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Types of quantum field thories
There are at least two things that are called quantum anomalies in the context of quantum field theory
anomalous action functional: the action functional (in path integral quantization) is not a globally well defined function, but instead a section of a line bundle on configuration space;
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.
There are two major kinds of action functionals that may be anomalous in that they are not actually functions/functionals on the configuration space of fields, but just sections of some line bundle:
theories with fermions (see e.g. spinors in Yang-Mills theory) whose action functional is given by a Dirac operator, or else other fields whose action functional is given by a Fredholm operator.
gauge theories with higher degree gauge fields (differential cocycles of higher degree.)
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 $8k+2$ 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 trivializable. Its non-trivializability is the fermionic anomaly .
More in detail (Freed 86), the path integral over an Lagrangian of the form $(\overline \phi, D \phi)$ for
a Fredholm operator computes the determinant of that operator. Formally this is a section of the determinant line bundle over the remaining fields
where the left hand side makes sense and the equivalence holds for $V$ and $W$ 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.
under construction
Let
be a (well defined) action functional. Write $P$ for its resolved covariant phase space in dg-geometry and
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 $P$ a priori only carries a presymplectic structure. But by BV-theory there exists an equivalent (homotopical) derived action functional $S_\Psi^{BV} : P \to \mathbb{R}$ such that $S_\Psi^{BV}$ does induce a genuine symplectic structure on the derived space $P$.
For ordinary Poisson manifolds and hence symplectic manifolds Maxim Kontsevich‘s theorem says that their deformation quantization always exist. But if $S$ is the action functional of a gauge theory then $P$ 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 $P$ exists only if the first and second infinity-Lie algebroid cohomology of $P$ vanishes:
These two cohomology groups
are called the gauge anomaly of the system. Only if they vanish does the quantization of the gauge theory encoded by $S$ exist.
More concretely, the function algebra on $P$ is a graded-commutative dg-algebra equipped with a graded Poisson bracket $\{-,-\}_{BV}$ and an element $Q \in C^\infty(P)$ (the BV-BRST charge) whose Hamiltonian vector field is the derivation that is the differential of the dg-algebra $C^\infty(X)$. If the gauge anomaly does not vanish, then, while the deformation quantization of the graded algebra $C^\infty(P)$ to a non commutative graded algebra with commutator $[-,-]$ will exist, it may happen that the image $S$ of $Q$ under the quantization no longer satisfies the quantum master equation $[S,S] = \hbar \Delta S$.
Therefore the derivation $[S,-]$ will not define a quantized differential and therefore the quantization of the graded-commutative dg-algebra $C^\infty(P)$ 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 $X$ 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 $X$ has a sufficiently high lift through the Whitehead tower of $O(n)$.
Spin structure the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has Spin structure
This is a classical result. A concrete derivation is in
String structure the worldsheet anomaly for the spinning string/superstring in heterotic string theory vanishes (essentially) when $X$ has String structure
This is originally due to Killingback and Witten. A commented list of literature is here. Recently Ulrich Bunke gave the rigorous proof
in terms of differential cohomology in general and differential string structures in particular.
The 2d CFT on the worldsheet of the bosonic string (in flat space, without further background fields) has an anomaly unless the dimensional target space is $d = 26$.
This is discussed as a condition of trivialization of a bundle in (Freed 86, section 2). A brief summary is stated this comment on MO.
For more see at conformal anomaly for more.
see at Freed-Witten anomaly.
see at Diaconescu-Moore-Witten anomaly
see at M5-brane anomaly
For the moment see Liouville cocycle.
The original articles on anomalous action functionals are
Luis Alvarez-Gaumé, Edward Witten, Gravitational Anomalies, Nucl. Phys. B234 (1984) 269 (doi:10.1016/0550-3213(84)90066-X, spire:192309)
Luis Alvarez-Gaumé, Paul Ginsparg, The structure of gauge and gravitational anomalies, Ann. Phys. 161 (1985) 423. (doi:10.1016/0003-4916(85)90087-9, spire:202565)
Edward Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197. (doi:10.1007/BF01212448, euclid:1103943444)
Review
Paolo Di Vecchia, Green-Schwarz anomaly cancellation (2010) (slides pdf)
Jeffrey Harvey, TASI 2003 Lectures on Anomalies (arXiv:hep-th/0509097, spire:692082)
The mathematical formulation of this in terms of index theory is due to
Michael Atiyah, Isadore Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984) (doi:10.1073/pnas.81.8.2597, jstor:23378)
Jean-Michel Bismut, Daniel Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles , Comm. Math. Phys. 106 (1986), no. 1, 159–176 (doi:10.1007/BF01210930, euclid:1104115586)
Jean-Michel Bismut, Daniel Freed, The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem , Comm. Math. Phys. 107 (1986), no. 1, 103–163 (doi:10.1007/BF01206955, euclid:1104115934)
and a clear comprehensive account of the situation (topological anomaly, geometric anomaly) is in
Raphael Flauger, Anomalies and the Atiyah-Singer Index Theorem (pdf)
Daniel Freed, Determinants, torsion, and strings, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. (doi:10.1007/BF01221001, euclid:1104116145)
Slick formulation of these anomalies as invertible topological field theories is discussed in
A physicists’ monograph is
A clear description of the quantum anomalies for higher gauge theories is in
As an application of this, a detailed discussion of the cancellation of the anomaly of the supergravity C-field in 11-dimensional supergravity 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
Discussion in the context of extended topological field theory includes
Anomaly field theories are discussed in
The original work on the chiral anomaly is due to
Stephen Adler. Axial-Vector Vertex in Spinor Electrodynamics Physical Review 177 (5): 2426. (1969)
John Bell, Roman Jackiw, A PCAC puzzle: π0→γγ in the σ-model“. Il Nuovo Cimento A 60: 47. (1969)
See also
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.
Mikio Nakahara, Chapter 13 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
General discussion in the context of BRST-BV formalism (breaking of the quantum master equation by quantum corrections) is discussed in
W. Troost, P. van Nieuwenhuizen, A. van Proeyen, Anomalies and the Batalin-Vilkovisky lagrangian formalism (web)
Paul Howe, Ulf Lindström, 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)
Glenn Barnich, Classical and quantum aspects of the
extended antifield formalism_ (arXiv:hep-th/0011120)
The fact that the anomaly 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 $W(3)$ 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)
Discussion in the context of AQFT with functional analysis taken into account is in
section 5.3.3. in Katarzyna Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory PhD thesis, Hamburg (2011) (arXiv:1111.5130)
Klaus Fredenhagen, Katarzyna Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory (arXiv:1110.5232)
An interpretation of gauge anomalies as failures of Hamiltonians to have self-adjoint extensions is in
Curtis Callan, Jeffrey Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B250 (1985) 427-436 (doi:10.1016/0550-3213(85)90489-4, spire:15691)
Edward Witten, Kazuya Yonekura, Anomaly Inflow and the $\etas$-Invariant (arXiv:1909.08775, spire:1755070)