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Articoli di riviste sul tema "Mirror symmetry"

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Autore: Grafiati

Pubblicato: 4 giugno 2021

Ultima modifica: 14 marzo 2023

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1

Ma, Zhi Yong. "Research on Concept System of Rotation-Mirror Symmetry in Mechanical Systems". Applied Mechanics and Materials 201-202 (ottobre 2012): 7–10. http://dx.doi.org/10.4028/www.scientific.net/amm.201-202.7.

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Mechanical rotation-mirror symmetry is grouped by rotation symmetry and mirror symmetry, and belongs to mechanical static structure symmetry. Collecting and Analyzing a lot of rotation-mirror symmetric instances, and referring to the researches on concept systems of rotation symmetry and mirror symmetry, the concept system of rotation-mirror symmetry was established. The concept system is classified by discrete mirror and continuous mirror rotation-mirror symmetry, unidirectional rotation and bidirectional rotation rotation-mirror symmetry, directed rotation and deflecting rotation rotation-mirror symmetry, entire rotation and partial rotation rotation-mirror symmetry. The concept system can completely contain all kinds of existence of rotation-mirror symmetry in mechanical systems.

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2

Takahashi, Nobuyoshi. "Log Mirror Symmetry and Local Mirror Symmetry". Communications in Mathematical Physics 220, n. 2 (luglio 2001): 293–99. http://dx.doi.org/10.1007/pl00005567.

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3

Blumenhagen, Ralph, Rolf Schimmrigk e Andreas Wiβkirchen. "(0,2) Mirror symmetry". Nuclear Physics B 486, n. 3 (febbraio 1997): 598–628. http://dx.doi.org/10.1016/s0550-3213(96)00698-0.

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4

Gross, Mark. "Topological mirror symmetry". Inventiones mathematicae 144, n. 1 (aprile 2001): 75–137. http://dx.doi.org/10.1007/s002220000119.

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5

Wan, Daqing. "Arithmetic Mirror Symmetry". Pure and Applied Mathematics Quarterly 1, n. 2 (2005): 369–78. http://dx.doi.org/10.4310/pamq.2005.v1.n2.a7.

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6

Zhang, Jun, e Gabriel Khan. "Statistical mirror symmetry". Differential Geometry and its Applications 73 (dicembre 2020): 101678. http://dx.doi.org/10.1016/j.difgeo.2020.101678.

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7

Ma, Zhi Yong. "Research on Concept System of Mechanical Glide Symmetry". Applied Mechanics and Materials 151 (gennaio 2012): 433–37. http://dx.doi.org/10.4028/www.scientific.net/amm.151.433.

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As one kind of mechanical static structure symmetry, glide symmetry is grouped by mirror symmetry and translation symmetry. Glide symmetry is widely exists in mechanical systems, and plays an important role in realizing the technical, economic and social performances of mechanical products. On the basis of research on the concept systems of mirror symmetry, translation symmetry and glide symmetric instances, and taking the characters of the different combined types of symmetry benchmarks as the standard, the concept system of mechanical glide symmetry was established, which can be the foundation of further researches on the application laws of glide symmetry in mechanical systems.

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8

MELKEMI, MAHMOUD, FREDERIC CORDIER e NICKOLAS S. SAPIDIS. "A PROVABLE ALGORITHM TO DETECT WEAK SYMMETRY IN A POLYGON". International Journal of Image and Graphics 13, n. 01 (gennaio 2013): 1350002. http://dx.doi.org/10.1142/s0219467813500022.

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This paper deals with the problem of detecting "weak symmetry" in a polygon, which is a special bijective and continuous mapping between the vertices of the given polygon. An application of this work is the automatic reconstruction of 3D polygons symmetric with respect to a plane from free-hand sketches of weakly-symmetric 2D polygons. We formalize the weak-symmetry notion and highlight its many properties which lead to an algorithm detecting it. The closest research work to the proposed approach is the detection of skewed symmetry. Skewed symmetry detection deals only with reconstruction of planar mirror-symmetric 3D polygons while our method is able to identify symmetry in projections of planar as well as nonplanar mirror-symmetric 3D polygons.

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9

Giveon, Amit, e Edward Witten. "Mirror symmetry as a gauge symmetry". Physics Letters B 332, n. 1-2 (luglio 1994): 44–50. http://dx.doi.org/10.1016/0370-2693(94)90856-7.

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10

DUNDEE, B., J. PERKINS e G. CLEAVER. "OBSERVABLE/HIDDEN SECTOR BROKEN SYMMETRY FOR SYMMETRIC BOUNDARY CONDITIONS". International Journal of Modern Physics A 21, n. 16 (30 giugno 2006): 3367–85. http://dx.doi.org/10.1142/s0217751x06031090.

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A four-dimensional heterotic string model of free fermionic construction is presented wherein mirror symmetry breaking between observable and hidden sector gauge groups occurs in spite of mirror symmetry between observable and hidden sector worldsheet fermion boundary conditions. The differentiation is invoked by an asymmetry in GSO projections necessarily resulting from the symmetry of the free fermionic boundary conditions. In the specific examples shown, an expected nonchiral Pati–Salam mirror universe model is transformed into a chiral model with enhanced hidden sector gauge symmetry and reduced observable sector gauge symmetry: [ SU (4)C ⊗ SU (2)L ⊗ SU (2)R]O ⊗ [ SU (4)C ⊗ SU (2)L ⊗ SU (2)R]H, is necessarily transformed into a chiral [ SU (4)C ⊗ SU (2)L]O ⊗ [ SO (10) ⊗ SU (2)R]H model because of an unavoidable asymmetry in GSO projections.

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11

Sintsov, E. V., S. S. Goncharenko e L. V. Aleksandrova. "Mirror Symmetry in Music". Leonardo Music Journal 8 (1998): 78. http://dx.doi.org/10.2307/1513411.

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12

LeBrasseur, Nicole. "Mirror symmetry in division". Journal of Cell Biology 177, n. 2 (9 aprile 2007): 186. http://dx.doi.org/10.1083/jcb.1772rr4.

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13

Brunner, Ilka, e Kentaro Hori. "Orientifolds and Mirror Symmetry". Journal of High Energy Physics 2004, n. 11 (4 novembre 2004): 005. http://dx.doi.org/10.1088/1126-6708/2004/11/005.

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14

Aganagic, Mina, e Cumrun Vafa. "Mirror symmetry and aG2flop". Journal of High Energy Physics 2003, n. 05 (27 maggio 2003): 061. http://dx.doi.org/10.1088/1126-6708/2003/05/061.

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15

Yao, Carl. "Magnetism and mirror symmetry". American Journal of Physics 63, n. 6 (giugno 1995): 520–23. http://dx.doi.org/10.1119/1.17863.

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16

Gu, Pei-Hong. "Mirror left–right symmetry". Physics Letters B 713, n. 4-5 (luglio 2012): 485–89. http://dx.doi.org/10.1016/j.physletb.2012.06.042.

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17

Ferrara, Sergio, Jeffrey A. Harvey, Andrew Strominger e Cumrun Vafa. "Second-quantized mirror symmetry". Physics Letters B 361, n. 1-4 (novembre 1995): 59–65. http://dx.doi.org/10.1016/0370-2693(95)01074-z.

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18

KONISHI, EIJI. "PLANAR HOMOLOGICAL MIRROR SYMMETRY". International Journal of Modern Physics A 22, n. 29 (20 novembre 2007): 5351–68. http://dx.doi.org/10.1142/s0217751x07037202.

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In this paper, we formulate a planar limited version of the B-side in homological mirror symmetry that formularizes Chern–Simons-type topological open string field theory using homotopy associative algebra (A∞ algebra). This formulation is based on the works by Dijkgraaf and Vafa. We show that our formularization includes gravity/gauge theory correspondence which originates in the AdS/CFT duality of Dijkgraaf–Vafa theory.

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19

Aganagic, Mina, e Cumrun Vafa. "Mirror symmetry and supermanifold". Advances in Theoretical and Mathematical Physics 8, n. 6 (2004): 939–54. http://dx.doi.org/10.4310/atmp.2004.v8.n6.a1.

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20

Leung, Naichung Conan. "Mirror Symmetry Without Corrections". Communications in Analysis and Geometry 13, n. 2 (2005): 287–331. http://dx.doi.org/10.4310/cag.2005.v13.n2.a2.

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21

Sheridan, Nick. "Versality in mirror symmetry". Current Developments in Mathematics 2017, n. 1 (2017): 37–86. http://dx.doi.org/10.4310/cdm.2017.v2017.n1.a2.

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22

Gammage, Benjamin, e David Nadler. "Mirror symmetry for honeycombs". Transactions of the American Mathematical Society 373, n. 1 (10 settembre 2019): 71–107. http://dx.doi.org/10.1090/tran/7909.

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23

Gukov, Sergei. "Quantization via mirror symmetry". Japanese Journal of Mathematics 6, n. 2 (dicembre 2011): 65–119. http://dx.doi.org/10.1007/s11537-011-1033-2.

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24

Parente, Carlos Benedicto Ramos, Vera Lucia Mazzocchi, José Marcos Sasaki e Lisandro Pavie Cardoso. "Mirror symmetries in multiple diffraction patterns of face-centred cubic crystals". Journal of Applied Crystallography 45, n. 4 (14 luglio 2012): 621–26. http://dx.doi.org/10.1107/s0021889812026830.

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In this work, a study of the mirror symmetries appearing in multiple diffraction patterns of face-centred cubic crystals is carried out. Several different X-ray and neutron multiple diffraction patterns have been simulated for different face-centred cubic structures. The patterns were plotted in circular plots which showed that two types of symmetry mirrors coexist in the patterns: isomorphic and anamorphic mirrors. The number and types of mirrors depend on then-fold symmetry of the scattering vector associated with the primary reflection. Forneven, onlynisomorphic mirrors appear in the patterns. Fornodd,nisomorphic mirrors are formed intercalated betweennanamorphic mirrors.

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25

Okino, Shinya, e Masato Nagata. "Asymmetric travelling waves in a square duct". Journal of Fluid Mechanics 693 (6 gennaio 2012): 57–68. http://dx.doi.org/10.1017/jfm.2011.455.

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AbstractTwo types of asymmetric solutions are found numerically in square-duct flow. They emerge through a symmetry-breaking bifurcation from the mirror-symmetric solutions discovered by Okino et al. (J. Fluid Mech., vol. 657, 2010, pp. 413–429). One of them is characterized by a pair of streamwise vortices and a low-speed streak localized near one of the sidewalls and retains the shift-and-reflect symmetry. The bifurcation nature as well as the flow structure of the solution show striking resemblance to those of the asymmetric solution in pipe flow found by Pringle & Kerswell (Phys. Rev. Lett., vol. 99, 2007, A074502), despite the geometrical difference between their cross-sections. The solution seems to be embedded in the edge state of square-duct flow identified by Biau & Bottaro (Phil. Trans. R. Soc. Lond. A, vol. 367, 2009, pp. 529–544). The other solution deviates slightly from the mirror-symmetric solution from which it bifurcates: the shift-and-rotate symmetry is retained but the mirror symmetry is broken.

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26

Ma, Zhi Yong. "Research on Existences and Concept System of Mirror Symmetry in Static Structure of Mechanical Products". Advanced Materials Research 421 (dicembre 2011): 806–9. http://dx.doi.org/10.4028/www.scientific.net/amr.421.806.

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Mirror symmetry is a most common phenomenon in static structure of mechanical products, and plays an important role in realizing the function of mechanical systems, improving their performance, and meeting their design restrictions. On the basis of analyzing plentiful instances, the existences of mirror symmetry were studied. By taking the independence property of symmetry components, the type of symmetry benchmark and the number of symmetry benchmark as standards, the concept system of mirror symmetry in static structure of mechanical products was established, and can provides a theoretical basis for the further comprehensive research on the application laws of mirror symmetry in mechanical systems.

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27

Friedenberg, Jay, Preston Martin, Naomi Uy e Mackenzie Kvapil. "The aesthetics of frieze patterns: Effects of symmetry, motif, and element size". i-Perception 13, n. 5 (settembre 2022): 204166952211311. http://dx.doi.org/10.1177/20416695221131112.

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Frieze patterns appear frequently in architectural designs and ornamental patterning but their aesthetic qualities have never been studied experimentally. In the first experiment, 39 undergraduates used a seven-point rating scale to assess the perceived beauty of the seven basic frieze types presented at a horizontal orientation. The friezes consisted of individual curved and linear motifs as well as random textures. Friezes that filled the entire pattern region and which contained emergent global features were preferred the most. In a second experiment, we utilized horizontal texture friezes that were completely filled and which varied in size and number of elements. Participants preferred patterns with larger features, probably because they make detection of the symmetric transformations more visible. The frieze with the greatest number of symmetries was preferred most but symmetric complexity by itself could not completely account for the predicted preference ordering. In both studies, friezes containing horizontal mirrors (translation, 180° rotation, horizontal mirror, vertical mirror, and glide reflection and translation, horizontal mirror, and glide reflection) were preferred far more than any other condition. Horizontal symmetry may enhance perceived beauty in these cases because it runs parallel to and so emphasizes the overall frieze orientation.

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28

Cowin, S. C., e M. M. Mehrabadi. "Anisotropic Symmetries of Linear Elasticity". Applied Mechanics Reviews 48, n. 5 (1 maggio 1995): 247–85. http://dx.doi.org/10.1115/1.3005102.

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The objective of this paper is to present a development of the anisotropic symmetries of linear elasticity theory based on the use of a single symmetry element, the plane of mirror symmetry. In this presentation the thirteen distinct planes of mirror symmetry are catalogued. Traditional presentations of the anisotropic elastic symmetries involve all the crystallographic symmetry elements which include the center of symmetry, the n-fold rotation axis and the n-fold inversion axis as well as the plane of mirror symmetry. It is shown that the crystal system symmetry groups, as opposed to the crystal class symmetry groups, of the elastic crystallographic symmetries can be generated by the appropriate combinations of the orthogonal transformations corresponding to each of the thirteen distinct planes of mirror symmetry. It is also shown that the restrictions on the elastic coefficients appearing in Hooke’s law follow in a simple and straightforward fashion from orthogonal transformations based on a small subset of the small catalogue of planes of mirror symmetry.

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29

Locher, Paul J., e Johan Wagemans. "Effects of Element Type and Spatial Grouping on Symmetry Detection". Perception 22, n. 5 (maggio 1993): 565–87. http://dx.doi.org/10.1068/p220565.

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The influence of local and global attributes of symmetric patterns on the perceptual salience of symmetry was investigated. After tachistoscopic viewing, subjects discriminated between symmetric and either random patterns (experiment 1) or their perturbed counterparts (experiment 2) created by replacing one third of the mirror element-pairs of symmetric stimuli with ‘random’ elements. In general, it was found that perceptibility of symmetry, measured by response time and detection accuracy, was not influenced in a consistent way by type of pattern element (dots or line segments oriented vertically, horizontally, obliquely, or in all three orientations about the symmetry axis). Nor did axis orientation (vertical, horizontal, oblique), advance knowledge of axis orientation, practice effects, or subject sophistication differentially affect detection. A highly salient global percept of symmetry emerged, on the other hand, when elements were clustered together within a pattern, or grouped in symmetric pairs along a single symmetry axis or two orthogonal axes. Results suggest that mirror symmetry is detected preattentively, presumably by some kind of integral code which emerges from the interaction between display elements and the way they are organized spatially. It is proposed that symmetry is coded and signalled by the same spatial grouping processes as those responsible for construction of the full primal sketch.

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30

GHOSH, RANJAN KUMAR, e SUMATHI RAO. "FEW-ANYON SPECTRA IN A HARMONIC OSCILLATOR POTENTIAL". Modern Physics Letters B 10, n. 11 (10 maggio 1996): 515–22. http://dx.doi.org/10.1142/s0217984996000560.

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We study the problem of four and five anyons in a harmonic oscillator potential and make some exact statements regarding its energy spectra. We show that the spectra exhibit some mirror symmetric (reflection symmetric about the semionic statistics point θ=π/2) features analogous to the mirror symmetry in the two and three anyon spectra.

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31

Gravel, P., e C. Gauthier. "Mirror symmetry and conformal flatness in general Relativity". International Journal of Mathematics and Mathematical Sciences 2004, n. 41 (2004): 2205–8. http://dx.doi.org/10.1155/s0161171204309038.

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Using symmetry arguments only, we show that every spacetime with mirror-symmetric spatial sections is necessarily conformally flat. The general form of the Ricci tensor of such spacetimes is also determined.

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32

Hochstein, Shaul. "The Eyes Wide Shut Illusion". Perception 47, n. 9 (12 luglio 2018): 985–90. http://dx.doi.org/10.1177/0301006618786863.

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The new “eyes wide shut” illusion uses a standard enlarging (shaving or makeup) mirror. Close one eye and look at the closed eye in the mirror; the eye should take up most of the mirror. Switch eyes to see the other closed eye. Switch back-and-forth a few times, then open both eyes. You see an open eye. Which eye is it? To find out, close one eye. Whichever you close, that’s the eye you see. How can this be possible? The brain is fusing two images of the two eyes! The illusion depends on (a) binocular fusion: The brain combines two images to a single percept; (b) symmetry: Mirrors don’t affect appearance of left–right symmetric objects and the eyes are sufficiently left–right symmetric for the brain to combine them. Why aren’t the lingering asymmetries sufficient to prevent fusion? (c) Only vision with scrutiny affords conscious access to scene details. Consistent with reverse hierarchy theory, vision at a glance grants conscious perception of the gist of the scene, integrating images of nonperfectly symmetric eyes.

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33

Böhm, Janko, Kathrin Bringmann, Arne Buchholz e Hannah Markwig. "Tropical mirror symmetry for elliptic curves". Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, n. 732 (1 novembre 2017): 211–46. http://dx.doi.org/10.1515/crelle-2014-0143.

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Abstract Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

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34

NAVILIAT-CUNCIC, OSCAR. "Mirror symmetry and fundamental interactions". European Review 13, S2 (22 agosto 2005): 13–27. http://dx.doi.org/10.1017/s1062798705000633.

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Our modern representation of elementary processes considers that all physical phenomena result from the interplay of four fundamental interactions: the gravitational attraction, the electromagnetic force, the strong interaction and the weak interaction. Formally, the quantum mechanical description of elementary processes introduces the concept of discrete symmetry, illustrated for instance by space and time inversions. Discrete symmetries play a central role in the elaboration of theories and models, and have profound consequences in the predictions of these theories. For nearly 50 years, it has been observed that, of the four fundamental interactions, only the weak interaction violates mirror symmetry, and all observations so far indicate that it does so in a so-called maximal way. Despite overwhelming evidence of mirror-symmetry breaking, the search for a possibly underlying left–right symmetry has been pursued for many years by dedicated experiments. In this paper we review the context of mirror symmetry breaking in the weak interaction, we describe its interpretation in the framework of the standard model of particle physics and describe current efforts to identify the restoration of the left–right symmetry.

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35

Abouzaid, Mohammed. "Homological mirror symmetry without correction". Journal of the American Mathematical Society 34, n. 4 (24 maggio 2021): 1059–173. http://dx.doi.org/10.1090/jams/973.

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Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.

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36

Batyrev, V., e E. Materov. "Toric Residues and Mirror Symmetry". Moscow Mathematical Journal 2, n. 3 (2002): 435–75. http://dx.doi.org/10.17323/1609-4514-2002-2-3-435-475.

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Lian, Bong H., e S. T. Yau. "Differential equations from mirror symmetry". Surveys in Differential Geometry 5, n. 1 (1999): 510–26. http://dx.doi.org/10.4310/sdg.1999.v5.n1.a7.

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38

Gross, Mark, e Bernd Siebert. "Theta functions and mirror symmetry". Surveys in Differential Geometry 21, n. 1 (2016): 95–138. http://dx.doi.org/10.4310/sdg.2016.v21.n1.a3.

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39

Alexandrov, Sergei, e Frank Saueressig. "Quantum mirror symmetry and twistors". Journal of High Energy Physics 2009, n. 09 (25 settembre 2009): 108. http://dx.doi.org/10.1088/1126-6708/2009/09/108.

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40

Ruddat, Helge. "Perverse curves and mirror symmetry". Journal of Algebraic Geometry 26, n. 1 (7 giugno 2016): 17–42. http://dx.doi.org/10.1090/jag/666.

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41

Tomasiello, Alessandro. "Topological mirror symmetry with fluxes". Journal of High Energy Physics 2005, n. 06 (28 giugno 2005): 067. http://dx.doi.org/10.1088/1126-6708/2005/06/067.

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42

Feng, Bo, e Amihay Hanany. "Mirror symmetry by O3-planes". Journal of High Energy Physics 2000, n. 11 (22 novembre 2000): 033. http://dx.doi.org/10.1088/1126-6708/2000/11/033.

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43

Dyatlov, I. T. "Spontaneous violation of mirror symmetry". Physics of Atomic Nuclei 78, n. 8 (novembre 2015): 956–62. http://dx.doi.org/10.1134/s1063778815080037.

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44

Bocklandt, Raf. "Toric systems and mirror symmetry". Compositio Mathematica 149, n. 11 (28 agosto 2013): 1839–55. http://dx.doi.org/10.1112/s0010437x1300701x.

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AbstractIn their paper [Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280], Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak del Pezzo surface a toric system, which defines a new toric surface. We interpret this construction as an instance of mirror symmetry and extend it to a duality on the set of toric weak del Pezzo surfaces equipped with a cyclic full strongly exceptional sequence.

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45

Yang, Hyun Seok. "Mirror symmetry in emergent gravity". Nuclear Physics B 922 (settembre 2017): 264–79. http://dx.doi.org/10.1016/j.nuclphysb.2017.07.003.

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Yang, Wenzhe. "Deligne's conjecture and mirror symmetry". Nuclear Physics B 962 (gennaio 2021): 115245. http://dx.doi.org/10.1016/j.nuclphysb.2020.115245.

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47

Borisov, Lev A. "Vertex Algebras and Mirror Symmetry". Communications in Mathematical Physics 215, n. 3 (gennaio 2001): 517–57. http://dx.doi.org/10.1007/s002200000312.

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48

Bayer, Gregor W. "Quantum Computation Violates Mirror Symmetry". Quantum Information Processing 5, n. 1 (febbraio 2006): 25–30. http://dx.doi.org/10.1007/s11128-005-0010-1.

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49

Solomon, Jake P. "Involutions, obstructions and mirror symmetry". Advances in Mathematics 367 (giugno 2020): 107107. http://dx.doi.org/10.1016/j.aim.2020.107107.

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Sibilla, Nicolò, David Treumann e Eric Zaslow. "Ribbon graphs and mirror symmetry". Selecta Mathematica 20, n. 4 (28 marzo 2014): 979–1002. http://dx.doi.org/10.1007/s00029-014-0149-7.

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