Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Hình thức chuẩn của các bản đồ tương đương trong không gian vô hạn chiều
Tóm tắt
Các định lý về hình thức chuẩn địa phương cho các bản đồ tương đương mượt mà giữa các đa tạp vô hạn chiều đã được thiết lập. Những kết quả về hình thức chuẩn này là mới ngay cả trong các chiều hữu hạn. Chứng minh được lấy cảm hứng từ sự rút gọn Lyapunov-Schmidt cho các hệ thống động lực học và từ phương pháp Kuranishi cho không gian moduli. Nó sử dụng định lý cắt cho các đa tạp Fréchet như một công cụ kỹ thuật chính. Do đó, không gian moduli trừu tượng thu được bằng cách phân loại một tập mức của bản đồ tương đương theo hành động của nhóm mang cấu trúc của một không gian Kuranishi, tức là các không gian moduli như vậy được mô hình hóa địa phương trên thương số của tập zero của một bản đồ mượt mà bởi một nhóm compact. Các kết quả chung được áp dụng cho không gian moduli của các instanton tự đối lập, không gian moduli Seiberg-Witten và không gian moduli của các đường cong pseudoholomorphic.
Từ khóa
#hình thức chuẩn #bản đồ tương đương #không gian vô hạn chiều #đa tạp Fréchet #không gian Kuranishi #không gian moduli #instanton tự đối lập #đường cong pseudoholomorphic.Tài liệu tham khảo
Abbati, M.C., Cirelli, R., Manià, A., Michor, P.: Smoothness of the action of the gauge transformation group on connections. J. Math. Phys. 27(10), 2469–2474 (1986)
Abbati, M.C., Cirelli, R., Manià, A.: The orbit space of the action of gauge transformation group on connections. J. Geom. Phys. 6(4), 537–557 (1989)
Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes, I. Ann. Math. 86(2), 374–407 (1967)
Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A. 308(1505), 523–615 (1993)
Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362(1711), 425–461 (1978)
Bochner, S.: Compact groups of differentiable transformations. Ann. Math. 46(3), 372–381 (1945)
Cervera, V., Mascaró, F., Michor, P.W.: The action of the diffeomorphism group on the space of immersions. Differ. Geom. Appl. 1(4), 391–401 (1991)
Chang, K.C.: Methods in Nonlinear Analysis: Springer Monographs in Mathematics. Springer, Berlin (2005)
Chossat, P., Lewis, D., Ortega, J.-P., Ratiu, T.S.: Bifurcation of relative equilibria in mechanical systems with symmetry. Adv. Appl. Math. 31(1), 10–45 (2003). arXiv:math/9912232
Cirelli, R., Manià, A.: The group of gauge transformations as a Schwartz-Lie group. J. Math. Phys. 26(12), 3036–3041 (1985)
Conn, J.F.: Normal forms for smooth poisson structures. Ann. Math. 121(3), 565 (1985)
Diez, T.: Slice Theorem for Fréchet Group Actions and Covariant Symplectic Field Theory. MA Thesis. Universität Leipzig (2013). arXiv:1405.2249
Diez, T.: Normal form of equivariant maps and singular symplectic reduction in infinite dimensions with applications to gauge field theory. PhD thesis. Universität Leipzig (2019). arXiv:1909.00744 [math.SG]
Diez, T.: Singular symplectic reduction in infinite dimensions (in preparation)
Diez, T., Huebschmann, J.: Yang–Mills moduli spaces over an orientable closed surface via Fréchet reduction. J. Geom. Phys. 132, 393–414 (2018). arXiv:1704.01982
Diez, T., Rudolph, G.: Slice theorem and orbit type stratification in infinite dimensions. Differential. Geom. Appl. 65, 176–211 (2019). arXiv:1812.04698 [math.DG]
Diez, T., Rudolph, G.: Singular symplectic cotangent bundle reduction of gauge field theory. J. Math. Phys. 61, 092902 (2020). arXiv:1812.04707 [math-ph]
Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Differential Geom. 18(2) (1983), 279–315
Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford University Press, Oxford Mathematical Monographs (1997)
Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Springer, Berlin Heidelberg (1999)
Ebin, D.G.: The manifold of Riemannian metrics. In: Proceedings of Symposia in Pure Mathematics, vol. 15, American Mathematical Society, pp. 11–40 (1970)
Edwards, R.E.: Functional Analysis. Theory and applications. Holt, Rinehart and Winston, New York-Toronto-London (1965)
Eliashberg, Y., Givental, A., Hofer, H.: Introduction to Symplectic Field Theory (2000). arXiv:math/0010059
Freed, D.S., Uhlenbeck, K.K.: Instantons and Four-Manifolds. Mathematical Sciences Research Institute Publications, Springer (1984)
Fukaya, K., Oh, Y.-G., Ohta, H., On, K.: Construction of Kuranishi Structures on the Moduli Spaces of Pseudo Holomorphic Disks: I (2017). arXiv:1710.01459 [math]
Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Technical details on Kuranishi structure and virtual fundamental chain (2012). arXiv:1209.4410 [math]
Fukaya, K., Ono, K.: arnold conjecture and Gromov–Witten invariant. Topology 38(5), 933–1048 (1999)
Gerstenberger, A.: A version of scale calculus and the associated Fredholm theory (2016). arXiv:1602.07108 [math]
Glöckner, H.: Finite order differentiability properties, fixed points and implicit functions over valued fields (2005). arXiv:math/0511218
Glöckner, H.: Implicit functions from topological vector spaces to Banach spaces. Israel J. Math. 155, 205–252 (2006)
Glöckner, H.: Fundamentals of submersions and immersions between infinite-dimensional manifolds (2015). arXiv:1502.05795 [math.DG]
Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)
Guillemin, V., Lerman, E., Sternberg, S.: Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press (1996)
Hamilton, R.S.: Deformation theory of foliations. Preprint available from Cornell University (1978)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc 7, 65–223 (1982)
Harte, R.: Invertibility And Singularity For Bounded Linear Operators. CRC Press (1987)
Hertsch, A., Rudolph, G., Schmidt, M.: On the gauge orbit types for theories with classical compact gauge group. Rep. Math. Phys. 66(3), 331–353 (2010)
Hertsch, A., Rudolph, G., Schmidt, M.: Gauge Orbit Types for Theories with Gauge Group O(n), SO(n) or Sp(n). Ann. Henri Poincaré 12(2), 351–395 (2011). arXiv:0812.0228
Hiltunen, S.: Implicit functions from locally convex spaces to Banach spaces. Studia Math. 134(3), 235–250 (1999)
Hofer, H., Wysocki, K., Zehnder, E.: Polyfold and Fredholm Theory (2017). arXiv:1707.08941 [math]
Hofer, H.H., Wysocki, K., Zehnder, E.: Polyfold and fredholm theory I: basic theory in M-polyfolds. Mem. Amer. Math. Soc. 248(1179) (2017). arXiv:1407.3185 [math]
Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Pseudo-Differential Operators (2007)
Joyce, D.: A new definition of Kuranishi space (2014). arXiv:1409.6908v3 [math.DG]
Kondracki, W., Rogulski, J.: On the stratification of the orbit space for the action of automorphisms on connections. Dissertationes Math. 250 (1986)
Köthe, G.: Topological Vector Spaces I. Springer, Berlin Heidelberg (1983)
Kuranishi, M.: New proof for the existence of locally complete families of complex structures. In: Aeppli, A., Calabi, E., Röhrl, H (eds.), Proceedings of the Conference on Complex Analysis, Springer, Berlin, pp. 142–154 (1965)
Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191, Springer, New York (1999)
Marcut, I.: Rigidity around Poisson submanifolds. Acta Mathematica 213(1), 137–198 (2014)
Margalef-Roig, J., Dominguez, E.O.: Differential Topology. North Holland, North-Holland Mathematics Studies (1992)
Marsden, J.: Darboux’s Theorem Fails for Weak Symplectic Forms. Proc. Amer. Math. Soc. 32(2), 590–592 (1972)
Marsden, J.E., Ratiu, T., Abraham, R.: Manifolds, Tensor Analysis, and Applications, 3rd edn. Applied Mathematical Sciences. Springer, New York (2002)
McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology, 2nd edn. American Mathematical Society, 726 pp (2012)
McDuff, D., Wehrheim, K.: Kuranishi atlases with trivial isotropy—the 2013 state of affairs (2015). arXiv:1208.1340 [math]
McDuff, D., Wehrheim, K.: The fundamental class of smooth Kuranishi atlases with trivial isotropy. J. Topol. Anal. 10(01), 71–243 (2018). arXiv:1508.01560 [math]
Michor, P.W.: Manifolds of Differentiable Mappings. Birkhauser (1980)
Moore, E.H.: On the Reciprocal of the General Algebraic Matrix. Bull. Amer. Math. Soc. 26, 394–395 (1920)
Neeb, K.-H.: Towards a Lie theory of locally convex groups. Japan. J. Math. 1(2), 291–468 (2006)
Nicolaescu, L.: Notes on Seiberg-Witten theory. American Mathematical Society, Providence, R.I (2000)
Oh, Y.-G.: Symplectic Topology and Floer Homology, Cambridge University Press, Cambridge, 420 pp (2015)
Oh, Y.-G., Ohta, H., Ono, K., Fukaya, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part II. Studies in Advanced Mathematics, American Mathematical Society (2009)
Ortega, J.P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction, Birkhäuser, Boston (2004)
Palais, R.S.: On the Existence of Slices for Actions of Non-Compact Lie Groups. Ann. of Math. 73(2), 295–323 (1961)
Palais, R.S.: Foundations of global non-linear analysis. Benjamin, New York (1968)
Penrose, R.: A generalized inverse for matrices. Math. Proc. Cambridge Philos. Soc. 51(03), 406–413 (1955)
Pflaum, M.J., Wilkin, G.: Equivariant control data and neighborhood deformation retractions (2017). arXiv:1706.09539
Rao, C.R.: A Note on a Generalized Inverse of a Matrix with Applications to Problems in Mathematical Statistics. J. Roy. Statist. Soc. Ser. B 24(1), 152–158 (1962)
Rudolph, G., Schmidt, M., Volobuev, I.P.: Classification of gauge orbit types for SU(n)-gauge theories. Math. Phys. Anal. Geom. 5(3), 201–241 (2002). arXiv:0003044 [math-ph]
Rudolph, G., Schmidt, M., Volobuev, I.P.: On the gauge orbit space stratification: a review. J. Phys. A: Math. Gen. 35(28), R1–R50 (2002). arXiv:0203027 [hep-th]
Rudolph, G., Schmidt, M., Volobuev, I.P.: Partial ordering of gauge orbit types for SUn-gauge theories. J. Geom. Phys. 42.1–42.2, 106–138 (2002). arXiv:0009018 [math-ph]
Rudolph, G., Schmidt, M: Differential Geometry and Mathematical Physics, Part II: Fibre Bundles, Topology and Gauge Fields, Springer (2017)
Schaefer, H.H.: Über singuläre Integralgleichungen und eine Klasse von Homomorphismen in lokalkonvexen Räumen. Math. Z. 66(1), 147–163 (1956)
Schaefer, H.H.: On the Fredholm alternative in locally convex linear spaces. Studia Math. 18(3), 229–245 (1959)
Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(\text{N}=2\) supersymmetric Yang-Mills theory. Nuclear Physics B 426(1), 19–52 (1994)
Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \(\text{N} = 2\) supersymmetric QCD. Nuclear Physics B 431(3), 484–550 (1994)
Sjöstrand, J., Zworski, M.: Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier (Grenoble) 57(7), 2095–2141 (2007)
Smale, S.: An Infinite Dimensional Version of Sard’s Theorem. Amer. J. Math. 87(4), 861–866 (1965)
Subramaniam, T.N.: Slices for the Actions of Smooth Tame Lie Groups. PhD Thesis, Brandeis University (1984)
Subramaniam, T.N.: Slices for actions of infinite-dimensional groups. In: Differential Analysis in Infinite Dimensional Spaces, vol. 54, American Mathematical Society, pp. 65–77 (1986)
Taubes, C.H.: Self-dual Yang-Mills connections on non-self-dual 4-manifolds. J. Differential Geom. 17(1), 139–170 (1982)
Teichmann, J.: A Frobenius Theorem on Convenient Manifolds. Monatshefte für Mathematik 134(2), 159–167 (2001)
Yang, D.: A Choice-independent Theory of Kuranishi Structures and the Polyfold–kuranishi Correspondence. PhD Thesis. New York University (2014)
