Bootstrapping \( \mathcal{N}=3 \) superconformal theories
Received: February
Published for SISSA by Springer
Open Access 0 1 2 6
c The Authors. 0 1 2 6
0 JGU Mainz , Staudingerweg 7, 55128 Mainz , Germany
1 Stony Brook University , Stony Brook, NY 11794-3636 , U.S.A
2 Notkestrasse 85 , D-22607 Hamburg , Germany
3 Simons Center for Geometry and Physics
4 PRISMA Cluster of Excellence, Institut fur Physik
5 DESY Hamburg, Theory Group
6 equations , to appear
We initiate the bootstrap program for N = 3 superconformal eld theories (SCFTs) in four dimensions. The problem is considered from two fronts: the protected subsector described by a 2d chiral algebra, and crossing symmetry for half-BPS operators whose superconformal primaries parametrize the Coulomb branch of N = 3 theories. With the goal of describing a protected subsector of a family of N = 3 SCFTs, we propose a new 2d chiral algebra with super Virasoro symmetry that depends on an arbitrary parameter, identi ed with the central charge of the theory. Turning to the crossing equations, we work out the superconformal block expansion and apply standard numerical bootstrap techniques in order to constrain the CFT data. We obtain bounds valid for any theory but also, thanks to input from the chiral algebra results, we are able to exclude solutions with N = 4 supersymmetry, allowing us to zoom in on a speci c N = 3 SCFT.
Conformal and W Symmetry; Extended Supersymmetry
Bootstrapping
1 Introduction 2
N = 3 chiral algebras
Generalities of N = 3 chiral algebras
2.2 [3; 0] chiral algebras
Fixing OPE coe cients
Crossing equations
Numerical results
Superconformal Ward identities
Superconformal blocks
Superconformal blocks for the non-chiral channel.
Superconformal blocks for the chiral channel.
From the chiral algebra to numerics
Fixing the chiral algebra contributions
Determination of the function fR(x)
Numerical methods
The case R = 2
The case R = 3
Central charge bounds
Bounding OPE coe cients
Dimension bounds
A.1 Decomposition in N = 2 multiplets
B OPEs of the chiral algebra
C Conformal blocks and generalized free
C.1 Conformal block conventions
C.2 Generalized free theory example
D Short contributions to crossing
D.1 Explicit expressions for Fshort
D.2 Summation for Hshort
The study of superconformal symmetry has given invaluable insights into quantum
theory, and in particular into the nature of strong-coupling dynamics. The presence of
supersymmetry gives us additional analytical tools and allows for computations that are
otherwise hard to perform. A cursory look at the superconformal literature in four
dimenperconformal theories are not restricted to just Lagrangian examples, and this has inspired
recent papers that revisit the status of N = 3 SCFTs.
Assuming these theories exist, the authors of [1] studied several of their properties.
They found in particular that the a and c anomaly coe cients are always the same, that
marginal deformations and are therefore always isolated, and also, in stark contrast with
SCFT is a vector multiplet, the low energy theory at a generic point on the moduli space
must involve vector multiplets, and the types of short multiplets whose expectations values
by N = 3.
studying N D3-branes in the presence of an S-fold plane, which is a generalization of the
standard orientifold construction that also includes the S-duality group.1 The classi cation
SCFTs. In [5] yet another generalization was considered, in which in addition to including
the S-duality group in the orientifold construction, one also considers T-duality. This
background is known as a U-fold, and the study of M5-branes on this background leads to
Coulomb branch) through their Coulomb branch geometries [6{9] has recovered the known
by starting from N
discrete gauging. Note that, as emphasized in [4, 10], gauging by a discrete symmetry does
not change the local dynamics of the theory on R4, only the spectrum of local and non-local
operators. In particular, the central charges and correlation functions remain the same.
Of the class of theories constructed in [4], labeled by the number N of D3-branes and
arise as discretely gauged versions of N
= 4. The non-trivial N
= 3 SCFT with the
with central charge given by 1152 . This corresponds to a rank one theory with Coulomb
branch parameter of scaling dimension three. Since the Coulomb branch operators of
number of D3-branes, higher rank versions of this minimal theory can be obtained . More
and have an N dimensional Coulomb branch.
to study by standard eld theoretical approaches. Apart from the aforementioned papers,
the S-fold (and generalizations thereof) constructions. On the other hand, one would like to
obtain more information on the spectrum of the currently known theories. In this paper we
SCFTs by studying the operators that parametrize the Coulomb branch. These operators
on the simplest case of Coulomb branch operators of dimension three.
The bootstrap approach does not rely on any Lagrangian or perturbative description
of the theory. It depends only on (...truncated)