Conformal manifolds: ODEs from OPEs
HJE
Conformal manifolds: ODEs from OPEs
Connor Behan 0 1
W Symmetry, Nonperturbative Effects
0 Stony Brook , NY 11794 , U.S.A
1 C.N. Yang Institute for Theoretical Physics, Stony Brook University
The existence of an exactly marginal deformation in a conformal field theory is very special, but it is not well understood how this is reflected in the allowed dimensions and OPE coefficients of local operators. To shed light on this question, we compute perturbative corrections to several observables in an abstract CFT, starting with the beta function. This yields a sum rule that the theory must obey in order to be part of a conformal manifold. The set of constraints relating CFT data at different values of the coupling can in principle be written as a dynamical system that allows one to flow arbitrarily far. We begin the analysis of it by finding a simple form for the differential equations when the spacetime and theory space are both one-dimensional. A useful feature we can immediately observe is that our system makes it very difficult for level crossing to occur.
Conformal Field Theory; Field Theories in Lower Dimensions; Conformal and
1 Introduction 2
Two-loop constraints
3
Evolution equations
2.1
2.2
2.3
3.1
3.2
3.3
Ambiguities in the sum rule
The alternating sign
Realizations
One dimension Avoided level crossing The compact free boson
4
Conclusion
1
Introduction
beyond the conformal group SO(d + 1, 1). In particular, all known examples in d ≥ 3 are
supersymmetric. The reason for this could simply be better analytic control which makes
it easier to discover new theories, or there could be a fundamental obstruction to
nonsupersymmetric conformal manifolds. It is therefore worthwhile to check if there are some
universal features of the operator algebra that we can associate with the presence of exactly
marginal operators. Just as the modern bootstrap [
3
] seeks to determine whether a putative
set of local operators can belong to a consistent conformal theory, there may be a test that
can narrow down the space of CFTs to the space of conformal manifolds.
The original references in this subject proved non-renormalization theorems to discover
conformal manifolds [
4–8
]. To some extent, they did so by making explicit reference to a
Lagrangian. Interestingly, some of these manifolds turn out to be strongly coupled at all
points. There is also a growing body of work developing the non-perturbative
understanding of these theories through the superconformal algebra [9–12]. The short multiplets to
1Continuous families of CFTs can arise in other ways as well. One example is the procedure in [
1, 2
]
for constructing a line of nonlocal fixed points. Liouville theory may also be seen as a fixed line as one
the associated marginal operator and its coupling by Oˆ and g respectively. An infinite
family of constraints follows from setting β(g), the running of the coupling, to zero. The
two-loop term becomes a sum rule for even-spin CFT data analogous to the one in [
3
].
The local operators in a conformal manifold, even-spin or otherwise, obey many
additional restrictions that require more work to state. Although there is no known way to tell
if a set of scaling dimensions and OPE coefficients {Δi, λijk} is part of a conformal
manifold, the framework of conformal perturbation theory holds promise in telling us whether
two such sets can consistently be part of the same conformal manifold. The key is that
when there is a unique operator of each dimension, a set of differential equations exists for
evolving {Δi(g), λijk(g)} from one value of g to another. Subtleties arise when there is
degeneracy and especially when there is more than one marginal operator. In this case, there
is a non-trivial Zamolodchikov metric and the curvatures built up from it become
interesting observables that affect how the equations for ddΔgi and dλijk must be defined [14, 15].
dg
Even after we limit ourselves to a single marginal operator, these equations can only be
written down once the appropriate conformal block expansions are known. This yields
conformal block requirements that are much steeper than those in other CFT techniques.
For comparison, we note that recent studies of the analytic bootstrap use conformal blocks
with small external spin that only need to be evaluated in certain limits [16–19]. Bounds
from spinning correlators, recently found with the numerical bootstrap, use the full
expressions, but again the external spin is at most 2 [20–23]. The flow equations for conformal
manifolds couple blocks of all internal and external Lorentz representations. For this reason
they seem to be prohibitive in d ≥ 3.
Nevertheless, we will see shortly that the system can in fact be analyzed sensibly in
d = 1. The main result we have derived from this is that there is no level crossing for
operators of the same symmetry. The absence of level crossing has long been predicted
on general grounds but it remains a challenge to se (...truncated)