Connections on the state-space over conformal field theories

Abstract

Motivated by the problem of background independence of closed string field theory we study geometry on the infinite vector bundle of local fields over the space of conformal field theories (CFT's). With any connection we can associate an excluded domain D for the integral of marginal operators, and an operator one-form ω µ . The pair (D, ω µ ) determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which ω µ 's can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, D, and ω µ . Among these connections three are of particular interest. A flat, metric compatible connection Γ, and connections c andc with non-vanishing curvature, with the latter metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either c orc, however, allows us to construct a CFT in the state space of another CFT a finite distance away. The construction is given in the form of perturbation theory manifestly free of divergences.

Aspects of Berry phase in QFT

“…As we describe in the next subsection 3.3 for a linearly polarized photon this will have the effect of a rotation of the polarization plane. Slow variation of θ (and e in general) refers to the conditions required by the adiabatic theorem m| dH dt |k 12) where ∆T km is the characteristic time of transition between the states k, m.…”

“…The above results in 2d N = (2, 2) and 4d N = 2 SCFTs are examples of a more general relation between the Berry connection for states of a CFT on R × S d−1 and the connection on the space of operators that is naturally defined in conformal perturbation theory [11,12]. In section 8 we present a general formal argument based on the operatorstate correspondence that exhibits the equivalence of the two connections in any CFT with a non-trivial conformal manifold and for any set of states/operators.…”

“…In general, different regularization schemes lead to different notions of connection on the conformal manifold. In [12] several possibilities were discussed in detail. One of them requires the introduction of small cutoff balls around the operator insertions so that two operators can never collide in the regulated expression.…”

“…Once the notion of a connection is available the comparison of two operators at nearby points of the conformal manifold becomes feasible and standard geometric notions, like that of a covariant derivative ∇ µ and parallel transport, immediately apply. A natural definition of such a connection in conformal perturbation theory, which follows directly from the dynamics of the CFT, has been discussed in many works in the past (see for example [11] for an early discussion, [12] for an extensive discussion in twodimensional CFTs, as well as [33]). The curvature of this connection (denoted A µ )…”

“…= ± are the two helicities of the photon modes and e ( k) the polarization vectors. 12 The creation and annihilation modes a, a † obey canonical commutation relations.…”