Chiral Anomalies

Abstract

Anomalies can be viewed as a breaking of some Noether symmetry through the effects of the vacuum. In relativistic quantum field theory (QFT), such a (classical) symmetry is broken by field quantization; cf. Holstein (Am J Phys 61(2):142–147, 1993, Am J Phys 82(6):591–596, 2014), Van Holten (Aspects of BRST quantization. Topology and geometry in physics, 2005) for rather recent reviews.

Appendix: Gravitational Chern–Simons and Pontryagin Terms

When the constant Dirac matrices $\gamma _\alpha$ satisfying $\gamma _\alpha \,\gamma _\beta +\gamma _\beta \gamma _\alpha =2\mathrm{o}_{\alpha \beta }$ saturate the index of the orthonormal coframe one-form $\vartheta ^\alpha =E_j{}^\alpha dx^j$ and its Hodge dual $\eta ^\alpha :={}^*\vartheta ^\alpha$, we obtain a basis of Clifford-algebra-valued exterior forms via

$$\begin{aligned} \gamma :=\gamma _\alpha \vartheta ^\alpha \,,\qquad {}^*\gamma =\gamma ^\alpha \eta _\alpha \,. \end{aligned}$$

(12.4.5)

In terms of the Clifford-algebra-valued connection $\varGamma := {i\over 4} \varGamma _i{}^{\alpha \beta }\,\sigma _{\alpha \beta } dx^i$, the $SL(2,\mathbb {C})$-covariant exterior derivative is given by $D=d+ \varGamma \wedge$, where ${\sigma }_{\alpha \beta }= \frac{i}{2} (\gamma _\alpha \gamma _\beta -\gamma _\beta \gamma _\alpha )$ are the Lorentz generators entering in the two-form $\sigma :={i\over 2}\gamma \wedge \gamma = {1\over 2}\,{\sigma }_{\alpha \beta } \,\vartheta ^\alpha \wedge \vartheta ^\beta$.

Differentiation of these basic variables leads to the Clifford-algebra-valued torsion and curvature two-forms

$$\begin{aligned} \varTheta :=D\gamma =T^{\alpha }\gamma _{\alpha }\;, \qquad \varOmega := d\varGamma +\varGamma \wedge \varGamma = {i\over 4}R^{\alpha \beta }\,\sigma _{\alpha \beta } \end{aligned}$$

(12.4.6)

in RC geometry, respectively. The Chern–Simons term for the Lorentz connection reads

$$\begin{aligned} C_\mathrm{RR} := - \mathrm{Tr}\, \left( {\varGamma }\wedge {\varOmega } - {1\over 3} {\varGamma }\wedge {\varGamma }\wedge {\varGamma } \right) \, . \end{aligned}$$

(12.4.7)

The corresponding Pontryagin topological term can be obtained by exterior differentiation:

$$\begin{aligned} dC_\mathrm{RR}= & {} - \mathrm{Tr}\, \left( {\varOmega }\wedge {\varOmega }\right) \nonumber \\= & {} {1\over 2}R^{\{\}}_{\alpha \beta }\wedge R^{\{\}\alpha \beta } \nonumber \\+ & {} \frac{1}{12}d\left[ \,^*{\mathcal A}\wedge R^{\{\}} -\frac{1}{3}{\mathcal A}\wedge d{\mathcal A} +\frac{1}{9} \,^*{\mathcal A}\wedge ^*({\mathcal A} \wedge \,^*{\mathcal A})\right] . \end{aligned}$$

(12.4.8)

The latter contains (Mielke & Romero  2006), among others, a term proportional to the curvature scalar $R:=\,^*(R^{\alpha \beta }\wedge \eta _{\beta \alpha })$ and the axial torsion piece $d{\mathscr {A}}\wedge d{\mathscr {A}}$ of the axial anomaly with a relative factor 9, as required by the supersymmetric path integral (Mavromatos 1988).

Since the coframe is the “soldered” translational part (Tresguerres & Mielke 2000) of the Cartan connection, a related translational CS term arises:

$$\begin{aligned} C_\mathrm{TT} := {1\over {8\ell ^2}} \mathrm{Tr}\, ( {\gamma } \wedge {\varTheta } )= {1\over {2\ell ^2}}\, {\vartheta ^\alpha }\wedge T_{\alpha }\,. \end{aligned}$$

(12.4.9)

By exterior differentiation we obtain the four-form of Nieh & Yan (1982):

$$\begin{aligned} dC_\mathrm{TT} = {1\over {8\ell ^2}} \mathrm{Tr}\, (\varTheta \wedge \varTheta -4i\varOmega \wedge \sigma ) ={1\over {2\ell ^2}} \left( T^\alpha \wedge T_\alpha +R_{\alpha \beta }\wedge \vartheta ^\alpha \wedge \vartheta ^\beta \right) \,. \end{aligned}$$

(12.4.10)

It is crucial to note that a fundamental length $\ell$ necessarily occurs here for dimensional reasons. This can also be understood by a de Sitter-type gauge approach, in which the $\mathfrak {sl}(5,\mathbb {R})$-valued connection ${\hat{\varGamma }} =\varGamma +(\vartheta ^\alpha L^4{}_\alpha + \vartheta _\beta L^\beta {}_4{})/\ell$ is expanded into the dimensionless linear connection $\varGamma$ plus the coframe $\vartheta ^\alpha = E_i{}^\alpha \, dx^i$, which carries canonical dimension [length]. The corresponding Pontryagin term ${\hat{C}}_\mathrm{RR}$ splits via

$$\begin{aligned} {\hat{C}}_\mathrm{RR} =C_\mathrm{RR} -2 C_\mathrm{TT} \end{aligned}$$

(12.4.11)

into a linear term and a translational CS term; see footnote 31 of Hehl et al. (1995).