# American Institute of Mathematical Sciences

March  2016, 8(1): 13-34. doi: 10.3934/jgm.2016.8.13

## Symplectic reduction at zero angular momentum

 1 Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 N. Charles St, Baltimore, MD 21218, United States 2 Departamento de Matemática Aplicada, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, CEP: 21941-909 - Rio de Janeiro, Brazil 3 Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, United States

Received  April 2015 Revised  August 2015 Published  February 2016

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.
Citation: Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13
##### References:
 [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.  Google Scholar [2] A. Beauville, Symplectic singularities, Invent. Math., 139 (2000), 541-549. doi: 10.1007/s002229900043.  Google Scholar [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.  Google Scholar [4] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., 88 (1987), 65-68. doi: 10.1007/BF01405091.  Google Scholar [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.  Google Scholar [6] H. 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Soc., 143 (2015), 4583-4589. doi: 10.1090/proc/12806.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Huebschmann, Kähler spaces, nilpotent orbits, and singular reduction, Mem. Amer. Math. Soc., 172 (2004), vi+96pp. doi: 10.1090/memo/0814.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math., 77 (1984), 547-552. doi: 10.1007/BF01388838.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] C. Procesi and G. Schwarz, Inequalities defining orbit spaces, Invent. Math., 81 (1985), 539-554. doi: 10.1007/BF01388587.  Google Scholar [24] G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 37-135.  Google Scholar [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.  Google Scholar [26] G. W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4), 28 (1995), 253-305.  Google Scholar [27] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2), 141 (1995), 87-129. doi: 10.2307/2118628.  Google Scholar [28] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2), 134 (1991), 375-422. doi: 10.2307/2944350.  Google Scholar [29] R. P. Stanley, Hilbert functions of graded algebras, Advances in Math., 28 (1978), 57-83. doi: 10.1016/0001-8708(78)90045-2.  Google Scholar [30] R. Terpereau, Schémas de Hilbert Invariants et Théorie Classique Des Invariants, Thesis (Ph.D.)-Université de Grenoble, 2012. Available from: arXiv:1211.1472. Google Scholar [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.  Google Scholar [32] K. Watanabe, Certain invariant subrings are Gorenstein. I, Osaka J. Math., 11 (1974), 1-8.  Google Scholar [33] K. Watanabe, Certain invariant subrings are Gorenstein. II, Osaka J. Math., 11 (1974), 379-388.  Google Scholar [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.  Google Scholar [35] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.  Google Scholar [36] Wolfram Research, Mathematica edition: Version 7.0,, , ().   Google Scholar

show all references

##### References:
 [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.  Google Scholar [2] A. Beauville, Symplectic singularities, Invent. Math., 139 (2000), 541-549. doi: 10.1007/s002229900043.  Google Scholar [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.  Google Scholar [4] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., 88 (1987), 65-68. doi: 10.1007/BF01405091.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] H. Flenner, Rationale quasihomogene Singularitäten, Arch. Math. (Basel), 36 (1981), 35-44. doi: 10.1007/BF01223666.  Google Scholar [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.  Google Scholar [10] D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,, 2012. Available from: , ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. Huebschmann, Kähler spaces, nilpotent orbits, and singular reduction, Mem. Amer. Math. Soc., 172 (2004), vi+96pp. doi: 10.1090/memo/0814.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] F. Kirwan, Convexity properties of the moment mapping. III, Invent. Math., 77 (1984), 547-552. doi: 10.1007/BF01388838.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] C. Procesi and G. Schwarz, Inequalities defining orbit spaces, Invent. Math., 81 (1985), 539-554. doi: 10.1007/BF01388587.  Google Scholar [24] G. W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 37-135.  Google Scholar [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.  Google Scholar [26] G. W. Schwarz, Lifting differential operators from orbit spaces, Ann. Sci. École Norm. Sup. (4), 28 (1995), 253-305.  Google Scholar [27] R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2), 141 (1995), 87-129. doi: 10.2307/2118628.  Google Scholar [28] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2), 134 (1991), 375-422. doi: 10.2307/2944350.  Google Scholar [29] R. P. Stanley, Hilbert functions of graded algebras, Advances in Math., 28 (1978), 57-83. doi: 10.1016/0001-8708(78)90045-2.  Google Scholar [30] R. Terpereau, Schémas de Hilbert Invariants et Théorie Classique Des Invariants, Thesis (Ph.D.)-Université de Grenoble, 2012. Available from: arXiv:1211.1472. Google Scholar [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.  Google Scholar [32] K. Watanabe, Certain invariant subrings are Gorenstein. I, Osaka J. Math., 11 (1974), 1-8.  Google Scholar [33] K. Watanabe, Certain invariant subrings are Gorenstein. II, Osaka J. Math., 11 (1974), 379-388.  Google Scholar [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.  Google Scholar [35] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.  Google Scholar [36] Wolfram Research, Mathematica edition: Version 7.0,, , ().   Google Scholar
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