virasoro algebra). Even in the simplest case, when the chiral net B on the boundary coincides with A (sometimes known as "the Cardy case"), the associated bulk QFT contains apart from the full 2D stress-energy tensor more ("nonchiral") local fields that factorize into chiral fields with braid group statistics.…”