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In Fermion Path Integrals And Topological Phases Witten showed that for a massless Dirac fermion in $2+1$ dimensions

$$S[\bar{\psi},\psi]=\int d^{3}x\bar{\psi}iD\!\!\!\!/_{A}\psi,$$

where $A$ is a $U(1)$ background gauge field, the partition function should be

$$\mathcal{Z}=|\det(iD\!\!\!\!/_{A})|\exp\left\{-\frac{i\pi\eta(A)}{2}\right\},$$

where $\eta(A)$ is the APS eta-invariant introduced in Spectral asymmetry and Riemannian Geometry. This function measures the asymmetry of the spectrum of the Dirac operator. In a sloppy language, $\eta(A)$ equals to the number of positive modes of $iD\!\!\!\!/_{A}$ minus the number of negative modes of $iD\!\!\!\!/_{A}$.

The phase factor comes from Pauli-Villars regularization. To be specific, since the formal expression of the partition function

$$\mathcal{Z}=\prod_{\lambda\in\mathrm{Spec}}\lambda,$$

where each $\lambda$ is an eigenvalue of the Hermitian operator $iD\!\!\!\!/_{A}$, is an infinite product, the overall sign of the partition function is not defined. One can regularize it by adding the Lagrangian a Pauli-Villars regulator

$$\mathcal{L}_{\mathrm{reg}}=\lim_{M\rightarrow\infty}\left(\bar{\chi}iD\!\!\!\!/_{A}\chi+iM\bar{\chi}\chi\right),$$

where $\chi$ is a ghost scalar satisfying the Dirac equation. Then, using the formula

$$\mathrm{Arg}(zw)=\mathrm{Arg}(z)+\mathrm{Arg}(w)\,\,\,\mathrm{mod}\,\,2\pi$$

one finds that the regularized partition function indeed becomes

$$\mathcal{Z}=|\det(iD\!\!\!\!/_{A})|\exp\left\{-\frac{i\pi\eta(A)}{2}\right\}.$$

However, there is still a problem when one performs a large gauge transformation. Under a large gauge transformation, there can be a net spectral flow of the Dirac operator. This was introduced in Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions.

To be specific, let us start from $A$. Under a large gauge transformation $\Phi$, $A$ is transformed to $A^{\Phi}$. Then, one can interpolate the two and make a gauge field

$$A_{s}=(1-s)A+sA^{\Phi},$$

where $s\in[0,1]$ parametrizes the spectral flow under a large gauge transformation. Here, $A$ and $A^{\Phi}$ are gauge equivalent.

For example, if one of the positive modes of $iD\!\!\!\!/_{A}$ flows through a zero-mode to the negative spectrum, then $\eta(A)$ would jump by $\pm 2$.

enter image description here

As a result, the sign of the partition function would flip accordingly. However, one would expect that the spectrum at $s=0$ is identical to the spectrum at $s=1$ since the two are related via a gauge transformation.

The existence of the spectral flow is guaranteed by the APS-index theorm. The number of eigenmodes flowing through $0$ equals to a Dirac index $\Delta$ in four dimensional spacetime. i.e. one views the $2+1$ dimensional spacetime as the boundary of a $3+1$ dimensional bulk. In other words, under a large gauge transformation, the partition transforms as

$$\mathcal{Z}\rightarrow\mathcal{Z}(-1)^{\Delta}.$$

However, there is no obvious reason why the index in the bulk can be an even number. This was also discussed in Anomalies and Odd Dimensions.

Does this mean that the massless Dirac fermion in $2+1$ dimensions suffer from a gauge anomaly?

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    $\begingroup$ This is called the parity anomaly(en.wikipedia.org/wiki/Parity_anomaly). The usual statement is that it is not possible to preserve P and the gauge symmetry - that is, there is no gauge-invariant, parity-preserving regulator. The non-invariance under large gauge transformation is not a problem; usually one doesn't consider large gauge transformations to be gauge transformation (the terminology is horrible, i know). $\endgroup$ – Lorenz Mayer Jan 16 '19 at 15:06
  • $\begingroup$ @LorenzMayer I know that it has a parity anomaly. But could you please elaborate why non-invariance under large gauge transformation isn't a problem? $\endgroup$ – Libertarian Monarchist Bot Jan 16 '19 at 17:25
  • $\begingroup$ I cannot give a general answer, otherwise i would have formulated one, but i felt it would be helpful pointing in some directions which seem important. One thing one can immediately say is that one CAN, consistently, use only the small gauge transformations as gauge group, because they are a connected component and a subgroup in all gauge transformations. Now, why SHOULD one do this - this is not clear to me either. There are these lectures by Andrew Strominger arxiv.org/pdf/1703.05448.pdf (cf section 2.7); if i find a more conclusive answer, i'll let you know. $\endgroup$ – Lorenz Mayer Jan 17 '19 at 15:33
  • $\begingroup$ @LorenzMayer Thank you for recommending me the lecture notes. Would you please tell me what software does Andrew Strominger use to draw those colourful pictures in the lecture notes? $\endgroup$ – Libertarian Monarchist Bot Jan 17 '19 at 15:52
  • $\begingroup$ I would very much like to know this as well. $\endgroup$ – Lorenz Mayer Jan 17 '19 at 18:11
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The eta invariant of a 2+1 Dirac operator coupled to a non-Abelian gauge field background is up to a constant (depending on the metric, which will be assumed to be fixed) an induced Chern-Simons action. The Coefficient of the Chern-Simons term is equal (for each flavor) to $\frac{|m|}{m}$, where $m$ is the fermion's mass.

A modern account of this fact is given by Witten in Fermion Path Integrals And Topological Phases (equation 2.51). A traditional field theoretical evaluation of the fermion partition function is given for example by Ojima.

Since $\frac{|m|}{m}$ is integer, the Chern-Simons action is invariant under large gauge transformation as well as for small gauge transformations. As, Witten explains, the partition function's nonvanishing phase is indeed an indication of an anomaly which can in the modern account be considered as a time-reversal anomaly, as the theory cannot be quantizaed in a $\mathrm{T}$ invariant manner due to the emergent Chern-Simons term.

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  • $\begingroup$ The coefficient of the Chern-Simons is half an integer and the phase written in terms of Chern-Simons action isn't gauge invariant. $\endgroup$ – Libertarian Monarchist Bot Jan 21 '19 at 20:24
  • $\begingroup$ Thank you Mr David Bar Moshe. I know that it has a parity anomaly but my question is about gauge anomaly. The eta invariant jumps by 2 under the spectral flow. $\endgroup$ – Libertarian Monarchist Bot Jan 21 '19 at 20:26
  • $\begingroup$ @The Last Knight of Silk Road sorry for the sloppy and incorrect answer. I'll try to correct everything in the next few days. $\endgroup$ – David Bar Moshe Jan 23 '19 at 7:04

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