Types of quantum field thories
There are at least two things that are called quantum anomalies in the context of quantum field theory
anomalous symmetry (gauge anomaly): a symmetry of the action functional does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the action of a group on classical phase space is not preserved by deformation quantization.
The path integral for a quantum field theory with fermions can be decomposed into a fermionic path integral (see there for more details) over the fermionic fields followed by that over the bosonic fields. The former, a Berezin integral, is typically well defined for a fixed configuration of the bosonic fields, but does not produce a well defined function on the space of all bosonic fields: but a twisted function , a section of some line bundle called a determinant line bundle or, in dimensions, its square root, the Pfaffian line bundle.
So to even start making sense of the remaining path integral over the bosonic degree of freedom, this determinant line bundle or the corresponding Pfaffian line bundle has to be trivializale. Its non-trivializability is the fermionic anomaly .
where the left hand side makes sense and the equivalence holds for and finite dimensional, and where the right hand side is the definition of the expression for general Fredholm operators. ((Freed 86, 1.))
In more detail this determinant line bundle also carries a connection on a bundle. To make the formal path integral, which is a section of this bundle, into an actual function, one this bundle with connection needs to be trivializable and trivialized. The obstruction to this is the anomaly.
For the moment see Green-Schwarz mechanism for more.
for the BV-action functional, both as given by BRST-BV formalism.
If the action functional is local (comes from a Lagrangian on a jet bundle) the covariant phase space a priori only carries a presymplectic structure. But by BV-theory there exists an equivalent (homotopical) derived action functional such that does induce a genuine symplectic structure on the derived space .
For ordinary Poisson manifolds and hence symplectic manifolds Maxim Kontsevich’s theorem says that their deformation quantization always exist. But if is the action functional of a gauge theory then is in general a nontrivial derived infinity-Lie algebroid (its function algebra has “ghosts” and “ghosts of ghost”: the Chevalley-Eilenberg algebra generators) and the theorem does not apply. Instead, the quantization of the derived symplectic space exists only if the first and second infinity-Lie algebroid cohomology of vanishes:
These two cohomology groups
More concretely, the function algebra on is a graded-commutative dg-algebra equipped with a graded Poisson bracket and an element (the BV-BRST charge) whose Hamiltonian vector field is the derivation that is the differential of the dg-algebra . If the gauge anomaly does not vanish, then, while the deformation quantization of the graded algebra to a non commutative graded algebra with commutator will exist, it may happen that the image of under the quantization no longer satisfies the quantum master equation .
Therefore the derivation will not define a quantized differential and therefore the quantization of the graded-commutative dg-algebra will only be a noncommutative algebra, not a non-commutative dg-algebra, hence will not be functions on a non-commutative space in derived geometry.
The sigma-model for a supersymmetric fundamental brane on a target space has an anomaly coming from the nontriviality of Pfaffian line bundles associated with the fermionic fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the tangent bundle of has a sufficiently high lift through the Whitehead tower of .
Spin structure the worldline anomaly for the spinning particle/superparticle vanishes when has Spin structure
This is a classical result. A concrete derivation is in
For more see at conformal anomaly for more.
see at Freed-Witten anomaly.
For the moment see Liouville cocycle.
The original articles on anomalous action functionals are
A survey of these results is in the slides
The mathematical formulation of this in terms of index theory is due to
and a clear comprehensive account of the situation (topological anomaly, geometric anomaly) is in
A physicists’ monograph is
A clear description of the quantum anomalies for higher gauge theories is in
The role of spin structures as the anomaly cancellation condition for the spinning particle is discussed in
The anomaly line bundles for self-dual higher gauge theory is discussed in
L. Faddeev and S. Shatashvili, “Algebraic and Hamiltonian Methods in the theory of Nonabelian Anomalies,” Theor. Math. Fiz., 60 (1984) 206; english transl. Theor. Math. Phys. 60 (1984) 770.
B. Zumino, “Chiral anomalies and differential geometry,” in Relativity, Groups and Topology II, proceedings of the Les Houches summer school, B.S. DeWitt and R. Stora, eds. North-Holland, 1984.
W. Troost, P. van Nieuwenhuizen, A. van Proeyen, Anomalies and the Batalin-Vilkovisky lagrangian formalism (web)
P.S. Howe, U. Lindström and P. White, Anomalies And Renormalization In The BRST-BV Framework , Phys. Lett. B246 (1990) 430.
J. Paris, W. Troost, Higher loop anomalies and their consistency conditions in nonlocal regularization , Nucl. Phys. B482 (1996) 373 (arXiv:hep-th/9607215)
The fact that the anomly sits in degree-1 BRST cohomology corresponds to the consistency condition discussed in
Discussion of special applications in
F. De Jonghe, J. Paris and W. Troost, The BPHZ renormalised BV master equation and Two-loop Anomalies in Chiral Gravities , Nucl. Phys. B476 (1996) 559 (arXiv:hep-th/9603012)
J. Paris, Nonlocally regularized antibracket - antifield formalism and anomalies in chiral gravity , Nucl. Phys. B450 (1995) 357 (arXiv:hep-th/9502140)
R. Amorim, N.R.F.Braga, R. Thibes, Axial and gauge anomalies in the field antifield quantization of the generalized Schwinger model (arXiv:hep-th/9712014)