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Journal articles on the topic 'Lagrangian embeddings'

Author: Grafiati

Published: 14 March 2023

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1

Yoshiyasu, Toru. "On Lagrangian embeddings into the complex projective spaces." International Journal of Mathematics 27, no. 05 (May 2016): 1650044. http://dx.doi.org/10.1142/s0129167x16500440.

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We prove that for any closed orientable connected [Formula: see text]-manifold [Formula: see text] and any Lagrangian immersion of the connected sum [Formula: see text] either into the complex projective [Formula: see text]-space [Formula: see text] or into the product [Formula: see text] of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg–Murphy’s [Formula: see text]-principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm–Eliashberg–Murphy–Smith’s [Formula: see text]-principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds.

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2

Müller, Stefan. "C0-characterization of symplectic and contact embeddings and Lagrangian rigidity." International Journal of Mathematics 30, no. 09 (August 2019): 1950035. http://dx.doi.org/10.1142/s0129167x19500356.

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We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

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3

Bykov, Dmitri. "Haldane limits via Lagrangian embeddings." Nuclear Physics B 855, no. 1 (February 2012): 100–127. http://dx.doi.org/10.1016/j.nuclphysb.2011.10.005.

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4

Biran, P. "Lagrangian barriers and symplectic embeddings." Geometric and Functional Analysis 11, no. 3 (August 2001): 407–64. http://dx.doi.org/10.1007/pl00001678.

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5

Hofer, Helmut. "Lagrangian embeddings and critical point theory." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 2, no. 6 (November 1985): 407–62. http://dx.doi.org/10.1016/s0294-1449(16)30394-8.

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6

Cieliebak, K., and K. Mohnke. "Punctured holomorphic curves and Lagrangian embeddings." Inventiones mathematicae 212, no. 1 (November 27, 2017): 213–95. http://dx.doi.org/10.1007/s00222-017-0767-8.

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7

Han, Qing, and Guofang Wang. "Hessian surfaces and local Lagrangian embeddings." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 35, no. 3 (May 2018): 675–85. http://dx.doi.org/10.1016/j.anihpc.2017.07.003.

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8

KASUYA, NAOHIKO, and TORU YOSHIYASU. "ON LAGRANGIAN EMBEDDINGS OF PARALLELIZABLE MANIFOLDS." International Journal of Mathematics 24, no. 09 (August 2013): 1350073. http://dx.doi.org/10.1142/s0129167x13500730.

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We prove that for any closed parallelizable n-manifold Mn, if the dimension n ≠ 7, or if n = 7 and the Kervaire semi-characteristic χ½(M7) is zero, then Mn can be embedded in the Euclidean space ℝ2n with a certain symplectic structure as a Lagrangian submanifold. By the results of Gromov and Fukaya, our result gives rise to symplectic structures of ℝ2n(n ≥ 3) which are not conformally equivalent to open domains in standard ones.

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9

Ramos, Vinicius Gripp Barros. "Symplectic embeddings and the Lagrangian bidisk." Duke Mathematical Journal 166, no. 9 (June 2017): 1703–38. http://dx.doi.org/10.1215/00127094-0000011x.

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10

Biran, Paul, and Kai Cieliebak. "Lagrangian embeddings into subcritical Stein manifolds." Israel Journal of Mathematics 127, no. 1 (December 2002): 221–44. http://dx.doi.org/10.1007/bf02784532.

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11

Ekholm, Tobias, Thomas Kragh, and Ivan Smith. "Lagrangian exotic spheres." Journal of Topology and Analysis 08, no. 03 (June 8, 2016): 375–97. http://dx.doi.org/10.1142/s1793525316500199.

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Let [Formula: see text]. We prove that the cotangent bundles [Formula: see text] and [Formula: see text] of oriented homotopy [Formula: see text]-spheres [Formula: see text] and [Formula: see text] are symplectomorphic only if [Formula: see text], where [Formula: see text] denotes the group of oriented homotopy [Formula: see text]-spheres under connected sum, [Formula: see text] denotes the subgroup of those that bound a parallelizable [Formula: see text]-manifold, and where [Formula: see text] denotes [Formula: see text] with orientation reversed. We further show that if [Formula: see text] and [Formula: see text] admits a Lagrangian embedding in [Formula: see text], then [Formula: see text]. The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere [Formula: see text] into the plumbing [Formula: see text] of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

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12

Ritter, Alexander F. "Novikov-symplectic cohomology and exact Lagrangian embeddings." Geometry & Topology 13, no. 2 (January 8, 2009): 943–78. http://dx.doi.org/10.2140/gt.2009.13.943.

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13

Cresson, Jacky, Isabelle Greff, and Charles Pierre. "Discrete Embeddings for Lagrangian and Hamiltonian Systems." Acta Mathematica Vietnamica 43, no. 3 (March 26, 2018): 391–413. http://dx.doi.org/10.1007/s40306-018-0257-0.

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14

Bouza-Allende, Gemayqzel, and Jurgen Guddat. "A note on embeddings for the Augmented Lagrange Method." Yugoslav Journal of Operations Research 20, no. 2 (2010): 183–96. http://dx.doi.org/10.2298/yjor1002183b.

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Nonlinear programs (P) can be solved by embedding problem P into one parametric problem P(t), where P(1) and P are equivalent and P(0), has an evident solution. Some embeddings fulfill that the solutions of the corresponding problem P(t) can be interpreted as the points computed by the Augmented Lagrange Method on P. In this paper we study the Augmented Lagrangian embedding proposed in [6]. Roughly speaking, we investigated the properties of the solutions of P(t) for generic nonlinear programs P with equality constraints and the characterization of P(t) for almost every quadratic perturbation on the objective function of P and linear on the functions defining the equality constraints.

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15

Yoshiyasu, Toru. "On Lagrangian embeddings of closed nonorientable 3–manifolds." Algebraic & Geometric Topology 19, no. 4 (August 16, 2019): 1619–30. http://dx.doi.org/10.2140/agt.2019.19.1619.

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16

Abouzaid, Mohammed. "Framed bordism and Lagrangian embeddings of exotic spheres." Annals of Mathematics 175, no. 1 (January 1, 2012): 71–185. http://dx.doi.org/10.4007/annals.2012.175.1.4.

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17

Ouchi, Genki. "Lagrangian embeddings of cubic fourfolds containing a plane." Compositio Mathematica 153, no. 5 (March 23, 2017): 947–72. http://dx.doi.org/10.1112/s0010437x16008307.

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We prove that a very general smooth cubic fourfold containing a plane can be embedded into an irreducible holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired irreducible holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.

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18

Mohnke, Klaus. "Lagrangian embeddings in the complement of symplectic hypersurfaces." Israel Journal of Mathematics 122, no. 1 (December 2001): 117–23. http://dx.doi.org/10.1007/bf02809894.

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19

NOHARA, YUICHI. "PROJECTIVE EMBEDDINGS AND LAGRANGIAN FIBRATIONS OF KUMMER VARIETIES." International Journal of Mathematics 20, no. 05 (May 2009): 557–72. http://dx.doi.org/10.1142/s0129167x09005418.

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It is known that holomorphic sections of an ample line bundle L (and its tensor power Lk) over an Abelian variety A are given by theta functions. Moreover, a natural basis of the space of holomorphic sections of L or Lk is related to a certain Lagrangian fibration of A. In our previous paper, we studied projective embeddings of A defined by these basis for Lk. For a natural torus action on the ambient projective space, it is proved that its moment map, restricted to A, approximates the Lagrangian fibration of A for large k, with respect to the "Gromov–Hausdorff topology". In this paper, we prove that the same is true for the Kummer variety associated to A.

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20

Nohara, Yuichi. "Projective Embeddings and Lagrangian Fibrations of Abelian Varieties." Mathematische Annalen 333, no. 4 (September 8, 2005): 741–57. http://dx.doi.org/10.1007/s00208-005-0685-8.

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21

Matessi, Diego. "Isometric Embeddings of Families of Special Lagrangian Submanifolds." Annals of Global Analysis and Geometry 29, no. 3 (May 2006): 197–220. http://dx.doi.org/10.1007/s10455-005-9008-2.

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22

Ganatra, Sheel, and Daniel Pomerleano. "A log PSS morphism with applications to Lagrangian embeddings." Journal of Topology 14, no. 1 (February 12, 2021): 291–368. http://dx.doi.org/10.1112/topo.12183.

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23

Chekanov, Yu V. "Invariant Finsler metrics on the space of Lagrangian embeddings." Mathematische Zeitschrift 234, no. 3 (July 2000): 605–19. http://dx.doi.org/10.1007/pl00004814.

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24

Bruzzo, U., and F. Pioli. "Complex Lagrangian embeddings of moduli spaces of vector bundles." Differential Geometry and its Applications 14, no. 2 (March 2001): 151–56. http://dx.doi.org/10.1016/s0926-2245(00)00040-1.

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25

Cresson, Jacky, Agnieszka B. Malinowska, and Delfim F. M. Torres. "Time scale differential, integral, and variational embeddings of Lagrangian systems." Computers & Mathematics with Applications 64, no. 7 (October 2012): 2294–301. http://dx.doi.org/10.1016/j.camwa.2012.03.003.

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26

Oh, Yong-Geun. "Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings." Mathematical Research Letters 4, no. 6 (1997): 895–905. http://dx.doi.org/10.4310/mrl.1997.v4.n6.a9.

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27

Nemirovski, Stefan Yu. "Lefschetz pencils, Morse functions, and Lagrangian embeddings of the Klein bottle." Izvestiya: Mathematics 66, no. 1 (February 28, 2002): 151–64. http://dx.doi.org/10.1070/im2002v066n01abeh000375.

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28

Mironov, A. E., and T. E. Panov. "Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings." Functional Analysis and Its Applications 47, no. 1 (March 2013): 38–49. http://dx.doi.org/10.1007/s10688-013-0005-0.

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29

PELLING, SIMON, and ALICE ROGERS. "MULTISYMPLECTIC BRST." International Journal of Geometric Methods in Modern Physics 10, no. 08 (August 7, 2013): 1360012. http://dx.doi.org/10.1142/s0219887813600128.

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After briefly describing Hamiltonian BRST methods and the multisymplectic approach to field theory, a symmetric geometric Lagrangian is studied by extending the BRST method to the multisymplectic setting. This work uses ideas first introduced by Hrabak [On the multisymplectic origin of the nonabelian deformation algebra of pseudoholomorphic embeddings in strictly almost Kähler ambient manifolds, and the corresponding BRST algebra, preprint (1999), arXiv: math-ph/9904026].

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30

Shevchishin, Vsevolod V. "Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups." Izvestiya: Mathematics 73, no. 4 (August 28, 2009): 797–859. http://dx.doi.org/10.1070/im2009v073n04abeh002465.

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31

Gadbled, Agnès. "Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle." Annales de l’institut Fourier 59, no. 3 (2009): 1135–75. http://dx.doi.org/10.5802/aif.2460.

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32

Bruzzo, U., and F. Pioli. "Erratum to: “Complex Lagrangian embeddings of moduli spaces of vector bundles” [Differential Geom. Appl. 14 (2001) 151–156]." Differential Geometry and its Applications 15, no. 2 (September 2001): 201. http://dx.doi.org/10.1016/s0926-2245(01)00058-4.

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33

BAKAS, IOANNIS. "CONSERVATION LAWS AND GEOMETRY OF PERTURBED COSET MODELS." International Journal of Modern Physics A 09, no. 19 (July 30, 1994): 3443–72. http://dx.doi.org/10.1142/s0217751x94001369.

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We present a Lagrangian description of the SU(2)/U(1) coset model perturbed by its first thermal operator. This is the simplest perturbation that changes sign under Krammers-Wannier duality. The resulting theory, which is a two-component generalization of the sine-Gordon model, is then taken in Minkowski space. For negative values of the coupling constant g, it is classically equivalent to the O(4) nonlinear σ model reduced in a certain frame. For g>0, it describes the relativistic motion of vortices in a constant external field. Viewing the classical equations of motion as a zero curvature condition, we obtain recursive relations for the infinitely many conservation laws by the abelianization method of gauge connections. The higher spin currents are constructed entirely using an off-critical generalization of the W∞ generators. We give a geometric interpretation to the corresponding charges in terms of embeddings. Applications to the chirally invariant U(2) Gross-Neveu model are also discussed.

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34

Viterbo, C. "A new obstruction to embedding Lagrangian tori." Inventiones Mathematicae 100, no. 1 (December 1990): 301–20. http://dx.doi.org/10.1007/bf01231188.

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35

Krumholz, Mark R., Christopher F. McKee, and Richard I. Klein. "Embedding Lagrangian Sink Particles in Eulerian Grids." Astrophysical Journal 611, no. 1 (August 10, 2004): 399–412. http://dx.doi.org/10.1086/421935.

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36

Purbhoo, Kevin. "A marvellous embedding of the Lagrangian Grassmannian." Journal of Combinatorial Theory, Series A 155 (April 2018): 1–26. http://dx.doi.org/10.1016/j.jcta.2017.08.012.

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37

OSTROVER, YARON. "A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS." Communications in Contemporary Mathematics 05, no. 05 (October 2003): 803–11. http://dx.doi.org/10.1142/s0219199703001154.

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We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

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38

Cresson, Jacky. "Fractional embedding of differential operators and Lagrangian systems." Journal of Mathematical Physics 48, no. 3 (March 2007): 033504. http://dx.doi.org/10.1063/1.2483292.

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39

Krause, Katharina, and Wim Klopper. "Communication: A simplified coupled-cluster Lagrangian for polarizable embedding." Journal of Chemical Physics 144, no. 4 (January 28, 2016): 041101. http://dx.doi.org/10.1063/1.4940895.

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40

HARIKUMAR, E., and M. SIVAKUMAR. "HAMILTONIAN vs LAGRANGIAN EMBEDDING OF A MASSIVE SPIN-ONE THEORY INVOLVING TWO-FORM FIELD." International Journal of Modern Physics A 17, no. 03 (January 30, 2002): 405–16. http://dx.doi.org/10.1142/s0217751x02006006.

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We consider the Hamiltonian and Lagrangian embedding of a first-order, massive spin-one, gauge noninvariant theory involving antisymmetric tensor field. We apply the BFV–BRST generalized canonical approach to convert the model to a first class system and construct nilpotent BFV–BRST charge and a unitarizing Hamiltonian. The canonical analysis of the Stückelberg formulation of this model is presented. We bring out the contrasting feature in the constraint structure, specifically with respect to the reducibility aspect, of the Hamiltonian and the Lagrangian embedded model. We show that to obtain manifestly covariant Stückelberg Lagrangian from the BFV embedded Hamiltonian, phase space has to be further enlarged and show how the reducible gauge structure emerges in the embedded model.

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41

Ohnita, Yoshihiro. "Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces." Complex Manifolds 6, no. 1 (January 1, 2019): 303–19. http://dx.doi.org/10.1515/coma-2019-0016.

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AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

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42

Cresson, Jacky, and Isabelle Greff. "Non-differentiable embedding of Lagrangian systems and partial differential equations." Journal of Mathematical Analysis and Applications 384, no. 2 (December 2011): 626–46. http://dx.doi.org/10.1016/j.jmaa.2011.06.008.

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43

Ekholm, Tobias, and Ivan Smith. "Nearby Lagrangian fibers and Whitney sphere links." Compositio Mathematica 154, no. 4 (February 20, 2018): 685–718. http://dx.doi.org/10.1112/s0010437x17007692.

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Let $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

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44

ABREU, E. M. C., A. C. R. MENDES, C. NEVES, W. OLIVEIRA, and F. I. TAKAKURA. "DUALITY THROUGH THE SYMPLECTIC EMBEDDING FORMALISM." International Journal of Modern Physics A 22, no. 21 (August 20, 2007): 3605–20. http://dx.doi.org/10.1142/s0217751x07036932.

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In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.

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45

Lekili, Yankı, and James Pascaleff. "Floer cohomology of -equivariant Lagrangian branes." Compositio Mathematica 152, no. 5 (December 17, 2015): 1071–110. http://dx.doi.org/10.1112/s0010437x1500771x.

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Building on Seidel and Solomon’s fundamental work [Symplectic cohomology and$q$-intersection numbers, Geom. Funct. Anal. 22 (2012), 443–477], we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$, where $\mathfrak{g}\subset SH^{1}(M)$ is a sub-Lie algebra of the symplectic cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic) homogeneous space $G/P$, homological mirror symmetry predicts that there is an embedding of $\mathfrak{g}$ in $SH^{1}(M)$. This allows us to study a mirror theory to classical constructions of Borel, Weil and Bott. We give explicit computations recovering all finite-dimensional irreducible representations of $\mathfrak{sl}_{2}$ as representations on the Floer cohomology of an $\mathfrak{sl}_{2}$-equivariant Lagrangian brane and discuss generalizations to arbitrary finite-dimensional semisimple Lie algebras.

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46

Pavšič, M. "Einstein's gravity from a first order lagrangian in an embedding space." Physics Letters A 116, no. 1 (May 1986): 1–5. http://dx.doi.org/10.1016/0375-9601(86)90344-0.

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47

HONG, SOON-TAE, YONG-WAN KIM, YOUNG-JAI PARK, and K. D. ROTHE. "SYMPLECTIC EMBEDDING AND HAMILTON–JACOBI ANALYSIS OF PROCA MODEL." Modern Physics Letters A 17, no. 08 (March 14, 2002): 435–51. http://dx.doi.org/10.1142/s0217732302006746.

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Following the symplectic approach we show how to embed the Abelian Proca model into a first-class system by extending the configuration space to include an additional pair of scalar fields, and compare it with the improved Dirac scheme. We obtain in this way the desired Wess–Zumino and gauge fixing terms of BRST-invariant Lagrangian. Furthermore, the integrability properties of the second-class system described by the Abelian Proca model are investigated using the Hamilton–Jacobi formalism, where we construct the closed Lie algebra by introducing operators associated with the generalized Poisson brackets.

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48

HONG, SOON-TAE. "SCHRÖDINGER REPRESENTATION AND SYMPLECTIC EMBEDDING OF TOPOLOGICAL SOLITONS." Modern Physics Letters A 20, no. 32 (October 20, 2005): 2455–65. http://dx.doi.org/10.1142/s0217732305018645.

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Exploiting the SU(2) Skyrmion Lagrangian with second-class constraints associated with Lagrange multiplier and collective coordinates, we convert the second-class system into the first-class one in the Batalin–Fradkin–Tyutin embedding through the introduction of Stückelberg coordinates. In the enlarged phase space possessing the Stückelberg coordinates, we perform the "canonical" quantization to describe the Schrödinger representation of the SU(2) Skyrmion, so that we can assign via the homotopy class π4( SU (2))=Z2 half integers to the isospin quantum number for the solitons. The symplectic embedding and the Becchi–Rouet–Stora–Tyutin symmetries involved in the SU(2) Skyrmion are also investigated.

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49

EKHOLM, TOBIAS, JOHN ETNYRE, and MICHAEL SULLIVAN. "ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS." International Journal of Mathematics 16, no. 05 (May 2005): 453–532. http://dx.doi.org/10.1142/s0129167x05002941.

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We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ2 to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into ℂn and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in ℤ2 otherwise.

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50

Shapere, Alfred D., and Frank Wilczek. "Regularizations of time-crystal dynamics." Proceedings of the National Academy of Sciences 116, no. 38 (August 14, 2019): 18772–76. http://dx.doi.org/10.1073/pnas.1908758116.

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We demonstrate that nonconvex Lagrangians, as contemplated in the theory of time crystals, can arise in the effective description of conventional, physically realizable systems. Such embeddings resolve dynamical singularities which arise in the reduced description. Microstructure featuring intervals of fixed velocity interrupted by quick resets—“Sisyphus dynamics”—is a generic consequence. In quantum mechanics, this microstructure can be blurred, leaving entirely regular behavior.

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