Nonlinear and Noncommutative Mathematics: New Developments and Applications in Quantum PhysicsView this Special Issue
Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics
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.
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  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 is a group and an infinite-dimensional manifold with smooth group operations Such a Lie group 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 , we talk about Banach, Hilbert, or Frechet Lie groups, respectively.
The Lie algebra of a Lie group is defined as left invariant vector fields on (tangent space at the identity ). The isomorphism is given (as in finite dimensions) by and the Lie bracket on is induced by the Lie bracket of left invariant vector fields
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 . 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 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 is a finite-dimensional Lie group, the exponential map is defined as follows. To each , we assign the corresponding left invariant vector field defined by (2.3). We take the flow of and define . The exponential map is a local diffeomorphism from a neighborhood of zero in onto a neighborhood of the identity in ; hence defines canonical coordinates on the Lie group . This is not true in infinite dimensions.(3)If are smooth Lie group homomorphisms (i.e., ) with , then locally . This is not true in infinite dimensions.(4)If is a continuous group homomorphism between finite-dimensional Lie groups, then is smooth. This is not true in infinite dimensions.(5)If is any finite-dimensional Lie algebra, then there exists a connected finite-dimensional Lie group with as its Lie algebra; that is, . This is not true in infinite dimensions.(6)If is a finite-dimensional Lie group and is a closed subgroup, then is a Lie subgroup (i.e., Lie group and submanifold). This is not true in infinite dimensions.(7)If is a finite-dimensional Lie group with Lie algebra and is a subalgebra, then there exists a unique connected Lie subgroup with as its Lie algebra; that is, . This is not true in infinite dimensions.
Some classical examples of finite-dimensional Lie groups are the matrix groups , , , , , , and with smooth group operations given by matrix multiplication and matrix inversion. The Lie algebra bracket is the commutator with exponential map given by .
2.3. Examples of Infinite-Dimensional Lie Groups
2.3.1. The Vector Groups
Let be a Banach space and take with , , and , which makes into an Abelian Lie group; that is, . For the Lie algebra we have . For the corresponding left invariant vector field is given by , ; that is, . Hence the Lie algebra with the trivial Lie bracket is Abelian. For the exponential map we get .
2.3.2. The General Linear Group
Let be a Banach space and the space of bounded linear operators . Then is a Banach space with the operator norm , and the group of all invertible elements is open in . So is a smooth Lie group with ,, and . Its Lie algebra is with the commutator bracket and exponential map .
2.3.3. The Abelian Gauge Group
Let be a finite-dimensional manifold and let (smooth functions on ). With group operation being addition, that is, , , and . is an Abelian (addition is smooth) Frechet Lie group with Lie algebra with trivial bracket , and . If we complete these spaces in the -norm, (denoted by ), then is a Banach-Lie group, and if we complete in the -Sobolev norm with then is a Hilbert-Lie group.
2.3.4. The Abelian Gauge Group
Let be a finite-dimensional manifold and let with group operation being multiplication; that is, , , and . For , is open in , and if is compact, then is a Banach-Lie group. If , then is closed under multiplication, and if is compact, then is a Hilbert-Lie group.
2.3.5. Loop Group
We generalize the Abelian example (see Section 2.3.4) by replacing with any finite-dimensional (non-Abelian) Lie group . Let with pointwise group operations , , and , where “” and “” are the operations in . If then is a Banach-Lie group. Let denote the Lie algebra of , then the Lie algebra of is with pointwise Lie bracket , , the latter bracket being the Lie bracket in . The exponential map defines the exponential map , , which is a local diffeomorphism. The same holds for if .
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.
As a special case of example mentioned in Section 2.3.5 we take , the circle. Then is called a loop group and 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,  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 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 ) and let be the group of all smooth diffeomorphisms on , with group operation being composition; that is, , , and . For diffeomorphisms, 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 ) to Frechet spaces; see the study by Keller in . One possibility is to generalize the notion of differentiability. For example, if we use the so-called differentiability, then becomes a Lie group with differentiable group operations. These notions of differentiability are difficult to apply to concrete examples. Another possibility is to complete in the Banach -norm, , or in the Sobolev -norm, . Then and become Banach and Hilbert manifolds, respectively. Then we consider the inverse limits of these Banach- and Hilbert-Lie groups, respectively: becomes a so-called ILB- (Inverse Limit of Banach) Lie group, or with the Sobolev topologies becomes a so-called ILH- (Inverse Limit of Hilbert) Lie group. See the study by Omori in  for details. Nevertheless, the group operations are not smooth, but have the following differentiability properties. If we equip the diffeomorphism group with the Sobolev -topology, then becomes a Hilbert manifold if and the group multiplication is differentiable; hence for , is only continuous on . The inversion is differentiable; hence for , is only continuous on . The same differentiability properties of and hold in the topology.
The Lie algebra of is given by being the space of smooth vector fields on . Note that the space of all vector fields is a Lie algebra only for vector fields, but not for or vector fields if , , because one loses derivatives by taking brackets.
The exponential map on the diffeomorphism group is given as follows. For any vector field , take its flow , then define , the flow at time . The exponential map 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
Several subgroups of have important applications.
2.4.2. Group of Volume-Preserving Diffeomorphisms
Let be a volume on and the group of volume-preserving diffeomorphisms. is a closed subgroup of with Lie algebra being the space of divergence-free vector fields on . is a Lie subalgebra of .
Remark 2.1. We cannot apply the finite-dimensional theorem that if is Lie algebra then there exists a Lie group whose Lie algebra it is; nor the one that if is a closed subgroup then it is an Lie subgroup.
Nevertheless is an ILH-Lie group.
2.4.3. Symplectomorphism Group
Let be a symplectic 2-form on and the group of canonical transformations (or symplectomorphisms). is a closed subgroup of with Lie algebra being the space of locally Hamiltonian vector fields on . is a Lie subalgebra of . Again 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 be a principal bundle with being a finite-dimensional Lie group (structure group) acting on from the right , , and .
The Gauge group is the group of gauge transformations defined by is a group under composition, hence a subgroup of the diffeomorphism group . Since a gauge transformation preserves fibers, we can realize each such via , where satisfies , for , . Let is a group under pointwise multiplication, hence a subgroup of the loop group (see Section 2.4.3), which extends to a Hilbert-Lie group if equipped with the -Sobolev topology. We give the induced topology and extend it to a Hilbert-Lie group denoted by . Another interpretation is that is isomorphic to the space of sections of the associated vector bundle . Completed in the Sobolev topology, we get .
Let denote the Lie algebra of . Then the Lie algebra of is a subalgebra of the loop algebra under pointwise bracket in , the finite-dimensional Lie algebra of ; that is, for any the bracket is defined by , . Then is the subalgebra of -invariant -valued functions on ; that is,
The Lie algebra (running out of symbols) of the gauge group is the Lie subalgebra of consisting of all -invariant vertical vector fields on ; that is, with commutator bracket .
On the other hand, the Lie algebra of is being the space of sections of the associated vector bundle with pointwise bracket.
We have three versions of gauge groups: , and . They are all group isomorphic. There is a natural group isomorphism defined by , , which preserves the product . Identifying with , we can avoid the troubles with diffeomorphism groups and we can extend to a Hilbert-Lie group . So is actually a Hilbert-Lie group in the classical sense; that is, the group operations are . Also the three Lie algebras , , and are canonically isomorphic. Indeed, for define by ; and for define by .
On the other hand, for define by that is, is the fundamental vector field on , generated by . is invariant if and only if .
To topologize , we complete in the -Sobolev norm. If , then are isomorphic Hilbert-Lie algebras.
There is a natural exponential map , which is a local diffeomorphism. Let be the finite-dimensional exponential map. Then define Or in terms of , .
We have the following theorem (Schmid ).
Theorem 2.2. For , is a smooth Hilbert-Lie group with Lie algebra and smooth exponential map, which is a local diffeomorphism,
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 , Marsden , as well as Marsden and Ratiu . For the general theory of infinite-dimensional manifolds and global analysis, see, for example, the studies by Bourbaki , Lang , as well as Palais .
3.1. Hamilton’s Equations on Poisson Manifolds
A Poisson manifold is a manifold (in general infinite-dimensional) equipped with a bilinear operation , called Poisson bracket, on the space of smooth functions on satisfying the following.(i) is a Lie algebra; that is, is bilinear, skew symmetric and satisfies the Jacobi identity for all .(ii) satisfies the Leibniz rule; that is, is a derivation in each factor: , for all .
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 we define the Hamiltonian vector field by It follows from (ii) that indeed defines a derivation on , hence a vector field on . Hamilton’s equations of motion for a function with Hamiltonian (energy function) are then defined by the flow (integral curves) of the vector field ; that is, We then call a Hamiltonian system on with energy (Hamiltonian function) .
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 with coordinates with the standard Poisson bracket for any two functions , given by Then the classical Hamilton’s equations are where . 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 and the Hamiltonian (energy) is Then Hamilton’s equations are
3.2.2. Infinite-Dimensional Classical Field Theory
Let be a Banach space and its dual space with respect to a pairing (i.e., is a symmetric, bilinear, nondegenerate function). On we have the canonical Poisson bracket for , , , and , given by where the functional derivatives , are the “duals” under the pairing of the partial gradients . The corresponding Hamilton’s equations are
As a special case in finite dimensions, if , so that and , and the pairing is the standard inner product in , 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 and (densities) and the pairing . We take the Hamiltonian to be , where is some function on . Then Hamilton’s (3.7) become which imply the wave equation Different choices of give different wave equations; for example, for we get the linear wave equation . For we get the Klein-Gordon equation . So these wave equations and the Klein-Gordon equation are infinite-dimensional Hamiltonian systems on .
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 is the cotangent bundle (phase space) of a manifold (configuration space). If is an -dimensional manifold, then is a -Poisson manifold locally isomorphic to whose Poisson bracket is locally given by (3.3) and Hamilton’s equations are locally given by (3.4). If is an infinite-dimensional Banach manifold, then is a Poisson manifold locally isomorphic to 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 . That means that is equipped with a symplectic structure which is a closed (d), (weakly) nondegenerate 2-form on the manifold . Then for any the corresponding Hamiltonian vector field is defined by d and the canonical Poisson bracket is given by For example, on the canonical symplectic structure is given by , where . The same formula for holds locally in for any finite-dimensional (Darboux’s Lemma). For the infinite-dimensional example , the symplectic form is given by . Again these two formulas for are identical if .
Remark 3.3. (A) If is a finite-dimensional symplectic manifold, then is even dimensional.
(B) If the Poisson bracket is nondegenerate, then comes from a symplectic form ; 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 be a Lie group with Lie algebra left invariant vector fields on , and let denote the Lie bracket (commutator) on . Let be the dual of a with respect to a pairing . Then for any and , the Lie-Poisson bracket is defined by where are the "duals" of the gradients under the pairing . Note that the Lie-Poisson bracket is degenerate in general; for example, for the vector space 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 (locally given by (3.3) and (3.6)) and then restricting it to the fiber at the identity . In this sense the Lie-Poisson bracket (3.10) is induced from the canonical Poisson bracket on . 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 is the configuration space of a free rigid body. Identifying the Lie algebra with , where is the vector product on , and , the Lie-Poisson bracket translates into For any , we have ; hence . With the Hamiltonian we get Hamilton’s equation as 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 or , or on the dual of a Lie algebra . Different, noncanonical Poisson brackets can arise from symmetries. Assume that a Lie group is acting in a Hamiltonian way on the Poisson manifold . That means that we have a smooth map such that the induced maps are canonical transformations, for each . In terms of Poisson manifolds, a canonical transformation is a smooth map that preserves the Poisson bracket. So the action of on is a Hamiltonian action if , for all . For any the canonical transformations generate a Hamiltonian vector field on and a momentum map given by , which is equivariant.
If a Hamiltonian system is invariant under a Lie group action, that is, , then we obtain a reduced Hamiltonian system on a reduced phase space (reduced Poisson manifold). We recall the following Marsden-Weinstein reduction theorem .
Theorem 3.5 (Reduction Theorem). For a Hamiltonian action of a Lie group on a Poisson manifold , there is an equivariant momentum map and for every regular the reduced phase space carries an induced Poisson structure ( being the isotropy group). Any -invariant Hamiltonian on defines a Hamiltonian on the reduced phase space , and the integral curves of the vector field project onto integral curves of the induced vector field on the reduced space .
Example 3.6 (Rigid Body). The rigid body discussed above can be viewed as an example of this reduction theorem. If and is acting on by the cotangent lift of the left translation , then the momentum map is given by and the reduced phase space is isomorphic to the coadjoint orbit through . Each coadjoint orbit carries a natural symplectic structure , and in this case, the reduced Lie-Poisson bracket on the coadjoint orbit is induced by the symplectic form on as in (3.9). Furthermore and the induced Poisson bracket on are identical with the Lie-Poisson bracket restricted to the coadjoint orbit . For the rigid body we apply this construction to .
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 discussed in Section 2.3.3 as symmetry group.
Let be the electric and magnetic fields on , then Maxwell’s equations for a charge density are Let be the magnetic potential such that . As configuration space we take , vector fields (potentials) on , so , and as phase space we have with the standard pairing and canonical Poisson bracket given by (3.6), which becomes As Hamiltonian we take the total electromagnetic energy
Then Hamilton’s equations in the canonical variables and are and So the first two equations of Maxwell’s equations (4.1) are Hamilton’s equations; we get the third one automatically from the potential and we obtain the 4th equation through the following symmetry (gauge invariance). The Lie group acts on by , The lifted action to becomes , and has the momentum map : With and , we identify elements of with charge densities. The Hamiltonian is invariant; that is, . Then the reduced phase space for is and the reduced Hamiltonian is The reduced Poisson bracket becomes for any functions on and a straightforward computation shows that 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 are equivalent to the equations of geodesics on See the study by Marsden et al. in  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  for details.
Maxwell-Vlasov’s equations for a plasma density generating the electric and magnetic fields and are the following set of equations: This coupled nonlinear system of evolution equations is an infinite-dimensional Hamiltonian system of the form on the reduced phase space ( 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 : and with Hamiltonian
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 For the MHD model, .
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 .
The Korteweg-deVries (KdV) equation is an infinite-dimensional Hamiltonian system with the Lie group of invertible Fourier integral operators as symmetry group. Gardner found that with the bracket and Hamiltonian satisfies the KdV equation (4.14) if and only if
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 of invertible Fourier integral operators on the circle . We briefly summarize the following.
A Fourier integral operators on a compact manifold is an operator locally given by where is a phase function with certain properties and the symbol belongs to a certain symbol class. A pseudodifferential operator is a special kind of Fourier integral operators, locally of the form Denote by and the groups under composition (operator product) of invertible Fourier integral operators and invertible pseudodifferential operators on , respectively. We have the following results.
Both groups and 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 and are the Lie algebras of all pseudodifferential operators under the commutator bracket. Moreover, is a smooth infinite-dimensional principal fiber bundle over the diffeomorphism group of canonical transformations with structure group (gauge group) .
For the KdV equation we take the special case where . Then the Gardner bracket (4.15) is the Lie-Poisson bracket on the coadjoint orbit of through the Schrodinger operator . Complete integrability of the KdV equation follows from the infinite system of conserved integral in involution given by ; in particular the Hamiltonian (4.16) equals .
See the study by Adams et al. in [34, 35] for details.
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 -bundle , with being a compact, orientable Riemannian manifold (e.g., and a compact non-Abelian gauge group with Lie algebra . Let be the infinite-dimensional affine space of connection 1-forms on . So each is a -valued, equivariant 1-form on (also called vector potential) and defines the covariant derivative of any field by . The curvature 2-form (or field strength) is a -valued 2-form and is defined as . They are locally given by and , where
In pure Yang-Mills theory the action functional is given by and the Yang-Mills equations become globally With added fermionic field interaction, the action becomes where is a section of the spin bundle and, 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 of gauge transformations of the principal -bundle is given by which is a smooth Hilbert-Lie group with smooth group operations .
We only sketch here what role this infinite-dimensional gauge group 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 acts on via pullback , , , or under the isomorphism (see Section 2.4.4) , we have acting on by . Hence the covariant derivative transforms as , and the action on the field is
The action functional (the Yang-Mills functional) is , locally given by . This action is gauge invariant , , so the Yang-Mills functional is defined on the orbit space The space is in general not a manifold since the action of on is not free. If we restrict to irreducible connections, then is a smooth infinite-dimensional manifold and is an infinite-dimensional principal fiber bundle with structure group .
For self-dual connections (instantons) on a compact 4-manifold, the moduli space is a smooth finite-dimensional manifold. Self-dual connections absolutely minimize the Yang-Mills action integral The Feynman path integral quantizes the action and we get the probability amplitude for any gauge-invariant functional .
Let be the group of gauge transformations. So is a diffeomorphism over ; that is, , , . Then acts on and by and . The action functionals are gauge invariant:
5.1.2. Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD)
In classical field theory, one considers a Lagrangian of the fields , , and and the corresponding action functional . The variational principle then leads to the Euler-Lagrange equations of motion
In QED and QCD the Lagrangian is more complicated of the form where is a potential 1-form (boson), and the field strength is given by . In QED the gauge group of the principal bundle is , and in QCD we have . The Dirac -matrices are where are the Pauli matrices (canonical basis of ) and is the Pauli adjoint with , is the electron mass, is the electron charge, and is a coupling constant.
5.1.3. The Equations of Motion
The variational principle of the Lagrangian (5.10) with respect to the fields and gives the corresponding Euler-Lagrange equations of motion. They describe, for instance, the motion of an electron (fermion, spinor) in an electromagnetic field , interacting with a bosonic field . We get, from the variational principle, which are Maxwell’s equations for .
In the free case, that is, when , we get , the vacuum Maxwell equations.
For these equations become , the Yang-Mills equations. Moreover, , which are Dirac’s equations, where . In the free case, that is, when , we get , 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 is given by , where is a constant and . The classical Noether current of this symmetry is given by which is conserved; that is, .
This conservation law breaks down after quantization; one gets This value is called the chiral anomaly.
The quantization is given by the Feynman path integral: which computes the expectation value of the function . This is an integral over two infinite-dimensional spaces: the gauge orbit space and the fermionic Berezin integral over the spin space . These integrals are mathematically not defined but physicists compute them by gauge fixing; that is, fixing a section , (e.g., , the Lorentz gauge) and then integrating over the section . 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 via the Faddeev-Popov procedure. The such obtained effective Lagrangian is no longer gauge invariant. This effective Lagrangian has the form in QCD:
We can write this globally as where is the Faddeev-Popov determinant, acting like the Jacobian of the global gauge variation over the section . Writing this term in the exponent of the action functional like a “fermionic Gaussian integral” leads to the Faddeev-Popov ghost fields in the form .
The effective Lagrangian is NOT gauge invariant but has a new symmetry, called BRST symmetry.
5.3. BRST Symmetry
Named after Becchi et al.  and Tyutin who discovered this invariance in 1975-76, the BRST operator is given as follows: Note that the BRST operator mixes bosons () and fermions (). This is an example of supersymmetry which we will discuss in Section 6. Also, the BRST operator is nilpotent; that is, . The question arises whether this operator 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 be the Chevalley-Eilenberg complex of the Lie algebra of infinitesimal gauge transformations, with respect to the induced adjoint representation on local forms , with boundary operator Then with , one has and the following.(1)For , then .(2)For , then , the Maurer-Cartan form.(3)The chiral anomaly (given by (5.11)) is represented as cohomology class of this complex .
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  complex and the corresponding cohomology.
Let be a Lie group with Lie algebra and let be a representation of on the vector space . Denote by the space of -valued -cochains on and define the coboundary operator by We have , and define the Lie algebra cohomology of with respect to as . This is called the Chevalley-Eilenberg cohomology  of the Lie algebra with respect to the representation .
The Noether current induced by the chiral symmetry (after quantization) for the free case (), that is, for pure Yang-Mills becomes See (5.11).
Note the similarity with the Chern-Simon Lagrangian
We are going to derive a representation of the chiral anomaly in the BRST cohomology that is .
The question is “if , does there exist a local functional , such that ? That is, is BRST -exact? The answer in general is NO; that is, represents a nontrivial cohomology class. This class is given by the Chern-Weil homotopy.
Let and . For , let and define the Chern-Simons form we get We write as sum of homogeneous terms in ghost number (upper index) and degree (lower index) Let .
Theorem 5.2 (see Schmid ). The form satisfies the Wess-Zumino consistency condition and represents the chiral anomaly .
We have an explicit form of the anomaly in dimensions: So for the non-Abelian anomaly in 2 dimensions becomes