Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

Abstract

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

1. Introduction

Lie groups play an important role in physical systems both as phase spaces and as symmetry groups. Infinite-dimensional Lie groups occur in the study of dynamical systems with an infinite number of degrees of freedom such as PDEs and in field theories. For such infinite-dimensional dynamical systems, diffeomorphism groups and various extensions and variations thereof, such as gauge groups, loop groups, and groups of Fourier integral operators, occur as symmetry groups and phase spaces. Symmetries are fundamental for Hamiltonian systems. They provide conservation laws (Noether currents) and reduce the number of degrees of freedom, that is, the dimension of the phase space.

The topics selected for review aim to illustrate some of the ways infinite-dimensional geometry and global analysis can be used in mathematical problems of physical interest. The topics selected are the following.(1)Infinite-Dimensional Lie Groups.(2)Lie Groups as Symmetry Groups of Hamiltonian Systems.(3)Applications.(4)Gauge Theories, the Standard Model, and Gravity.(5)SUSY (supersymmetry).

2. Infinite-Dimensional Lie Groups

2.1. Basic Definitions

A general theory of infinite-dimensional Lie groups is hardly developed. Even Bourbaki [1] only develops a theory of infinite-dimensional manifolds, but all of the important theorems about Lie groups are stated for finite-dimensional ones.

An infinite-dimensional Lie group 𝒢 is a group and an infinite-dimensional manifold with smooth group operations 𝑚∶𝒢×𝒢→𝒢,𝑚(𝑔,ℎ)=𝑔⋅ℎ,𝐶^∞, (2.1)𝑖∶𝒢→𝒢,𝑖(𝑔)=𝑔⁻¹,𝐶^∞. (2.2) Such a Lie group 𝒢 is locally diffeomorphic to an infinite-dimensional vector space. This can be a Banach space whose topology is given by a norm ‖⋅‖, a Hilbert space whose topology is given by an inner product ⟨⋅,⋅⟩, or a Frechet space whose topology is given by a metric but not by a norm. Depending on the choice of the topology on 𝒢, we talk about Banach, Hilbert, or Frechet Lie groups, respectively.

The Lie algebra  𝔤 of a Lie group 𝒢 is defined as 𝔤={left invariant vector fields on 𝒢}≃𝑇_𝑒𝒢 (tangent space at the identity 𝑒). The isomorphism is given (as in finite dimensions) by𝜉∈𝑇_𝑒𝒢↦𝑋_𝜉∈𝔤,𝑋_𝜉(𝑔)∶=𝑇_𝑒𝐿_𝑔(𝜉), (2.3) and the Lie bracket on 𝔤 is induced by the Lie bracket of left invariant vector fields [𝜉,𝜂]=[𝑋_𝜉,𝑋_𝜂](𝑒),𝜉,𝜂∈𝑇_𝑒𝒢.

These definitions in infinite dimensions are identical with the definitions in finite dimensions. The big difference although is that infinite-dimensional manifolds, hence Lie groups, are not locally compact. For Frechet Lie groups, we have the additional nontrivial difficulty of the question how to define differentiability of functions defined on a Frechet space; see the study by Keller in [2]. Hence the very definition of a Frechet manifold is not canonical. This problem does not arise for Banach- and Hilbert-Lie groups; the differential calculus extends in a straightforward manner from ℝ^𝑛 to Banach and Hilbert spaces, but not to Frechet spaces.

2.2. Finite- versus Infinite-Dimensional Lie Groups

Infinite-dimensional Lie groups are NOT locally compact. This causes some deficiencies of the Lie theory in infinite dimensions. We summarize some classical results in finite dimensions which are NOT true in general in infinite dimensions as follows.(1)There is NO Implicit Function Theorem or Inverse Function Theorem in infinite dimensions! (except Nash-Moser-type theorems).(2)If 𝐺 is a finite-dimensional Lie group, the exponential map  exp∶𝔤→𝐺 is defined as follows. To each 𝜉∈𝔤, we assign the corresponding left invariant vector field 𝑋_𝜉 defined by (2.3). We take the flow 𝜑_𝜉(𝑡) of 𝑋_𝜉 and define exp(𝜉)=𝜑_𝜉(1). The exponential map is a local diffeomorphism from a neighborhood of zero in 𝔤 onto a neighborhood of the identity in 𝐺; hence exp defines canonical coordinates on the Lie group 𝐺. This is not true in infinite dimensions.(3)If 𝑓₁,𝑓₂∶𝐺₁→𝐺₂ are smooth Lie group homomorphisms (i.e., 𝑓_𝑖(𝑔⋅ℎ)=𝑓_𝑖(𝑔)⋅𝑓_𝑖(ℎ),𝑖=1,2) with 𝑇_𝑒𝑓₁=𝑇_𝑒𝑓₂, then locally 𝑓₁=𝑓₂. This is not true in infinite dimensions.(4)If 𝑓∶𝐺→𝐻 is a continuous group homomorphism between finite-dimensional Lie groups, then 𝑓 is smooth. This is not true in infinite dimensions.(5)If 𝔤 is any finite-dimensional Lie algebra, then there exists a connected finite-dimensional Lie group 𝐺 with 𝔤 as its Lie algebra; that is, 𝔤≃𝑇_𝑒𝐺. This is not true in infinite dimensions.(6)If 𝐺 is a finite-dimensional Lie group and 𝐻⊂𝐺 is a closed subgroup, then 𝐻 is a Lie subgroup (i.e., Lie group and submanifold). This is not true in infinite dimensions.(7)If 𝐺 is a finite-dimensional Lie group with Lie algebra 𝔤 and 𝔥⊂𝔤 is a subalgebra, then there exists a unique connected Lie subgroup 𝐻⊂𝐺 with 𝔥 as its Lie algebra; that is, 𝔥≃𝑇_𝑒𝐻. This is not true in infinite dimensions.

Some classical examples of finite-dimensional Lie groups are the matrix groups 𝐺𝐿(𝑛), 𝑆𝐿(𝑛), 𝑂(𝑛), 𝑆𝑂(𝑛), 𝑈(𝑛), 𝑆𝑈(𝑛), and 𝑆𝑝(𝑛) with smooth group operations given by matrix multiplication and matrix inversion. The Lie algebra bracket is the commutator [𝐴,𝐵]=𝐴𝐵−𝐵𝐴 with exponential map given by exp(𝐴)=∑^∞_𝑖=0(1/𝑖!)𝐴^𝑖=𝑒^𝐴.

2.3. Examples of Infinite-Dimensional Lie Groups

2.3.1. The Vector Groups 𝒢=(𝑉,+)

Let 𝑉 be a Banach space and take 𝒢=𝑉 with 𝑚(𝑥,𝑦)=𝑥+𝑦, 𝑖(𝑥)=−𝑥, and 𝑒=0, which makes 𝒢 into an Abelian Lie group; that is, 𝑚(𝑥,𝑦)=𝑚(𝑦,𝑥). For the Lie algebra we have 𝔤≃𝑇_𝑒𝑉≃𝑉. For 𝑢∈𝑇_𝑒𝑉 the corresponding left invariant vector field 𝑋_𝑢 is given by 𝑋_𝑢(𝑣)=𝑢, ∀𝑣∈𝑉; that is, 𝑋_𝑢=const. Hence the Lie algebra 𝔤=𝑉 with the trivial Lie bracket [𝑢,𝑣]=0 is Abelian. For the exponential map we get exp∶𝔤=𝑉→𝒢=𝑉,exp=𝑖𝑑_𝑉.

2.3.2. The General Linear Group 𝒢=(𝐺𝐿(𝑉),∘)

Let 𝑉 be a Banach space and 𝐿(𝑉,𝑉) the space of bounded linear operators 𝐴∶𝑉→𝑉. Then 𝐿(𝑉,𝑉) is a Banach space with the operator norm ‖𝐴‖=sup_{‖𝑥‖≤1}‖𝐴(𝑥)‖, and the group 𝒢=𝐺𝐿(𝑉) of all invertible elements is open in 𝐿(𝑉,𝑉). So 𝐺𝐿(𝑉) is a smooth Lie group with 𝑚(𝑓,𝑔)=𝑓∘𝑔,𝑖(𝑓)=𝑓⁻¹, and 𝑒=𝑖𝑑_𝑉. Its Lie algebra is 𝔤=𝐿(𝑉,𝑉) with the commutator bracket [𝐴,𝐵]=𝐴𝐵−𝐵𝐴 and exponential map exp𝐴=𝑒^𝐴.

2.3.3. The Abelian Gauge Group 𝒢=(𝐶^∞(𝑀),+)

Let 𝑀 be a finite-dimensional manifold and let 𝒢=𝐶^∞(𝑀) (smooth functions on 𝑀). With group operation being addition, that is, 𝑚(𝑓,𝑔)=𝑓+𝑔, 𝑖(𝑓)=−𝑓, and 𝑒=0. 𝒢 is an Abelian 𝐶^∞ (addition is smooth) Frechet Lie group with Lie algebra 𝔤=𝑇_𝑒𝐶^∞(𝑀)≃𝐶^∞(𝑀), with trivial bracket [𝜉,𝜂]=0, and exp=𝑖𝑑. If we complete these spaces in the 𝐶^𝑘-norm, 𝑘<∞ (denoted by 𝒢^𝑘), then 𝒢^𝑘 is a Banach-Lie group, and if we complete in the 𝐻^𝑠-Sobolev norm with 𝑠>(1/2)dim𝑀 then 𝒢^𝑠 is a Hilbert-Lie group.

2.3.4. The Abelian Gauge Group 𝒢=(𝐶^∞(𝑀,ℝ−{0}),⋅)

Let 𝑀 be a finite-dimensional manifold and let 𝒢=𝐶^∞(𝑀,ℝ−{0}), with group operation being multiplication; that is, 𝑚(𝑓,𝑔)=𝑓⋅𝑔, 𝑖(𝑓)=𝑓⁻¹, and 𝑒=1. For 𝑘<∞, 𝐶^𝑘(𝑀,ℝ−{0}) is open in 𝐶^∞(𝑀,ℝ), and if 𝑀 is compact, then 𝐶^𝑘(𝑀,ℝ−{0}) is a Banach-Lie group. If 𝑠>(1/2)dim𝑀, then 𝐻^𝑠(𝑀,ℝ−{0}) is closed under multiplication, and if 𝑀 is compact, then 𝐻^𝑠(𝑀,ℝ−{0}) is a Hilbert-Lie group.

2.3.5. Loop Group 𝒢=(𝐶^𝑘(𝑀,𝐺),⋅)

We generalize the Abelian example (see Section 2.3.4) by replacing ℝ−{0} with any finite-dimensional (non-Abelian) Lie group 𝐺. Let 𝒢=𝐶^𝑘(𝑀,𝐺) with pointwise group operations 𝑚(𝑓,𝑔)(𝑥)=𝑓(𝑥)⋅𝑔(𝑥), 𝑥∈𝑀, and 𝑖(𝑓)(𝑥)=(𝑓(𝑥))⁻¹, where “⋅” and “(⋅)⁻¹” are the operations in 𝐺. If 𝑘<∞ then 𝐶^𝑘(𝑀,𝐺) is a Banach-Lie group. Let 𝑔 denote the Lie algebra of 𝐺, then the Lie algebra of 𝒢=𝐶^𝑘(𝑀,𝐺) is 𝔤=𝐶^𝑘(𝑀,𝑔), with pointwise Lie bracket [𝜉,𝜂](𝑥)=[𝜉(𝑥),𝜂(𝑥)], 𝑥∈𝑀, the latter bracket being the Lie bracket in 𝑔. The exponential map exp∶𝑔→𝐺 defines the exponential map EXP∶𝔤=𝐶^𝑘(𝑀,𝑔)→𝒢=𝐶^𝑘(𝑀,𝐺), EXP(𝜉)=exp∘𝜉, which is a local diffeomorphism. The same holds for 𝐻^𝑠(𝑀,𝐺) if 𝑠>(1/2)dim𝑀.

Applications of these infinite-dimensional Lie groups are in gauge theories and quantum field theory, where they appear as groups of gauge transformations. We will discuss these in Section 5.

Special Case: 𝒢=(𝐶^𝑘(𝑆¹,𝐺),⋅)
As a special case of example mentioned in Section 2.3.5 we take 𝑀=𝑆¹, the circle. Then 𝒢=𝐶^𝑘(𝑆¹,𝐺)=ℒ^𝑘(𝐺) is called a loop group and 𝔤=𝐶^𝑘(𝑆¹,𝑔)=𝔩^𝑘(𝑔) is its loop algebra. They find applications in the theory of affine Lie algebras, Kac-Moody Lie algebras (central extensions), completely integrable systems, soliton equations (Toda, KdV, KP), and quantum field theory; see, for example, [3] and Section 5. Central extensions of loop algebras are examples of infinite-dimensional Lie algebras which need not have a corresponding Lie group.
Certain subgroups of loop groups play an important role in quantum field theory as groups of gauge transformations. We will discuss these in Section 2.4.4.

2.4. Diffeomorphism Groups

Among the most important “classical” infinite-dimensional Lie groups are the diffeomorphism groups of manifolds. Their differential structure is not the one of a Banach Lie group as defined above. Nevertheless they have important applications.

Let 𝑀 be a compact manifold (the noncompact case is technically much more complicated but similar results are true; see the study by Eichhorn and Schmid in [4]) and let 𝒢=Diff^∞(𝑀) be the group of all smooth diffeomorphisms on 𝑀, with group operation being composition; that is, 𝑚(𝑓,𝑔)=𝑓∘𝑔, 𝑖(𝑓)=𝑓⁻¹, and 𝑒=𝑖𝑑_𝑀. For 𝐶^∞ diffeomorphisms, Diff^∞(𝑀) is a Frechet manifold and there are nontrivial problems with the notion of smooth maps between Frechet spaces. There is no canonical extension of the differential calculus from Banach spaces (which is the same as for ℝ^𝑛) to Frechet spaces; see the study by Keller in [2]. One possibility is to generalize the notion of differentiability. For example, if we use the so-called 𝐶^∞_Γ differentiability, then 𝒢=Diff^∞(𝑀) becomes a 𝐶^∞_Γ Lie group with 𝐶^∞_Γ differentiable group operations. These notions of differentiability are difficult to apply to concrete examples. Another possibility is to complete Diff^∞(𝑀) in the Banach 𝐶^𝑘-norm, 0≤𝑘<∞, or in the Sobolev 𝐻^𝑠-norm, 𝑠>(1/2)dim𝑀. Then Diff^𝑘(𝑀) and Diff^𝑠(𝑀) become Banach and Hilbert manifolds, respectively. Then we consider the inverse limits of these Banach- and Hilbert-Lie groups, respectively:Diff^∞(𝑀)=lim_←Diff^𝑘(𝑀) (2.4) becomes a so-called ILB- (Inverse Limit of Banach) Lie group, or with the Sobolev topologiesDif𝑓^∞(𝑀)=lim_←Diff^𝑠(𝑀) (2.5) becomes a so-called ILH- (Inverse Limit of Hilbert) Lie group. See the study by Omori in [5] for details. Nevertheless, the group operations are not smooth, but have the following differentiability properties. If we equip the diffeomorphism group with the Sobolev 𝐻^𝑠-topology, then Diff^𝑠(𝑀) becomes a 𝐶^∞ Hilbert manifold if 𝑠>(1/2)dim𝑀 and the group multiplication𝑚∶Diff^𝑠+𝑘(𝑀)×Diff^𝑠(𝑀)→Diff^𝑠(𝑀) (2.6) is 𝐶^𝑘 differentiable; hence for 𝑘=0, 𝑚 is only continuous on Dif𝑓^𝑠(𝑀). The inversion𝑖∶Diff^𝑠+𝑘(𝑀)→Diff^𝑠(𝑀) (2.7) is 𝐶^𝑘 differentiable; hence for 𝑘=0, 𝑖 is only continuous on Diff^𝑠(𝑀). The same differentiability properties of 𝑚 and 𝑖 hold in the 𝐶^𝑘 topology.

The Lie algebra of Diff^∞(𝑀) is given by 𝔤=𝑇_𝑒Diff^∞(𝑀)≃Vec^∞(𝑀) being the space of smooth vector fields on 𝑀. Note that the space Vec(𝑀) of all vector fields is a Lie algebra only for 𝐶^∞ vector fields, but not for 𝐶^𝑘 or 𝐻^𝑠 vector fields if 𝑘<∞, 𝑠<∞, because one loses derivatives by taking brackets.

The exponential map on the diffeomorphism group is given as follows. For any vector field 𝑋∈Vec^∞(𝑀), take its flow 𝜑_𝑡∈Diff^∞(𝑀), then define EXP∶Vec^∞(𝑀)→Diff^∞(𝑀)∶𝑋↦𝜑₁, the flow at time 𝑡=1. The exponential map EXP is NOT a local diffeomorphism; it is not even locally surjective.

We see that the diffeomorphism groups are not Lie groups in the classical sense, but what we call nested Lie groups. Nevertheless they have important applications as we will see.

2.4.1. Subgroups of Diff^∞(𝑀)

Several subgroups of Diff^∞(𝑀) have important applications.

2.4.2. Group of Volume-Preserving Diffeomorphisms

Let 𝜇 be a volume on 𝑀 and 𝒢=Diff^∞_𝜇(𝑀)={𝑓∈Diff^∞(𝑀)∣𝑓^∗𝜇=𝜇} (2.8) the group of volume-preserving diffeomorphisms. Diff^∞_𝜇(𝑀) is a closed subgroup of Diff^∞(𝑀) with Lie algebra 𝔤=Vec^∞_𝜇(𝑀)={𝑋∈Vec^∞(𝑀)∣div_𝜇𝑋=0} (2.9) being the space of divergence-free vector fields on 𝑀. Vec^∞_𝜇(𝑀) is a Lie subalgebra of Vec^∞(𝑀).

Remark 2.1. We cannot apply the finite-dimensional theorem that if Vec^∞_𝜇(𝑀) is Lie algebra then there exists a Lie group whose Lie algebra it is; nor the one that if Diff^∞_𝜇(𝑀)⊂Diff(𝑀) is a closed subgroup then it is an Lie subgroup.
Nevertheless Diff^∞_𝜇(𝑀) is an ILH-Lie group.

2.4.3. Symplectomorphism Group

Let 𝜔 be a symplectic 2-form on 𝑀 and 𝒢=Diff^∞_𝜔(𝑀)={𝑓∈xDiff^∞(𝑀)∣𝑓^∗𝜔=𝜔} (2.10) the group of canonical transformations (or symplectomorphisms). Diff^∞_𝜔(𝑀) is a closed subgroup of Diff^∞(𝑀) with Lie algebra 𝔤=Vec^∞_𝜔(𝑀)={𝑋∈Vec^∞(𝑀)∣𝐿_𝑋𝜔=0} (2.11) being the space of locally Hamiltonian vector fields on 𝑀. Vec^∞_𝜔(𝑀) is a Lie subalgebra of Vec^∞(𝑀). Again Diff^∞_𝜔(𝑀) is an ILH-Lie group.

2.4.4. Group of Gauge Transformations

The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups which are not only ILH-Lie groups but actually Hilbert-Lie groups.

Let 𝜋∶𝑃→𝑀 be a principal 𝐺 bundle with 𝐺 being a finite-dimensional Lie group (structure group) acting on 𝑃 from the right 𝑝∈𝑃, 𝑔∈𝐺, and 𝑝⋅𝑔∈𝑃.

The Gauge group 𝒢 is the group of gauge transformations defined by𝒢={𝜙∈Diff^∞(𝑃);𝜙(𝑝⋅𝑔)=𝜙(𝑝)⋅𝑔,𝜋(𝜙(𝑝))=𝜋(𝑝)}. (2.12)𝒢 is a group under composition, hence a subgroup of the diffeomorphism group Diff^∞(𝑃). Since a gauge transformation 𝜙∈𝒢 preserves fibers, we can realize each such 𝜙∈𝒢 via 𝜙(𝑝)=𝑝⋅𝜏(𝑝), where 𝜏∶𝑃→𝐺 satisfies 𝜏(𝑝⋅𝑔)=𝑔⁻¹𝜏(𝑝)𝑔, for 𝑝∈𝑃, 𝑔∈𝐺. LetGau(𝑃)={𝜏∈𝐶^∞(𝑃,𝐺);𝜏(𝑝⋅𝑔)=𝑔⁻¹𝜏(𝑝)𝑔}. (2.13)Gau(𝑃) is a group under pointwise multiplication, hence a subgroup of the loop group 𝐶^∞(𝑃,𝐺) (see Section 2.4.3), which extends to a Hilbert-Lie group if equipped with the 𝐻^𝑠-Sobolev topology. We give Gau(𝑃) the induced topology and extend it to a Hilbert-Lie group denoted by Gau^𝑠(𝑃). Another interpretation is that Gau(𝑃) is isomorphic to 𝐶^∞(𝐴𝑑𝑃) the space of sections of the associated vector bundle 𝐴𝑑(𝑃)=𝑃×_𝐺𝐺. Completed in the 𝐻^𝑠 Sobolev topology, we get Gau^𝑠(𝑃)≃𝐻^𝑠(𝐴𝑑𝑃).

Let 𝐠 denote the Lie algebra of 𝐺. Then the Lie algebra  𝔤𝔞𝔲(𝑃) of Gau(𝑃) is a subalgebra of the loop algebra 𝐻^𝑠(𝑃,𝐠) under pointwise bracket in 𝐠, the finite-dimensional Lie algebra of 𝐺; that is, for any 𝜉,𝜂∈𝐻^𝑠(𝑃,𝐠) the bracket is defined by [𝜉,𝜂]_{𝔤𝔞𝔲(𝑃)}(𝑝)=[𝜉(𝑝),𝜂(𝑝)]_𝐠, 𝑝∈𝑃. Then 𝔤𝔞𝔲^𝑠(𝑃) is the subalgebra of 𝐴𝑑 -invariant 𝐠-valued functions on 𝑃; that is,𝔤𝔞𝔲(𝑃)={𝜉∈𝐶^∞(𝑃,𝐠);𝜉(𝑝⋅𝑔)=𝐴𝑑_𝑔−1𝜉(𝑝)}. (2.14)

The Lie algebra 𝔩𝔦𝔢𝒢 (running out of symbols) of the gauge group 𝒢 is the Lie subalgebra of 𝔛^∞(𝑃) consisting of all 𝐺-invariant vertical vector fields 𝑋 on 𝑃; that is,𝔩𝔦𝔢𝒢={𝑋∈𝔛^∞(𝑃);𝑅^∗_𝑔𝑋=𝑋,𝑋(𝑝)∈𝐠,𝑔∈𝐺,𝑝∈𝑃} (2.15) with commutator bracket [𝑋₁,𝑋₂]=𝑋₁𝑋₂−𝑋₂𝑋₁∈𝔩𝔦𝔢𝒢.

On the other hand, the Lie algebra of 𝐶^∞(𝐴𝑑𝑃) is 𝐶^∞(𝑎𝑑(𝑃)) being the space of sections of the associated vector bundle 𝑎𝑑(𝑃)≡(𝑃×_𝐺𝑔)→𝑀 with pointwise bracket.

We have three versions of gauge groups: 𝒢,Gau(𝑃), and 𝐶^∞(𝐴𝑑𝑃). They are all group isomorphic. There is a natural group isomorphism Gau(𝑃)→𝒢∶𝜏↦𝜙 defined by 𝜙(𝑝)=𝑝⋅𝜏(𝑝), 𝑝∈𝑃, which preserves the product 𝜏₁⋅𝜏₂↦𝜙₁∘𝜙₂. Identifying 𝒢 with Gau(𝑃), we can avoid the troubles with diffeomorphism groups and we can extend 𝒢 to a Hilbert-Lie group 𝒢^𝑠. So 𝒢^𝑠 is actually a Hilbert-Lie group in the classical sense; that is, the group operations are 𝐶^∞. Also the three Lie algebras 𝔩𝔦𝔢𝒢, 𝔤𝔞𝔲(𝑃), and 𝐶^∞(𝑎𝑑𝑃) are canonically isomorphic. Indeed, for 𝑠∈𝐶^∞(𝑎𝑑𝑃) define 𝜉∈𝔤𝔞𝔲(𝑃)𝜉∶𝑃→𝑔 by 𝜉(𝑝⋅𝑎)∶=𝐴𝑑_𝑎−1𝜉(𝑝); and for 𝜉∈𝔤𝔞𝔲(𝑃) define 𝑠∈𝐶^∞(𝑎𝑑𝑃) by 𝑠(𝜋(𝑝))∶=[𝑝,𝜉(𝑝)].

On the other hand, for 𝜉∈𝔤𝔞𝔲(𝑃) define 𝑍_𝜉∈𝔩𝔦𝔢𝒢 by 𝑍_𝜉𝑑(𝑝)=∣𝑑𝑡^𝑡=0𝑅(𝑝,exp𝑡𝜉(𝑝))(=𝜉(𝑝)^∗(𝑝)), (2.16) that is, 𝑍_𝜉 is the fundamental vector field on 𝑃, generated by 𝜉∈𝐠. 𝑍_𝜉 is invariant if and only if 𝜉(𝑝⋅𝑔)=𝐴𝑑_𝑔−1𝜉(𝑝).

To topologize 𝔩𝔦𝔢𝒢, we complete 𝐶^∞(𝑎𝑑𝑃) in the 𝐻^𝑠-Sobolev norm. If 𝑠>(1/2)dim𝑀, then 𝔩𝔦𝔢𝒢^𝑠≃𝐻^𝑠(𝑎𝑑𝑃)≃𝔤𝔞𝔲^𝑠(𝑃) are isomorphic Hilbert-Lie algebras.

There is a natural exponential map Exp∶𝔤𝔞𝔲(𝑃)→Gau(𝑃), which is a local diffeomorphism. Let exp∶𝐠→𝐺  be the finite-dimensional exponential map. Then defineExp∶𝔤𝔞𝔲^𝑠(𝑃)→Gau^𝑠(𝑃)∶(Exp𝜉)(𝑝)=exp(𝜉(𝑝)),𝜉∈𝔤𝔞𝔲^𝑠(𝑃). (2.17) Or in terms of 𝒢, Exp∶𝔩𝔦𝔢𝒢^𝑠→𝒢^𝑠∶(Exp𝜉)(𝑝)=𝑝⋅exp(𝜉_𝑝).

We have the following theorem (Schmid [6]).

Theorem 2.2. For 𝑠>(1/2)dim𝑀, 𝒢^𝑠≃Gau^𝑠(𝑃)≃𝐻^𝑠(𝐴𝑑𝑃) (2.18) is a smooth Hilbert-Lie group with Lie algebra 𝔩𝔦𝔢𝒢^𝑠≃𝔤𝔞𝔲^𝑠(𝑃)≃𝐻^𝑠(𝑎𝑑𝑃) (2.19) and smooth exponential map, which is a local diffeomorphism, EXP∶𝔩𝔦𝔢𝒢^𝑠→𝒢^𝑠∶(EXP𝜉)(𝑝)=𝑝⋅exp(𝜉(𝑝)). (2.20)

See [1–5, 7–19].

3. Lie Groups as Symmetry Groups of Hamiltonian Systems

A short introduction and “crash course” to geometric mechanics can be found in the studies by Abraham and Marsden [20], Marsden [21], as well as Marsden and Ratiu [22]. For the general theory of infinite-dimensional manifolds and global analysis, see, for example, the studies by Bourbaki [9], Lang [14], as well as Palais [18].

3.1. Hamilton’s Equations on Poisson Manifolds

A Poisson manifold is a manifold 𝑃 (in general infinite-dimensional) equipped with a bilinear operation {⋅,⋅}, called Poisson bracket, on the space 𝐶^∞(𝑃) of smooth functions on 𝑃 satisfying the following.(i)(𝐶^∞(𝑃),{⋅,⋅}) is a Lie algebra; that is, {⋅,⋅}∶𝐶^∞(𝑃)×𝐶^∞(𝑃)→𝐶^∞(𝑃) is bilinear, skew symmetric and satisfies the Jacobi identity {{𝐹,𝐺},𝐻}+{{𝐻,𝐹},𝐺}+{{𝐺,𝐻},𝐹}=0 for all 𝐹,𝐺,𝐻∈𝐶^∞(𝑃).(ii){⋅,⋅} satisfies the Leibniz rule; that is, {⋅,⋅} is a derivation in each factor: {𝐹⋅𝐺,𝐻}=𝐹⋅{𝐺,𝐻}+𝐺⋅{𝐹,𝐻}, for all 𝐹,𝐺,𝐻∈𝐶^∞(𝑃).

The notion of Poisson manifolds was rediscovered many times under different names, starting with Lie, Dirac, Pauli, and others. The name Poisson manifold was coined by Lichnerowicz.

For any 𝐻∈𝐶^∞(𝑃) we define the Hamiltonian vector field  𝑋_𝐻 by𝑋_𝐻(𝐹)={𝐹,𝐻},𝐹∈𝐶^∞(𝑃). (3.1) It follows from (ii) that indeed 𝑋_𝐻 defines a derivation on 𝐶^∞(𝑃), hence a vector field on 𝑃. Hamilton’s equations of motion for a function 𝐹∈𝐶^∞(𝑃) with Hamiltonian 𝐻∈𝐶^∞(𝑃) (energy function) are then defined by the flow (integral curves) of the vector field 𝑋_𝐻; that is,𝐹̇=𝑋_𝐻(𝐹)={𝐹,𝐻},wherė=𝑑.𝑑𝑡 (3.2) We then call 𝐹 a Hamiltonian system on 𝑃 with energy (Hamiltonian function) 𝐻.

3.2. Examples of Poisson Manifolds and Hamilton’s Equations

Poisson manifolds are a generalization of symplectic manifolds on which Hamilton’s equations have a canonical formulated.

3.2.1. Finite-Dimensional Classical Mechanics

For finite-dimensional classical mechanics we take 𝑃=ℝ^2𝑛 with coordinates (𝑞¹,…,𝑞^𝑛,𝑝₁,…,𝑝_𝑛) with the standard Poisson bracket for any two functions 𝐹(𝑞^𝑖,𝑝_𝑖), 𝐻(𝑞^𝑖,𝑝_𝑖) given by{𝐹,𝐻}=^𝑛_𝑖=1𝜕𝐹𝜕𝑝_𝑖𝜕𝐻𝜕𝑞^𝑖−𝜕𝐻𝜕𝑝_𝑖𝜕𝐹𝜕𝑞^𝑖. (3.3) Then the classical Hamilton’s equations are𝑞̇^𝑖={𝑞^𝑖,𝐻}=𝜕𝐻𝜕𝑝_𝑖,𝑝̇_𝑖={𝑝_𝑖,𝐻}=−𝜕𝐻𝜕𝑞^𝑖, (3.4) where 𝑖=1,…,𝑛. This finite-dimensional Hamiltonian system is a system of ordinary differential equations for which there are well-known existence and uniqueness theorems; that is, it has locally unique smooth solutions, depending smoothly on the initial conditions.

Example 3.1 (Harmonic Oscillator). As a concrete example we consider the harmonic oscillator. Here 𝑃=ℝ² and the Hamiltonian (energy) is 𝐻(𝑞,𝑝)=(1/2)(𝑞²+𝑝²). Then Hamilton’s equations are 𝑞̇=𝑝,𝑝̇=−𝑞. (3.5)

3.2.2. Infinite-Dimensional Classical Field Theory

Let 𝑉 be a Banach space and 𝑉^∗ its dual space with respect to a pairing ⟨⋅,⋅⟩∶𝑉×𝑉^∗→ℝ (i.e., ⟨⋅,⋅⟩ is a symmetric, bilinear, nondegenerate function). On 𝑃=𝑉×𝑉^∗ we have the canonical Poisson bracket for 𝐹, 𝐻∈𝐶^∞(𝑃), 𝜑∈𝑉, and 𝜋∈𝑉^∗, given by{𝐹,𝐻}=⟨𝛿𝐹,𝛿𝜋𝛿𝐻𝛿𝜑⟩−⟨𝛿𝐻,𝛿𝜋𝛿𝐹𝛿𝜑⟩, (3.6) where the functional derivatives 𝛿𝐹/𝛿𝜋∈𝑉, 𝛿𝐹/𝛿𝜑∈𝑉^∗ are the “duals” under the pairing ⟨⋅,⋅⟩ of the partial gradients 𝐷₁𝐹(𝜋)∈𝑉^∗,𝐷₂𝐹(𝜑)∈𝑉^∗∗≃𝑉. The corresponding Hamilton’s equations are𝜑̇={𝜑,𝐻}=𝛿𝐻,̇𝛿𝜋𝜋={𝜋,𝐻}=−𝛿𝐻.𝛿𝜑 (3.7)

As a special case in finite dimensions, if 𝑉≃ℝ^𝑛, so that 𝑉^∗≃ℝ^𝑛 and 𝑃=𝑉×𝑉^∗≃ℝ^2𝑛, and the pairing is the standard inner product in ℝ^𝑛, then the Poisson bracket (3.6) and Hamilton’s equations (3.7) are identical with (3.3) and (3.4), respectively.

Example 3.2 (Wave Equations). As a concrete example we consider the wave equations. Let 𝑉=𝐶^∞(ℝ³) and 𝑉^∗=Den(ℝ³) (densities) and the 𝐿² pairing ⟨𝜑,𝜋⟩=∫𝜑(𝑥)𝜋(𝑥)𝑑𝑥. We take the Hamiltonian to be 𝐻(𝜑,𝜋)=∫((1/2)𝜋²+(1/2)|∇𝜑|²+𝐹(𝜑))𝑑𝑥, where 𝐹 is some function on 𝑉. Then Hamilton’s (3.7) become 𝜑̇=𝜋,𝜋̇=∇²𝜑−𝐹^′(𝜑),where^′=𝑑,𝑑𝜑 (3.8) which imply the wave equation 𝜕²𝜑/𝜕𝑡²=∇²𝜑−𝐹^′(𝜑). Different choices of 𝐹 give different wave equations; for example, for 𝐹=0 we get the linear wave equation 𝜕²𝜑/𝜕𝑡²=∇²𝜑. For 𝐹=(1/2)𝑚𝜑 we get the Klein-Gordon equation ∇²𝜑−𝜕²𝜑/𝜕𝑡²=𝑚𝜑. So these wave equations and the Klein-Gordon equation are infinite-dimensional Hamiltonian systems on 𝑃=𝐶^∞(ℝ³)×Den(ℝ³).

3.2.3. Cotangent Bundles

The finite-dimensional examples of Poisson brackets (3.3) and Hamilton’s (3.4) and the infinite-dimensional examples (3.6) and (3.7) are the local versions of the general case where 𝑃=𝑇^∗𝑄 is the cotangent bundle (phase space) of a manifold 𝑄 (configuration space). If 𝑄 is an 𝑛-dimensional manifold, then 𝑇^∗𝑄 is a 2𝑛-Poisson manifold locally isomorphic to ℝ^2𝑛 whose Poisson bracket is locally given by (3.3) and Hamilton’s equations are locally given by (3.4). If 𝑄 is an infinite-dimensional Banach manifold, then 𝑇^∗𝑄 is a Poisson manifold locally isomorphic to 𝑉×𝑉^∗ whose Poisson bracket is given by (3.6) and Hamilton’s equations are locally given by (3.7).

3.2.4. Symplectic Manifolds

All the examples above are special cases of symplectic manifolds (𝑃,𝜔). That means that 𝑃 is equipped with a symplectic structure 𝜔 which is a closed (d𝜔=0), (weakly) nondegenerate 2-form on the manifold 𝑃. Then for any 𝐻∈𝐶^∞(𝑃) the corresponding Hamiltonian vector field 𝑋_𝐻 is defined by d𝐻=𝜔(𝑋_𝐻,⋅) and the canonical Poisson bracket is given by{𝐹,𝐻}=𝜔(𝑋_𝐹,𝑋_𝐻),𝐹,𝐻∈𝐶^∞(𝑃). (3.9) For example, on ℝ^2𝑛 the canonical symplectic structure 𝜔 is given by 𝜔=∑^𝑛_𝑖=1d𝑝_𝑖∧d𝑞^𝑖=𝑑𝜃, where 𝜃=∑^𝑛_𝑖=1𝑝_𝑖∧d𝑞^𝑖. The same formula for 𝜔 holds locally in 𝑇^∗𝑄 for any finite-dimensional 𝑄 (Darboux’s Lemma). For the infinite-dimensional example 𝑃=𝑉×𝑉^∗, the symplectic form 𝜔 is given by 𝜔((𝜑₁,𝜋₁),(𝜑₂,𝜋₂))=⟨𝜑₁,𝜋₂⟩−⟨𝜑₂,𝜋₁⟩. Again these two formulas for 𝜔 are identical if 𝑉=ℝ^𝑛.

Remark 3.3. (A) If 𝑃 is a finite-dimensional symplectic manifold, then 𝑃 is even dimensional.
(B) If the Poisson bracket {⋅,⋅} is nondegenerate, then {⋅,⋅} comes from a symplectic form 𝜔; that is, {⋅,⋅} is given by (3.9).

3.2.5. The Lie-Poisson Bracket

Not all Poisson brackets are of the form given in the above examples (3.3), (3.6), and (3.9); that is, not all Poisson manifolds are symplectic manifolds. An important class of Poisson bracket is the so-called Lie-Poisson bracket. It is defined on the dual of any Lie algebra. Let 𝐺 be a Lie group with Lie algebra 𝔤=𝑇_𝑒𝐺≃{left invariant vector fields on 𝐺}, and let [⋅,⋅] denote the Lie bracket (commutator) on 𝔤. Let 𝔤^∗ be the dual of a 𝔤 with respect to a pairing ⟨⋅,⋅⟩∶𝔤^∗×𝔤→ℝ. Then for any 𝐹,𝐻∈𝐶^∞(𝔤^∗) and 𝜇∈𝔤^∗, the Lie-Poisson bracket is defined by{𝐹,𝐻}(𝜇)=±⟨𝜇,[𝛿𝐹,𝛿𝜇𝛿𝐻𝛿𝜇]⟩, (3.10) where 𝛿𝐹/𝛿𝜇,𝛿𝐻/𝛿𝜇∈𝔤 are the "duals" of the gradients 𝐷𝐹(𝜇),𝐷𝐻(𝜇)∈𝔤^∗∗≃𝔤 under the pairing ⟨⋅,⋅⟩. Note that the Lie-Poisson bracket is degenerate in general; for example, for 𝐺=𝑆𝑂(3) the vector space 𝔤^∗ is 3 dimensional, so the Poisson bracket (3.10) cannot come from a symplectic structure. This Lie-Poisson bracket can also be obtained in a different way by taking the canonical Poisson bracket on 𝑇^∗𝐺 (locally given by (3.3) and (3.6)) and then restricting it to the fiber at the identity 𝑇^∗_𝑒𝐺=𝔤^∗. In this sense the Lie-Poisson bracket (3.10) is induced from the canonical Poisson bracket on 𝑇^∗𝐺. It is induced by the symmetry of left multiplication as we will discuss in Section 3.3.

Example 3.4 (Rigid Body). A concrete example of the Lie-Poisson bracket is given by the rigid body. Here 𝐺=𝑆𝑂(3) is the configuration space of a free rigid body. Identifying the Lie algebra (𝔰𝔬(3),[⋅,⋅]) with (ℝ³,×), where × is the vector product on ℝ³, and 𝔤^∗=𝔰𝔬(3)^∗≃ℝ³, the Lie-Poisson bracket translates into {𝐹,𝐻}(𝑚)=−𝑚⋅(∇𝐹×∇𝐻). (3.11) For any 𝐹∈𝐶^∞(𝔰𝔬(3)^∗), we have (𝑑𝐹/𝑑𝑡)(𝑚)=∇𝐹⋅𝑚̇={𝐹,𝐻}(𝑚)=−𝑚⋅(∇𝐹×∇𝐻)=∇𝐹⋅(𝑚×∇𝐻); hence 𝑚̇=𝑚×∇𝐻. With the Hamiltonian 𝐻=(1/2)(𝑚²₁/𝐼²₁+𝑚²₂/𝐼²₂+𝑚²₃/𝐼²₃) we get Hamilton’s equation as 𝑚̇₁=𝐼₂−𝐼₃𝐼₂𝐼₃𝑚₂𝑚₃,𝑚̇₂=𝐼₃−𝐼₁𝐼₃𝐼₁𝑚₃𝑚₁,𝑚̇₃=𝐼₁−𝐼₂𝐼₁𝐼₂𝑚₁𝑚₂. (3.12) These are Euler’s equations for the free rigid body.

3.3. Reduction by Symmetries

The examples we have discussed so far are all canonical examples of Poisson brackets, defined either on a symplectic manifold (𝑃,𝜔) or 𝑇^∗𝑄, or on the dual of a Lie algebra 𝔤^∗. Different, noncanonical Poisson brackets can arise from symmetries. Assume that a Lie group 𝐺 is acting in a Hamiltonian way on the Poisson manifold (𝑃,{⋅,⋅}). That means that we have a smooth map 𝜑∶𝐺×𝑃→𝑃∶𝜑(𝑔,𝑝)=𝑔⋅𝑝 such that the induced maps 𝜑_𝑔=𝜑(𝑔,⋅)∶𝑃→𝑃 are canonical transformations, for each 𝑔∈𝐺. In terms of Poisson manifolds, a canonical transformation is a smooth map that preserves the Poisson bracket. So the action of 𝐺 on 𝑃 is a Hamiltonian action if 𝜑^∗_𝑔{𝐹,𝐻}={𝜑^∗_𝑔𝐹,𝜑^∗_𝑔𝐻}, for all 𝐹,𝐻∈𝐶^∞(𝑃),𝑔∈𝐺. For any 𝜉∈𝔤 the canonical transformations 𝜑_{exp(𝑡𝜉)} generate a Hamiltonian vector field 𝜉_𝐹 on 𝑃 and a momentum map 𝐽∶𝑃→𝔤^∗ given by 𝐽(𝑥)(𝜉)=𝐹(𝑥), which is 𝐴𝑑^∗ equivariant.

If a Hamiltonian system 𝑋_𝐻 is invariant under a Lie group action, that is, 𝐻(𝜑_𝑔(𝑥))=𝐻(𝑥), then we obtain a reduced Hamiltonian system on a reduced phase space (reduced Poisson manifold). We recall the following Marsden-Weinstein reduction theorem [23].

Theorem 3.5 (Reduction Theorem). For a Hamiltonian action of a Lie group 𝐺 on a Poisson manifold (𝑃,{⋅,⋅}), there is an equivariant momentum map 𝐽∶𝑃→𝔤^∗ and for every regular 𝜇∈𝔤^∗ the reduced phase space 𝑃_𝜇≡𝐽⁻¹(𝜇)/𝐺_𝜇 carries an induced Poisson structure {⋅,⋅}_𝜇 (𝐺_𝜇 being the isotropy group). Any 𝐺-invariant Hamiltonian 𝐻 on 𝑃 defines a Hamiltonian 𝐻_𝜇 on the reduced phase space 𝑃_𝜇, and the integral curves of the vector field 𝑋_𝐻 project onto integral curves of the induced vector field 𝑋_𝐻𝜇 on the reduced space 𝑃_𝜇.

Example 3.6 (Rigid Body). The rigid body discussed above can be viewed as an example of this reduction theorem. If 𝑃=𝑇^∗𝐺 and 𝐺 is acting on 𝑇^∗𝐺 by the cotangent lift of the left translation 𝑙_𝑔∶𝐺→𝐺,𝑙_𝑔(ℎ)=𝑔ℎ, then the momentum map 𝐽∶𝑇^∗𝐺→𝔤^∗ is given by 𝐽(𝛼_𝑔)=𝑇^∗_𝑒𝑅_g(𝛼_𝑔) and the reduced phase space (𝑇^∗𝐺)_𝜇=𝐽⁻¹(𝜇)/𝐺_𝜇 is isomorphic to the coadjoint orbit 𝒪_𝜇 through 𝜇∈𝔤^∗. Each coadjoint orbit 𝒪_𝜇 carries a natural symplectic structure 𝜔_𝜇, and in this case, the reduced Lie-Poisson bracket {⋅,⋅}_𝜇 on the coadjoint orbit 𝒪_𝜇 is induced by the symplectic form 𝜔_𝜇 on 𝒪_𝜇 as in (3.9). Furthermore 𝑇^∗𝐺/𝐺≃𝔤^∗ and the induced Poisson bracket {⋅,⋅}_𝜇 on 𝒪_𝜇 are identical with the Lie-Poisson bracket restricted to the coadjoint orbit 𝒪_𝜇⊂𝔤^∗. For the rigid body we apply this construction to 𝐺=𝑆𝑂(3).

See [1, 8, 10, 17, 19–31].

4. Applications

We now discuss some infinite-dimensional examples of reduced Hamiltonian systems.

4.1. Maxwell’s Equations

Maxwell’s equations of electromagnetism are a reduced Hamiltonian system with the Lie group 𝒢=(𝐶^∞(𝑀),+) discussed in Section 2.3.3 as symmetry group.

Let 𝐸,𝐵 be the electric and magnetic fields on ℝ³, then Maxwell’s equations for a charge density 𝜌 are𝐸̇=curl𝐵,𝐵̇=−curl𝐸,div𝐵=0,div𝐸=𝜌. (4.1)(4.2) Let 𝐴 be the magnetic potential such that 𝐵=−curl𝐴. As configuration space we take 𝑉=Vec(ℝ³), vector fields (potentials) on ℝ³, so 𝐴∈𝑉, and as phase space we have 𝑃=𝑇^∗𝑉≃𝑉×𝑉^∗∋(𝐴,𝐸), with the standard 𝐿² pairing ⟨𝐴,𝐸⟩=∫𝐴(𝑥)𝐸(𝑥)𝑑𝑥, and canonical Poisson bracket given by (3.6), which becomes{𝐹,𝐻}(𝐴,𝐸)=(𝛿𝐹𝛿𝐴𝛿𝐻−𝛿𝐸𝛿𝐻𝛿𝐴𝛿𝐹𝛿𝐸)𝑑𝑥. (4.3) As Hamiltonian we take the total electromagnetic energy 1𝐻(𝐴,𝐸)=2(|curl𝐴|²+|𝐸|²)𝑑𝑥. (4.4)

Then Hamilton’s equations in the canonical variables 𝐴 and 𝐸 are 𝐴̇=𝛿𝐻/𝛿𝐸=𝐸⇒𝐵̇=−curl𝐸 and 𝐸̇=−𝛿𝐻/𝛿𝐴=−curlcurl𝐴=curl𝐵. So the first two equations of Maxwell’s equations (4.1) are Hamilton’s equations; we get the third one automatically from the potential div𝐵=−divcurl𝐴=0 and we obtain the 4th equation div𝐸=𝜌 through the following symmetry (gauge invariance). The Lie group 𝒢=(𝐶^∞(ℝ³),+) acts on 𝑉 by 𝜑⋅𝐴=𝐴+∇𝜑, 𝜑∈𝒢,𝐴∈𝑉. The lifted action to 𝑉×𝑉^∗ becomes 𝜑⋅(𝐴,𝐸)=(𝐴+∇𝜑,𝐸), and has the momentum map 𝐽∶𝑉×𝑉^∗→𝔤^∗≃{chargedensities}:𝐽(𝐴,𝐸)=div𝐸. (4.5) With 𝔤=𝐶^∞(ℝ³) and 𝔤^∗=Den(ℝ³), we identify elements of 𝔤^∗ with charge densities. The Hamiltonian 𝐻 is 𝒢 invariant; that is, 𝐻(𝜑⋅(𝐴,𝐸))=𝐻(𝐴+∇𝜑,𝐸)=𝐻(𝐴,𝐸). Then the reduced phase space for 𝜌∈𝔤^∗ is (𝑉×𝑉^∗)_𝜌=𝐽⁻¹(𝜌)/𝐺={(𝐸,𝐵)∣div𝐸=𝜌,div𝐵=0} and the reduced Hamiltonian is𝐻_𝜌1(𝐸,𝐵)=2(|𝐸|²+|𝐵|²)𝑑𝑥. (4.6) The reduced Poisson bracket becomes for any functions 𝐹,𝐻 on (𝑉×𝑉^∗)_𝜌{𝐹,𝐻}_𝜌(𝐸,𝐵)=(𝛿𝐹𝛿𝐸⋅curl𝛿𝐻−𝛿𝐵𝛿𝐻𝛿𝐸⋅curl𝛿𝐹𝛿𝐵)𝑑𝑥, (4.7) and a straightforward computation shows that𝐹̇={𝐹,𝐻_𝜌}_𝜌⇔𝐸̇=curl𝐵,𝐵̇=−curl𝐸,div𝐵=0,div𝐸=𝜌. (4.8) So Maxwell’s equations (4.1), (4.2) are an infinite-dimensional Hamiltonian system on this reduced phase space with respect to the reduced Poisson bracket.

4.2. Fluid Dynamics

Euler’s equations for an incompressible fluid𝜕𝑢𝜕𝑡+𝑢⋅∇𝑢=−∇𝑝,div𝑢=0 (4.9) are equivalent to the equations of geodesics on Diff^∞_𝜇(𝑀). See the study by Marsden et al. in [15] for details.

4.3. Plasma Physics

The Maxwell-Vlasov’s equations are a reduced Hamiltonian system on a more complicated reduced space. See the study by Marsden et al. in [32] for details.

Maxwell-Vlasov’s equations for a plasma density 𝑓(𝑥,𝑣,𝑡) generating the electric and magnetic fields 𝐸 and 𝐵 are the following set of equations:𝜕𝑓𝜕𝑡+𝑣⋅𝜕𝑓𝜕𝑥+(𝐸+𝑣×𝐵)𝜕𝑓𝜕𝑣=0,𝜕𝐵𝜕𝑡=−curl𝐸,𝜕𝐸𝜕𝑡=curl𝐵−𝐽_f,𝐽_𝑓=currentdensity,div𝐸=𝜌_𝑓,𝜌_𝑓=chargedensity,div𝐵=0. (4.10) This coupled nonlinear system of evolution equations is an infinite-dimensional Hamiltonian system of the form 𝐹̇={𝐹,𝐻}_𝜌𝑓 on the reduced phase spaceℳ𝒱=(𝑇^∗Diff^∞_𝜔(ℝ⁶)×𝑇^∗𝑉)/𝐶^∞(ℝ⁶) (4.11) (𝑉 being the same space as in the example of Maxwell’s equations) with respect to the following reduced Poisson bracket, which is induced via gauge symmetry from the canonical Poisson bracket on 𝑇^∗Diff^∞_𝜔(ℝ⁶)×𝑇^∗𝑉:{𝐹,𝐺}_𝜌𝑓(𝑓,𝐸,𝐵)=𝑓{𝛿𝐹,𝛿𝑓𝛿𝐺+(𝛿𝑓}𝑑𝑥𝑑𝑣𝛿𝐹𝛿𝐸⋅curl𝛿𝐺−𝛿𝐵𝛿𝐺𝛿𝐸⋅curl𝛿𝐹𝛿𝐵)𝑑𝑥𝑑𝑣+(𝛿𝐹⋅𝛿𝐸𝜕𝑓𝜕𝑣𝛿𝐺−𝛿𝑓𝛿𝐺⋅𝛿𝐸𝜕𝑓𝜕𝑣𝛿𝐹+𝜕𝛿𝑓)𝑑𝑥𝑑𝑣𝑓𝐵⋅(𝜕𝑣𝛿𝐹×𝜕𝛿𝑓𝜕𝑣𝛿𝐺𝛿𝑓)𝑑𝑥𝑑𝑣, (4.12) and with Hamiltonian1𝐻(𝑓,𝐸,𝐵)=2𝑣²1𝑓(𝑥,𝑣,𝑡)𝑑𝑣+2(|𝐸|²+|𝐵|²)𝑑𝑥. (4.13)

More complicated plasma models are formulated as Hamiltonian systems. For example, for the two-fluid model the phase space is a coadjoint orbit of the semidirect product (⋉) of the group 𝒢=Diff^∞(ℝ⁶)⋉(𝐶^∞(ℝ⁶)×𝐶^∞(ℝ⁶)). For the MHD model, 𝒢=Diff^∞(ℝ⁶)⋉(𝐶^∞(ℝ⁶)×Ω²(ℝ³)).

4.4. The KdV Equation and Fourier Integral Operators

There are many known examples of PDEs which are infinite-dimensional Hamiltonian systems, such as the Benjamin-Ono, Boussinesq, Harry Dym, KdV, KP equations, and others. In many cases the Poisson structures and Hamiltonians are given ad hoc on a formal level. We illustrate this with the KdV equation, where at least one of the three known Hamiltonian structures is well understood [33].

The Korteweg-deVries (KdV) equation𝑢_𝑡+6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0 (4.14) is an infinite-dimensional Hamiltonian system with the Lie group of invertible Fourier integral operators as symmetry group. Gardner found that with the bracket{𝐹,𝐺}=₀^2𝜋𝛿𝐹𝜕𝛿𝑢𝜕𝑥𝛿𝐺𝛿𝑢𝑑𝑥 (4.15) and Hamiltonian𝐻(𝑢)=₀^2𝜋(𝑢³+12𝑢³_𝑥)𝑑𝑥 (4.16)𝑢 satisfies the KdV equation (4.14) if and only if 𝑢̇={𝑢,𝐻}. (4.17)

The question is where this Poisson bracket (4.15) and Hamiltonian (4.16) come from? We showed [33–35] that this bracket is the Lie-Poisson bracket on a coadjoint orbit of Lie group 𝒢=FIO of invertible Fourier integral operators on the circle 𝑆¹. We briefly summarize the following.

A Fourier integral operators on a compact manifold 𝑀 is an operator 𝐴∶𝐶^∞(𝑀)→𝐶^∞(𝑀) (4.18) locally given by𝐴(𝑢)(𝑥)=(2𝜋)^−𝑛𝑒^{𝑖𝜑(𝑥,𝑦,𝜉)}𝑎(𝑥,𝜉)𝑢(𝑦)𝑑𝑦𝑑𝜉, (4.19) where 𝜑(𝑥,𝑦,𝜉) is a phase function with certain properties and the symbol 𝑎(𝑥,𝜉) belongs to a certain symbol class. A pseudodifferential operator is a special kind of Fourier integral operators, locally of the form𝑃(𝑢)(𝑥)=(2𝜋)^−𝑛𝑒^{𝑖(𝑥−𝑦)⋅𝜉}𝑝(𝑥,𝜉)𝑢(𝑦)𝑑𝑦𝑑𝜉. (4.20) Denote by FIO and Ψ𝐷𝑂 the groups under composition (operator product) of invertible Fourier integral operators and invertible pseudodifferential operators on 𝑀, respectively. We have the following results.

Both groups Ψ𝐷𝑂 and FIO are smooth infinite-dimensional ILH-Lie groups. The smoothness properties of the group operations (operator multiplication and inversion) are similar to the case of diffeomorphism groups (2.6), (2.7). The Lie algebras of both ILH-Lie groups Ψ𝐷𝑂 and FIO are the Lie algebras of all pseudodifferential operators under the commutator bracket. Moreover, FIO is a smooth infinite-dimensional principal fiber bundle over the diffeomorphism group of canonical transformations Diff^∞_𝜔(𝑇^∗𝑀−{0}) with structure group (gauge group) Ψ𝐷𝑂.

For the KdV equation we take the special case where 𝑀=𝑆¹. Then the Gardner bracket (4.15) is the Lie-Poisson bracket on the coadjoint orbit of FIO through the Schrodinger operator 𝑃∈Ψ𝐷𝑂. Complete integrability of the KdV equation follows from the infinite system of conserved integral in involution given by 𝐻_𝑘=Trace(𝑃^𝑘); in particular the Hamiltonian (4.16) equals 𝐻=𝐻₂.

See the study by Adams et al. in [34, 35] for details.

See [10, 15, 31–40].

5. Gauge Theories, the Standard Model, and Gravity

Here we will encounter various infinite-dimensional Lie groups and algebras such as diffeomorphism groups, loop groups, groups of gauge transformations, and their cohomologies.

5.1. Gauge Theories: Yang-Mills, QED, and QCD

Consider a principal 𝐺-bundle 𝜋∶𝑃→𝑀, with 𝑀 being a compact, orientable Riemannian manifold (e.g., 𝑀=𝑆⁴,𝑇⁴) and 𝐺 a compact non-Abelian gauge group with Lie algebra 𝐠. Let 𝒜 be the infinite-dimensional affine space of connection 1-forms on 𝑃. So each 𝐴∈𝒜 is a 𝐠-valued, equivariant 1-form on 𝑃 (also called vector potential) and defines the covariant derivative of any field 𝜑 by 𝐷_𝐴𝜑=𝑑𝜑+(1/2)[𝐴,𝜑]. The curvature 2-form 𝐹_𝐴 (or field strength) is a 𝐠-valued 2-form and is defined as 𝐹_𝐴=𝐷_𝐴𝐴=𝑑𝐴+(1/2)[𝐴,𝐴]. They are locally given by 𝐴=𝐴_𝜇𝑑𝑥^𝜇 and 𝐹=(1/2)𝐹_𝜇𝜈𝑑𝑥^𝜇∧𝑑𝑥^𝜈, where 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇+[𝐴_𝜇,𝐴_𝜈].

In pure Yang-Mills theory the action functional is given by1𝑆(𝐴)=2‖𝐹_𝐴‖²=12_𝑀Tr(𝐹_𝜇𝜈𝐹^𝜇𝜈), (5.1) and the Yang-Mills equations become globally𝑑∗𝐹_𝐴=0. (5.2) With added fermionic field 𝜓 interaction, the action becomes^{1𝑆(𝐴,𝜓)=}₂^‖𝐹_𝐴^‖2+⟨/^𝜕_𝐴^{𝜓,𝜓⟩,} (5.3) where 𝜓 is a section of the spin bundle 𝒮𝑝𝑖𝑛^±(𝑀) and, /^𝜕_𝐴^{∶𝒮𝑝𝑖𝑛±(𝑀)→𝒮𝑝𝑖𝑛∓(𝑀)} is the induced Dirac operator.

5.1.1. Gauge Invariance

In gauge theories the symmetry group is the group of gauge transformations. The diffeomorphism subgroups that arise in gauge theories as gauge groups behave nicely because they are isomorphic to subgroups of loop groups, as discussed in Section 2.4.4.

The group 𝒢 of gauge transformations of the principal 𝐺-bundle 𝜋∶𝑃→𝑀 is given by 𝒢={𝜙∈Diff^∞(𝑃);𝜙(𝑝⋅𝑔)=𝜙(𝑝)⋅𝑔,𝜋𝜙(𝑝)=𝜋(𝑝)}≅{𝜏∈𝐶^∞(𝑃,𝐺);𝜏(𝑝⋅𝑔)=𝑔⁻¹𝜏(𝑝)𝑔}=Gau(𝑃) (5.4) which is a smooth Hilbert-Lie group with smooth group operations [6].

We only sketch here what role this infinite-dimensional gauge group 𝒢 plays in these quantum field theories. A good reference for this topic is the study by Deligne et al. in [41, 42].

The gauge group 𝒢 acts on 𝒜 via pullback 𝜙∈𝒢, 𝐴∈𝒜, 𝜙⋅𝐴=(𝜙⁻¹)^∗𝐴∈𝒜, or under the isomorphism (see Section 2.4.4) 𝒢≅Gau(𝑃), 𝜙⇔𝜏 we have Gau(𝑃) acting on 𝒜 by 𝜏⋅𝐴=𝜏𝐴𝜏⁻¹+𝜏𝑑𝜏⁻¹. Hence the covariant derivative transforms as 𝐷_𝜏⋅𝐴=𝜏𝐷_𝐴𝜏⁻¹, and the action on the field is 𝜏⋅𝐹_𝐴∶=𝐹_𝜏⋅𝐴=𝜏𝐹_𝐴𝜏⁻¹.

The action functional (the Yang-Mills functional) is 𝑆(𝐴)=‖𝐹_𝐴‖², locally given by ‖𝐹_𝐴‖²=(1/2)∫_𝑀Tr(𝐹_𝜇𝜈𝐹^𝜇𝜈). This action is gauge invariant 𝑆(𝜙⋅𝐴)=𝑆(𝐴), 𝜙∈𝒢, so the Yang-Mills functional is defined on the orbit space ℳ=𝒜/𝒢. The space ℳ is in general not a manifold since the action of 𝒢 on 𝒜 is not free. If we restrict to irreducible connections, then ℳ is a smooth infinite-dimensional manifold and 𝒜→ℳ is an infinite-dimensional principal fiber bundle with structure group 𝒢.

For self-dual connections 𝐹_𝐴=∗𝐹_𝐴 (instantons) on a compact 4-manifold, the moduli space ℳ={𝐴∈𝒜;𝐴self-dual}/𝒢 is a smooth finite-dimensional manifold. Self-dual connections absolutely minimize the Yang-Mills action integral 𝐘𝐌(𝐴)=_Ω‖𝐹_𝐴‖²,Ω⊂𝑀compact. (5.5) The Feynman path integral quantizes the action and we get the probability amplitude 𝑊(𝑓)=_𝒜/𝒢𝑒^{−𝑆(𝐴)}𝑓(𝐴)𝒟(𝐴) (5.6) for any gauge-invariant functional 𝑓(𝐴).

Let 𝒢 be the group of gauge transformations. So 𝜙∈𝒢⇔𝜙∶𝑃→𝑃 is a diffeomorphism over 𝑖𝑑_𝑀; that is, 𝜙(𝑝⋅𝑔)=𝜙(𝑝)⋅𝑔, 𝑝∈𝑃, 𝑔∈𝐺. Then 𝒢 acts on 𝒜 and 𝒮𝑝𝑖𝑛^±(𝑀) by 𝜙⋅𝐴=(𝜙⁻¹)^∗𝐴 and 𝜙⋅𝜓=(𝜙⁻¹)^∗𝜓. The action functionals 𝑆 are gauge invariant:Yang-Mills:𝑆(𝜙⋅𝐴)=𝑆(𝐴),𝐴∈𝒜,𝜙∈𝒢,QED:𝑆(𝜙⋅𝐴,𝜙⋅𝜓)=𝑆(𝐴,𝜓),𝐴∈𝒜,𝜓∈𝒮𝑝𝑖𝑛^±(𝑀),𝜙∈𝒢. (5.7)(5.8)

5.1.2. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)

In classical field theory, one considers a Lagrangian ℒ(𝜙_𝑖,𝜕_𝜇𝜙_𝑖) of the fields 𝜑_𝑖∶ℝ^𝑛→ℝ, 𝑖=1,…,𝑘, and 𝜕_𝜇=𝜕/𝜕𝑥_𝜇 and the corresponding action functional 𝑆=∫ℒ(𝜙_𝑖,𝜕_𝜇𝜙_𝑖)𝑑^𝑛𝑥. The variational principle 𝛿𝑆=0 then leads to the Euler-Lagrange equations of motion𝜕ℒ𝜕𝜙_𝑖−𝜕_𝜇𝜕ℒ𝜕(𝜕_𝜇𝜙_𝑖)=0. (5.9)

In QED and QCD the Lagrangian is more complicated of the form1ℒ(𝐴,𝜓,𝜑)=−4𝑔²Tr𝐹_𝜇𝜈𝐹^𝜇𝜈−𝑖𝜓[𝛾^𝜇(𝜕_𝜇+𝑖𝑒𝐴_𝜇)+𝑚]𝜓+(𝐷^𝜇_𝐴𝜑)^†(𝐷^𝜇_𝐴𝜑)−𝑚²𝜑^†𝜑, (5.10) where 𝐴_𝜇(𝑥) is a potential 1-form (boson), and the field strength 𝐹 is given by 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇+[𝐴_𝜇,𝐴_𝜈]. In QED the gauge group of the principal bundle is 𝐺=𝑈(1), and in QCD we have 𝐺=𝑆𝑈(2). The Dirac 𝛾-matrices are 𝛾^𝑖=(^0−𝜎𝑖_𝜎𝑖0), where 𝜎_𝑖 are the Pauli matrices (canonical basis of 𝔰𝔲(2)) and 𝜓=𝜓^†𝛾^𝑜 is the Pauli adjoint with 𝛾^𝑜=(⁰¹₁₀), 𝑚 is the electron mass, 𝑒 is the electron charge, and 𝑔 is a coupling constant.

5.1.3. The Equations of Motion

The variational principle of the Lagrangian (5.10) with respect to the fields 𝐴,𝜓, and 𝜑 gives the corresponding Euler-Lagrange equations of motion. They describe, for instance, the motion of an electron 𝜓(𝑥) (fermion, spinor) in an electromagnetic field 𝐹, interacting with a bosonic field 𝜑. We get, from the variational principle, 𝛿𝑆/𝛿𝐴_𝜇=0⇒𝜕_𝜇𝐹^𝜇𝜈=𝑒𝜓𝛾^𝜈𝜓, which are Maxwell’s equations for 𝐺=𝑈(1).

In the free case, that is, when 𝜓=0, we get 𝜕_𝜇𝐹^𝜇𝜈=0, the vacuum Maxwell equations.

For 𝐺=𝑆𝑈(2) these equations become 𝐷^𝜇𝐹_𝜇𝜈=0, the Yang-Mills equations. Moreover, ^{𝛿𝑆/𝛿𝜓=0⇒𝑖(}/^𝜕_𝐴^{−𝑚)𝜓=0}, which are Dirac’s equations, where /^𝜕_𝐴^{=𝛾𝜇(𝜕}_𝜇^{+𝑖𝑒𝐴}_𝜇^{)=𝛾𝜇𝐷𝜇}_𝐴. In the free case, that is, when 𝐴=0, we get ^𝑖(/^𝜕_𝐴^{−𝑚)𝜓=0}, the classical Dirac equation.

5.1.4. Chiral Symmetry

The chiral symmetry is the symmetry that leads to anomalies and the BRST invariance. In QCD the chiral symmetry of the Fermi field 𝜓 is given by 𝜓↦𝑒^{𝑖𝛽𝛾5}𝜓, where 𝛽 is a constant and 𝛾₅=𝑖𝛾^𝑜𝛾¹𝛾²𝛾³. The classical Noether current of this symmetry is given by 𝐽_𝜇=𝜓𝛾_𝜇𝛾₅𝜓 which is conserved; that is, 𝜕^𝜇𝐽_𝜇=0.

This conservation law breaks down after quantization; one gets𝜕^𝜇𝐽_𝜇=2im𝜓𝛾₅𝑔𝜓−²8𝜋²Tr𝐹_𝜇𝜈𝐹^𝜇𝜈≡𝜔≠0. (5.11) This value 𝜔 is called the chiral anomaly.

5.2. Quantization

The quantization is given by the Feynman path integral:_{𝒜/𝒢×𝒮𝑝𝑖𝑛}𝑒^{𝑖𝑆(𝐴,𝜓)}ℱ(𝐴,𝜓)𝒟𝐴𝒟𝜓=⟨ℱ(𝐴,𝜓)⟩ (5.12) which computes the expectation value ⟨ℱ(𝐴,𝜓)⟩ of the function ℱ(𝐴,𝜓). This is an integral over two infinite-dimensional spaces: the gauge orbit space 𝒜/𝒢 and the fermionic Berezin integral over the spin space 𝒮𝑝𝑖𝑛^±(𝑀). These integrals are mathematically not defined but physicists compute them by gauge fixing; that is, fixing a section 𝜎∶𝒜/𝒢→𝒜, (e.g., 𝜎(𝐴)=𝜕_𝜇𝐴^𝜇=0, the Lorentz gauge) and then integrating over the section 𝜎. Such a section does not exist globally, but only locally (Gribov ambiguity!). The effect of such a gauge fixing is that one gets extra terms in the Lagrangian (gauge-fixing terms) and one has to introduce new fields, so-called ghost fields 𝜂 via the Faddeev-Popov procedure. The such obtained effective Lagrangian is no longer gauge invariant. This effective Lagrangian has the form in QCD:ℒ_eff1(𝐴,𝜓,𝜂)=2Tr(𝐹_𝜇𝜈𝐹^𝜇𝜈+1)kineticenergy2𝛼Tr(𝜕_𝜇𝐴^𝜇)²gauge-fixingterm−𝑔𝜕_𝜇𝜂𝐷^𝜇_𝐴𝜂ghostterm+⋯interactionterms. (5.13)

We can write this globally asℒ_eff=12‖𝐹_𝐴‖²+12‖𝜎(𝐴)‖²+𝜂ℳ𝜂+⋯, (5.14) where ℳ=(𝛿/𝛿𝜙)(𝜎(𝜙⋅𝐴)) is the Faddeev-Popov determinant, acting like the Jacobian of the global gauge variation 𝛿/𝛿𝜙 over the section 𝜎. Writing this term in the exponent of the action functional like a “fermionic Gaussian integral” leads to the Faddeev-Popov ghost fields 𝜂,𝜂 in the form detℳ=∫𝑒^{−𝜂ℳ𝜂}𝑑𝜂𝑑𝜂.

The effective Lagrangian ℒ_eff is NOT gauge invariant but has a new symmetry, called BRST symmetry.

5.3. BRST Symmetry

Named after Becchi et al. [43] and Tyutin who discovered this invariance in 1975-76, the BRST operator 𝔰 is given as follows:1𝔰𝐴=𝑑𝜂+[𝐴,𝜂]𝔰𝜂=−2ℒ[𝜂,𝜂]^effisBRSTinvariant. (5.15) Note that the BRST operator 𝔰 mixes bosons (𝐴) and fermions (𝜂). This is an example of supersymmetry which we will discuss in Section 6. Also, the BRST operator 𝔰 is nilpotent; that is, 𝔰²=0. The question arises whether this operator 𝔰 is the coboundary operator of some kind of cohomology. The affirmative answer is given by the following theorem (Schmid [6, 44]).

Theorem 5.1. Let 𝒞^𝑞,𝑝(𝔩𝔦𝔢𝒢,Ω_loc) be the Chevalley-Eilenberg complex of the Lie algebra 𝔩𝔦𝔢𝒢 of infinitesimal gauge transformations, with respect to the induced adjoint representation on local forms Ω_loc, with boundary operator 𝛿_loc∶𝒞^𝑞,𝑝(𝔩𝔦𝔢𝒢,Ω_loc)→𝒞^𝑞+1,𝑝(𝔩𝔦𝔢𝒢,Ω_loc),𝛿²_loc=0. (5.16) Then with 𝔰∶=(−1)^𝑝+1/(𝑞+1)𝛿_loc, one has 𝔰²=0 and the following.(1)For 𝑞=0,𝑝=1,𝐴∈𝒜⊂𝒞^0,1, then 𝔰𝐴=d𝜂+[𝐴,𝜂].(2)For 𝑞=1,𝑝=0,𝜂∈𝒞^1,0, then 𝔰𝜂=−(1/2)[𝜂,𝜂], the Maurer-Cartan form.(3)The chiral anomaly 𝜔 (given by (5.11)) is represented as cohomology class of this complex [𝜔]∈ℋ^1,0_BRST(𝔩𝔦𝔢𝒢,Ω_loc).

5.3.1. The Chevalley-Eilenberg Cohomology

We are now going to explain the previous theorem, in particular the general definition of the Chevalley-Eilenberg [45] complex and the corresponding cohomology.

Let 𝐺 be a Lie group with Lie algebra 𝔤 and let 𝜎 be a representation of 𝔤 on the vector space 𝑊. Denote by 𝒞^𝑞(𝔤,𝑊) the space of 𝑊-valued 𝑞-cochains on 𝔤 and define the coboundary operator 𝛿∶𝒞^𝑞(𝔤,𝑊)→𝒞^𝑞+1(𝔤,𝑊) by𝛿Φ(𝜉_𝑜,…,𝜉_𝑞)=^𝑞_𝑖=0(−1)^𝑖𝜎(𝜉_𝑖)Φ(𝜉_𝑜,…,𝜉̂_𝑖,…,𝜉_𝑞)+_𝑖<𝑗(−1)^𝑖+𝑗Φ(𝜎(𝜉_𝑖)𝜉_𝑗,…,𝜉̂_𝑖,…,𝜉̂_𝑗,…,𝜉_𝑞). (5.17) We have 𝛿²=0, and define the Lie algebra cohomology of 𝔤 with respect to (𝜎,𝑊) as ℋ^∗(𝔤,𝑊)=ker𝛿/im𝛿. This is called the Chevalley-Eilenberg cohomology [45] of the Lie algebra 𝔤 with respect to the representation 𝜎.

5.3.2. Anomalies

The Noether current induced by the chiral symmetry (after quantization) for the free case (𝜓=0), that is, for pure Yang-Mills becomes𝜕^𝜇𝐽_𝜇𝑔=−²8𝜋²𝜀^{𝜇𝜈𝜌𝜏}Tr𝐹_𝜇𝜈𝐹_𝜌𝜏1=−4𝜋²𝜀^{𝜇𝜈𝜌𝜏}Tr𝜕_𝜇(𝐴_𝜈𝜕_𝜌𝐴_𝜏+23𝐴_𝜇𝐴_𝜌𝐴_𝜏)=𝜔≠0anomaly. (5.18) See (5.11).

Note the similarity with the Chern-Simon Lagrangianℒ(𝐴)=_𝑀2Tr(𝐴𝑑𝐴+3𝐴³). (5.19)

We are going to derive a representation of the chiral anomaly 𝜔 in the BRST cohomology that is [𝜔]∈ℋ^1,0_BRST(𝔩𝔦𝔢𝒢,Ω_loc).

The question is “if 𝔰𝜔=0, does there exist a local functional 𝐹(𝐴), such that 𝜔=𝔰(𝐹(𝐴))? That is, is 𝜔 BRST 𝔰-exact? The answer in general is NO; that is, 𝜔 represents a nontrivial cohomology class. This class is given by the Chern-Weil homotopy.

Let 𝐴=𝐴+𝜂∈𝒞^0,1×𝒞^1,0 and 𝐹≡𝑠𝐴+𝐴²=𝐹_𝐴. For 𝑡∈[0,1], let 𝐹_𝑡=𝑡𝐹+(𝑡²−𝑡)𝐴² and define the Chern-Simons form𝜔_2𝑞−1≡𝑞¹₀Tr(𝐴𝐹_𝑡^𝑞−1)𝑑𝑡, (5.20) we get𝔰𝜔_2𝑞−1=Tr𝐹^𝑞. (5.21) We write 𝜔_2𝑞−1 as sum of homogeneous terms in ghost number (upper index) and degree (lower index) 𝜔_2𝑞−1=𝜔⁰_2𝑞−1+𝜔¹_2𝑞−2+𝜔²_2𝑞−3+⋯+𝜔₀^2𝑞−1. Let 𝜔(𝑋,𝐴)=∫_𝑀𝜔¹_2𝑞−2(𝑋).

Theorem 5.2 (see Schmid [46]). The form 𝜔(𝑋,𝐴)=∫_𝑀∫¹₀𝐴𝐹_𝑡^𝑞−1(𝑋)𝑑𝑡 satisfies the Wess-Zumino consistency condition (𝔰𝜔)(𝑋₀,𝑋₁,𝐴)=0 and represents the chiral anomaly [𝜔]∈ℋ^1,0_BRST(𝔩𝔦𝔢𝒢,Ω_loc).

We have an explicit form of the anomaly in (2𝑞−2) dimensions: 𝜔¹_2𝑞−2=𝑞(𝑞−1)¹₀(1−𝑡)Tr(𝜂𝛿_loc(𝐴𝐹_𝑡^𝑞−2))𝑑𝑡. (5.22) So for 𝑞=2 the non-Abelian anomaly in 2 dimensions becomes 𝜔¹₂=Tr(𝜂𝛿_loc𝐴), and for 𝑞=3 the non-Abelian anomaly in 4 dimensions becomes𝜔¹₄=Tr(𝜂𝛿_loc(𝐴𝛿_loc1𝐴+2𝐴³)). (5.23)

5.4. The Standard Model

The standard model is a Yang-Mills gauge theory. Recall that the free Yang-Mills equations are 𝐷^∗_𝐴𝐹=0, where 𝐴 is a connection 1-form (vector potential), and 𝐹 is the associated curvature 2-form (field) on the principal bundle 𝑃. The connection 𝐴 defines the covariant derivative 𝐷_𝐴 and the curvature 𝐹 given by 𝐹=𝐷_𝐴𝐴=𝑑𝐴+(1/2)[𝐴,𝐴], or locally 𝐹_𝜇𝜈=𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇+[𝐴_𝜇,𝐴_𝜈], and 𝐷^𝜇𝐹_𝜇𝜈=0. Again the connection 𝐴 is the fundamental object.

For different choices of the gauge Lie group 𝐺, we obtain the 3 theories that make up the standard model. For 𝐺=𝑈(1) on a trivial bundle (i.e., global symmetry, which gives charge conservation) the curvature 2-form 𝐹 is simply the electromagnetic field, and the Yang-Mills equations 𝐷^∗_𝐴𝐹=0 are Maxwell’s equations  𝑑𝐹=0, locally 𝜕_𝜇𝐹^𝜇𝜈=0. For 𝐺=𝑈(1) as local gauge group we get the quantum mechanical symmetry and the equations of motion are Dirac’s equations. Combing the two, we get QED as a 𝑈(1) gauge theory. For 𝐺=𝑆𝑈(𝑁) we get the full non-Abelian Yang-Mills equations 𝐷^∗_𝐴𝐹=0. For weak interactions with 𝐺=𝑆𝑈(2) and combining the two (spontaneous symmetry breaking, Higgs), we get the Glashow-Weinberg-Salam model as 𝑆𝑈(2)×𝑈(1) Yang-Mills theory of electroweak interactions. For 𝐺=𝑆𝑈(3) we obtain the Yang-Mills equations 𝐷^∗_𝐴𝐹=0 for strong interactions and the equations of motion for QCD. Finally that standard model is a 𝑆𝑈(3)×𝑆𝑈(2)×𝑈(1) gauge theory governed by the corresponding Yang-Mills equations 𝐷^∗_𝐴𝐹=0. Recall that 𝐹 is the curvature in the corresponding principal bundle determined by the connection 𝐴.

For interactions, all the relevant fields involved can be considered as sections of corresponding associated vector bundles induced by representations of the gauge groups, for example, the Dirac operator on the associated spin bundle (induced by the spin representation of 𝑆𝑈(2)) acting on spinors (sections of this bundle). The vector potentials are the corresponding connection 1-forms and the Yang-Mills fields are the corresponding curvature 2-forms on these bundles over spacetime.

Again we do not need the metric and the curvature is determined by the potential, so the potential is the fundamental object.

5.5. Gravity

5.5.1. Stop Looking for Gravitons

Stop looking for the graviton, not because it had been found but because it does not exist. The graviton is supposed to be the particle that communicates the gravitational force. But the gravitational force is not a fundamental force. Gravity is geometry. One might as well search for the Corioliston for the coriolis force or the Centrifugiton for the centrifugal force.

Since Einstein in the 1920s, physicists have tried to unify what are considered the four fundamental forces, namely, electromagnetism, weak and strong nuclear forces, and the gravitational force. In the 1970s, the three nongravitational forces were unified in the standard model. At high enough energy (about 10¹⁵ GeV) they become the same force.

Since then, with all the string theory, SUSY, branes, and extra dimensions, the gravitational force could not be incorporated into GUT that includes all 4 forces and no graviton has been found experimentally. The reason is simple: not many people, including Einstein himself, take/took the general theory of relativity seriously enough, according to which we know that the gravitational force does not exist as fundamental force but as geometry! We do not feel it. What we feel is the resistance of the solid ground on which we stand. In general relativity, free-falling objects follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics because of the mechanical resistance of matter. Newton’s apple falls downward because the spacetime in which we exist is curved. The “gravitational force” is not a force but it is the geometry of spacetime as Einstein observed in [47, page 137]:

“Die Koeffizienten (𝑔_𝜇𝜈) dieser Mertik beschreiben in Bezug auf das gewählte Koordinatensystem zugleich das Gravitationsfeld.”

(“The coefficients (𝑔_𝜇𝜈) of this metric with respect to the chosen coordinate system describe at the same time the gravitational field”) [47, page 146]:

“Aus pysikalischen Gründen bestand die Überzeugung, dass das metrische Feld zugleich das Gravitationsfeld sei.”

(“For physical reasons there was the conviction that the metric field was at the same time the gravitational field”).

Therefore GUT, the grand unified theory had been completed since the 1970s with the standard model. Since the gravitational force does not exist as a fundamental force, there is nothing more to unify as forces. If we want to unify all four theories, then it has to be done in a geometric way. The equations governing gravity as well as the standard model are all curvature equations, Einstein’s equation, and the Yang-Mills equations.

5.5.2. Einstein’s Vacuum Field Equations

Let (𝑀,𝑔) be spacetime with Lorentzian metric 𝑔. Then Einstein’s vacuum field equations are Ric=0, (5.24) where Ric is the Ricci curvature of the Lorentz metric 𝑔. These are the Euler-Lagrange equations for the Lagrangian ℒ(𝑔)=∫𝑅(𝑔)𝜇(𝑔), where 𝜇(𝑔)=−det𝑔𝑑⁴𝑥 and 𝑅(𝑔) is the scalar curvature of 𝑔.

Or in general, locally, in terms of the stress-energy tensor 𝑇_𝜇𝜈, Einstein’s equations are 𝐺_𝜇𝜈=𝜅𝑇_𝜇𝜈 with the Einstein tensor 𝐺_𝜇𝜈=𝑅_𝜇𝜈−(1/2)𝑔_𝜇𝜈𝑅. The stress-energy tensor 𝑇_𝜇𝜈 is the conserved Noether current corresponding to spacetime translation invariance.

The Levi-Civita connection Γ of the Riemannian metric 𝑔 is given by Γ^𝜆_𝜇𝜈=12𝑔^𝜆𝜎(𝜕𝑔_𝜎𝜇𝜕𝑥^𝜈+𝜕𝑔_𝜎𝜈𝜕𝑥^𝜇−𝜕𝑔_𝜇𝜈𝜕𝑥^𝜎). (5.25)

The curvature tensor 𝑅 and the Ricci curvature Ric in Einstein’s field equations are completely determined by the connection Γ.

First the curvature tensor 𝑅 is locally given by 𝑅^𝜆_𝜇𝜈𝜅=(𝜕Γ^𝜆_𝜇𝜈𝜕𝑥^𝜅−𝜕Γ^𝜆_𝜇𝜅𝜕𝑥^𝜈)+(Γ^𝜂_𝜇𝜈Γ^𝜆_𝜅𝜂−Γ^𝜂_𝜇𝜅Γ^𝜆_𝜈𝜂). (5.26) Taking its trace, we get the Rici tensor Ric as (Ric)^𝜇𝜈=𝑅^𝜆_𝜇𝜆𝜈.

So we can express Einstein’s equations completely in terms of the connection (potential) Γ; we do not need the metric 𝑔; also the curvature 𝑅 is determined by the potential Γ. So the potential Γ is the fundamental object.

The free motion in spacetime is along geodesic curves 𝛾(𝑡) which again are expressed in terms of the connection by 𝛾̈^𝛼+Γ^𝛼_𝛽𝜈𝛾̇^𝛽𝛾̇^𝜈=0. (5.27)

5.5.3. Symmetry

In general relativity the diffeomorphism group plays the role of a symmetry group of coordinate transformations. Then the vacuum Einstein’s field equations Ric(𝑔)=0 are invariant under coordinate transformations, that is, under the action of Diff^∞(𝑀). Denote by ℳ the space of all metrics 𝑔 on 𝑀. Then, Einstein’s field equations Ric(𝑔)=0 are a Hamiltonian system on the reduced space 𝑃=𝑀/Diff^∞(𝑀); see the study by Marsden et al. in [15] for details.

5.6. Conclusions

The relation between the connection Γ occurring in Einstein’s equations and the connection 𝐴 in Yang-Mills equations is as follows. 𝐴 is a Lie algebra-valued 1-form on a principal 𝐺-bundle (𝑃,𝜋,𝑀) over spacetime (𝑀,𝑔) (or any associated vector bundle given by representations of 𝐺). The Levi-Civita connection Γ is a connection 1-form in this sense on the tangent bundle 𝑇𝑀 (frame bundle) with G=𝐺𝐿(𝑛). So in this sense general relativity and the standard model are Yang-Mills gauge theories.

Therefore all four theories, electromagnetism, weak interaction, strong interaction, and gravity, are unified as curvature equations in vector bundles over spacetime. Different interactions require different bundles.

There is no hierarchy problem because there is no fundamental gravitational force. The question why gravitational interaction is so much weaker than electroweak and strong interactions is meaningless, comparing apples with oranges. Why are so many physicists still talking about gravitational force? It is like as if we are still talking about “sun rise” and “sun set”, 500 years after Copernicus! only worse; these are serious scientists trying to unify all four “forces” to a TOE.

I am not saying that there are no open problems in physics. Of course there is still the problem of unifying quantum mechanics and general relativity on a geometric level (not as forces). The question is “how does spacetime look at the Planck scale? Do we have to modify spacetime to incorporate quantum mechanics or quantum mechanics to accommodate spacetime, or both? We need a theory of quantum gravity. There are several theories in the developing stage that promise to accomplish this. (i)Superstring theory by E. Witten et al. (ii)Discrete spacetime at Planck length by R. Loll in "Causal dynamical triangulation" and by J. Ambjorn, J. Jurkiewicz, and R. Loll in "The Universe from Scratch" [arXiv: hep-th/0509010].(iii)Spacetime quantization: loop quantum gravity by L. Smolin in “Three Roads to Quantum Gravity” (London: Weidenfeld and Nicholson, 2000) and by S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin in “Quantum Gravity and the Standard Model”, preprint 2006.(iv)Geometric formulation of quantum mechanics by A. Ashtekar and T. A. Schilling [arXiv: gr-qc/9706069].(v)Deterministic quantum mechanics at Planck scale by G. t’Hooft in “Quantum Gravity as a Dissipative Deterministic System.” (vi)Branes and new dimensions: parallel universes by L. Randell et al., D. Deutsch, PS. In the brane world, gravity is again singled out as the only force not confined to one brane. (vii)Noncommutative Geometry. A. Connes describes the standard model form general relativity. (viii)The most recent new development is by Verlinde [48]. He agrees that gravity is not a fundamental force, but explains it as an emergent force (entropic force) caused by a change in the amount of information (entropy) associated with the positions of bodies of matter.

See [6, 11, 27, 36, 41–59].

6. SUSY (Supersymmetry)

Supersymmetry (SUSY) is an important idea in quantum filed theory and string theories. The BRST symmetry we described in Section 5.3 is an example of SUSY. Now we give a summary of a mathematical description of super Hamiltonian systems on supersymplectic supermanifolds, state a generalization of the Marsden-Weinstein reduction theorem in this context, and illustrate the method with examples. This is a very technical topic, so we only give a brief sketch; for details see the studies Glimm in [60] and Tuynman in [61].

The classical Marsden-Weinstein reduction theorem is a geometrical result stating that if a Lie group 𝐺 acts on a symplectic manifold 𝑃 by symplectomorphisms and admits an equivariant momentum map 𝐽∶𝑃→𝔤^∗, then, for any regular value 𝜇∈𝔤 of 𝐽, the quotient 𝑃_𝜇=𝐽⁻¹(𝜇)/𝐺_𝜇 of the preimage 𝐽⁻¹(𝜇) by the isotropy group 𝐺_𝜇 of 𝜇 has a natural symplectic structure. A dynamical interpretation of the Marsden-Weinstein reduction theorem gives the following. If a given Hamiltonian 𝐻∈𝐶^∞(𝑃) is invariant under the action of the group 𝐺, then it projects to a reduced Hamiltonian 𝐻_𝜇 on the reduced space 𝑃_𝜇. The integral curves of the Hamiltonian vector field 𝑋_𝐻 project to the integral curves of 𝑋_𝐻𝜇. In this sense, one has reduced the system by symmetries. This reduction procedure unifies many methods and results concerning the use of symmetries in classical mechanics, some dating back to the time of Euler and Lagrange.

In [60], Glimm generalizes this result to the setting of supermanifolds, using the analytic construction of supercalculus and supergeometry due to DeWitt [62] and Tuynman [61]. The general idea of supermanifolds and superanalysis is to do geometry and analysis over a graded algebra of even supernumbers rather than ℝ. In “supermathematics,” we have even and odd variables. Two variables 𝑎,𝑏 are called even, or commuting, or bosonic if 𝑎⋅𝑏=𝑏⋅𝑎, whereas 𝜉,𝜒 are called odd, or anticommuting, or fermionic if 𝜉⋅𝜒=−𝜒⋅𝜉. The problem is doing analysis with even and odd variables.

Many classes of differential equations have extensions that involve odd variables. These are called superized versions, or supersymmetric extensions. An active area of research is in particular the construction of supersymmetric integrable systems.

As an example, consider the Korteweg-de Vries equation 𝑢_𝑡=−𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥. A possible supersymmetric extension is the following system of an even variable 𝑢(𝑡,𝑥) and an odd variable 𝜉(𝑡,𝑥):𝑢_𝑡=−𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥−3𝜉𝜉_𝑥𝑥,𝜉_𝑡=−𝜉_𝑥𝑥𝑥+3𝑢𝜉_𝑥+3𝜉𝑢_𝑥. (6.1)

There is a different way of writing system (6.1). For this, one considers the so-called 2|1-dimensional superspace. This is a space with coordinates (𝑥,𝑡,𝜗), where 𝑥 and 𝑡 are even numbers as before and 𝜗 is an odd number. We can now gather the components 𝑢(𝑥,𝑡) and 𝜉(𝑥,𝑡) into a superfield Φ(𝑡,𝑥,𝜗) defined on superspace viaΦ(𝑡,𝑥,𝜗)=𝜉(𝑡,𝑥)+𝜗⋅𝑢(𝑡,𝑥). (6.2) The function Φ(𝑡,𝑥,𝜗) takes as values odd numbers; it is thus called an odd function. Note how the right-hand side can be seen as a Taylor series in 𝜗; indeed, all higher powers of 𝜗 are zero because 𝜗 is anticommuting. One defines the odd differential operator 𝐷=𝜕/𝜕𝜗+𝜗⋅(𝜕/𝜕𝑥), acting on superfields. A computation yields that system (6.1) is equivalent toΦ_𝑡=−𝐷⁶Φ+3𝐷²Φ𝐷Φ. (6.3) (Note that 𝐷²=𝜕_𝑥.) System (6.1) is called the component formulation; (6.3) is called the superspace formulation. In our example at hand, a justification for calling the system an “extension” of KdV would be that if one takes (6.3) and writes it in component form (6.1), one recovers the “usual” KdV by setting 𝜉 to zero. Also, it can be shown that (6.3) is invariant under transformations Φ(𝑡,𝑥,𝜗)↦Φ(𝑡,𝑥−𝜂𝜗,𝜗+𝜂), where 𝜂 is an odd parameter. The infinitesimal version of this transformation is 𝛿Φ=𝜂(𝜕_𝜗−𝜗𝜕_𝑥)Φ. This transformation is called a “supersymmetry’’ in the present context. In components, it reads𝛿𝑢=𝜂𝜉_𝑥,𝛿𝜉=𝜂𝑢. (6.4) These equations illustrate a characterization of supersymmetries: supersymmetries (as opposed to regular symmetries) “mix” even and odd variables.

One needs some concept of supermanifold even if one only works with the component formulation. For example, one has implicitly in system (6.1) the space of all (𝑢(𝑡,𝑥),𝜉(𝑡,𝑥)) on which the equations are defined; this is some kind of superspace itself. Also, there is some kind of supersubmanifold of those (𝑢,𝜉) which solve the equations.

In [60], Glimm proves a comprehensive result on supersymplectic reduction. He uses an analytic-geometric approach to the theory of supermanifolds, and not the Kostant theory of graded manifolds [49]. The Poisson bracket induced by odd supersymplectic forms is not a super Lie bracket on the space of supersmooth functions. This stands of course in contrast to both the usual ungraded case and the super case with even supersymplectic forms. While this makes the algebraic approach conceptually more difficult, no such problems arise in the analytic approach. Also, we do not require that the action be free and proper, but have the weaker requirement that the quotient space only has a manifold structure.

There are different approaches to supermanifolds. The Algebraic approach (Kostant [49], Berezin-Leĭtes, late 1970s) takes “superfunctions” as fundamental object. A graded manifold is a pair (𝑀,𝐴), where 𝑀 is a conventional manifold and 𝑈↦𝐴(𝑈) is the following sheaf over 𝑀:𝐴(𝑈)≃𝐶^∞(𝑈)⊗ℝ^𝑛,𝑈⊆𝑀. (6.5) So a superfunction 𝑓∈𝐴(𝑈) can be written as 𝑓=𝑓_∅+_𝑖𝑓_{𝑖}𝜗^𝑖+_𝑖<𝑗𝑓_{𝑖,𝑗}𝜗^𝑖𝜗^𝑗+⋯. (6.6)

The Geometric approach (DeWitt [62]) takes points as fundamental objects. Let 𝒜 be the ring generated by 𝐿 generators 𝜃¹,𝜃²,…,𝜃^𝐿 (Grassmann numbers) with relations 𝜃^𝑖𝜃^𝑗=−𝜃^𝑗𝜃^𝑖.

An element is written as𝑎=𝑎_{∅=B𝑎(“Body")}+_𝑖𝑎_{𝑖}𝜃^𝑖+_𝑖<𝑗𝑎_{𝑖,𝑗}𝜃^𝑖𝜃^𝑗+⋯_{nilpotentpart}(𝑎_𝜔∈ℝ),𝑎∈𝒜₀⇔𝑎=𝑎_∅+𝑎_{𝑖,𝑗}𝜃^𝑖𝜃^𝑗+termswitheven#of𝜃s,𝑎∈𝒜₁⇔𝑎=𝑎_{𝑖}𝜃^𝑖+𝑎_{{𝑖,𝑗,𝑘}}𝜃^𝑖𝜃^𝑗𝜃^𝑘+termswithodd#of𝜃s. (6.7)

For calculus we “replace” reals ℝ by Grassmann numbers 𝒜. A DeWitt supermanifold is a topological space 𝑀 which is locally superdiffeomorphic to 𝒜=^{𝑚∣𝑛def}(𝒜^𝑚⊕𝑛)₀=𝒜^𝑚₀×𝒜^𝑛₁.

The two approaches are equivalent. There is a one-to-one correspondence between isomorphism classes of 𝑚|𝑛-dimensional DeWitt supermanifolds whose body is a fixed smooth 𝑚-dimensional manifold 𝑋 and isomorphism classes of 𝑚|𝑛-dimensional graded manifolds over 𝑋.

The DeWitt topology of 𝒜^𝑚∣𝑛 is defined as follows.𝑈⊆𝒜^𝑚∣𝑛 is open if and only if B(𝑈) is open in 𝐵𝒜^𝑚⊕𝑛=ℝ^𝑚,𝑈=𝐵⁻¹(𝐵(𝑈)).

Smooth functions 𝒜^𝑚∣𝑛→𝒜 are defined as follows. Let 𝑈⊆𝒜^𝑚∣𝑛 be an open set. A function 𝑓∶𝑈→𝒜 is called smooth if there is a collection of smooth real functions defined on 𝐵𝑈⊆ℝ^𝑚:𝑓_{𝑖1…𝑖𝑛}∶𝐵𝑈→ℝ,for𝑖₁,…,𝑖_𝑛=0,1 (6.8) such that 𝑓(𝑥₁,…,𝑥_𝑚,𝜉₁,…,𝜉_𝑛)=¹_{𝑖1,…,𝑖𝑛=0}𝜉^𝑖1₁…𝜉^𝑖𝑛_𝑛⋅𝑍𝑓_{𝑖1…𝑖𝑛}(𝑥₁,…,𝑥_𝑚), (6.9) where 𝑍𝑔 is defined as follows. If 𝐱∈𝒜^𝑚∣0 has the decomposition 𝐱=𝐵𝐱+𝐧, then𝑍𝑔(𝐱)=^∞_𝑘=11𝐷𝑘!^𝑘𝑔(𝐵𝐱)[𝐧,…,𝐧]. (6.10)

Note. Smooth functions map the body to the body!

Superdifferential Geometry
Versions of the inverse function theorem and implicit function theorem are still valid. Concepts of tangent space, vector fields, flows, and Lie groups can be developed as in the ungraded case.

Lie Supergroups
A Lie supergroup is a supermanifold 𝐺 that is a group and for which the group operations of multiplication and inversion are smooth. If a Lie supergroup 𝐺 acts freely and properly on a supermanifold 𝑀, then the quotient 𝑀/𝐺 can be given the structure of a supermanifold such that the projection 𝜋∶𝑀→𝑀/𝐺 is a surjective submersion. The structure of 𝑇(𝑀/𝐺) is given by the following.

Theorem 6.1. Let 𝜑∶𝑀×𝐺→𝑀 be an action. Suppose that 𝑀/𝐺 has a supermanifold structure such that 𝜋∶𝑀→𝑀/𝐺 is a surjective submersion; that is, 𝑇𝜋(𝑝) is onto for every 𝑝∈𝑀. For any 𝑝∈𝑀, one has ker𝑇𝜋(𝑝)={𝔛_𝑀(𝑝)∣𝔛∈𝔤} (6.11) and this is a proper subspace of 𝑇_𝑝𝑀. In particular, 𝑇_[𝑝](𝑀/𝐺)≃𝑇_𝑝𝑀/{𝔛_𝑀(𝑝)∣𝔛∈𝔤}. (6.12)

Remark 6.2. There are examples where 𝑀/𝐺 does not have a supermanifold structure and {𝔛_𝑀(𝑝)∣𝔛∈𝔤} fails to be a free submodule.

Supersymplectic Structures
A supersymplectic supermanifold (𝑀,𝜔) is a supermanifold 𝑀 together with a closed (i.e., 𝑑𝜔=0) nondegenerate homogeneous left 2-form 𝜔∈Ω²_𝐿(𝑀).

Examples 6.3. (1) 𝒜^2𝑚∣𝑛 with coordinates (𝑞^𝑖,𝑝^𝑖,𝜉^𝑗), 𝜔=𝑑𝑞^𝑖∧𝑑𝑝^𝑖+12𝑑𝜉^𝑗∧𝑑𝜉^𝑗 (6.13) defines an even supersymplectic form.
(2) On 𝒜^𝑚∣𝑚 with coordinates (𝑥^𝑖,𝜉^𝑗), 𝜔=𝑑𝑥^𝑖∧𝑑𝜉^𝑖 (6.14) defines an odd supersymplectic form.

Let 𝜔∈Ω²_𝐿(𝑀) be a 2-form on the supermanifold 𝑀 and 𝑝∈𝑀. Then 𝜔(𝑝)∈Alt²_𝐿(𝑇_𝑝𝑀) is nondegenerate if and only if the real 2-form 𝐵𝜔(𝐵𝑝)=𝜔(𝐵𝑝)∣_{𝐵(𝑇𝐵𝑝𝑀)}∈Alt²(𝐵(𝑇_𝐵𝑝𝑀)) is nondegenerate.

Hamiltonian Supermechanics
A smooth vector field 𝑋∈𝔛(𝑀) is called (globally) Hamiltonian if there is some function 𝐻∈𝐶^∞(𝑀,𝒜) such that 𝑖_𝑋𝜔=𝑑𝐻. For 𝑓,𝑔∈𝐶^∞(𝑀,𝒜) define the Super-Poisson bracket by {𝑓,𝑔}=⟨𝑋_𝑓,𝑋_𝑔∣𝜔⟩∈𝐶^∞(𝑀,𝒜). (6.15)

Fact 6. If 𝜔 is even, then 𝐶^∞(𝑀,𝒜) is a Lie superalgebra with respect to {⋅,⋅}. This is false if 𝜔 is odd.

Momentum Maps
Let 𝜑∶𝑀×𝐺→𝑀 be an action of the Lie supergroup 𝐺 on the supersymplectic manifold (𝑀,𝜔) which preserves 𝜔. Recall that, in the ungraded case, a momentum map is an ℝ-linear map 𝐽̂∶𝔤→𝐶^∞(𝑀) such that 𝑋̂𝐽_(𝔛)=𝔛_𝑀.

Superversion. 𝐽̂∈𝐶^∞(𝔤×𝑀,𝒜) is a momentum map for the action of 𝐺 on 𝑀 if 𝐽̂ is leftlinear in the first argument and ⟨(0,𝑣)∣𝑑𝐽(̂𝔛,𝑥)⟩=⟨𝑣,𝔛_𝑀(𝑥)∣𝜔(𝑥)⟩ (6.16) for all 𝑥∈𝑀, for all 𝑣∈𝑇_𝑥𝑀. Instead of 𝐽̂∈𝐶^∞(𝔤×𝑀,𝒜), one can consider 𝐉∶𝑀→𝔤^∗ defined through ⟨𝔛∣𝐉(𝑥)⟩=𝐽(̂𝔛,𝑥).

Theorem 6.4. Let 𝐻∈𝐶^∞(𝑀,𝒜) be a Hamiltonian with vector field 𝑋_𝐻 such that [𝑋_𝐻,𝑋_𝐻]=0. If 𝐻 is 𝐺-invariant, then 𝐉 is preserved by the flow 𝜙 of 𝑋_𝐻; that is, 𝐉∘𝜙_𝑡,𝜏=𝐉∀(𝑡,𝜏)∈𝒜^1∣1(wheredefined). (6.17)

Suppose that the momentum map is 𝐴𝑑^∗-equivariant. Let 𝜇∈𝐵𝔤^∗_𝑝(𝜔) be a regular value of 𝐉. Let 𝐺_𝜇 be the isotropy group of 𝜇. Suppose that the quotient space 𝑃_𝜇=𝐉⁻¹(𝜇)/𝐺_𝜇 can be given a supermanifold structure such that the projection 𝜋_𝜇∶𝐉⁻¹(𝜇)→𝑃_𝜇 is a surjective submersion.

Superreduction Theorem
The supermanifold 𝑃_𝜇 has a unique supersymplectic form 𝜔_𝜇∈Ω²_𝐿(𝑃_𝜇) with the property 𝜋^∗_𝜇𝜔_𝜇=𝚤^∗_𝜇𝜔, (6.18) where 𝜋_𝜇∶𝐉⁻¹(𝜇)→𝑃_𝜇 is the projection and 𝚤_𝜇∶𝐉⁻¹(𝜇)→𝑀 is the inclusion. The form 𝜔_𝜇 has the same parity as 𝜔.

Example 6.5 (The Bose-Fermi Oscillator). We consider the phase space 𝑃=𝒜^2∣2 with supersymplectic form 𝜔=𝑑𝑝∧𝑑𝑞+(1/2)(𝑑𝜉₁∧𝑑𝜉₁+𝑑𝜉₂∧𝛿𝜉₂). The Hamiltonian 𝐻(𝑞,𝑝,𝜉₁,𝜉₂)=(1/2)(𝑝²+𝑞²)−𝜉₁𝜉₂ defines the Hamiltonian vector field 𝑋_𝐻=𝑝𝜕_𝑞−𝑞𝜕_𝑝−𝜉₂𝜕_𝜉1+𝜉₁𝜕_𝜉2.

The SUSY algebra of the Bose-Fermi oscillator is given as follows. The Lie supergroup of invertible 2|2-supermatrices acts on 𝒜^2∣2 via 𝒜^2∣2×𝐺𝐿(2|2)→𝒜^2∣2,(𝐪,G)↦𝐪^𝑆𝑇𝐺. (6.19)

The algebra of Bose-Fermi supersymmetry is the intersection of the Lie superalgebras of the stabilizer of 𝜔 and the stabilizer of 𝐻:𝔟𝔣(2|2)=𝔰𝔱𝔞𝔟(𝐻)∩𝔰𝔱𝔞𝔟(𝜔). (6.20)

The SUSY algebra of the Bose-Fermi oscillator 𝔟𝔣(2|2) is generated by 𝐴₁,𝐴₂,𝐶₁,𝐶₂, where 𝐴_𝑖=00𝐞^𝑇_𝑖𝜸⁰00−𝐞^𝑇_𝑖𝐞_𝑖−𝜸⁰𝐞_𝑖𝟎𝐶for𝑖=1,2,₁=12−𝜸⁰𝟎𝟎𝜸⁰,𝐶₂=12𝜸⁰𝟎𝟎𝜸⁰,𝜸⁰=.0−110 (6.21)

Momentum Map and Quotient
The action of 𝐵𝐹(2|2) on 𝒜^2∣2 admits a momentum map 𝐽̂∶𝔟𝔣(2|2)×𝒜^2∣2→𝒜,(𝑋,𝐪)↦12𝐪^𝑆𝑇𝑋Ω𝐪. (6.22) In components, we write 𝑋=∑_𝑖=1,2𝑎_𝑖𝐴_𝑖+𝑐_𝑖𝐶_𝑖. Then 𝐽(̂𝑋,𝑞,𝑝,𝝃)=(𝑐₂−𝑐₁)14(𝑝²+𝑞²)−(𝑐₂+𝑐₁)14𝝃^𝑇𝜸⁰𝝃−𝐚^𝑇(𝑝𝕀₂−𝑞𝜸⁰)𝝃, (6.23) where 𝐚=(𝑎₁,𝑎₂)^𝑇.

Now let 𝝁∈𝐵(𝔟𝔣(2|2)^∗₀)≃ℝ². The isotropy group of 𝝁 is the whole group 𝐵𝐹(2|2). The action of 𝐵𝐹(2|2) on 𝐉⁻¹(𝝁) is transitive; that is, the quotient space 𝐉⁻¹(𝝁)/𝐵𝐹(2|2) is a single point.

Example 6.6 (Wess-Zumino Model in 2+1-Dimensional Spacetime). Let 𝒮(𝒜) be the Schwartz space of functions ℝ²→𝒜. Consider the phase space 𝒮(𝒜)^2∣2=[𝒮(𝒜)₀]×[𝒮(𝒜)₀]×[𝒮(𝒜)₁]²∋(𝜙,𝜋,𝜓) with supersymplectic form 𝜔=𝑑𝜋∧𝑑𝜙+(1/2)𝑑𝜓∧𝑑𝜓 and Hamiltonian 1ℋ(𝜙,𝜋,𝜓)=2_ℝ2(‖∇𝜙‖²+𝜋²+𝜓𝔤^𝑖𝑑_𝑖𝜓)𝑑²𝑥, (6.24) where 𝔤¹=(⁰¹₁₀), 𝔤²=(¹⁰₀₋₁), and 𝜓=𝜓^𝑇𝛾⁰.

Then Hamilton’s equations are the following:𝜙̇=𝛿ℋ̇𝛿𝜋=𝜋,𝜋=−𝛿ℋ̇𝛿𝜙=Δ𝜙,𝜓=𝛿ℋ𝛿𝜓=𝔤⁰𝔤^𝑖𝜕_𝑖𝜓. (6.25) This is equivalent to (𝜕²₀−𝜕²₁−𝜕²₂)𝜙=0, 𝔤^𝜇𝜕_𝜇𝜓=0 which are the massless Klein-Gordon and Dirac equations in 2+1-dimensional spacetime.

Remark 6.7. The well-known SUSY algebra from the Lagrangian description can be “exported” to the Hamiltonian setup.

We reduce by an Abelian subgroup of the SUSY group:𝒮(𝒜)^2∣2×𝒜^2∣0→𝒮(𝒜)^2∣2,(Φ,𝑟)↦[𝑆_𝑟(Φ)](𝑥)=_𝑘1𝐷𝑘!^𝑘Φ(𝑥+𝐵𝑟)[𝑛,…,𝑛], (6.26) where 𝑟=𝐵𝑟+𝑛, is a “superspatial shift”. The momentum map becomes𝐉(Φ)=((𝜋𝜕_𝑖1Φ+2𝜓^𝑇𝜕_𝑖𝜓))_𝑖=1,2. (6.27)

We determine the reduced phase space. The center of mass of Φ=(𝜙,𝜋,𝜓)∈𝒮(𝒜)^2∣2 is 𝒞(Φ)=(1/𝑀(Φ))∫_ℝ2𝑥⋅|Φ|²𝑑²𝑥∈𝒜^2∣0, where 𝑀(Φ)=∫|Φ|²𝑑²𝑥∈𝒜₀. We may identify the quotient space 𝐉⁻¹(𝝁)/𝒜^2∣0 with the subset𝑃_𝜇={Φ∈𝐉⁻¹(𝜇)∣𝒞(Φ)=0}⊆𝒮(𝒜)^2∣2. (6.28)

Finally the reduced equations become𝜙̇=𝜋+𝐹^𝑖(𝜙,𝜋,𝜓)𝜕_𝑖̇𝜙,𝜋=Δ𝜙+𝐹^𝑖(𝜙,𝜋,𝜓)𝜕_𝑖̇𝜋,𝜓=𝔤⁰𝔤^𝑖𝜕_𝑖𝜓+𝐹^𝑖(𝜙,𝜋,𝜓)𝜕_𝑖𝜓, (6.29) where 𝐹^𝑖(Φ)=(1/𝑀(Φ))∫𝑥^𝑖⋅(𝜙𝜋+𝜋Δ𝜙−𝜓^𝑇𝔤^𝑗𝜕_𝑗𝜓)𝑑²𝑥.

See [41, 42, 49, 60–63].