Parity Anomalies and Background Contact Terms

In three dimensions, a gauge-invariant regulator for a Dirac fermion fixes a parity-odd phase of its determinant. For an odd collection of appropriately charged fermions, no counterterm preserves both large-gauge invariance and parity. The same phase appears for nondynamical flavor and $R$ fields as a Chern–Simons contact term, so it is part of every precise duality dictionary—even when no dynamical Chern–Simons term was written in the classical action.

Required background. We use the normalization of supersymmetric Yang–Mills, Chern–Simons, BF, and matter actions and the origin of anomalies in regulated fermion measures. Helpful background. The distinction between a local density and a globally defined phase is analogous to the discussion of theta periodicity and vacuum sectors.

The regulated determinant and the half-level shift

Work on a closed oriented spin three-manifold. For compact $U(1)$ fields $A_I$, normalized so every fermion charge $Q_I$ is integral, write

$$S_{\rm ct}=\frac{1}{4\pi}\sum_{I,J}\kappa_{IJ} \int A_I\wedge dA_J,$$

with a symmetric matrix $\kappa_{IJ}$. If a two-component complex fermion of real mass $M$ and charges $Q_I$ is integrated out, our joint mass-sign, orientation, and Chern–Simons convention assigns

$$\Delta\kappa_{IJ} =\frac12Q_IQ_J\,\operatorname{sgn}(M).$$

This is the low-momentum parity-odd part of the determinant. Reversing the regulator shifts the answer by an allowed integer local counterterm; it cannot change the relative one-unit jump between the two mass signs. Mass differences and fractional parts modulo allowed counterterms are invariant Niemi and Semenoff 1983, pp. 2077–2080; Redlich 1984, pp. 2366–2374.

One-loop origin of the half level

The factor $1/2$ can be seen without evaluating the full determinant. Expand the Euclidean current two-point function at external momentum much smaller than $|M|$. The parity-odd trace contains one power of $M$, one power of the external momentum, and three gamma matrices. In the convention above it gives

$$\Pi^{\mu\nu}_{IJ,\mathrm{odd}}(p) =2Q_IQ_JM\,I(M)\, \epsilon^{\mu\nu\rho}p_\rho +O\!\left(\frac{p^3}{M^2}\right),$$

where the finite scalar integral is

$$I(M)=\int\frac{\mathrm d^3\ell}{(2\pi)^3} \frac{1}{(\ell^2+M^2)^2} =\frac{1}{8\pi|M|}.$$

Hence

$$\Pi^{\mu\nu}_{IJ,\mathrm{odd}}(p) =\frac{Q_IQ_J}{4\pi}\operatorname{sgn}(M) \epsilon^{\mu\nu\rho}p_\rho+\cdots .$$

The quadratic expansion of $S_{\rm ct}$ has coefficient $\kappa_{IJ}/(2\pi)$, so comparison gives $\Delta\kappa_{IJ}=Q_IQ_J\operatorname{sgn}(M)/2$. The loop fixes this chamber-to-chamber change, while the ultraviolet definition fixes the integer origin of $\kappa$. This separates the calculable infrared response from the regulator choice Closset et al. 2012, Appendix A, pp. 28–30.

There is a sign translation to make when using that reference directly. Its Euclidean mass parameter $m$ and contact coefficient obey $\kappa_{\mathrm{IR}}-\kappa_{\mathrm{UV}}=-\operatorname{sgn}(m)/2$. For comparison with that appendix, the source-to-site dictionary is $M=-m$, so the same result is the positive-sign formula displayed here. Changing a regulator is an integer shift; changing the definition of the mass parameter is the sign crosswalk.

For a non-Abelian simple factor, with the fundamental $SU(N)$ trace normalized by $\operatorname{Tr}_{\boldsymbol N}T^aT^b=\frac12\delta^{ab}$, a Dirac fermion in $R$ shifts

$$\Delta k=\operatorname{sgn}(M)T(R), \qquad T(\boldsymbol N)=\frac12.$$

Thus a fundamental shifts the familiar level by $\pm\frac12$. The formula must be applied to every simple and Abelian factor, including mixed Abelian terms. Quotient gauge groups can impose stronger integrality conditions than their Lie algebra suggests.

For a simply connected $SU(N)$ factor in this trace convention, complex massless matter requires $k_{\rm bare}+\sum_iT(R_i)\in\mathbb Z$. This compact formula is not a substitute for the global quantization law when the group is quotiented or has Abelian factors. The sum must also include every massive fermion that is removed on the way to a bosonic infrared description. In particular, integrating out a Chern–Simons-massive adjoint gaugino can make the supersymmetric ultraviolet level differ from the level of the residual bosonic topological theory. A notation such as $G_k$ is incomplete unless it says which of those levels $k$ denotes.

At zero mass the two parity-related limits differ by one unit of the appropriate contact term. For a dynamical compact $U(1)$ field coupled to fermions of charges $q_i$, large-gauge invariance requires

$$k_{\rm bare}+\frac12\sum_iq_i^2\in\mathbb Z$$

in this convention. If $\frac12\sum_iq_i^2$ is not integral, a gauge-invariant definition chooses a parity-breaking half-level counterterm. The combined fermion-plus-counterterm system is well defined; the half-level local polynomial alone is not an ordinary bosonic compact $U(1)$ Chern–Simons theory.

One chiral, two massive phases

Take a charge-one $\mathcal N=2$ chiral and choose $k_{\rm bare}=-\frac12$. The gauge-invariance condition is satisfied. Giving the chiral a real mass gives

$$k_{\rm eff}(M) =-\frac12+\frac12\operatorname{sgn}(M) =\begin{cases} 0,&M>0,\\ -1,&M<0. \end{cases}$$

The two matter-mass chambers therefore have different induced gauge levels. The $M<0$ chamber contains a level-$-1$ spin Chern–Simons sector and correlated background and gravitational responses. The $M>0$ theory has $k_{\rm eff}=0$, but that statement alone does not make the remaining dynamical gauge field an empty gapped phase: its infrared fate also depends on Maxwell, FI, monopole, and global-symmetry data. The level difference is nevertheless the elementary parity-odd response test behind many three-dimensional dualities.

The dimension-labelled anomaly and contact comparison places this induced-level result beside a two-dimensional c-extremization fixture without identifying the two notions of anomaly.

Background contact terms are observables modulo integers

For a conserved current $j_I^\mu$, use the site Fourier convention $j(p)=\int \mathrm d^3x\,e^{+ip\cdot x}j(x)$. The Euclidean position-space contact term $i\kappa_{IJ}\epsilon^{\mu\nu\rho}\partial_\rho\delta^{(3)}(x)/(2\pi)$ then becomes

$$\langle j_I^\mu(p)j_J^\nu(-p)\rangle \supset \frac{\kappa_{IJ}}{2\pi} \epsilon^{\mu\nu\rho}p_\rho.$$

The factor of $i$ in position space and the derivative of the delta distribution combine to give the real momentum polynomial shown here; changing the Fourier phase changes this intermediate sign. If both $A_I$ and $A_J$ are background fields, an allowed local Chern–Simons counterterm shifts $\kappa_{IJ}$ by a quantized amount. Consequently the fractional class of $\kappa_{IJ}$ is a scheme-independent observable, while its integer representative records a scheme choice that must be made consistently on both sides of a duality Closset et al. 2012, §§1–2, pp. 2–8.

The distinction changes when a field is dynamical. A badly quantized dynamical level makes the path integral ill defined, rather than merely changing a convention. A proposed duality must therefore separate:

• levels for dynamical fields, which must satisfy the exact global quantization law;
• fractional background contacts, which are invariant data;
• integer background contacts, which may be shifted but must be stated in a common scheme;
• invertible or transparent spin sectors, which can carry response even without nontrivial local operators.

Flavor, R, and gravitational terms in an N=2 chiral

Let a chiral multiplet have scalar $R$-charge $r$ and flavor charges $Q_A$. Its fermion has $R$-charge $r-1$. In a chamber where its real mass is $M$, the fermion shifts

$$\begin{aligned} \Delta\kappa_{AB}&=\frac12Q_AQ_B\operatorname{sgn}(M),\\ \Delta\kappa_{AR}&=\frac12Q_A(r-1)\operatorname{sgn}(M),\\ \Delta\kappa_{RR}&=\frac12(r-1)^2\operatorname{sgn}(M). \end{aligned}$$

Gauge indices can replace either $A$ or $B$, yielding mixed gauge–flavor and gauge–$R$ levels. These mixed terms determine, among other things, the flavor and $R$ charges of monopole operators through Chern–Simons Gauss laws.

For definiteness define

$$S_{\rm grav}=\frac{\kappa_g}{192\pi} \int\operatorname{Tr}\!\left( \omega\wedge d\omega+\frac23\omega\wedge\omega\wedge\omega \right).$$

In the same regulator convention, one massive complex two-component fermion shifts $\kappa_g$ by $\operatorname{sgn}(M)$. A fermion in a representation $R$ contributes $\dim(R)\operatorname{sgn}(M)$, with any additional flavor multiplicity included. Other references absorb factors into the definition of gravitational Chern–Simons; a comparison is meaningful only after translating the normalization. Supersymmetry packages flavor, $R$, and gravitational terms into supersymmetric counterterms, which is why an isolated comparison of flavor levels can miss a mismatch Closset et al. 2012, §§3–4.

A phase-by-phase duality check

Suppose theories $A$ and $B$ are claimed to share an infrared fixed point. Couple every common global current to the same background field, choose one counterterm convention, and deform by generic real masses. In every chamber:

1. list the sign of each fermion mass eigenvalue;
2. sum all induced dynamical, mixed, background, and gravitational levels;
3. integrate out fields and solve for the vacuum selected by D- and F-terms;
4. identify the residual topological field theory, including global form and transparent lines;
5. compare the background response after applying the proposed symmetry map.

Matching only the rank and absolute Chern–Simons level is insufficient. A sign mismatch changes Hall response; an omitted integer counterterm can change the claimed action of time reversal; an omitted gravitational term changes the framing anomaly. Level/rank duality may be needed to recognize two residual topological theories as equivalent, but its spin and line-operator qualifications must be retained.

The chapter’s monopole–contact–duality map shows how this regulator record feeds the monopole and infrared-dictionary checks without treating $k_{\rm eff}=0$ as a complete phase diagnosis.

Boundaries, orientation reversal, and Pin structures

Parity or time reversal reverses orientation and sends a Chern–Simons action to its negative. A theory can be invariant only if its complete response—including dynamical, background, and gravitational terms—is equivalent to the orientation-reversed response, possibly after adding an allowed counterterm or permuting symmetries.

On a manifold with boundary, a bulk Chern–Simons variation is an anomaly inflow term. Boundary conditions or boundary degrees of freedom must cancel it. On unorientable manifolds, “time-reversal invariant” requires a Pin$^+$ or Pin$^-$ refinement and cannot be inferred from an oriented-manifold calculation alone. The formulas on this page establish oriented spin data; they do not automatically define the unorientable extension.

Common pitfalls

Dropping integer background terms. Their fractional parts carry the anomaly, but the integer representative still fixes the convention in which a duality and its deformation map are written. State it before comparing partition-function phases.

Applying $\frac12\operatorname{sgn}M$ without charges or multiplicities. The shift is a quadratic form in every gauge and background charge. Non-Abelian representations contribute their full Dynkin index.

Calling the two signs of a fermion mass “the same trivial phase.” They differ by quantized gauge and gravitational response. That difference is often the sharpest test of a proposed bosonization or mirror relation.

Copying a sign from another source without its mass convention. A source may reverse the sign of the fermion bilinear, the orientation, or the Chern–Simons functional. Translate one complete massive chamber—not only the symbol $\operatorname{sgn}(M)$—and verify that the difference between the two chambers remains one quantized unit.

Exercises

1. Two Dirac fermions have charges $(1,1)$ and $(1,-1)$ under background $U(1)_A\times U(1)_B$, and both masses are positive. Compute the induced matrix $\Delta\kappa$.

Solution

Each fermion contributes one half of the outer product of its charge vector. Hence

$$\Delta\kappa=\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix} +\frac12 \begin{pmatrix}1&-1\\-1&1\end{pmatrix} =\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$

The mixed contact cancels, while each diagonal contact is shifted by one.

2. A compact $U(1)$ theory has three charge-one Dirac fermions. What congruence must $k_{\rm bare}$ satisfy? What are the effective levels when all three masses are positive or all are negative?

Solution

Gauge invariance requires $k_{\rm bare}+3/2\in\mathbb Z$, so $k_{\rm bare}$ is half-integral in the chosen regulator presentation. The two chambers have $k_{\rm eff}=k_{\rm bare}+3/2$ and $k_{\rm eff}=k_{\rm bare}-3/2$; they differ by three integral units.

3. A chiral multiplet has gauge charge $1$, flavor charge $2$, and scalar $R$-charge $r=0$. Compute the change in all gauge, flavor, and $R$ contact terms when its real mass is taken from negative to positive. Also compute the change in $\kappa_g$.

Solution

The chiral fermion has charge vector $(1,2,-1)$ under $(U(1)_{\rm gauge},U(1)_F,U(1)_R)$. Since the coefficient changes from $-1/2$ to $+1/2$, the chamber difference is the outer product of this vector with itself:

$$\kappa(+)-\kappa(-)= \begin{pmatrix} 1&2&-1\\ 2&4&-2\\ -1&-2&1 \end{pmatrix}.$$

Thus, for example, the gauge–flavor contact changes by $2$ and the gauge–$R$ contact by $-1$. The gravitational coefficient changes from $-1$ to $+1$, so $\kappa_g(+)-\kappa_g(-)=2$. Every entry is integral, as it must be for the difference between two massive chambers.

References

• Closset, C., Dumitrescu, T. T., Festuccia, G., Komargodski, Z., and Seiberg, N. (2012), “Comments on Chern–Simons Contact Terms in Three Dimensions,” Journal of High Energy Physics 2012(09), 091. doi:10.1007/JHEP09(2012)091. Open PDF
• Niemi, A. J., and Semenoff, G. W. (1983), “Axial-Anomaly-Induced Fermion Fractionization and Effective Gauge-Theory Actions in Odd-Dimensional Space-Times,” Physical Review Letters 51, 2077–2080. doi:10.1103/PhysRevLett.51.2077
• Redlich, A. N. (1984), “Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions,” Physical Review D 29, 2366–2374. doi:10.1103/PhysRevD.29.2366

Next steps

Use these induced levels when computing the gauge, flavor, and $R$ charges of monopole operators, and retain the full response when testing Aharony and Giveon–Kutasov dualities.