68 research outputs found
Comments on twisted indices in 3d supersymmetric gauge theories
We study three-dimensional supersymmetric gauge theories on
with a topological twist along , a genus-
Riemann surface. The twisted supersymmetric index at genus and the
correlation functions of half-BPS loop operators on can be computed
exactly by supersymmetric localization. For , this gives a simple UV
computation of the 3d Witten index. Twisted indices provide us with a clean
derivation of the quantum algebra of supersymmetric Wilson loops, for any
Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe
equations for the theory on . This also provides a
powerful and simple tool to study 3d Seiberg dualities.
Finally, we study A- and B-twisted indices for supersymmetric
gauge theories, which turns out to be very useful for quantitative studies of
three-dimensional mirror symmetry. We also briefly comment on a relation
between the twisted indices and the Hilbert series of
moduli spaces.Comment: 66 pages plus appendix; v2: corrected typos and added reference
Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)
These lecture notes are an introduction to toric geometry. Particular focus
is put on the description of toric local Calabi-Yau varieties, such as needed
in applications to the AdS/CFT correspondence in string theory. The point of
view taken in these lectures is mostly algebro-geometric but no prior knowledge
of algebraic geometry is assumed. After introducing the necessary mathematical
definitions, we discuss the construction of toric varieties as holomorphic
quotients. We discuss the resolution and deformation of toric Calabi-Yau
singularities. We also explain the gauged linear sigma-model (GLSM) Kahler
quotient construction.Comment: Based on lectures given at the Modave Summer School in Mathematical
Physics 2008. 35 pages. v2: Added reference
Supersymmetric partition functions and the three-dimensional A-twist
We study three-dimensional supersymmetric gauge theories on
, an oriented circle bundle of degree over a closed
Riemann surface, . We compute the supersymmetric
partition function and correlation functions of supersymmetric loop operators.
This uncovers interesting relations between observables on manifolds of
different topologies. In particular, the familiar supersymmetric partition
function on the round can be understood as the expectation value of a
so-called "fibering operator" on with a topological twist.
More generally, we show that the 3d supersymmetric partition
functions (and supersymmetric Wilson loop correlation functions) on
are fully determined by the two-dimensional A-twisted
topological field theory obtained by compactifying the 3d theory on a circle.
We give two complementary derivations of the result. We also discuss
applications to F-maximization and to three-dimensional supersymmetric
dualities.Comment: 84 pages plus appendix, 8 figures; v2: added reference
B-branes and supersymmetric quivers in 2d
We study 2d supersymmetric quiver gauge theories that
describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY)
singularities. On general grounds, the holomorphic sector of these
theories---matter content and (classical) superpotential interactions---should
be fully captured by the topological -model on the CY. By studying a
number of examples, we confirm this expectation and flesh out the dictionary
between B-brane category and supersymmetric quiver: the matter content of the
supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext
groups of coherent sheaves), while the superpotential interactions are encoded
in the algebra satisfied by the morphisms. This provides us with a
derivation of the supersymmetric quiver directly from the CY geometry. We
also suggest a relation between triality of gauge theories
and certain mutations of exceptional collections of sheaves. 0d
supersymmetric quivers, corresponding to D-instantons probing CY
singularities, can be discussed similarly.Comment: 63 pages plus appendix, 21 figures; v2: corrected typos and added
reference. JHEP version; v3: added referenc
supersymmetric indices and the four-dimensional A-model
We compute the supersymmetric partition function of
supersymmetric gauge theories with an -symmetry on , a principal elliptic fiber bundle of degree
over a genus- Riemann surface, . Equivalently, we compute the
generalized supersymmetric index , with the
supersymmetric three-manifold as the spatial slice. The
ordinary supersymmetric index on the round three-sphere is
recovered as a special case. We approach this computation from the point of
view of a topological -model for the abelianized gauge fields on the base
. This -model---or -twisted two-dimensional
gauge theory---encodes all the information about the
generalized indices, which are viewed as expectations values of some
canonically-defined surface defects wrapped on inside . Being defined by compactification on the torus, the -model also enjoys
natural modular properties, governed by the four-dimensional 't Hooft
anomalies. As an application of our results, we provide new tests of Seiberg
duality. We also present a new evaluation formula for the three-sphere index as
a sum over two-dimensional vacua.Comment: 91 pages including appendices; v2: corrected typos and added
references, JHEP versio
A-twisted correlators and Hori dualities
The Hori-Tong and Hori dualities are infrared dualities between
two-dimensional gauge theories with supersymmetry, which
are reminiscent of four-dimensional Seiberg dualities. We provide additional
evidence for those dualities with , , and
gauge groups, by matching correlation functions of Coulomb branch operators on
a Riemann surface , in the presence of the topological -twist. The
theories studied, denoted by and , can be understood
as orbifolds of an theory. The correlators of these
theories on with are obtained by computing correlators with
-twisted boundary conditions and summing them up with weights
determined by the orbifold projection.Comment: 45 pages plus appendix; v2: updated bibliography and acknowledgement
Graded quivers and B-branes at Calabi-Yau singularities
A graded quiver with superpotential is a quiver whose arrows are assigned
degrees , for some integer , with relations
generated by a superpotential of degree . Ordinary quivers ( often
describe the open string sector of D-brane systems; in particular, they capture
the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type
IIB string theory, in the guise of 4d supersymmetric quiver
gauge theories. It was pointed out recently that graded quivers with and
similarly describe systems of D-branes at CY 4-fold and 5-fold
singularities, as 2d and 0d gauge theories,
respectively. In this work, we further explore the correspondence between
-graded quivers with superpotential, , and CY -fold
singularities, . For any , the open string sector of the
topological B-model on can be described in terms of a
graded quiver. We illustrate this correspondence explicitly with a few infinite
families of toric singularities indexed by , for which we
derive "toric" graded quivers associated to the geometry, using several
complementary perspectives. Many interesting aspects of supersymmetric quiver
gauge theories can be formally extended to any ; for instance, for one
family of singularities, dubbed , that generalizes
the conifold singularity to , we point out the existence of a formal
"duality cascade" for the corresponding graded quivers.Comment: 82 pages, 20 figure
Grothendieck lines in 3d SQCD and the quantum K-theory of the Grassmannian
We revisit the 3d GLSM computation of the equivariant quantum K-theory ring
of the complex Grassmannian from the perspective of line defects. The 3d GLSM
onto is a circle compactification of the 3d
supersymmetric gauge theory with gauge group and fundamental chiral multiplets, for any choice of the
Chern-Simons levels in the `geometric window'. For
and , the twisted chiral ring generated by the
half-BPS lines wrapping the circle has been previously identified with the
quantum K-theory ring QK. We identify new half-BPS line defects in the
UV gauge theory, dubbed Grothendieck lines, which flow to the structure sheaves
of the (equivariant) Schubert varieties of . They are defined by coupling
supersymmetric gauged quantum mechanics of quiver type to the
3d GLSM. We explicitly show that the 1d Witten index of the defect worldline
reproduces the Chern characters for the Schubert classes, which are written in
terms of double Grothendieck polynomials. This gives us a physical realisation
of the Schubert-class basis for QK. We then use 3d -model techniques
to explicitly compute QK as well as other K-theoretic enumerative
invariants such as the topological metric. We also consider the 2d/0d limit of
our 3d/1d construction, which gives us local defects in the 2d GLSM, the
Schubert defects, that realise equivariant quantum cohomology classes.Comment: 50 pages plus 3 appendice
The -plane of rank-one 4d KK theories
The simplest non-trivial 5d superconformal field theories (SCFT) are the
famous rank-one theories with flavour symmetry. We study their -plane,
which is the one-dimensional Coulomb branch of the theory on . The total space of the Seiberg-Witten (SW) geometry -- the
SW curve fibered over the -plane -- is described as a rational elliptic
surface with a singular fiber of type at infinity. A classification
of all possible Coulomb branch configurations, for the theories and their
4d descendants, is given by Persson's classification of rational elliptic
surfaces. We show that the global form of the flavour symmetry group is encoded
in the Mordell-Weil group of the SW elliptic fibration. We study in detail many
special points in parameters space, such as points where the flavour symmetry
enhances, and/or where Argyres-Douglas and Minahan-Nemeschansky theories
appear. In a number of important instances, including in the massless limit,
the -plane is a modular curve, and we use modularity to investigate aspects
of the low-energy physics, such as the spectrum of light particles at strong
coupling and the associated BPS quivers. We also study the gravitational
couplings on the -plane, matching the infrared expectation for the couplings
and to the UV computation using the Nekrasov partition function.Comment: 137 pages plus appendix, many figures. v2: added references,
corrected typos and imprecision
On the Witten index of 3d unitary SQCD with general CS levels
We consider unitary SQCD, a three-dimensional supersymmetric
Chern-Simons-matter theory consisting of one vector
multiplet coupled to fundamental and antifundamental chiral
multiplets, where and parameterise generic CS levels for
. We study the moduli space of
vacua of this theory with , for generic values of the parameters and with a non-zero Fayet-Ilopoulos parameter turned on. We uncover
a rich pattern of vacua including Higgs, topological and hybrid phases. This
allows us to derive a closed-form formula for the flavoured Witten index of
unitary SQCD for any , generalising previously known results for
either or . Finally, we analyse the vacuum structure of recently
proposed infrared-dual gauge theories and we match vacua across the dualities,
thus providing intricate new checks of those dualities. Incidentally, we also
discuss a seemingly new level/rank duality for pure CS theories with
gauge group.Comment: 37 pages+appendix; v2: corrected typos and references; v3: small
clarifications after feedback from SciPost referee
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