whose space of objects is the space of configurations/histories of a given physical system;
whose morphisms describe the gauge transformations between these configurations/histories;
whose -morphisms describe the -fold gauge-of-gauge transformations.
The generators of the BRST complex are called
We discuss the BRST complex for a field theory such as Yang-Mills theory or Chern-Simons theory whose configuration space is one of connections on -principal bundles for some Lie group – the gauge group .
For simplicity of the exposition we first restrict attention to the cases where the underlying bundles are trivial.
In that case, over a smooth manifold ,
Eventually we want to say that the full configuration space is therefore the groupoid of Lie algebra valued 1-forms on , but we will be interested now only in the infinitesimal approximation to the gauge transformations. Since the infinitesimal approximation of a Lie group is its Lie algebra, these infinitesimal gauge transformations are given by smooth functions from to the Lie algebra of , relating field configurations by
To see in more detail what this equation means, we choose a basis for the vector space that underlies the Lie algebra. In terms of this basis the Lie bracket is defined by its structure constants defined by
In terms of this, a field configuration decomposes into components defined by
and the equation (1) for infinitesimal gauge transformations reads equivalently
The BRST complex is a cochain complex of functions – on this configuration space and on these gauge transformations.
A typical smooth function on the space of field configurations is the function that evaluates a field configuration at some point on some vector and picks the resulting component of . We write for this
Analogously we have functions on the space of gauge transformations that evaluate at a point and pick the component of . These we write
Moreover, there exists smooth functions of this kind that evaluate not or themselves, but some derivative of them.
The general kind of functions on the configuration space and on the space of gauge transformation that we want to consider can be thought of as being generated by such local functions .
But since the are functions on gauge transformations, which are 1-morphisms in the groupoid of Lie algebra valued forms, while the are functions on ordinary elements of this groupoid (0-morphisms) we declare them to be in degree 1, too.
In other words, we regard the smooth functions on field configurations and on gauge transformations as forming a -graded algebra, where
functions like , etc. are in degree 0;
functions like , etc. are in degree 1;
functions like are in degree 2;
Moreover, we declare the elements in degree 1 to anticommute with each other
So far this describes a graded algebra of functions. What is still missing is the information encoded in equation (1), which links the gauge transformations with the field configurations that they act on.
By that formula, if we evaluate the functional not on a field itself, but the gauge transformation of this field by some , then the result is not but . Therefore on gauge transformed fields the functional is replaced by
A similarly analysis shows that
This operation of “shifting functionals of fields and gauge transformations by infinitesimal gauge transformations” promotes the graded algebra constructed so far to a differential graded algebra. This is the BRST-complex of the given system.
In a precise sense, this differential graded algebra encodes the infinitesimal groupoid of Lie algebra valued 1-forms that it is an algebra of functions on in the same way that an algebra of functions on some manifold characterizes that manifold.
Using the above discussion, we can now deduce the observables on the configurations of the gauge theory in question.
First notice that if we had no gauge transformations, then configuration space would be an ordinary manifold , and an observable would be simply a function
on that space, namely, an assignment of a number to each field configuration (say the energy or momentum, assigned to each field configuration).
In the case at hand now the configuration space is not quite the manifold , but is that manifold equipped with the information of the infinitesimal gauge transformations in . One way to say this is that the configuration space now is the action Lie algebroid and that an observable is hence a function
So we need to formulate these maps between generalized manifolds algebraically. This is straightforward for the familiar case of manifolds:
And in fact such a morphism of algebras characterizes the smooth function that it comes from. Therefore we can speak of morphisms of manifolds dually in terms of their algebras of functions. An observable on some configuration manifold is therefore equivalently an algebra homomorphism
This may be most familiar from algebraic geometry, where one considers bare rings of functions, without any smooth structure. For the application to physics we usually want to keep track of the smooth structure. This works seamlessly if we simply replace the notion of algebra by that of smooth algebra .
In any case, we can now grasp the way in which the configuration space of a gauge theory is not an ordinary manifold in terms of its dual function algebra: the BRST complex that we have described above is the function algebra on that configuration space, but it is not a plain algebra (or smooth algebra), but in fact a dg-algebra, in that it carries a grading and a differential. One therefore also says that the configuration space of a gauge theory is a dg-manifold .
Therefore, if we write for our BRST complex, we find that an observable now is a homomorphism
But, clearly, this is now to be regarded as a homomorphism of dg-algebras. We may regard any ordinary algebra, such as , as a dg-algebra by taking all its elements to be in degree 0 and taking its differential to vanish (to take the value 0 on all elements).
Since a homomorphism of dg-algebras preserves the grading and intertwines the differential, this implies two things.
The morphism needs to send every element of to an element of degree-0 in our BRST complex. As we have seen, these elements of degree 0 formed the algebra of functions . So underlying the dg-algebra homomorphism is an ordinary algebra homomorphism . By duality, this is equivalently simply a single element
namely the image of the identity function.
This morphism of algebras found this way still needs to respect the differentials in order to qualify as a homomorphism of dg-algebras. Since the differential of vanishes, this gives the condition
In conclusion, this shows that the observables in the BRST complex are precisely the BRST-closed elements in degree 0. These are precisely those functions on the space of field configurations which are invariant under the gauge transformations of the fields.
gauge field: models and components
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
The idea of “ghost” fields was introduced in
and expanded on in
A canonical textbook reference on the BRST complex is (chapter 8 of)
Discussion with more emphasis on the applications to quantum field theory of interest is in lecture 3 of
The perspective on the BRST complex as a formal dual to a space in dg-geometry is relatively clearly stated in section 2 of
For more along these lines see BV-BRST formalism.