Article Contents
Article Contents

# Symplectic reduction at zero angular momentum

• We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $O_n$ on $k$ copies of $T^\ast\mathbb{R}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $O_n$- and $SO_n$-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when $n \leq k$, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of $k$, the ring of regular functions on the symplectic quotient is graded Gorenstein.
Mathematics Subject Classification: Primary: 53D20, 13A50; Secondary: 57S15, 37J15, 20G20.

 Citation:

•  [1] J. M. Arms, M. J. Gotay and G. Jennings, Geometric and algebraic reduction for singular momentum maps, Adv. Math., 79 (1990), 43-103.doi: 10.1016/0001-8708(90)90058-U. [2] A. Beauville, Symplectic singularities, Invent. Math., 139 (2000), 541-549.doi: 10.1007/s002229900043. [3] L. Bos and M. J. Gotay, Reduced canonical formalism for a particle with zero angular momentum, in XIIIth International Colloquium on Group Theoretical Methods in Physics (College Park, Md., 1984), World Sci. Publishing, Singapore, 1984, 83-91. [4] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., 88 (1987), 65-68.doi: 10.1007/BF01405091. [5] D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. [6] H. Derksen and G. Kemper, Computational Invariant Theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002.doi: 10.1007/978-3-662-04958-7. [7] C. Farsi, H.-C. Herbig and C. Seaton, On orbifold criteria for symplectic toric quotients, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), Paper 032, 33pp.doi: 10.3842/SIGMA.2013.032. [8] H. Flenner, Rationale quasihomogene Singularitäten, Arch. Math. (Basel), 36 (1981), 35-44.doi: 10.1007/BF01223666. [9] M. J. Gotay, Reduction of homogeneous Yang-Mills fields, J. Geom. Phys., 6 (1989), 349-365.doi: 10.1016/0393-0440(89)90009-0. [10] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, 2012. Available from: http://www.math.uiuc.edu/Macaulay2/. [11] H.-C. Herbig, D. Herden and C. Seaton, On compositions with $x^2/(1-x)$, Proc. Amer. Math. Soc., 143 (2015), 4583-4589.doi: 10.1090/proc/12806. [12] H.-C. Herbig and G. W. Schwarz, The Koszul complex of a moment map, J. Symplectic Geom., 11 (2013), 497-508.doi: 10.4310/JSG.2013.v11.n3.a9. [13] H.-C. Herbig, G. W. Schwarz and C. Seaton, When is a symplectic quotient an orbifold?, Adv. Math., 280 (2015), 208-224.doi: 10.1016/j.aim.2015.04.016. [14] H.-C. Herbig and C. Seaton, An impossibility theorem for linear symplectic circle quotients, Rep. Math. Phys., 75 (2015), 303-331.doi: 10.1016/S0034-4877(15)00019-1. [15] H.-C. Herbig and C. Seaton, The Hilbert series of a linear symplectic circle quotient, Exp. Math., 23 (2014), 46-65.doi: 10.1080/10586458.2013.863745. [16] J. Huebschmann, Singularities and Poisson geometry of certain representation spaces, in Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001, 119-135. [17] J. Huebschmann, Kähler spaces, nilpotent orbits, and singular reduction, Mem. Amer. Math. Soc., 172 (2004), vi+96pp.doi: 10.1090/memo/0814. [18] C. Huneke, Tight closure, parameter ideals, and geometry, in Six Lectures on Commutative Algebra, Mod. Birkhäuser Class., Birkhäuser Verlag, Basel, 2010, 187-239.doi: 10.1007/978-3-0346-0329-4_3. [19] G. Kempf and L. Ness, The length of vectors in representation spaces, in Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., 732, Springer, Berlin, 1979, 233-243. [20] F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math., 77 (1984), 547-552.doi: 10.1007/BF01388838. [21] E. Lerman, R. Montgomery and R. Sjamaar, Examples of singular reduction, in Symplectic Geometry, London Math. Soc. Lecture Note Ser., 192, Cambridge Univ. Press, Cambridge, 1993, 127-155. [22] K. McGerty and T. Nevins, Derived equivalence for quantum symplectic resolutions, Selecta Math. (N.S.), 20 (2014), 675-717.doi: 10.1007/s00029-013-0142-6. [23] C. Procesi and G. Schwarz, Inequalities defining orbit spaces, Invent. Math., 81 (1985), 539-554.doi: 10.1007/BF01388587. [24] G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 37-135. [25] G. W. Schwarz, The topology of algebraic quotients, in Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988), Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989, 135-151. [26] G. W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4), 28 (1995), 253-305. [27] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2), 141 (1995), 87-129.doi: 10.2307/2118628. [28] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2), 134 (1991), 375-422.doi: 10.2307/2944350. [29] R. P. Stanley, Hilbert functions of graded algebras, Advances in Math., 28 (1978), 57-83.doi: 10.1016/0001-8708(78)90045-2. [30] R. Terpereau, Schémas de Hilbert Invariants et Théorie Classique Des Invariants, Thesis (Ph.D.)-Université de Grenoble, 2012. Available from: arXiv:1211.1472. [31] È. B. Vinberg and V. L. Popov, Invariant Theory, in Algebraic Geometry. IV, A translation of Algebraic Geometry, 4 (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [MR1100483], Translation edited by A. N. Parshin and I. R. Shafarevich, Encyclopaedia of Mathematical Sciences, 55, Springer-Verlag, Berlin, 1994, 123-278.doi: 10.1007/978-3-662-03073-8. [32] K. Watanabe, Certain invariant subrings are Gorenstein. I, Osaka J. Math., 11 (1974), 1-8. [33] K. Watanabe, Certain invariant subrings are Gorenstein. II, Osaka J. Math., 11 (1974), 379-388. [34] K. Watanabe, Rational singularities with $k^*$-action, in Commutative Algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math., 84, Dekker, New York, 1983, 339-351. [35] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. [36] Wolfram Research, Mathematica edition: Version 7.0, http://www.wolfram.com/mathematica/.