A
bstract
By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2
,
0) theory on a three-manifold
M
3
. This generalization ...is applicable to both the 3d
N
= 2 and
N
= 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by
M
3
. This is carried out in detail for
M
3
a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on
M
3
, which matches the Witten index computation that takes the higher-form symmetries into account.
Trifectas for TN in 5d Eckhard, Julius; Schäfer-Nameki, Sakura; Wang, Yi-Nan
The journal of high energy physics,
07/2020, Volume:
2020, Issue:
7
Journal Article
Peer reviewed
Open access
A
bstract
The trinions
T
N
are a class of 5d
N
= 1 superconformal field theories (SCFTs) realized as M-theory on ℂ
3
/ℤ
N
×
ℤ
N
. We apply to
T
N
, as well as closely-related SCFTs that are obtained ...by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view — toric and gluing of compact surfaces as well as combined fiber diagrams — to magnetic quivers and Hasse diagrams.
(5d RG-flow) trees in the tropical rain forest van Beest, Marieke; Bourget, Antoine; Eckhard, Julius ...
The journal of high energy physics,
03/2021, Volume:
2021, Issue:
3
Journal Article
Peer reviewed
Open access
A
bstract
5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates ...a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in
1
, which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries.
A
bstract
We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is ...motivated by a dual, tropical curve decomposition of the (
p, q
) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.
An N=1 3d-3d correspondence Eckhard, Julius; Schäfer-Nameki, Sakura; Wong, Jin-Mann
The journal of high energy physics,
07/2018, Volume:
2018, Issue:
7
Journal Article
Peer reviewed
Open access
A
bstract
M5-branes on an associative three-cycle
M
3
in a
G
2
-holonomy manifold give rise to a 3d
N
=
1
supersymmetric gauge theory,
T
N
=
1
M
3
. We propose an
N
=
1
3d-3d correspondence, based on ...two observables of these theories: the Witten index and the
S
3
-partition function. The Witten index of a 3d
N
=
1
theory
T
N
=
1
M
3
is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on
M
3
. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on
M
3
. Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2
,
0) theory. We also consider a correspondence for the
S
3
-partition function of the
T
N
=
1
M
3
theories, by determining the dimensional reduction of the M5-brane theory on
S
3
. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on
M
3
, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic
G
2
-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the
S
3
-partition function of
T
N
=
1
M
3
is given by the Witten-Reshetikhin-Turaev invariant of
M
3
.
(5d RG-flow) trees in the tropical rain forest van Beest, Marieke; Bourget, Antoine; Eckhard, Julius ...
The journal of high energy physics,
03/2021, Volume:
2021, Issue:
3
Journal Article
Peer reviewed
A bstract 5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates ...a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in 1, which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries.
A bstract We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is ...motivated by a dual, tropical curve decomposition of the ( p, q ) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.
An $$ \mathcal{N}=1 $$ 3d-3d correspondence Eckhard, Julius; Schäfer-Nameki, Sakura; Wong, Jin-Mann
The journal of high energy physics,
07/2018, Volume:
2018, Issue:
7
Journal Article
Peer reviewed
Open access
A bstract M5-branes on an associative three-cycle M 3 in a G 2 -holonomy manifold give rise to a 3d $$ \mathcal{N}=1 $$ N = 1 supersymmetric gauge theory, $$ {T}_{\mathcal{N}=1}\left{M}_3\right $$ T ...N = 1 M 3 . We propose an $$ \mathcal{N}=1 $$ N = 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S 3 -partition function. The Witten index of a 3d $$ \mathcal{N}=1 $$ N = 1 theory $$ {T}_{\mathcal{N}=1}\left{M}_3\right $$ T N = 1 M 3 is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on M 3 . The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M 3 . Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2 , 0) theory. We also consider a correspondence for the S 3 -partition function of the $$ {T}_{\mathcal{N}=1}\left{M}_3\right $$ T N = 1 M 3 theories, by determining the dimensional reduction of the M5-brane theory on S 3 . The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M 3 , whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G 2 -manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S 3 -partition function of $$ {T}_{\mathcal{N}=1}\left{M}_3\right $$ T N = 1 M 3 is given by the Witten-Reshetikhin-Turaev invariant of M 3 .
Supersymmetric gauge theories and superconformal field theories have long played a central role in string theory. In this thesis, which is split into two parts, we explore multiple facets of this ...rich class of theories. In the first part we establish a convenient way to describe 5d SCFTs and gauge theories — generalised toric polygons (GTPs). Based on insights from (p, q) 5-brane-webs we derive a dictionary between a 5d theory TP and its associated GTP P. We provide an algorithm to describe the Higgs branch of TP by computing its magnetic quiver. Furthermore, we identify the Hasse diagram and thereby the entire foliation structure of the Higgs branch. We apply this technique to a large class of 5d SCFTs with a weakly coupled SU(N) gauge theory description. In this way we both recover known results and derive a series of new magnetic quivers. In the second part we turn to the 3d-3d correspondence, that relates observables of supersymmetric 3d gauge theories TM3 to partition functions of topological field theories on M3. We study the sensitivity of this setup to the gauging of higher-form symmetries. We find a refinement of the 3d-3d correspondence for the Witten index, associated to the global structure of the gauge group of TM3. For M3 a Seifert manifold and gauge algebra g = su(2) we verify this explicitly by counting the solutions to the resulting Bethe equations. We complement this analysis by a refined counting of the flat connections on M3, accounting for the higher-form symmetries. Finally, we investigate the so far unknown 3d-3d correspondences for theories with N = 1 supersymmetry. We study the theories TN =1M3 and analyse their connection to the geometry of G2-manifolds. On the complementary side of the 3d-3d correspondence we determine the topological field theories whose partition functions compute the Witten index and the S 3 -partition function of TN =1M3. In the process we point out the relevance of a new generalisation of 3d Seiberg-Witten equations.
5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates a ...decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in arXiv:2008.05577, which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries.
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