Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Về phương trình Einstein trên các đa tạp và siêu đa tạp
Tóm tắt
Các phương trình Einstein (EE) là những điều kiện nhất định trên tensor Riemann trong không gian Minkowski thực M. Trong hình ảnh twistor, sau khi phức hóa và thu gọn, M trở thành Grassmannian $$Gr_2^4$$ của các không gian con 2-dimension trong không gian phức 4-dimension. Ở đây, chúng tôi trả lời cho những miền cổ điển nào được coi là các đa tạp có cấu trúc G mà có thể áp dụng các điều kiện tương tự như EE theo một nghĩa nào đó. Cuộc điều tra ở trên có tương đương trên các siêu miền: một đại lượng tương tự như tensor Riemann được định nghĩa cho bất kỳ siêu đa tạp nào có cấu trúc G với bất kỳ siêu nhóm Lie nào G. Chúng tôi cũng suy diễn các tương đương tương tự của EE trên các siêu đa tạp. Các tương đương của chúng tôi về EE không phải là những gì các nhà vật lý coi là SUGRA (siêu trọng lực), để biết về SUGRA xem [16, 34].
Từ khóa
Tài liệu tham khảo
Arnold V, Mathematical Methods of Classical Mechanics, Springer, 1980.
Berezin F, Representations of the Supergroup U(p, q), Funkcional. Anal. i Prilozhen. 10, Nr. 3 (1976), 70–71 (in Russian); Analysis with Anticommuting Variables, Kluwer, 1987.
Bernstein J and Leites D, Irreducible Representations of Type Q, Odd Trace and Odd Determinant, C. R. Acad. Bulg. Sci. 35, Nr. 3 (1982), 285–286.
Castellani L, D’Auria R and Fré P, Supergravity and Superstrings. A Geometric Perspective, Vol. 1–3, World Scientific, Teaneck, NJ, 1991.
Cheng S and Kac V, Generalized Spencer Cohomology and Filtered Deformations of Z-graded Lie Superalgebras, Adv. Theor. Math. Phys. 2 (1998), Nr. 5, 1141–1182.
Cheng S and Kac V, Structure of Some Z-graded Lie Superalgebras of Vector Fields, Transformation Groups 4 (1999), 219–272.
Deligne P, Etingof P, Freed D, Jeffrey L, Kazhdan D, Morgan J, Morrison D and Witten E (Editors), Quantum Fields and Strings: a Course for Mathematicians, Vol. 1, 2, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997, AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999.
Feigin B L, Leites D A and Serganova V V, Kac–Moody Superalgebras, in Group Theoretical Methods in Physics, Zvenigorod, 1982, Editors: Markov M and Manko V, Harwood Academic Publ., Chur, 1985, Vol. 1–3, 631–637.
Fuks (Fuchs) D, Cohomology of Infinite Dimensional Lie Algebras, Consultants Bureau, NY, 1986.
Galperin A, Ivanov E, Kalitzin S, Ogievetsky V and Sokatchev E, Unconstrained Off-Shell N = 3 Supersymmetric Yang–Mills Theory, Classical Quantum Gravity 2, Nr. 2 (1985), 155– 166; Corrigendum: Unconstrained N = 2 matter, Yang–Mills and Supergravity Theories in Harmonic Superspace, Classical Quantum Gravity 2, Nr. 1 (1985), 127; Unconstrained N = 2 Matter, Yang–Mills and Supergravity Theories in Harmonic Superspace, Classical Quantum Gravity 1, Nr. 5 (1984), 469–498.
Gomis J, París J and Samuel S, Antibracket, Antifields and Gauge-Theory Quantization, Phys. Rep. 259, Nr. 1–2 (1995), 145 pp.
Goncharov A, Infinitesimal Structures Related to Hermitian Symmetric Spaces, Funct. Anal. Appl. 15, Nr. 3 (1981), 23–24 (in Russian); a detailed version: Generalized Conformal Structures on Manifolds, in [28, Vol. 11] and Selecta Math. Soviet. 6, Nr. 4 (1987), 307–340.
Grozman P and Leites D, Supergravities and N-Extended Minkowski Superspaces for Any N, in Supersymmetries and Quantum Symmetries, Proc. International Conference in Memory of V Ogievetsky, June 1997, Editors: Wess J and Ivanov E, Lecture Notes in Physics, Vol. 524, Springer, 1999, 58–67.
Grozman P and Leites D, Mathematica-aided Study of Lie Algebras and their Cohomology. From Supergravity to Ballbearings and Magnetic Hydrodynamics, in The Second International Mathematica Symposium, Editor: Keränen V, Rovaniemi, 1997, 185–192.
Grozman P and Leites D, Defining Relations for Lie Superalgebras with Cartan Matrix, hep-th 9702073; Czech. J. Phys. 51, Nr. 1 (2001), 1–22.
Grozman P and Leites D, hep-th 0202115; An Unconventional Supergravity, in Proc. NATO Advanced Research Workshop “Noncommutative Structures in Mathematics and Physics”, Kyiv, Ukraine, September 24–28, 2000, Editors: Duplij S and Wess J, Kluwer, 2001, 41–48.
Grozman P, Leites D and Poletaeva E, Analogues of the Riemann, Penrose and Nijenhuis Tensors for Supermanifolds, MPIM-Bonn, preprint, 2002, to appear.
Grozman P, Leites D and Shchepochkina I, Lie Superalgebras of String Theories, hep-th 9702120; Acta Mathematica Vietnamica 26, Nr. 1 (2001), 27–63.
Shchepochkina I, The Analogs of the Riemann Tensor for Exceptional Structures on Supermanifolds, to appear.
Guillemin V, The Integrability Problem for G-Structures, Trans. of American Math. Soc. 116 (1964), 544–560.
Helgason S, Differential Geometry, Lie Groups and Symmetric Spaces, Acad. Press, 1978.
Kac V G, Lie Superalgebras, Adv. Math. 26 (1977), 8–96.
Kac V G, Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras, Commun. Alg. 5, Nr. 13 (1977), 1375–1400; Hogben L and Kac V, Erratum: Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras, Commun. Algebra 11, Nr. 10 (1983), 1155–1156.
Kac V G, Classification of Infinite-Dimensional Simple Linearly Compact Lie Superalgebras, Adv. Math. 139, Nr. 1 (1998), 1–55; Classification of Infinite-Dimensional Simple Groups of Supersymmetry and Quantum Field Theory, math.QA/9912233. GAFA 2000 (Tel Aviv, 1999); Geom. Funct. Anal. Special Volume, Part I (2000), 162–183.
Lambe L and Seiler W, Differential Equations, Spencer Cohomology, and Computing Resolutions, preprint-2002, Dept. of Math., Mannheim University.
Leites D, New Lie Superalgebras and Mechanics, Soviet. Math. Dokl. 18, Nr. 5 (1977), 1277– 1280.
Leites D, Introduction to Supermanifold Theory, Russian Math. Surveys, 33, Nr. 1 (1980), 1–55; an expanded version: Supermanifold Theory, Karelia Branch of the USSR Acad. of Sci., Petrozavodsk, 1983 (in Russian); a still more expanded version in English: [28].
Leites D (Editor), Seminar on Supermanifolds, Reports of Dept. of Math. of Stockholm Univ., Vol. 1–34, 1986–1992.
Leites D, Quantization and Supermanifolds. Appendix 3, in Schrödinger Equation, Editors: Berezin F and Shubin M, Kluwer, Dordrecht, 1991, 483–522.
Leites D, Representations of Lie Superalgebras, Theor. Math. Phys. 32, Nr. 2 (1982), 225– 228; Indecomposable Representations of Lie Superalgebras, math.RT 0202184; in Memorial Volume Dedicated to Misha Saveliev and Igor Luzenko, Editor: Sissakian A N, JINR, Dubna, 2000, 126–131.
Leites D, On Unconventional Integrations on Supermanifolds and Cross Ratio on Classical Superspaces, math.RT 0202194; in Supersymmetries and Quantum Symmetries, SQS’01, 22– 26 September, 2001, Karpacz, Editors: Krivonos S and Pashnev A, Elsever, to appear.
Leites D, The Riemann Tensor for Nonholonomic Manifolds. math.RT/0202213; Homology, Homotopy and Applications 4, Nr. 2 (2002), 397–407.
Leites D and Poletaeva E, Analogues of the Riemannian Structure on Supermanifolds, in Proceedings of the International Conference on Algebra,, August 1989, Novosibirsk, Editors: Bokut’ L A, Ershov Yu L and Kostrikin A I, Contemporary Math. AMS, 1992, Part 1, 603–612.
Leites D and Poletaeva E, Supergravities and Contact Type Structures on Supermanifolds, in Proceedings of the Second International Conference on Algebra, Barnaul, 1991, Editors: Bokut’ L A, Kostrikin A I, and Kutateladze S S, Contemp. Math., Vol. 184, Amer. Math. Soc., Providence, RI, 1995, 267–274.
Leites D and Shchepochkina I, Classification of Simple Vectorial Lie Superalgebras, to appear.
Leites D, Serganova V and Vinel G, Classical Superspaces and Related Structures, in Differential Geometric Methods in Theoretical Physics, Proc. DGM-XIX, Rapallo, 1990, Editors: Bartocci C, Bruzzo U and Cianci R, Lecture Notes in Phys., Vol. 375, Springer, Berlin, 1991, 286–297.
Manin Yu, Gauge Field Theory and Complex Geometry, 2nd ed., Springer, 1997.
Onishchik A L and Vinberg É B, Seminar on Algebraic Groups and Lie Groups, Springer, Berlin, 1990.
Poletaeva E, Structure Functions on the Usual and Exotic Symplectic and Periplectic Supermanifolds, in Differential Geometric Methods in Theoretical Physics, Proc. DGM-XIX, Rapallo, 1990, Editors: Bartocci C, Bruzzo U and Cianci R, Lecture Notes in Phys., Vol. 375, Springer, Berlin, 1991, 390–395.
Poletaeva E, Penrose’s Tensors on Super-Grassmannians, I. II, Math. Scand. 72, Nr. 2 (1993), 161–190, 191–211.
Poletaeva E, Analogues of Riemann Tensors for the Odd Metric on Supermanifolds, Acta Appl. Math. 31, Nr. 2 (1993), 137–169.
Rosly A A and Schwarz A S, Geometry of N = 1 Supergravity, Comm. Math. Phys. 95, Nr. 2 (1984), 161–184; Comm. Math. Phys. 96, Nr. 3 (1984), 285–309.
Schwarz A S, Supergravity, Complex Geometry and G-Structures, Comm. Math. Phys. 87, Nr. 1 (1982/83), 37–63.
Serganova V, Classification of Real Simple Lie Superalgebras and Symmetric Superspaces, Funct. Anal. Appl. 17, Nr. 3 (1983), 46–54.
Sternberg S, Lectures on Differential Geometry, 2nd edition, Chelsey, 1985.
Shander V N, Analogues of the Frobenius and Darboux Theorems for Supermanifolds, C. R. Acad. Bulgare Sci. 36, Nr. 3 (1983), 309–312.
Shchepochkina I, hep-th 9702121; Five Simple Exceptional Lie Superalgebras of Vector Fields, Funktsional. Anal. i Prilozhen. 33 (1999), Nr. 3, 59–72 (in Russian); translation in Funct. Anal. Appl. 33, Nr. 3 (1999), 3, 208–219; The Five Exceptional Simple Lie Superalgebras of Vector Fields and their Fourteen Regradings, Represent. Theory 3 (1999), 373–415.
Vershik A, Classical and Nonclassical Dynamics with Constraints, in Global Analysis — Studies and Applications, Editors: Borisovich Yu and Gliklikh Yu, Lecture Notes in Mathematics, Vol. 1108, Springer-Verlag, Berlin, 1984, 278–301.
Weil A, Théorie des points proches sur les variétés différentiables, Géométrie différentielle, Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, 1953, 111–117 (in French).
Witten E, An Interpretation of Classical Yang–Mills Theory, Phys. Lett. B77, Nr. 4–5 (1978), 394–400; Grand Unification with and without Supersymmetry, in Introduction to Supersymmetry in Particle and Nuclear Physics, Mexico City, 1981, Plenum, New York, 1984, 53–76.
Wess J and Zumino B, Supergauge Transformations in Four Dimensions, Nuclear Phys. B70 (1974), 39–50; Wess J, Supersymmetry-Supergravity, in Topics in Quantum Field Theory and Gauge Theories, Editor: de Azcárraga J A, Proc. VIII Internat. GIFT Sem. Theoret. Phys., Salamanca, 1977, Lecture Notes in Phys., Vol. 77, Springer, Berlin – New York, 1978, 81–125; Wess J and Zumino B, Superspace Formulation of Supergravity, Phys. Lett. B66, Nr. 4 (1977), 361–364; Wess J, Supersymmetry/Supergravity, in Concepts and Trends in Particle Physics, Schladming, 1986, Editors: Latal H and Mitter H, Schladming, 1986, Springer, Berlin, 1987, 29–58; Wess J, Introduction to Supersymmetric Theories, in Frontiers in Particle Physics ‘83, Dubrovnik, 1983, Editors: Šijački Dj, Bilić N, Dragović B and Popović D, World Sci. Publishing, Singapore, 1984, 104–131; Wess J and Bagger J, Supersymmetry and Supergravity, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1983.