Anomaly cancellation in three-dimensional noncommutative gauge theories

Abstract We construct a family of four-dimensional noncommutative deformations of U(1) gauge theory following a general scheme, recently proposed in JHEP 08 (2020) 041 for a class of coordinate-dependent noncommutative algebras. This class includes the $$ \mathfrak{su} $$ su (2), the $$ \mathfrak{su} $$ su (1, 1) and the angular (or λ-Minkowski) noncommutative structures. We find that the presence of a fourth, commutative coordinate x0 leads to substantial novelties in the expression for the deformed field strength with respect to the corresponding three-dimensional case. The constructed field theoretical models are Poisson gauge theories, which correspond to the semi-classical limit of fully noncommutative gauge theories. Our expressions for the deformed gauge transformations, the deformed field strength and the deformed classical action exhibit flat commutative limits and they are exact in the sense that all orders in the deformation parameter are present. We review the connection of the formalism with the L∞ bootstrap and with symplectic embeddings, and derive the L∞-algebra, which underlies our model.

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Wilson loop algebras and quantum K-theory for Grassmannians

Journal of High Energy Physics ◽

10.1007/jhep10(2020)036 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Hans Jockers ◽

Peter Mayr ◽

Urmi Ninad ◽

Alexander Tabler

Keyword(s):

Wilson Loop ◽

Gauge Theories ◽

Wilson Line ◽

Quantum Cohomology ◽

Three Dimensional ◽

Loop Algebra ◽

Loop Algebras ◽

Verlinde Algebra ◽

K Theory ◽

Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

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Three-dimensional 𝒩 = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review

International Journal of Modern Physics A ◽

10.1142/s0217751x19300114 ◽

2019 ◽

Vol 34 (23) ◽

pp. 1930011 ◽

Cited By ~ 4

Author(s):

Cyril Closset ◽

Heeyeon Kim

Keyword(s):

Correlation Functions ◽

Gauge Theories ◽

Three Dimensional ◽

Partition Functions ◽

Exact Results ◽

Strongly Coupled ◽

Seifert Manifolds ◽

Recent Developments ◽

Supersymmetric Gauge ◽

Chern Simons

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D [Formula: see text] supersymmetric Chern–Simons-matter theories (with an [Formula: see text]-symmetry) on half-BPS closed three-manifolds — including [Formula: see text], [Formula: see text], and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D [Formula: see text] 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.

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Omega vs. pi, and 6d anomaly cancellation

Journal of High Energy Physics ◽

10.1007/jhep05(2021)267 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Joe Davighi ◽

Nakarin Lohitsiri

Keyword(s):

Gauge Theories ◽

Necessary Conditions ◽

Homotopy Groups ◽

Mapping Torus ◽

Anomaly Cancellation ◽

Global Anomaly ◽

Gauge Anomalies ◽

Local Anomalies ◽

Dimensional Mapping

Abstract In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

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