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Symmetry and bifurcations of momentum mappings

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Abstract

The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.

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Abbreviations

(P,ω):

symplectic manifold

\(T_{x_0 } P\) :

tangent space toP atx 0εP

G,g:

Lie group, Lie algebra

g·x=Φ g (x):

action ofG onP

ξ P :

infinitesimal generator of the action onP corresponding toξεg

J:P→g*:

momentum mapping

\(dJ\left( {x_0 } \right):T_{x_0 } P \to \mathfrak{g}^ * \) :

differential ofJ atx 0

\(\mathbb{J}\) :

complex structure onP

\(S_{x_0 } \) :

slice for theG-action atx 0

\(I_{x_0 } \) :

isotropy group ofx 0; {g εG|gx 0=x 0}

\(\mathcal{L}_{x_0 } ,\mathfrak{s}_{x_0 } \) :

identity component of\(I_{x_0 } \), its Lie algebra [Eq. (10)]

«,»:

(weak) metric onP

〈,〉:

pairing between g* and g

\((,)_{x_0 } \) :

inner product of g* depending onx 0εP

\(dJ\left( {x_0 } \right)^ * :\mathfrak{g} \to T_{x_0 } P\) :

adjoint ofdJ(x 0) relative to 〈,〉 and «,»

\(dJ\left( {x_0 } \right)^\dag :\mathfrak{g}^ * \to T_{x_0 } P\) :

adjoint ofdJ(x 0) relative to (,) and «,»

\(T_{x_0 } P = Range\left[ {\mathbb{J} \circ dJ\left( {x_0 } \right)^ * } \right] \oplus Range\left[ {dJ\left( {x_0 } \right)^ * } \right] \oplus \left[ {\ker \left( {dJ\left( {x_0 } \right) \circ \mathbb{J}} \right) \cap \ker dJ\left( {x_0 } \right)} \right]\) :

Moncrief's decomposition

ℙ:g*→RangeddJ(x 0):

orthogonal projection

\(\mathcal{C} = J^{ - 1} \left( 0 \right)\) :

zero set ofJ (or constraint set)

\(\mathcal{C}_\mathbb{P} = \left( {\mathbb{P}J} \right)^{ - 1} \left( 0 \right)\) :

zero set of ℙ∘J

\(N_{x_0 } \) :

x's with the same orbit type asx 0

\(\mathcal{N}_{x_0 } \) :

\(N_{x_0 } \cap S_{x_0 } \) (Lemma 1)

\(\mathfrak{g}_{x_0 }^ * \) :

elements in g* with same symmetry asx 0 (Lemma 5)

f :

\(f = \left( {Id - \mathbb{P}} \right) \circ J:\mathcal{C}_\mathbb{P} \cap S_{x_0 } \to \ker dJ\left( {x_0 } \right)^\dag \) (Lemma 8)

Δ=dJ(x 0)∘dJ(x 0)† :

“elliptic” operator associated withJ

G=Δ−1∘ℙ:

Greens' function for Δ

F(x)=x+dJ(x 0)†∘G∘Q(h),h=x−x 0 :

Kuranishi map (Lemma 9)

\(C_{x_0 } \) :

homogeneous cone associated withd 2 J(x 0) (Theorem 3)

:

orthogonal projection onto kerdJ(x 0) (Theorem 3)

\(\mathfrak{h}\), ℋ:

a Lie subalgebra of\(\mathfrak{s}_{x_0 } \), its Lie group

:

points with symmetry (at least) ℋ (Theorem 4)

:

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Judith M. Arms

  2. Department of Mathematics, University of California, 94720, Berkeley, CA, USA

    Jerrold E. Marsden

  3. Department of Physics, Yale University, Yale Station, P.O. Box 2014, 06520, New Haven, CT, USA

    Vincent Moncrief

Authors
  1. Judith M. Arms
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  2. Jerrold E. Marsden
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  3. Vincent Moncrief
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Additional information

Communicated by A. Jaffe

This work was partially supported by the National Science Foundation. The second author was supported by a Killam Visiting fellowship at the University of Calgary during the completion of the paper

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Arms, J.M., Marsden, J.E. & Moncrief, V. Symmetry and bifurcations of momentum mappings. Commun.Math. Phys. 78, 455–478 (1981). https://doi.org/10.1007/BF02046759

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