1,177 research outputs found
Comments on twisted indices in 3d supersymmetric gauge theories
Get PDFWe study three-dimensional N = 2 supersymmetric gauge theories on Σg × S
1
with a topological twist along Σg, a genus-g Riemann surface. The twisted supersymmetric
index at genus g and the correlation functions of half-BPS loop operators on S
1
can be
computed exactly by supersymmetric localization. For g = 1, 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-Simonsmatter
theory, in terms of the associated Bethe equations for the theory on R
2 × S
1
. This
also provides a powerful and simple tool to study 3d N = 2 Seiberg dualities. Finally, we
study A- and B-twisted indices for N = 4 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 S
2 × S
1
twisted indices and the Hilbert series
of N = 4 moduli spaces
Three-dimensional N=2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review
Get PDFWe give a pedagogical introduction to the study of supersymmetric partition functions of 3D N=2 supersymmetric Chern–Simons-matter theories (with an R-symmetry) on half-BPS closed three-manifolds — including S3, S2×S1, and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D N=2 supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities
Seifert fibering operators in 3d theories
Get PDFWe study 3d supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q) with rational squashing parameter b ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.We study 3d supersymmetric gauge theories on closed oriented Seifert manifold---circle bundles over an orbifold Riemann surface---, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space with rational squashing parameter , comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks
supersymmetric indices and the four-dimensional A-model
Get PDFWe compute the supersymmetric partition function of = 1 supersymmetric gauge theories with an R-symmetry on , a principal elliptic fiber bundle of degree p over a genus-g Riemann surface, Σ . Equivalently, we compute the generalized supersymmetric index , with the supersymmetric three-manifold as the spatial slice. The ordinary = 1 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 A-model for the abelianized gauge fields on the base Σ . This A-model — or A-twisted two-dimensional = (2, 2) gauge theory — encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on T inside Σ × T. Being defined by compactification on the torus, the A-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.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
Supersymmetric partition functions and the three-dimensional A-twist
No full textWe study three-dimensional N = 2 supersymmetric gauge theories on M-g,M-p an oriented circle bundle of degree p over a closed Riemann surface, Sigma(g). We compute the M-g,M-p, 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 S-3 can be understood as the expectation value of a so-called "fibering operator" on S-2 x,S-1 with a topological twist. More generally, we show that the 3d N = 2 supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on M-g,M-p, 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.
Commentary upon the Gospel according to Saint Luke by Saint Cyril of Alexandria. Part 2
No full textAs the title of this work broadly indicates, it is the translation into English of St. Cyril of Alexandria’s commentary on the Gospel of Luke. This manuscript document had recently been acquired by Oxford University in Syriac. Payne Smith published an edition, but quickly realized that the work would largely go ignored if it were not translated into English. Few scholars of his day were as able to undertake this task as Payne Smith. Cyril represented the extremely influential Alexandrian school of early Christianity that gave the church much of the material that would lead eventually to the doctrine of the Trinity. Needless to comment, Cyril’s own interpretation of one of the Gospels focuses a crucial eye on a major source for understanding early Christianity. Scholars of the Christian Scriptures will find a useful cross-section of early interpretation here, and students of the major figures of the Alexandrian school will garner some of Cyril’s considerable insights into Scripture. This book retains its value to students of many specializations in Late Antiquity.
Robert Payne Smith (1819-1895) was a priest who had studied Classics at Pembroke College, Oxford University. He later became Regius Professor of Divinity at Oxford University. He was eventually appointed the Dean of Canterbury Cathedral. He was most noted for his Syriac lexicon entitled Thesaurus Syriacus.Translated into English from an ancient Syriac version
Commentary upon the Gospel according to Saint Luke by Saint Cyril, Patriarch of Alexandria. Part 1
No full textAs the title of this work broadly indicates, it is the translation into English of St. Cyril of Alexandria’s commentary on the Gospel of Luke. This manuscript document had recently been acquired by Oxford University in Syriac. Payne Smith published an edition, but quickly realized that the work would largely go ignored if it were not translated into English. Few scholars of his day were as able to undertake this task as Payne Smith. Cyril represented the extremely influential Alexandrian school of early Christianity that gave the church much of the material that would lead eventually to the doctrine of the Trinity. Needless to comment, Cyril’s own interpretation of one of the Gospels focuses a crucial eye on a major source for understanding early Christianity. Scholars of the Christian Scriptures will find a useful cross-section of early interpretation here, and students of the major figures of the Alexandrian school will garner some of Cyril’s considerable insights into Scripture. This book retains its value to students of many specializations in Late Antiquity.
Robert Payne Smith (1819-1895) was a priest who had studied Classics at Pembroke College, Oxford University. He later became Regius Professor of Divinity at Oxford University. He was eventually appointed the Dean of Canterbury Cathedral. He was most noted for his Syriac lexicon entitled Thesaurus Syriacus.Translated into English from an ancient Syriac version
’t Hooft anomalies and the holomorphy of supersymmetric partition functions
Get PDFWe study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, GF, for 2d N = (0, 2) and 4d N = 1 supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of GF ’t Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, ZT2 , and for a large class of 4d partition functions on half-BPS four-manifolds, ZM4 — in particular, for M 4 = S3 × S1 and M 4 = Σg × T2. We propose a new expression for ZMd−1×S1 , which differs from earlier holomorphic results by the introduction of a non-holomorphic “Casimir” pre-factor. The latter is fixed by studying the “high temperature” limit of the partition function. Our proposal agrees with the supersymmetric Ward identities, and with explicit calculations of the absolute value of the partition function using a gauge-invariant zeta-function regularization
Chiral flavors and M2-branes at toric CY4 singularities
No full textWe extend the stringy derivation of N=2 AdS4/CFT3 dualities to cases where the M-theory circle degenerates at complex codimension-two submanifolds of a toric
conical CY4. The type IIA backgrounds include D6-branes, and the dual N=2 quiver gauge theories contain chiral flavors. We provide a general recipe to derive the geometric moduli space of flavored versions of Abelian toric quiver gauge theories. The CY4 cone is reproduced thanks to a non-trivial quantum F-term relation between diagonal monopole
operators and bifundamental fields. We find new field theory duals to many geometries, including Q111
Comments on 3d Seiberg-like dualities
Get PDFWe study Seiberg-like dualities in three dimensional N=2 supersymmetric theories, emphasizing Chern-Simons terms for the global symmetry group, which affect contact terms in two-point functions of global currents and are essential to the duality map. We introduce new Seiberg-like dualities for Yang-Mills-Chern-Simons theories with unitary gauge groups with arbitrary numbers of matter fields in the fundamental and antifundamental representations. These dualities are derived from Aharony duality by real mass deformations. They allow to initiate the systematic study of Seiberg-like dualities in Chern-Simons quivers. We also comment on known Seiberg-like dualities for symplectic and orthogonal gauge groups and extend the latter to the Yang-Mills case. We check our proposals by showing that the localized partition functions on the squashed S^3 match between dual descriptions
- …
