Symplectic reduction at zero angular momentum

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

Symplectic reduction at zero angular momentum

Joshua Cape ^1, , Hans-Christian Herbig ^2, and Christopher Seaton ^3,

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

Abstract
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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.

Keywords: moment map, rational singularities., $O_n$-representation, angular momentum, Symplectic reduction.

Mathematics Subject Classification: Primary: 53D20, 13A50; Secondary: 57S15, 37J15, 20G2.

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.
[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/.

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.
[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/.

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