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.

The first anomaly of this kind was the triangle anomaly of [a1], [a2]. This anomaly is one of the many that involves chiral Fermions coupled to a $ U ( 1 ) $ gauge field.

Many other anomalies are now known: for example, anomalies are encountered in Yang–Mills theories, when one replaces the gauge group $ U ( 1 ) $ by a non-Abelian group $ G $, 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.

Local anomaly.

The discussion below involves Riemannian spin manifolds (cf. Spinor structure) throughout, so that space-time $ M $ is viewed as Euclidean (a Hamiltonian treatment is also possible).

For the moment it is assumed that the dimension $ n $ of $ M $ is even, so that the notion of chirality exists. Take a Yang–Mills theory (cf. also Yang–Mills field) with gauge field $ A $ coupled to chiral Fermions $ \psi $; $ A $ is a $ \mathfrak g $- valued 1-form, where $ \mathfrak g $ is the Lie algebra of the group $ G $. The corresponding action $ S $ of the quantum field theory is given by

$$ S \equiv S ( A, \psi ) = \left \| F \right \| ^ {2} + \left \langle {\psi, {\partial slash } _ {A} \psi } \right \rangle = $$

$$ = - { \mathop{\rm tr} } \int\limits _ { M } F \wedge \star F + $$

$$ + { \frac{1}{2} } \int\limits _ { M } {\overline \psi \; } \gamma _ \mu ( \partial ^ \mu + \Gamma ^ \mu + A ^ \mu ) ( 1 + \gamma _ {5} ) \psi $$

where $ {\partial slash } _ {A} = \gamma _ \mu ( \partial ^ \mu + \Gamma ^ \mu + A ^ \mu ) ( 1 + \gamma _ {5} ) /2 $ is the chiral Dirac operator, $ F $ is the Yang–Mills curvature $ 2 $- form and $ \Gamma = \Gamma _ \mu dx ^ \mu $ is the Levi-Civita connection $ 1 $- form.

To unearth a possible anomaly, one constructs its quantization via the partition function $ Z $ which is obtained by integrating over the space of Fermions and over $ {\mathcal A} $, the space of connections. $ Z $ is then expressed as the functional integral

$$ Z = \int\limits {\mathcal D} {\mathcal A} {\mathcal D} \psi {\mathcal D} {\overline \psi \; } { \mathop{\rm exp} } [ - \left \| F \right \| ^ {2} - \left \langle {\psi, {\partial slash } _ {A} \psi } \right \rangle ] . $$

One can carry out the Fermionic integration using the expression

$$ \int\limits {\mathcal D} \psi {\mathcal D} {\overline \psi \; } { \mathop{\rm exp} } [ - \left \langle {\psi, {\partial slash } _ {A} \psi } \right \rangle ] = \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } . $$

This allows one to write

$$ Z = \int\limits { {\mathcal D} {\mathcal A} } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } { \mathop{\rm exp} } [ - \left \| F \right \| ^ {2} ] $$

Now this action $ S $ is supposed to possess a gauge invariance under the group of gauge transformations $ {\mathcal G} $. If this is so, then $ Z $ is naturally expressed as an integral over the space of gauge orbits $ {\mathcal A}/ {\mathcal G} $. 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 $ \| F \| ^ {2} $ is manifestly gauge invariant, the same cannot be said of the Fermionic determinant $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) $. Indeed, if $ g $ is a gauge transformation under which $ A \mapsto A _ {g} = g ^ {- 1 } ( d + A ) g $, then, in general,

$$ { \mathop{\rm det} } ( {\partial slash } _ {A _ {g} } ^ {*} {\partial slash } _ {A _ {g} } ) \neq { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) . $$

One can now verify this by direct calculation. In addition, it can be shown that the variation of the Fermionic determinant $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) $ under a gauge transformation has a natural interpretation as a cohomology class in $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $.

Choose $ g = { \mathop{\rm exp} } ( tf ) $, where $ t $ is a real parameter and $ f $ is locally represented by an expression of the form $ f = t ^ {a} f ^ {a} ( x ) $, where $ \{ t ^ {a} \} $ is a basis for the spinor representation of $ \mathfrak g $. Taking the Fermionic integral with $ A = A _ {g} $ one finds that

$$ \left . { \frac{d}{dt } } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A _ {g ( t ) } } ^ {*} {\partial slash } _ {A _ {g ( t ) } } ) } \right | _ {t = 0 } = $$

$$ = \int\limits {\mathcal D} \psi {\mathcal D} {\overline \psi \; } { \mathop{\rm exp} } [ - \left \langle {\psi, {\partial slash } _ {A} \psi } \right \rangle ] F, $$

$$ F = \int\limits _ {S ^ {n} } f ^ {a} ( x ) \nabla _ \mu ^ {ab } j _ {5} ^ {b \mu } ( x ) $$

and

$$ j _ {5} ^ {b \mu } ( x ) = { \frac{1}{2} } {\overline \psi \; } \gamma ^ \mu ( 1 + \gamma _ {5} ) t ^ {b} \psi, $$

where $ \nabla _ \mu ^ {ab } $ denotes covariant derivative.

If $ G = U ( 1 ) $, then $ t ^ {a} $ is a one-dimensional matrix and the covariant derivative expression $ f ^ {a} ( x ) \nabla _ \mu ^ {ab } j _ {5} ^ {b \mu } ( x ) $ reduces to just $ f ( x ) \partial _ \mu j _ {5} ^ \mu ( x ) $. The presence of an anomaly then forces the non-conservation of the well-known $ U ( 1 ) $, or singlet, axial current

$$ j _ {5} ^ \mu ( x ) = { \frac{1}{2} } {\overline \psi \; } \gamma ^ \mu ( 1 + \gamma _ {5} ) \psi, $$

cf. [a1], [a2].

If, in the above calculation, the standard normalization factor $ \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } $ used to eliminate vacuum Feynman graphs is included, one obtains

$$ { \frac{1}{\sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } } } { \frac{d}{dt } } \left . \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A _ {g ( t ) } } ^ {*} {\partial slash } _ {A _ {g ( t ) } } ) } \right | _ {t = 0 } = $$

$$ = { \frac{\int\limits { {\mathcal D} \psi {\mathcal D} {\overline \psi \; } { \mathop{\rm exp} } [ - \left \langle {\psi, {\partial slash } _ {A} \psi } \right \rangle ] } \int\limits dx \nabla _ \mu ^ {ab } j _ \mu ^ {b5 } ( x ) f ^ {a} ( x ) }{\sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } } } . $$

The left-hand side now has the preferred structure and is the infinitesimal variation of $ { \mathop{\rm ln} } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } $, while the right-hand side is just the vacuum expectation value of $ \nabla _ \mu ^ {ab } j _ {5} ^ {b \mu } ( x ) $ smeared with an arbitrary $ f $. Hence if $ { \mathop{\rm ln} } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } $ really were gauge invariant, one could conclude that

$$ \nabla _ \mu ^ {ab } j _ {5} ^ {b \mu } ( x ) = 0. $$

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 $ { \mathop{\rm ln} } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } $ 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 $ { \mathop{\rm exp} } [ - \langle {\psi, {\partial slash } _ {A} \psi } \rangle ] $ of the Fermionic integral is gauge invariant, the lack of gauge invariance of $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) $ must occur somewhere in the integration process over $ \psi $ and $ {\overline \psi \; } $. This is the point of view of [a3], where it is shown that the Fermionic `measure' $ {\mathcal D} \psi {\mathcal D} {\overline \psi \; } $ 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 $ { \mathop{\rm det} } ( {\partial slash } _ {A _ {g} } ^ {*} {\partial slash } _ {A _ {g} } ) $, or more simply $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A _ {g} } ) $, and uses its infinitesimal variation to construct an element of $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $.

More precisely, one defines [a4]

$$ T _ {g} = {\partial slash } _ {A} ^ {*} {\partial slash } _ {A _ {g} } . $$

Then the natural cohomology class to define is $ [ \mu _ {1} ] $, where

$$ \mu _ {1} = { \frac{d ( { \mathop{\rm det} } T _ {g} ) }{ { \mathop{\rm det} } T _ {g} } } , [ \mu _ {1} ] \in H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) , $$

and $ d $ denotes the exterior derivative (cf. also Exterior form; Exterior algebra) acting on the infinite-dimensional space $ {\mathcal G} $. Now this $ 1 $- form $ \mu _ {1} $ would be exact and hence cohomologically trivial, provided one could write

$$ \mu _ {1} = d { \mathop{\rm ln} } { \mathop{\rm det} } T _ {g} . $$

But one cannot do this unless $ { \mathop{\rm ln} } { \mathop{\rm det} } T _ {g} $ exists. Recall now that a necessary condition for $ { \mathop{\rm ln} } f $ to exist, for a mapping $ f : W \rightarrow {\mathbf C \setminus \{ 0 \} } $, is that there is no loop $ \alpha $ in $ W $ whose image $ f ( \alpha ) $ circles the origin in $ \mathbf C $. When $ W $ is simply connected, this is avoided. But in the case under consideration, $ W $ is $ {\mathcal G} $ and if one takes $ M = S ^ {2n } $, then $ {\mathcal G} $ is weakly homotopic to the loop space $ \Omega ^ {2n } G $, so that

$$ \pi _ {1} ( {\mathcal G} ) = \pi _ {1} ( \Omega ^ {2n } G ) = \pi _ {2n + 1 } ( G ) \neq 0 $$

in general. For example, if $ M = S ^ {4} $ and $ G = U ( N ) $, then, from standard results on the homotopy of Lie groups,

$$ \pi _ {1} ( {\mathcal G} ) = \pi _ {5} ( U ( N ) ) = \left \{ \begin{array}{l} {0, \ N = 1, } \\ {\mathbf Z _ {2} , \ N = 2, } \\ {\mathbf Z, \ N \geq 3. } \end{array} \right . $$

Thus, for $ N \geq 3 $, $ [ \mu _ {1} ] $ is a non-trivial cohomology class in $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $.

The next step is to show how the non-triviality of $ [ \mu _ {1} ] $ is naturally related to the families index theorem.

Recall the partition function $ Z $, for which

$$ Z = \int\limits _ {\mathcal A} { {\mathcal D} {\mathcal A} } \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } { \mathop{\rm exp} } [ - \left \| F \right \| ^ {2} ] . $$

This expression contains both the Dirac operator $ {\partial slash } _ {A} $ and a sum over all $ A \in {\mathcal A} $. Thus it is natural to consider the family of elliptic operators given by $ {\partial slash } _ {A} $ as $ A $ varies throughout $ {\mathcal A} $.

Now, for the construction of $ { \mathop{\rm det} } T _ {g} $ one has to know whether the zero eigenvalue spaces $ { \mathop{\rm ker} } {\partial slash } _ {A} $ and $ { \mathop{\rm ker} } {\partial slash } _ {A} ^ {*} $ are non-empty or not. The dimensions of these vector spaces are $ n _ {+} $ and $ n _ {-} $, these being the numbers of positive and negative chirality zero mass Fermions, respectively. But $ { \mathop{\rm index} } {\partial slash } _ {A} $( for a fixed $ A $) is given by their difference, i.e.

$$ { \mathop{\rm index} } {\partial slash } _ {A} = n _ {+} - n _ {-} = k, $$

where $ k $ is an integer given by the usual characteristic class formula of the index theorem for the Dirac operator $ {\partial slash } _ {A} $; in four dimensions, $ - k $ is the familiar instanton number. Thus, a non-zero $ k $, which is the generic situation, produces an asymmetry in the massless chiral Fermion sector.

Hence, when $ A $ 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 $ Y $, the index of the family is given by an element of $ K ( Y ) $, the $ K $- theory of $ Y $.

In the present case, $ Y = {\mathcal A} $ and, denoting the index of the Dirac family by $ { \mathop{\rm Index} } {\partial slash } $, one obtains

$$ { \mathop{\rm Index} } {\partial slash } = \left \{ { { \mathop{\rm ker} } {\partial slash } _ {A} } : {A \in {\mathcal A} } \right \} - \left \{ { { \mathop{\rm ker} } {\partial slash } ^ {*} _ {A} } : {A \in {\mathcal A} } \right \} = $$

$$ = [ { \mathop{\rm ker} } {\partial slash } ] - [ { \mathop{\rm ker} } {\partial slash } ^ {*} ] . $$

Such a formal difference of (equivalence classes) of vector bundles defines an element of the $ K $- theory $ K ( {\mathcal A} ) $ of $ {\mathcal A} $, which immediately projects to an element of $ K ( {\mathcal A}/ {\mathcal G} ) $ because of the gauge invariance of $ {\partial slash } _ {A} $.

It is now natural to consider a certain determinant line bundle $ { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } $ associated with $ { \mathop{\rm Index} } {\partial slash } $. This bundle is defined by

$$ { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } = ( { \mathop{\rm det} } { \mathop{\rm ker} } {\partial slash } ) ^ {*} \otimes ( { \mathop{\rm det} } { \mathop{\rm ker} } {\partial slash } ^ {*} ) . $$

Since $ { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } $ is a line bundle, it is characterized topologically by its Chern class $ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) $; however, one can calculate $ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) $ 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

$$ [ \mu _ {1} ] \in H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) , $$

$$ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) \in H _ {\textrm{ de Rham } } ^ {2} ( { {\mathcal A}/G } ) . $$

Actually, it can be shown by transgression that

$$ [ \mu _ {1} ] \neq 0 \Rightarrow c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) \neq 0 . $$

In addition one can give formulas for $ \mu _ {1} $ and $ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) $. For example, if $ M = S ^ {4} $ and $ G = SU ( N ) $, then, if $ F _ {t} = t dA + t ^ {2} A \wedge A $, a formula for $ \mu _ {1} $ is the integral below restricted to the orbit $ {\mathcal G} \cdot A $:

$$ \mu _ {1} = - { \frac{i}{( 2 \pi ) ^ {3} } } \int\limits _ { 0 } ^ { 1 } dt \int\limits _ {S ^ {4} } { \mathop{\rm tr} } ( A \wedge F _ {t} \wedge F _ {t} ) . $$

The cohomology class $ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) $ is the one required for the anomaly. To see this, suppose that there is no anomaly. Then $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) $ is gauge invariant and the partition function descends to an integral over the orbit space $ { {\mathcal A}/G } $ on which also $ { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) $ projects to $ { \mathop{\rm det} } ( {\partial slash } ) $. But in this case $ { \mathop{\rm det} } ( {\partial slash } ) $ provides a global section of the line bundle $ { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } $, causing $ c _ {1} ( { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } ) $ to vanish. Thus, the presence of the anomaly is detected by the non-triviality of the bundle $ { \mathop{\rm det} } { \mathop{\rm Index} } {\partial slash } $ and one sees that the index theorem for the Dirac family $ \{ { {\partial slash } _ {A} } : {A \in {\mathcal A} } \} $ brings out succinctly the anomaly as an obstacle to the definition of a gauge-invariant determinant for $ {\partial slash } _ {A} $.

Note that if $ G = U ( 1 ) $, then $ \pi _ {1} ( {\mathcal G} ) $ vanishes; hence $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $ 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 $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $ 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 $ { \mathop{\rm Diff} } ^ {+} ( M ) $ is the group of (orientation-preserving) coordinate transformations of $ M $, then one can consider the change of the Dirac operator $ ( 1/2 ) \gamma _ \mu ( \partial ^ \mu + \Gamma ^ \mu ) ( 1 + \gamma _ {5} ) $ under a coordinate transformation $ h \in { \mathop{\rm Diff} } ^ {+} ( M ) $ rather than an element of $ g \in {\mathcal G} $. Recall that the $ \gamma $- matrices depend on a choice of metric; the same is therefore true of the Dirac operator. Hence, if $ { \mathop{\rm Met} } ( M ) $ denotes the space of metrics on $ M $ and $ \rho \in { \mathop{\rm Met} } ( M ) $, then one can display the metric dependence of the Dirac operator by writing it as $ {\partial slash } _ \rho $. In this context one now searches for an obstacle to the descent of the determinant $ { \mathop{\rm det} } ( {\partial slash } _ \rho ^ {*} {\partial slash } _ \rho ) $ from $ { \mathop{\rm Met} } ( M ) $ to the quotient $ { \mathop{\rm Met} } ( M ) / { \mathop{\rm Diff} } ^ {+} ( M ) $; actually, for technical reasons due to fixed points of the action of $ { \mathop{\rm Diff} } ^ {+} ( M ) $ on $ { \mathop{\rm Met} } ( M ) $, one replaces $ { \mathop{\rm Diff} } ^ {+} ( M ) $ by the (fixed-point-free) subgroup $ { \mathop{\rm Diff} } _ {0} ( M ) $ consisting of those elements of $ { \mathop{\rm Diff} } ^ {+} ( M ) $ that leave a basis at one point fixed. So, now $ { \mathop{\rm Met} } ( M ) / { \mathop{\rm Diff} } _ {0} ( M ) $ plays the role formerly played by $ {\mathcal A}/G $. An anomaly of this kind is known as a gravitational anomaly and lies in $ K ( { \mathop{\rm Met} } ( M ) / { \mathop{\rm Diff} } _ {0} ( M ) ) $.

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 $ {\overline \partial \; } $- operator on a compact Riemann surface $ \Sigma $. The partition function for the Bosonic string contains a power of $ { \mathop{\rm det} } ( {\overline \partial \; } ^ {*} {\overline \partial \; } ) $. The $ {\overline \partial \; } $- operator depends on the metric on $ \Sigma $ and $ { \mathop{\rm det} } ( {\overline \partial \; } ^ {*} {\overline \partial \; } ) $ should only depend on the conformal equivalence class of this metric. Now $ { \mathop{\rm Met} } ( \Sigma ) $, the space of metrics on $ \Sigma $, is acted on by diffeomorphisms and conformal transformations, i.e. by $ { \mathop{\rm Diff} } _ {0} ( \Sigma ) $ and $ C _ {+} ^ \infty ( \Sigma ) $, the space of positive functions on $ \Sigma $. The determinant $ { \mathop{\rm det} } ( {\overline \partial \; } ^ {*} {\overline \partial \; } ) $ should descend to a quotient of $ { \mathop{\rm Met} } ( \Sigma ) $ under this combined action. Failure to do this is an anomaly; this conformal anomaly vanishes when the string moves in $ d $ dimensions where $ d = 26 $. 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 $ \Sigma $.

Global anomaly.

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 $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) $ or $ H _ {\textrm{ de Rham } } ^ {1} ( { \mathop{\rm Diff} } ^ {+} ( M ) ) $ say, vanishes. Global anomalies enter when $ {\mathcal G} $ or $ { \mathop{\rm Diff} } ^ {+} ( M ) $ 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 $ Y $.

An example in which $ {\mathcal G} $ is disconnected is provided by a Yang–Mills theory (cf. also Yang–Mills field) with group $ G $. One can count the connected components of $ {\mathcal G} $ with $ \pi _ {0} ( {\mathcal G} ) $ and, doing this with $ M = S ^ {4} $ and $ G = { \mathop{\rm SU} } ( 2 ) $, one has

$$ \pi _ {0} ( {\mathcal G} ) = \pi _ {4} ( { \mathop{\rm SU} } ( 2 ) ) = \mathbf Z _ {2} , $$

i.e. $ \pi _ {0} ( {\mathcal G} ) \neq 0 $, also $ H _ {\textrm{ de Rham } } ^ {1} ( {\mathcal G} ) = 0 $, so that the local anomaly is zero. This means that there are global gauge transformations under which $ { \mathop{\rm det} } {\partial slash } _ {A} $ is not invariant — in fact they change the sign of $ \sqrt { { \mathop{\rm det} } ( {\partial slash } _ {A} ^ {*} {\partial slash } _ {A} ) } $.

An example in which one has a global gravitational anomaly is provided by $ M = S ^ {10 } $, since then

$$ \pi _ {0} ( { \mathop{\rm Diff} } ^ {+} ( S ^ {10 } ) ) = \mathbf Z _ {992 } $$

and this fact is also very closely tied to the existence of $ 992 $ distinct differentiable structures on $ S ^ {10 } $, i.e. to the existence of exotic spheres. The cancellation of the local and global anomalies in $ 10 $ dimensions for $ N = 1 $ supersymmetric string theories was first described in [a6], [a7].

The quantization of two-dimensional conformal field theories on a Riemann surface $ \Sigma $ involves a natural, projectively flat, connection on a vector bundle $ V $ over the moduli space of $ \Sigma $. One also chooses an integer $ k $, 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 $ { \mathop{\rm Diff} } ^ {+} ( \Sigma ) $ on the sections of $ V $. This then gives rise to an anomaly which is manifested as a shift in the level $ k $. Such a quantization arises in the quantum field theoretic formulation of Jones' knot polynomial.

A Hamiltonian approach to anomalies takes $ M $ to be the manifold of space rather than of space-time; hence, in the chiral examples, $ M $ 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 $ {D slash } _ {A} $ and realizes it as a Fredholm operator. A treatment using Fermionic Fock space is also possible.

A general reference is [a9]. See also Index formulas; Chiral anomaly.

References

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[a2]	J. Bell, R. Jackiw, "A PCAC puzzle in the -model" Nuovo Cim. , 60A (1969) pp. 47–61
[a3]	K. Fujikawa, "Path integral measure for gauge invariant Fermion theories" Phys. Rev. Lett. , 42 (1979) pp. 1195–1197
[a4]	M.F. Atiyah, I.M. Singer, "Dirac operators coupled to vector potentials" Proc. Nat. Acad. Sci. USA , 81 (1984) pp. 2597–2600 MR0742394 Zbl 0547.58033
[a5]	L. Bonora, P. Cotta-Ramusino, "Some Remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations" Comm. Math. Phys. , 87 (1983) pp. 589–603 MR0691046 Zbl 0521.53064
[a6]	M.B. Green, J.H. Schwarz, "Anomaly cancellations in supersymmetric gauge theory and superstring theory" Phys. Lett. , 149B (1984) pp. 117–122
[a7]	E. Witten, "Global gravitational anomalies" Comm. Math. Phys. , 100 (1985) pp. 197–229 MR0804460 Zbl 0581.58038
[a8]	N.J. Hitchin, "Flat connections and geometric quantisation" Comm. Math. Phys. , 131 (1990) pp. 347–380
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