∞-Lie theory

# Contents

## Idea

What is called the “BRST complex” in the physics literature is the qDGCA which is the Chevalley-Eilenberg algebra of the $L_\infty$-algebroid which is the differential version in Lie theory of the $\infty$-groupoid

• 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 $k$-morphisms describe the $k$-fold gauge-of-gauge transformations.

The generators of the BRST complex are called

The cochain cohomology of the BRST complex is called, of course, BRST cohomology.

## Examples

### Gauge theory of a Lie algebra valued connection

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 $G$-principal bundles for some Lie group $G$ – the gauge group .

#### The BRST complex

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 $X$,

• a field configuration is a differential form $A$ on $X$ with values in the Lie algebra $\mathfrak{g}$ of $G$,

$A \in \Omega^1(X, \mathfrak{g})$
• a gauge transformation $A_1 \stackrel{g}{\to} A_2$ between two field configurations $A_1$ and $A_2$ is given by a smooth function $g : X \to G$ such that

$A_2 = g^{-1} A_1 g + g^{-1} d_{dR} g \,.$

Eventually we want to say that the full configuration space is therefore the groupoid of Lie algebra valued 1-forms on $X$, 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 $\lambda : X \to \mathfrak{g}$ from $X$ to the Lie algebra of $G$, relating field configurations by

(1)$A_2 = [A_1, \lambda] + d \lambda \,.$

To see in more detail what this equation means, we choose a basis $\{t_a\}$ for the vector space $\mathfrak{g}$ that underlies the Lie algebra. In terms of this basis the Lie bracket is defined by its structure constants $\{C^a{}_{b c} \in \mathbb{R}\}$ defined by

$[t_b, t_c] = C^a{}_{b c} t_a \,.$

In terms of this, a field configuration decomposes into components $\{A^a \in \Omega^1(X)\}$ defined by

$A = A^a t_a \in \Omega^1(X,\mathfrak{g}) \simeq \Omega^1(X)\otimes \mathfrak{g}$

and the equation (1) for infinitesimal gauge transformations reads equivalently

$A_2^a = C^a{}_{b c} A_1^b \lambda^c + d_{dR} \lambda^a \,.$

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 $\Omega^1(X,\mathfrak{g})$ of field configurations is the function that evaluates a field configuration at some point $x \in X$ on some vector $\frac{\partial}{\partial x^\mu}$ and picks the resulting component of $t_a$. We write for this

$\mathbf{A}^a_\mu(x) : \Omega^1(X,\mathfrak{g}) \to \mathbb{R}$
$A \mapsto A^a_\mu(x) \,.$

Analogously we have functions on the space $\Omega^0(X,\mathfrak{g})$ of gauge transformations that evaluate at a point $x$ and pick the component of $t_a$. These we write

$\mathbf{c}^a (x) : \Omega^0(X, \mathfrak{g}) \to \mathbb{R}$
$\lambda \mapsto \lambda^a(x) \,.$

Moreover, there exists smooth functions of this kind that evaluate not $A$ or $\lambda$ 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 $\mathbf{c}^a(x)$ are functions on gauge transformations, which are 1-morphisms in the groupoid of Lie algebra valued forms, while the $\mathbf{A}^a(x)$ are functions on ordinary elements of this groupoid (0-morphisms) we declare them to be in degree 1, too.

( Fundamentally what is going in here is that we form “function algebras on ∞-stacks” where the ∞-stack in question is the action Lie algebroid $\Omega^1(X,\mathfrak{g})//\Omega^0(X, \mathfrak{g})$).

In other words, we regard the smooth functions on field configurations and on gauge transformations as forming a $\mathbb{N}$-graded algebra, where

• functions like $\mathbf{A}^a_\mu(x)$, $C^a{}_{b c}\mathbf{A}^a_\mu(x) \mathbf{A}^b_\nu(x) : \Omega^1(X,\mathfrak{g}) \to \mathbb{R}$ etc. are in degree 0;

• functions like $\mathbf{c}^a(x)$, $C^a{}_{b c}\mathbf{c}^a(x) \mathbf{A}^b_\mu(x) : \Omega^1(X,\mathfrak{g}) \to \mathbb{R}$ etc. are in degree 1;

• functions like $C^a{}_{b c}\mathbf{c}^a(x) \mathbf{c}(x)$ are in degree 2;

• etc.

Moreover, we declare the elements in degree 1 to anticommute with each other

$\mathbf{c}^a(x)\mathbf{c}^b(x) = - \mathbf{c}^b(x)\mathbf{c}^a(x) \,.$

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 $\mathbf{A}^a_\mu(x)$ not on a field $A$ itself, but the gauge transformation of this field by some $\lambda$, then the result is not $A^a_\mu(x)$ but $C^a{}_{b c} A^b_\mu(x)\lambda^c(x) + (d\lambda)_\mu^a(x)$. Therefore on gauge transformed fields the functional $\mathbf{A}^a_\mu(x)$ is replaced by

$\mathbf{A}^a_\mu(x) \mapsto d_{BSRT}\mathbf{A}^a_\mu(x) := C^a{}_{b c} \mathbf{A}^b_\mu(x) \mathbf{c}^c(x) + d \mathbf{c}^a \,.$

A similarly analysis shows that

$d_{BRST} \mathbf{c}^a(x) = -\frac{1}{2}C^a{}_{b c} \mathbf{c}^b(x) \mathbf{c}^c(x) \,.$

This operation $d_{BRST}$ 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.

#### Observables

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 $X$, and an observable would be simply a function

$O : X \to \mathbb{R}$

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 $\Omega^1(X, \mathfrak{g})$, but is that manifold equipped with the information of the infinitesimal gauge transformations in $\Omega^0(X,\mathfrak{g})$. One way to say this is that the configuration space now is the action Lie algebroid $\Omega^1(X,\mathfrak{g})//\Omega^0(X, \mathfrak{g})$ and that an observable is hence a function

$O : \Omega^1(X, \mathfrak{g})//\Omega^0(X,\mathfrak{g}) \to \mathbb{R} \,.$

One can understand what this means in terms of functors between groupoids. But the BRST complex provides an algebraic way of talking about this situation, and this is what we look at here.

So we need to formulate these maps between generalized manifolds algebraically. This is straightforward for the familiar case of manifolds:

any smooth function of smooth manifolds $f : X \to Y$ induces dually a homomorphism of their algebras of smooth functions, which goes the other way round

$f^* : C^\infty(Y) \to C^\infty(X) \,.$

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 $X$ is therefore equivalently an algebra homomorphism

$C^\infty(X) \leftarrow C^\infty(\mathbb{R}) : O^* \,.$

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 $C^\infty(\Omega^1(X, \mathfrak{g})//\Omega^0(X, \mathfrak{g}))$ for our BRST complex, we find that an observable now is a homomorphism

$C^\infty(\Omega^1(X, \mathfrak{g})//\Omega^0(X, \mathfrak{g})) \leftarrow C^\infty(\mathbb{R}) : O^* \,.$

But, clearly, this is now to be regarded as a homomorphism of dg-algebras. We may regard any ordinary algebra, such as $C^\infty(\mathbb{R})$, 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.

1. The morphism $O^*$ needs to send every element of $C^\infty(\mathbb{R})$ to an element of degree-0 in our BRST complex. As we have seen, these elements of degree 0 formed the algebra of functions $C^\infty(\Omega^1(X,\mathfrak{g}))$. So underlying the dg-algebra homomorphism $O^*$ is an ordinary algebra homomorphism $C^\infty(\mathbb{R}) \to C^\infty(\Omega^1(X,\mathfrak{g}))$. By duality, this is equivalently simply a single element

$O := O^*(id) \in C^\infty(\Omega^1(X,\mathfrak{g})) \,,$

namely the image of the identity function.

2. 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 $C^\infty(\mathbb{R})$ vanishes, this gives the condition

\begin{aligned} d_{BRST} O & := d_{BRST} O^*(id) \\ & = O^*(d_{C^\infty(\mathbb{R})} id) \\ & = O^*(0) \\ & = 0 \end{aligned} \,.

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.

The BRST complex described a homotopical quotient of a space by an infinitesimal action. Combined with a homotopical intersection, it is part of the BRST-BV complex.

gauge field: models and components

## References

The idea of “ghost” fields was introduced in

• Richard Feynman, Quantum theory of gravitation In: Acta physica polonica. vol 24, 1963, S. 697

and expanded on in

• Wheeler-Festschrift, Klauder (ed.): Magic without Magic . (1972)

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

Discussion of the BRST complex of the bosonic string/for 2d CFT includes

• Graeme Segal, p.114 and following of The definition of conformal field theory , preprint, 1988; also in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (pdf)

Discussion of the BRST complex for the superstring (hence with the corresponding Lie algebroid being actually a super Lie algebroid) is for instance in

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.

Revised on September 11, 2014 11:15:51 by Urs Schreiber (141.0.9.78)