# nLab quantum anomaly

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

There are at least two things that are called quantum anomalies in the context of quantum field theory

## Definition

### Anomalous action functional

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:

#### Fermionic anomalies

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 trivializale. 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

$D \;\colon\; V \longrightarrow W$

a Fredholm operator computes the determinant of that operator. Formally this is a section of the determinant line bundle over the remaining fields

$(det V)^\ast \otimes (det W) \simeq (det ker D)^\ast \otimes (det coker D) \,,$

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.

#### Higher gauge-theoreric anomalies

For the moment see Green-Schwarz mechanism for more.

### Anomalous symmetry

{AnomalousSymmetry}

under construction

Let

$S : C \to \mathbb{R}$

be a (well defined) action functional. Write $P$ for its resolved covariant phase space in dg-geometry and

$S^{BV} : P \to \mathbb{R}$

for the BV-action functional, both as given by BRST-BV formalism.

If the action functional is local (comes from a Lagrangian on jet space) 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

$Anom_{gauge} = H^1(CE(P)) \oplus H^2(CE(P))$

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.

## Examples

### Anomalous action functional

#### Spinning particles and super-branes

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

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)
• 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

• Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

in terms of differential cohomology in general and differential string structures in particular.

#### Conformal anomaly of the string

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.

#### Freed-Witten anomaly

see at Freed-Witten anomaly.

### Anomalous symmetry

#### Conformal anomaly

For the moment see Liouville cocycle.

## References

### Anomalous action functional

The original articles on anomalous action functionals are

A survey of these results is in the slides

• Paolo Di Vecchia, Green-Schwarz anomaly cancellation (2010) (pdf)

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)

• Jean-Michel Bismut and Daniel Freed, The analysis of elliptic families. I. Metrics and connections on determinant bundles , Comm. Math. Phys. 106 (1986), no. 1, 159–176.

• Jean-Michel Bismut and 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.

and a clear comprehensive account of the situation (topological anomaly, geometric anomaly) is in

• Daniel Freed, Determinants, torsion, and strings, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. (Euclid)

A physicists’ monograph is

• Reinhold A. Bertlmann, Anomalies in quantum field theory, Oxford Science Publ., 1996, 2000

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

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

The anomaly line bundles for self-dual higher gauge theory is discussed in

• Samuel Monnier, The anomaly line bundle of the self-dual field theory (arXiv:1109.2904)

### Gauge anomaly

• 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.

#### In BV-BRST formulation

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)

• 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)

• Glenn Barnich, Classical and quantum aspects of the extended antifield formalism (arXiv:hep-th/0011120)

The fact that the anomly sits in degree-1 BRST cohomology corresponds to the consistency condition discussed in

• Julius Wess and B. Zumino, Consequences Of Anomalous Ward Identities , Phys. Lett. B37 (1971) 95.

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 of

and

#### Other

An interpretation of gauge anomalies as failures of Hamiltonians to have self-adjoint extensions is in

### In finite-dimensional quantum mechanics

• A. P. Balachandran, Amilcar R. de Queiroz, Mixed states from anomalies, arxiv/1108.3898
• Carlos Alcalde, Daniel Sternheimer, Analytic vectors, anomalies and star representations, Lett. Math. Phys. 17 (1989), no. 2, 117–127. MR90h:22012, doi (the last section has also the field theory case)

Revised on December 23, 2013 05:52:36 by Urs Schreiber (89.204.139.158)