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"Symmetry"
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Noncommutative Homological Mirror Functor
We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer
theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the
constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain
non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting
results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and
symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo
surface as a noncommutative mirror to an elliptic orbifold.
eBook
Expression of Concern: Lin et al. A Perception Study for Unit Charts in the Context of Large-Magnitude Data Representation. Symmetry 2023, 15, 219
With this notice, the Symmetry Editorial Office states its awareness of concerns regarding the availability and validity of the data as well as the overall findings of the published manuscript [...]
Journal Article
Exploring 2-group global symmetries
A
bstract
We analyze four-dimensional quantum field theories with continuous 2-group global symmetries. At the level of their charges, such symmetries are identical to a product of continuous flavor or spacetime symmetries with a 1-form global symmetry
U
(1)
B
(1)
, which arises from a conserved 2-form current
J
B
(2)
. Rather, 2-group symmetries are characterized by deformed current algebras, with quantized structure constants, which allow two flavor currents or stress tensors to fuse into
J
B
(2)
. This leads to unconventional Ward identities, which constrain the allowed patterns of spontaneous 2-group symmetry breaking and other aspects of the renormalization group flow. If
J
B
(2)
is coupled to a 2-form background gauge field
B
(2)
, the 2-group current algebra modifies the behavior of
B
(2)
under background gauge transformations. Its transformation rule takes the same form as in the Green-Schwarz mechanism, but only involves the background gauge or gravity fields that couple to the other 2-group currents. This makes it possible to partially cancel reducible ’t Hooft anomalies using Green-Schwarz counterterms for the 2-group background gauge fields. The parts that cannot be cancelled are reinterpreted as mixed, global anomalies involving
U
(1)
B
(1)
, which receive contributions from topological, as well as massless, degrees of freedom. Theories with 2-group symmetry are constructed by gauging an abelian flavor symmetry with suitable mixed ’t Hooft anomalies, which leads to many simple and explicit examples. Some of them have dynamical string excitations that carry
U
(1)
B
(1)
charge, and 2-group symmetry determines certain ’t Hooft anomalies on the world sheets of these strings. Finally, we point out that holographic theories with 2-group global symmetries have a bulk description in terms of dynamical gauge fields that participate in a conventional Green-Schwarz mechanism.
Journal Article
Symmetry : a very short introduction
Ian Stewart demonstrates symmetry's deep implications, describing how symmetry's applications range across the entire field of mathematics and how symmetry governs the structure of crystals, innumerable types of pattern formation, and how systems change their state as parameters vary. Symmetry is also highly visual, with applications that include animal markings, locomotion, evolutionary biology, elastic buckling, waves, the shape of the Earth, and the form of galaxies -- Source other than Library of Congress.
Book
PT-Symmetry in 2 x 2 Matrix Polynomials Formed by Pauli Matrices
2 × 2 matrix polynomials of the form Pn(z) = Σnj=0 σjzj, for the cases n = 1,2,3 are constructed, and the nature of PT-symmetry is examined across different points z = (x, y) in the complex plane. The PT-symmetric properties of Pn(z) can be characterized by two functions, denoted by s(x, y) and h(x, y). If the trace of the matrix polynomial is real, then the points at which it can exhibit PT-symmetry are defined by the family of curves s(x, y) = 0. Additionally, at points where the function h(x, y) ≥ 0, the matrix polynomial exhibits unbroken PT-symmetry; otherwise, it exhibits broken PT-symmetry. The intersection points of the curves s(x, y) = 0 and h(x, y) = k, for a given k ∈ R, are shown to lie on an ellipse, hyperbola, two lines passing through the origin, or a straight line, depending on the nature of PTsymmetry of the matrix polynomial. The PT-symmetric behaviour of Pn(z) at the zeros of the matrix polynomial is also studied.
Journal Article
Symmetry : through the eyes of old masters
\"A large range of symmetries in art is presented through clear and aesthetically outstanding examples of historical ornaments. Compendious comments illustrate the selected photographic material by addressing the interested and specialist\"-- Provided by publisher.
Book
