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Representation of Geometric Objects by Path Integrals

桑田, 健 北海道大学

2022.03.24

概要

In this thesis, we study two problems by using physical models with supersymmetry and their path integral. Since this thesis deals with these topics from a common perspective, but in different ways, we have divided this thesis into two parts. One is deriving fixed-point theorem using path integral [25]. The other is the Euler number and Mathai-Quillen formalism in the Grassmann manifold [20]. This thesis is based on [20], [25] and [28].

In the part I, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from CP to a K¨ahler manifold M, with potential terms induced from a holomorphic vector field K on M [25]. The Bott residue formula represents the intersection number of Chern classes of holomorphic vector bundles on M as the sum of contributions from fixed point sets of K on M. Our strategy is to represent the integral of differential form on M by a correlation function and show that the correlation function is obtained by collecting contributions from the zero set of K. It is realized by showing the invariance of correlation function for the parameter of potential terms. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model.

In the part II, we provide a recipe for computing Euler number of Grassmann manifold G(k, N) by using Mathai-Quillen formalism [33] and Atiyah-Jeffrey construction [3]. Especially, we construct the path integral representation of Euler number of G(k, N) [20]. As a by-product, we construct free fermion realization of cohomology ring of G(k, N). It means that the cohomology ring of G(k, N) can be represented by fermionic fields that appear in our model. As an application, we calculate some integrals of cohomology classes by using fermion integrals [28]

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参考文献

[1] A. Alekseev, Notes on Equivariant localization, Vienna, Preprint ESI 744 (1999).

[2] L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. 90, no. 2, 161-173 (1983).

[3] M. F. Atiyah, L. Jeffrey, Topological Lagrangians and cohomology, J. Geom. Phys., 7, 119-136 (1990).

[4] C. Beasley, E. Witten, Residues and World-Sheet Instantons, arxiv:hep-th/0304115v1, (2003). (JHEP 0310:065,2003, DOI: https://doi.org/10.1088/1126-6708/2003/10/065)

[5] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag Berlin Heidelberg (1992).

[6] M. Blau, The Mathai–Quillen formalism and topological field theory, J. Geom. Phys., 11, 95-127 (1993).

[7] R. Bott, Vector fields and characteristic numbers, Michigan Math.J., 14, 231-244 (1967).

[8] R. Bott. A residue formula for holomorphic vector fields, J.Differential Geometry, 1, 311-330 (1967).

[9] R. Bott, L. Tu, Differential Forms in Algebraic Topology, (1st ed.), Springer-Verlag New York, (1982).

[10] U. Bruzzo, V. Rubtsov, On localization in holomorphic equivariant cohomology, arXiv:0910.2019v4, (2012). (centr.eur.j.math. 10, 1442–1454 (2012). DOI: https://doi.org/10.2478/s11533-012-0054-2)

[11] N. Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP1 into Gr(C n ), arXiv:hep-th/9808170, (1998).

[12] G. Camano-Garcia, Statistics on Stiefel manifolds, Retrospective Theses and Dissertations, Paper 1493, (2006).

[13] D. Friedan, P. Windey, Supersymmetric derivation of the Atiyah-Singer index and the chiral anomaly, Nuclear Phys. B 235, no. 3, 395-416 (1984).

[14] K. Fujii, Introduction to Grassmann manifolds and quantum computation, J. Appl. Math., 2, 371-405 (2002).

[15] K. Fujii, T. Kashiwa, S. Sakoda, Coherent states over Grassmann manifolds and the WKB exactness in path integral, Journal of Mathematical Physics, 37,567 (1996).

[16] W. Fulton, Intersection Theory, Springer-Verlag Berlin Hidelberg(1998).

[17] P. Griffiths; J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics. Wiley–Interscience, New York (1978).

[18] K. Hori, S. Katz, A. Klemn, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry, Clay Mathematics Monographs 1, American Mathematical Society, Providence, Clay Mathematics Institute, Cambridge, (2003).

[19] S. Imanishi, Computation of Euler number for the Grassmann manifold G(2, N) via Mathai-Quillen formalism, (in Japanese), Unpublished Master’s Thesis Department of Mathematics Graduate School of Science, Hokkaido University (2019).

[20] S. Imanishi, M. Jinzenji, K. Kuwata, Evaluation of Euler Number of Complex Grassmann Manifold G(k, N) via Mathai-Quillen Formalism, preprint, arXiv:2108.13623 [hep-th], (2021). (DOI: https://doi.org/10.48550/arXiv.2108.13623)

[21] T.Ikeda, H.Naruse, Modern Schubert calculus, from the special polynomial theory’s point of view (in Japanese, printed in Japan), Sugaku, 63(3), 313-337 (2011).

[22] M. Jinzenji, Introduction to Mirror Symmetry for Mathematical Physics ( For the Geometric Interpretation of Classical Mirror Symmetry), (in Japanese, printed in Japan), Saiensu sha (2014).

[23] M. Jinzenji, Classical Mirror Symmetry, SpringerBriefs in Mathematical Physics, 29, Springer, Singapore, (2018).

[24] M. Jinzenji, Introduction to Complex Geometry for Physics, ( manifolds, differential forms, vector bundles, sheafs, complex structures and its deformation), (in Japanese, printed in Japan), Saiensu sha (2019).

[25] M. Jinzenji, K. Kuwata, Holomorphic Vector Field and Topological Sigma Model on CP 1 World Sheet, International Journal of Modern Physics A, Vol. 35, No. 30, 2050192 (35 pages), (2020). (DOI: https://doi.org/10.1142/S0217751X20501924)

[26] S. Katz, (translation by Y. Shimizu), Enumarative Geometry and String Theory, (in Japanese, printed in Japan), Nippon hyouron sha (2011).

[27] K. Kuwata, Holomorphic vector field and topological sigma model, (in Japanese), Unpublished Master’s Thesis Department of Mathematics Graduate School of Science, Hokkaido University (2019).

[28] K. Kuwata, Schubert cycles and Grassmann variables, preprint, arXiv:2110.15940

[math.AG], (2021). (DOI: https://doi.org/10.48550/arXiv.2110.15940)

[29] S. Kobayashi, Complex Geometry 1 (Iwanami courses foundations of modern mathematics), (in Japanese, printed in Japan), Iwanami shoten (1997).

[30] J. M. F. Labastida, C. Lozano, Lectures on topological quantum field theory, AIP Conf. Proc., 419, 54-93 (1998), arXiv:hep-th/9709192.

[31] J.M.F.Labastida, P.M.Llatas, Potentials for topological sigma models, Physics Letters B 271(1991), 101-108.

[32] K. Liu. Holomorphic equivariant cohomology, Math.Ann. 303, 125-148 (1995).

[33] V. Mathai, D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology, 25, 85-110 (1986).

[34] M. Nakahara, (translation by M. Nakahara and K. Sakuma), Geometry, topology and physics I, (in Japanese, printed in Japan), Pearson Education Japan (2000).

[35] Richard J. Szabo, Equivariant Localization of Path Integrals, arXiv:hepth/9608068v1, (1996).

[36] S. Li, Supersymmetric Quantum Mechanics and Lefschetz fixed-point formula, arXiv:hep-th/0511101, (2005).

[37] A. Sako, Supersymmetric Gauge Theory and Geometry: Nonperturbative approach, (in Japanese, printed in Japan), Nippon hyoron sha (2007).

[38] E. Witten, Supersymmetry and Morse Theory, J.Differential Geometry 17(1982), 661-692.

[39] E. Witten, Topological quantum field theory, Commun.Math. Phys., 117, 353-386 (1988).

[40] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, arXiv:hep-th/9312104, (1993).

[41] W. Zhang, Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics-Vol.4, World Scientific Publishing (2001).

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