Punctures for theories of class \( {\mathcal{S}}_{\varGamma } \)
Received: February
Punctures for theories of class
Jonathan J. Heckman 0 1 3
Patrick Je erson 0 1 2
Tom Rudelius 0 1 2
Cumrun Vafa 0 1 2
0 Open Access , c The Authors
1 Chapel Hill , NC 27599 , U.S.A
2 Je erson Physical Laboratory, Harvard University
3 Department of Physics, University of North Carolina
With the aim of understanding compacti cations of 6D superconformal eld theories to four dimensions, we study punctures for theories of class S . The class S theories arise from M5-branes probing C2= , an ADE singularity. The resulting 4D theories descend from compacti cation on Riemann surfaces decorated with punctures. We show that for class S theories, a puncture is speci ed by singular boundary conditions for elds in the 5D quiver gauge theory obtained from compacti cation of the 6D theory on a cylinder geometry. We determine general boundary conditions and study in detail solutions with rst order poles. This yields a generalization of the Nahm pole data present for 1=2 BPS punctures for theories of class S. Focusing on speci c algebraic structures, we show how the standard discussion of nilpotent orbits and its connection to representations of su(2) generalizes in this broader context. metric Gauge Theory
Conformal Field Theory; Field Theories in Higher Dimensions; Supersym-
1 Introduction
2 Punctures and M5-branes 2.1 2.2 2.3
1=4 BPS punctures for class S theories
Punctures for class S theories
Flavor symmetries and mass parameters
3 Commuting nilpotent matrices
4 su(2)Q ansatz
Flavor symmetries
5 su(2)Q
su(2)Qe ansatz
Flavor symmetries
6 su(2)l directed paths ansatz
7 Conclusions
A Further examples
A.1 Three M5-branes probing an A1 singularity
A.2 One M5-brane probing a D4 singularity
Introduction
explanation of non-trivial strongly coupled phenomena in lower dimensions.
From this perspective, it is natural to consider compacti cations of 6D superconformal
compacti cations of these \master theories." Given the classi cation of (2; 0) and (1; 0) 6D
SCFTs via F-theory [2{5], the time is ripe to ask what new theories can be obtained via
compacti cation to lower dimensions- { in particular, four dimensions. This has already
been carried out for the (2; 0) theories compacti ed on Riemann surfaces, leading to 4D
choices of punctures for class S theories is still incomplete. Nonetheless, a subset called
\regular punctures" have been classi ed and are related to homomorphisms su(2) ! gADE
for class S theories of type gADE an ADE Lie algebra [20].
of these theories have been studied [21{30]. Much as in the case of the (2; 0) theories,
i.e. a choice of punctures on the compacti cation manifold.
nomenclature used for (2; 0) theories, we refer to these theories as \class S ," where
is a discrete ADE subgroup of SU(2) indicating the type singularity. For a preliminary
discussion of punctures in the case
= Zk, see [21].
These 6D theories provide examples of \conformal matter" [3], and form the building
blocks for more elaborate 6D SCFTs [5]. Already for this limited class, we nd a much
ries, leading to a rich class of novel 4D theories. We defer the challenging question of
classi cation to future work.
The basic idea is rather simple: studying the allowed supersymmetric punctures is
theories on a cylinder, viewed as a semi-in nite tube sticking out of the Riemann surface.
The semi-in nite tube can be viewed as S1
R 0. So we rst have to study the resulting
5D theory obtained by compactifying the (1; 0) theory on the S1 factor, in which we have
some singular behavior for elds in the R 0 factor. For the class of theories obtained from
M5-branes probing C2= , with
SU(2) an ADE discrete subgroup, the resulting 5D
system is an a ne ADE quiver gauge theory that admits a Lagrangian description. The
gauge algebra is:
gQuiver =
where N is the total number of M5-branes, the product on i runs over the nodes of the
corresponding a ne ADE Dynkin diagram, and di is the Dynkin index of a node in the
for each of the ADE subgroups. We use this Lagrangian description to determine the
allowed supersymmetric boundary conditions for
elds of the quiver theory with poles at
the origin of R 0. In this work we primarily focus on the case of elds with simple poles:
regular punctures.
In the special case where
is trivial, we recover the punctures of a (2; 0) theory.
by the equations:
where , Q and Qe are N
N matrices with complex entries, with
Hermitian. The special
natural generalization of this. As we show, these equations specify a pair of commuting
punctures for theories of class S, see [31].
boundary conditions we nd are most conveniently stated in terms of an algebra of N j j
N j j matrices with entries in C, where j j is the order of the discrete ADE subgroup
SU(2). Given
Hermitian and Q and Qe matrices with general complex entries,
the set of regular punctures P obeys the conditions of equation (1.2). To get a solution
for the quiver gauge theory, we project to the quiver basis of elds as dictated by the
D (...truncated)