Conformal surface defects in Maxwell theory are trivial

Abstract

We consider a free Maxwell field in four dimensions in the presence of a codimension two defect. Reflection positive, codimension two defects which preserve conformal symmetry in this context are very limited. We show only generalized free fields can appear in the defect operator product expansion of the bulk Maxwell field; in particular correlation functions of these defect operators can be evaluated via Wick’s Theorem.

Conformal boundary conditions for a 4d scalar field

“…Intuitively, this has to be the case since bosonic self-interactions are repulsive and tend to destabilize the vacuum. Note that the bound (11) does not depend on g and holds for any finite g. In the limit where g = 0 it matches the bound derived in [18], once we implement the relation h ther e = 2h her e . In the limit g → ∞ the bulk scalar field decouples and the resulting 3d theories are the ones where the quartic coupling flows to criticality.…”

“…The red and blue family correspond to fixed point where β ′ h < 0 and the cubic operator is marginally relevant, whereas for the green family β ′ h > 0 and the cubic operator is marginally irrelevant. Combining with the constraint from vacuum stability (46), 11 depicted in gray in fig. 9, only one of these solutions corresponds to unitary and stable interacting conformal boundary conditions.…”

“…Other interesting examples of perturbative constructions of boundary conditions for a free scalar field can be found in [7,8]. 3 Systematic studies of the space of higher co-dimension conformal defects for free theories can be found in [10] (for the free massless scalar) and in [11] for the 4d Maxwell field. in particular that, if they exist, unitary, local (in the sense of [9]) and interacting conformal boundary conditions near the Dirichlet end should be almost decoupled from the bulk.…”

Polyakov’s confinement mechanism for generalized Maxwell theory

“…References [5,[13][14][15][16] studied N f QED 3 and similar vector models in this holographic setup. Related examples include theories on manifolds with a boundary where bulk fields are effectively integrated out, e.g., free bulk scalar fields which interact on the boundary [17][18][19], quantum systems interacting with an environment [20][21][22], and the theory of four-dimensional photons that couple to electrons restricted to live on a three-dimensional surface -a system which admits a purely three-dimensional effective description involving a non-local Maxwell term [18,[23][24][25][26][27][28][29][30][31][32], and which has been usefully applied as a model of graphene [25,33].…”

Bootstrapping line defects with O(2) global symmetry

“…See[32,33] for an alternative numerical bootstrap method that does not require positivity 4. Some progress in this direction can be found in[29,30,34] in the context of conformal boundaries and defects for the free massless scalar field and in[35] in the context of surface defects for the 4d Maxwell field.5 See[36] and references therein for works that combine analytic functionals in 1d with the numerical conformal bootstrap, and[37] for a first single-correlator numerical bootstrap study of 1d O(2) symmetric CFTs 6. Here and below (whenever necessary) we will denote the global symmetry group as O(2)F , to be distinguished from the group of transverse rotations about the defect that is denoted as SO(2)T .…”