Anomalies (in quantization)
The term "anomaly" is not very precise, but here it will be taken to mean an obstruction to the passage from a classical theory to a corresponding quantum theory while, at the same time, maintaining the same invariance groups of the classical theory. A common feature of many of these anomalies is that this obstruction has a topological origin (see below). The topology involved is usually the cohomology of the invariance group of the theory.
Many other anomalies are now known: for example, anomalies are encountered in Yang–Mills theories, when one replaces the gauge group by a non-Abelian group , in gravity theory, where general coordinate transformations replace the gauge transformations of Yang–Mills theories, in string theory, where both kinds of transformations are present, and in two-dimensional conformal field theories.
Within this miscellany of types of anomaly one distinguishes two categories: local anomalies and global anomalies.
For the moment it is assumed that the dimension of is even, so that the notion of chirality exists. Take a Yang–Mills theory (cf. also Yang–Mills field) with gauge field coupled to chiral Fermions ; is a -valued 1-form, where is the Lie algebra of the group . The corresponding action of the quantum field theory is given by
where is the chiral Dirac operator, is the Yang–Mills curvature -form and is the Levi-Civita connection -form.
To unearth a possible anomaly, one constructs its quantization via the partition function which is obtained by integrating over the space of Fermions and over , the space of connections. is then expressed as the functional integral
One can carry out the Fermionic integration using the expression
This allows one to write
Now this action is supposed to possess a gauge invariance under the group of gauge transformations . If this is so, then is naturally expressed as an integral over the space of gauge orbits . An anomaly is said to have arisen if this is not the case.
Unfortunately, the supposition of gauge invariance requires justification and is generally false: Although the expression is manifestly gauge invariant, the same cannot be said of the Fermionic determinant . Indeed, if is a gauge transformation under which , then, in general,
One can now verify this by direct calculation. In addition, it can be shown that the variation of the Fermionic determinant under a gauge transformation has a natural interpretation as a cohomology class in .
Choose , where is a real parameter and is locally represented by an expression of the form , where is a basis for the spinor representation of . Taking the Fermionic integral with one finds that
where denotes covariant derivative.
If , then is a one-dimensional matrix and the covariant derivative expression reduces to just . The presence of an anomaly then forces the non-conservation of the well-known , or singlet, axial current
If, in the above calculation, the standard normalization factor used to eliminate vacuum Feynman graphs is included, one obtains
The left-hand side now has the preferred structure and is the infinitesimal variation of , while the right-hand side is just the vacuum expectation value of smeared with an arbitrary . Hence if really were gauge invariant, one could conclude that
The catch is that when an anomaly is present, the above equation is false.
A direct perturbative way of seeing this is to expand the infinitesimal variation of in terms of the coupling constant and calculate the resultant Feynman graphs. If one does this one finds at one loop the celebrated divergent triangle graph, whose non-vanishing implies the existence of the anomaly.
Alternatively, from the point of view of functional integration, since the integrand of the Fermionic integral is gauge invariant, the lack of gauge invariance of must occur somewhere in the integration process over and . This is the point of view of [a3], where it is shown that the Fermionic `measure' is not gauge invariant.
The previous calculation suggests the use of a de Rham technique to produce a topological interpretation of the anomaly: One takes , or more simply , and uses its infinitesimal variation to construct an element of .
More precisely, one defines [a4]
Then the natural cohomology class to define is , where
and denotes the exterior derivative (cf. also Exterior form; Exterior algebra) acting on the infinite-dimensional space . Now this -form would be exact and hence cohomologically trivial, provided one could write
But one cannot do this unless exists. Recall now that a necessary condition for to exist, for a mapping , is that there is no loop in whose image circles the origin in . When is simply connected, this is avoided. But in the case under consideration, is and if one takes , then is weakly homotopic to the loop space , so that
in general. For example, if and , then, from standard results on the homotopy of Lie groups,
Thus, for , is a non-trivial cohomology class in .
The next step is to show how the non-triviality of is naturally related to the families index theorem.
Recall the partition function , for which
This expression contains both the Dirac operator and a sum over all . Thus it is natural to consider the family of elliptic operators given by as varies throughout .
Now, for the construction of one has to know whether the zero eigenvalue spaces and are non-empty or not. The dimensions of these vector spaces are and , these being the numbers of positive and negative chirality zero mass Fermions, respectively. But (for a fixed ) is given by their difference, i.e.
where is an integer given by the usual characteristic class formula of the index theorem for the Dirac operator ; in four dimensions, is the familiar instanton number. Thus, a non-zero , which is the generic situation, produces an asymmetry in the massless chiral Fermion sector.
Hence, when varies, one ought to consider the index of a whole family of Dirac operators. The families index theorem provides the framework to do precisely this: for a family of elliptic operators parametrized by a space , the index of the family is given by an element of , the -theory of .
In the present case, and, denoting the index of the Dirac family by , one obtains
Such a formal difference of (equivalence classes) of vector bundles defines an element of the -theory of , which immediately projects to an element of because of the gauge invariance of .
It is now natural to consider a certain determinant line bundle associated with . This bundle is defined by
Since is a line bundle, it is characterized topologically by its Chern class ; however, one can calculate from the standard cohomology formula for the index of elliptic families. Summing up these cohomology calculations, one sees that two cohomology group elements have been constructed, namely
Actually, it can be shown by transgression that
In addition one can give formulas for and . For example, if and , then, if , a formula for is the integral below restricted to the orbit :
The cohomology class is the one required for the anomaly. To see this, suppose that there is no anomaly. Then is gauge invariant and the partition function descends to an integral over the orbit space on which also projects to . But in this case provides a global section of the line bundle , causing to vanish. Thus, the presence of the anomaly is detected by the non-triviality of the bundle and one sees that the index theorem for the Dirac family brings out succinctly the anomaly as an obstacle to the definition of a gauge-invariant determinant for .
Note that if , then vanishes; hence also vanishes. But it is known from the triangle graph that there is an anomaly in this theory. It is also known that there is no local counterterm ( "local" meaning: a polynomial in fields and their derivatives) which can cancel the triangle graph and remove the anomaly. Thus, the anomaly is, physically speaking, unremovable. Mathematically speaking, the vanishing of means that there is a counterterm, but it is non-local. This suggests defining a local cohomology theory next to the conventional cohomology theory, and that one should distinguish cases where the two cohomologies differ, cf. [a5].
Anomalies also occur in other contexts. If is the group of (orientation-preserving) coordinate transformations of , then one can consider the change of the Dirac operator under a coordinate transformation rather than an element of . Recall that the -matrices depend on a choice of metric; the same is therefore true of the Dirac operator. Hence, if denotes the space of metrics on and , then one can display the metric dependence of the Dirac operator by writing it as . In this context one now searches for an obstacle to the descent of the determinant from to the quotient ; actually, for technical reasons due to fixed points of the action of on , one replaces by the (fixed-point-free) subgroup consisting of those elements of that leave a basis at one point fixed. So, now plays the role formerly played by . An anomaly of this kind is known as a gravitational anomaly and lies in .
Anomalous determinants need not contain Dirac operators; other elliptic operators give rise to anomalies. One of the most celebrated is in string theory, where the anomaly concerns the -operator on a compact Riemann surface . The partition function for the Bosonic string contains a power of . The -operator depends on the metric on and should only depend on the conformal equivalence class of this metric. Now , the space of metrics on , is acted on by diffeomorphisms and conformal transformations, i.e. by and , the space of positive functions on . The determinant should descend to a quotient of under this combined action. Failure to do this is an anomaly; this conformal anomaly vanishes when the string moves in dimensions where . The anomaly can be seen to arise from the need, in string theory, to consider representations of the Virasoro group (cf. also Virasoro algebra), that is, to consider central extensions of the group of conformal transformations on .
The last kind of anomaly described in this article is a global anomaly. A global anomaly is well defined when the appropriate local anomaly, for example, an element of or say, vanishes. Global anomalies enter when or have more than one connected component. This means that they contain elements not continuously connected to the identity. Such discreteness is not detectable using methods of curvature and de Rham cohomology — these methods are only sensitive to objects in the tangent space. The situation is closely analogous to the calculation of torsion in homology and cohomology. Unfortunately, torsion calculations are typically more difficult than free cohomology calculations, for which the de Rham method is usually available. Index theory for families can still be used with the addition of some other ingredients, such as holonomy and spectral flow round loops on the appropriate family parameter space .
An example in which is disconnected is provided by a Yang–Mills theory (cf. also Yang–Mills field) with group . One can count the connected components of with and, doing this with and , one has
i.e. , also , so that the local anomaly is zero. This means that there are global gauge transformations under which is not invariant — in fact they change the sign of .
An example in which one has a global gravitational anomaly is provided by , since then
and this fact is also very closely tied to the existence of distinct differentiable structures on , i.e. to the existence of exotic spheres. The cancellation of the local and global anomalies in dimensions for supersymmetric string theories was first described in [a6], [a7].
The quantization of two-dimensional conformal field theories on a Riemann surface involves a natural, projectively flat, connection on a vector bundle over the moduli space of . One also chooses an integer , known as the level and constructs a suitable quotient of an infinite-dimensional affine space, cf. [a8]. The projective flatness arises because of the necessity to consider a central extension of the action of on the sections of . This then gives rise to an anomaly which is manifested as a shift in the level . Such a quantization arises in the quantum field theoretic formulation of Jones' knot polynomial.
A Hamiltonian approach to anomalies takes to be the manifold of space rather than of space-time; hence, in the chiral examples, would be odd dimensional and there is no splitting of the Dirac operator into two chiral pieces. Instead, one works with the full self-adjoint Dirac operator and realizes it as a Fredholm operator. A treatment using Fermionic Fock space is also possible.
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