Two-Dimensional QED, the Thirring Model, and Currents

The previous page treated currents and stress tensors as responses to background fields. A conserved $U(1)$ current is the response to a background gauge field $A_\mu$; the stress tensor is the response to a background metric $g_{\mu\nu}$. This page puts that idea into the simplest nontrivial laboratory: two-dimensional massless fermions.

Two dimensions are special for several reasons. A massless Dirac fermion splits into two chiral components. The vector and axial currents are related by Hodge duality. The gauge field has no propagating transverse photon polarization, yet its coupling to fermions is highly nontrivial. And a local four-fermion interaction, the Thirring interaction, is not irrelevant: it is a current-current operator of dimension two, hence marginal.

The guiding idea is this: QED₂ and the Thirring model can be organized around the same current-coupled field $A$. If $A$ has a Maxwell kinetic term, the theory is two-dimensional electrodynamics. If $A$ has only an algebraic quadratic term, integrating it out gives a local current-current interaction. This is a compact place to see gauge fields, auxiliary fields, Ward identities, anomalies, and conformal data sitting in one picture.

Required background. Lesson 19 supplies the distinction between background-field Ward identities and dynamical gauge equations. Helpful background. Lesson 10 introduces two-dimensional chiral spinors, while Lessons 15–16 review OPEs and conformal maps.

Chiral fermions and their currents

Light-cone coordinates, orientation, and Abelian charge

The Lorentzian discussion uses the site-wide mostly-minus metric and

$$x^\pm=x^0\pm x^1, \qquad \partial_\pm={1\over2}(\partial_0\pm\partial_1),$$

with $A_\pm$ the corresponding one-form components. We write $\psi_+$ and $\psi_-$ for the two chiral components of a massless Dirac fermion and normalize their first-order equations as

$$\partial_-\psi_+=0, \qquad \partial_+\psi_-=0.$$

After Wick rotation, the holomorphic CFT sections instead use independent Euclidean coordinates

$$z=x^1+ix^2, \qquad \bar z=x^1-ix^2, \qquad \partial={\partial\over\partial z}, \qquad \bar\partial={\partial\over\partial\bar z}.$$

The symbols $x^\pm$ and $(z,\bar z)$ encode analogous chiral factorizations, but they are not literally identified before analytic continuation.

The site-wide Abelian convention is

$$D_\mu=\partial_\mu-iA_\mu, \qquad \psi\mapsto e^{+i\alpha}\psi, \qquad A_\mu\mapsto A_\mu+\partial_\mu\alpha$$

for a unit-charged fermion. Relative to the opposite convention often used in two-dimensional treatments, this is the translation $A_\mu^{\rm site}=-A_\mu^{\rm source}$ and $J_\mu^{\rm site}=J_\mu^{\rm source}$. The curvature therefore changes sign, while the covariant operator $D_\mu$, gauge-invariant spectrum, and anomaly relation below remain unchanged after the same translation.

Numerical coefficients in anomaly equations depend on current and orientation conventions. We choose $\epsilon^{01}=+1$ and the common one-Dirac-fermion normalization

$$\partial_\mu J_A^\mu =-{1\over2\pi}\epsilon^{\mu\nu}F_{\mu\nu} =-{1\over\pi}F_{01}.$$

To avoid hiding factors of two in light-cone tensor components, define the anomaly source

$$\mathcal A(x)\equiv -{1\over\pi}F_{01}(x).$$

Our chiral-current normalization is then fixed by

$$\partial_\mu J_V^\mu=\partial_-J_+ + \partial_+J_-, \qquad \partial_\mu J_A^\mu=\partial_-J_+ - \partial_+J_-.$$

A massless Dirac fermion in two dimensions decomposes into two independent first-order systems. In light-cone notation the free action may be written as

$$S_0=\int d^2x\, \left( i\psi_+^\dagger\partial_-\psi_+ + i\psi_-^\dagger\partial_+\psi_- \right).$$

The first term propagates one chirality, and the second propagates the other. In Euclidean notation this is the same structure as

$$S_0={1\over2\pi}\int d^2z\, \left( \psi^\dagger\bar\partial\psi+ \bar\psi^\dagger\partial\bar\psi \right),$$

where the equations of motion say

$$\bar\partial\psi=0, \qquad \partial\bar\psi=0.$$

Thus $\psi$ is holomorphic and $\bar\psi$ is antiholomorphic away from insertions.

With the gauge convention above, the vector $U(1)$ symmetry rotates both chiral components by the same phase,

$$\psi_+\mapsto e^{+i\alpha}\psi_+, \qquad \psi_-\mapsto e^{+i\alpha}\psi_-.$$

Classically, the massless theory also has an axial symmetry,

$$\psi_+\mapsto e^{+i\beta}\psi_+, \qquad \psi_-\mapsto e^{-i\beta}\psi_-.$$

Equivalently, before anomalies and mass terms are considered, the two chiral components have independent $U(1)_+\times U(1)_-$ rotations.

The chiral number currents are

$$J_+=:\!\psi_+^\dagger\psi_+\!:, \qquad J_-=:\!\psi_-^\dagger\psi_-\!:.$$

The vector and axial currents are the sum and difference of these chiral currents. In covariant notation,

$$J_V^\mu=\bar\Psi\gamma^\mu\Psi, \qquad J_A^\mu=\bar\Psi\gamma^\mu\gamma^5\Psi.$$

In two dimensions the axial current is, up to sign conventions, the Hodge dual of the vector current:

$$J_A^\mu=\epsilon^{\mu\nu}J_{V,\nu}.$$

In light-cone components, vector conservation and axial conservation would say

$$\partial_-J_+ + \partial_+J_-=0, \qquad \partial_-J_+ - \partial_+J_-=0.$$

Together they imply separate chirality conservation,

$$\partial_-J_+=0, \qquad \partial_+J_-=0.$$

This is the clean classical picture: a right-moving current remains right-moving, and a left-moving current remains left-moving.

A two-dimensional massless Dirac fermion splits into two first-order chiral systems. The right-moving current $J_+$ couples to $A_-$, while the left-moving current $J_-$ couples to $A_+$.

Background gauge fields and Ward identities

A background Abelian gauge field couples to the vector current. In chiral notation the coupled action has the form

$$S[\psi,A] = \int d^2x\, \left[ \psi_+^\dagger(\partial_--iA_-)\psi_+ + \psi_-^\dagger(\partial_+-iA_+)\psi_- \right] +S_A[A].$$

Equivalently,

$$S[\psi,A]=S_0-i\int d^2x\,A_\mu J^\mu+S_A[A].$$

For the moment $A_\mu$ can be either a nondynamical source or a dynamical field. The distinction is not bookkeeping; it changes the meaning of the current equation.

If $A_\mu$ is a background source, then a local phase rotation of the fermions with $A_\mu$ held fixed gives

$$\delta\psi=+i\alpha(x)\psi, \qquad \delta\psi^\dagger=-i\alpha(x)\psi^\dagger.$$

For constant $\alpha$, the action is invariant. For spacetime-dependent $\alpha(x)$, the variation is

$$\delta S =+i\int d^2x\,J^\mu\partial_\mu\alpha =-i\int d^2x\,\alpha\,\partial_\mu J^\mu,$$

after integrating by parts. Thus the Ward identity expresses current conservation,

$$\boxed{\partial_\mu J^\mu=0}$$

inside correlation functions, with contact terms at charged insertions. In complex coordinates this becomes the contour statement that

$$Q_C={1\over2\pi i}\oint_C dz\,J(z)$$

generates the $U(1)$ phase rotation of operators inside $C$.

If instead the gauge field is transformed together with the fermion,

$$A_\mu\mapsto A_\mu+\partial_\mu\alpha,$$

then the local variation of the fermion kinetic term is cancelled by the variation of $-A_\mu J^\mu$. This is ordinary gauge invariance. The same formula has two interpretations: with fixed $A_\mu$, it is a source for a global-current Ward identity; with transformed $A_\mu$, it is a gauge redundancy.

Dynamical gauge fields and total current constraints

Now let $A_\mu$ be dynamical. Returning to Lorentzian signature, take

$$S_A[A]=-{1\over4e^2}\int d^2x\,F_{\mu\nu}F^{\mu\nu}, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

Together with the interaction $-\int d^2x\,A_\mu J^\mu$, the gauge-field equation of motion is

$${1\over e^2}\partial_\mu F^{\mu\nu}-J^\nu=0.$$

The manuscript calls the left-hand side the total gauge current:

$$\boxed{ J_{\mathrm{tot}}^\nu \equiv {1\over e^2}\partial_\mu F^{\mu\nu}-J^\nu=0. }$$

This notation is useful but must be read correctly. $J_{\mathrm{tot}}^\nu$ is the Euler–Lagrange operator for $A_\nu$, not a second gauge-invariant Noether current. It vanishes on the classical equations of motion and, quantum mechanically, inside correlators away from contact terms with $A_\nu$ insertions. It does not say that the matter current vanishes; it says that the matter source is balanced by the field-strength term. Taking the divergence gives

$$\partial_\nu J^\nu=0,$$

because $\partial_\nu\partial_\mu F^{\mu\nu}=0$ identically.

This is the gauge-theory analogue of a point made on the previous page for gravity. If the metric is treated as a background, diffeomorphism invariance gives the Ward identity

$$\nabla_\mu T^{\mu\nu}=0.$$

If the metric is dynamical, varying the full action with respect to $g_{\mu\nu}$ gives a gravitational Euler–Lagrange equation, which one may schematically package as

$$T_{\mu\nu}^{\mathrm{tot}}=0,$$

where gravitational and matter variations have been combined. This is not the assertion that the matter stress tensor vanishes. Diffeomorphism redundancy also prevents a unique gauge-invariant local gravitational energy density; suitable boundary charges and relational observables require separate constructions. Two-dimensional gauge theory is a useful analogue of the same distinction: source Ward identities and dynamical-field equations are close relatives, but not the same statement.

With $A_\mu$ fixed, localizing a global $U(1)$ rotation gives the Ward identity for $J^\mu$. When $A_\mu$ is dynamical, the gauge-field Euler–Lagrange equation balances the matter current against the field-strength term. The quantity labeled $J_{\mathrm{tot}}^\nu$ is an equation-of-motion operator, not an additional matter current.

QED₂ versus the Thirring interaction

The same current coupling can also produce the Thirring model. Instead of giving $A$ a Maxwell kinetic term, give it an algebraic quadratic action,

$$S_{\mathrm{aux}}[A,J] = \int d^2x\, \left( {1\over g}A_+A_- - A_+J_- - A_-J_+ \right).$$

The field $A$ is now auxiliary: it has no derivative term and therefore no propagation. Its equations of motion are algebraic,

$$A_+=gJ_+, \qquad A_-=gJ_-.$$

Substituting back gives

$$S_{\mathrm{aux}}[A_\star,J] =-g\int d^2x\,J_+J_-.$$

Thus the fermionic effective action becomes

$$\boxed{ S_{\mathrm{Th}} = \int d^2x\, \left( i\psi_+^\dagger\partial_-\psi_+ + i\psi_-^\dagger\partial_+\psi_- -gJ_+J_- \right). }$$

Changing the sign of $g$ or absorbing factors of $2$ into $J_\pm$ gives the equivalent common covariant form

$$\mathcal L_{\mathrm{Th}} = \bar\Psi i\gamma^\mu\partial_\mu\Psi -{g_T\over2}(\bar\Psi\gamma^\mu\Psi)(\bar\Psi\gamma_\mu\Psi).$$

In two dimensions the vector-current square is proportional to $J_+J_-$, which is why the chiral notation is so efficient.

The contrast with QED₂ is transparent:

$$\begin{array}{c|c|c} \text{choice of }S_A[A] & \text{field }A & \text{interaction after eliminating }A \\ \hline \displaystyle {1\over g}\int A_+A_- & \text{auxiliary} & \text{local }J_+J_- \\ \displaystyle {1\over4e^2}\int F_{\mu\nu}^2 & \text{gauge field} & \text{nonlocal current interaction} \end{array}$$

After Wick rotation and gauge fixing, the Euclidean Maxwell action is schematically

$$S_A={1\over2e^2}\int {d^2p\over(2\pi)^2}\, A_\mu(-p) \left(p^2\delta_{\mu\nu}-p_\mu p_\nu\right) A_\nu(p).$$

Integrating out $A$ gives a nonlocal interaction,

$$S_{\mathrm{eff}}[J] \sim {e^2\over2}\int {d^2p\over(2\pi)^2}\, J_\mu(-p) {1\over p^2} \left(\delta_{\mu\nu}-{p_\mu p_\nu\over p^2}\right) J_\nu(p),$$

modulo gauge-fixing conventions and contact terms. The auxiliary Thirring field has a constant kernel; the Maxwell field has a derivative kernel. Inverting the first gives a contact interaction. Inverting the second gives an inverse Laplacian.

Vector symmetry and axial anomaly

For a massless Dirac fermion, the classical $U(1)_+\times U(1)_-$ symmetry is larger than the vector gauge symmetry. Quantum mechanically, in a background gauge field, the vector symmetry is preserved but the axial symmetry is anomalous.

With the normalization fixed above,

$$\boxed{\partial_\mu J_V^\mu=0,}$$

while

$$\boxed{\partial_\mu J_A^\mu=-{1\over\pi}F_{01}.}$$

Using the chiral-current normalization stated in the convention note, write this as

$$\partial_-J_+ + \partial_+J_-=0,$$

and

$$\partial_-J_+ - \partial_+J_-=\mathcal A, \qquad \mathcal A\equiv -{1\over\pi}F_{01}.$$

Solving these two equations gives

$$\boxed{ \partial_-J_+={1\over2}\mathcal A, \qquad \partial_+J_-=-{1\over2}\mathcal A. }$$

Changing the orientation or exchanging the labels $+$ and $-$ flips the displayed signs, but the invariant statement is simple: a background electric field transfers charge between the two chiral sectors. Vector charge is conserved. Axial charge is not.

The two chiral currents would be separately conserved in the free massless theory. In a background gauge field, vector charge is conserved but the axial current is anomalous. In the site-wide Abelian convention, the anomaly source $\mathcal A=-F_{01}/\pi$ appears with opposite signs in the two chiral equations.

A compact way to understand the anomaly is to remember that a regulator must make a choice. Preserving vector gauge invariance is mandatory if $A_\mu$ is a gauge field. The price is that the axial transformation changes the fermion measure. In Fujikawa language the axial rotation has a nontrivial Jacobian. In diagrammatic language the one-loop current correlator cannot preserve both vector and axial Ward identities simultaneously. In operator language, spectral flow in an electric field moves levels between left- and right-moving sectors.

For QED₂ with one massless Dirac fermion, this anomaly is responsible for the Schwinger mechanism: the gauge-invariant electric field becomes a massive excitation with

$$m_\gamma^2={e^2\over\pi}$$

in this normalization. The model has no ordinary propagating photon before coupling to matter; after the fermion loop is included, the gauge-invariant electric field carries a mass scale. So “no transverse photon polarization” does not mean “no dynamics.”

Current algebra of a free chiral fermion

The holomorphic part of a free complex fermion has the short-distance OPE

$$\psi(z)\psi^\dagger(w)\sim {1\over z-w}.$$

Define the holomorphic $U(1)$ current

$$J(z)=:\!\psi^\dagger\psi\!:(z).$$

Wick contraction gives

$$J(z)\psi(w)\sim -{\psi(w)\over z-w}, \qquad J(z)\psi^\dagger(w)\sim +{\psi^\dagger(w)\over z-w}.$$

Thus $J$ measures the number charge: with $J=:\!\psi^\dagger\psi\!:$, annihilation and creation fields have charges $-1$ and $+1$, respectively. Reversing the overall sign of $J$ reverses all charges but leaves the physics unchanged. Double contraction gives the current-current OPE

$$\boxed{ J(z)J(w)\sim {1\over(z-w)^2}. }$$

More generally, an Abelian current algebra at level $k$ has

$$J(z)J(w)\sim {k\over(z-w)^2}.$$

The charge associated with a contour $C$ is

$$Q_C={1\over2\pi i}\oint_C dz\,J(z).$$

If $C$ surrounds an operator $O_q(w)$ with $U(1)$ charge $q$, then

$$J(z)O_q(w)\sim {q\over z-w}O_q(w),$$

and therefore

$$Q_C O_q(w)=qO_q(w).$$

This contour formulation is the chiral version of the Ward identity. It is also the bridge from current conservation to conformal field theory: once an operator has a prescribed singular OPE with the current, Ward identities follow by contour deformation.

The Thirring model as a marginal deformation

The massless Thirring interaction is

$$S_{\mathrm{int}}=g\int d^2z\,J(z)\bar J(\bar z),$$

where $J$ is holomorphic and $\bar J$ is antiholomorphic in the free theory. Since

$$\Delta[J]=1, \qquad \Delta[\bar J]=1,$$

the perturbing operator $J\bar J$ has total scaling dimension

$$\Delta[J\bar J]=2.$$

In two dimensions the action perturbation $\int d^2z\,J\bar J$ is therefore classically marginal. For the Abelian current algebra this deformation is exactly marginal: it changes the radius, or stiffness, of the equivalent compact boson. It does not generate a mass gap by itself, provided the resulting Gaussian kinetic term remains in its positive, unitary range.

What changes, then? Scaling dimensions of charged operators. Perturbatively, logarithms appear in correlation functions. A typical momentum-space fermion propagator has the schematic form

$$G_+(p) ={1\over ip_-} \left[ 1-{\gamma_\psi(g)\over2}\log{\Lambda^2\over p^2}+\cdots \right],$$

where $\Lambda$ is a short-distance cutoff and $\gamma_\psi(g)$ is an anomalous-dimension coefficient. Renormalization-group improvement turns the logarithm into a power,

$$\boxed{ G_+(p) \sim {1\over ip_-} \left({p^2\over\Lambda^2}\right)^{\gamma_\psi(g)/2}. }$$

Equivalently, in position space the power-law decay is changed. The theory remains scale invariant, but the exponents vary continuously with $g$.

The marginal current-current interaction produces logarithmic corrections in perturbation theory. Renormalization-group improvement reorganizes those logarithms into a power law, i.e. into an anomalous scaling dimension.

A useful operator explanation comes from the equation of motion. In chiral notation the Thirring action gives schematically

$$\partial_+\psi_- = gJ_+\psi_-, \qquad \partial_-\psi_+ = gJ_-\psi_+,$$

up to sign and normalization conventions. Because the current has an OPE with a double pole, solving the first-order equation effectively attaches a bosonic dressing to the fermion. Gaussian fluctuations of that dressing produce a logarithm; exponentiating the logarithm gives a new power law.

This is the same mechanism that appears in Luttinger liquids and in bosonization. The massless Thirring model is not merely “a four-fermion theory.” It is a line of conformal field theories whose local operator dimensions depend on the current-current coupling.

QED₂ as a nonlocal current-current theory

It is tempting to regard QED₂ as just another current-current deformation, because integrating out $A_\mu$ produces a quadratic functional of $J_\mu$. But the Maxwell kernel introduces an inverse Laplacian. In a transverse gauge one finds schematically

$$S_{\mathrm{QED}_2,\mathrm{eff}} = S_0 +{e^2\over2}\int d^2x\,d^2y\, J_\mu(x)G(x-y)P_{\mu\nu}^{\mathrm T}J_\nu(y),$$

where

$$-\partial^2G(x-y)=\delta^{(2)}(x-y), \qquad P_{\mu\nu}^{\mathrm T}=\delta_{\mu\nu}-{\partial_\mu\partial_\nu\over\partial^2}.$$

In two dimensions,

$$G(x)\sim -{1\over2\pi}\log(\mu |x|),$$

so the gauge-mediated interaction is long range. This is why QED₂ belongs in the same conversation as the Thirring model but is not the same deformation. The Thirring interaction changes the local stiffness of the current algebra. QED₂ imposes gauge constraints and, with massless fermions, generates a mass scale through the anomaly.

One way to summarize the difference is

$$\text{Thirring:}\quad K_A(p)\sim \text{constant}, \qquad \text{QED}_2:\quad K_A(p)\sim p^2.$$

Since integrating out $A$ uses $K_A^{-1}$, the first gives a local interaction and the second gives a nonlocal one.

From current Ward identities to conformal transformations

The last ingredient on this page is the transformation law for local operators under a conformal map. It will become the main language of the next pages.

A primary field with holomorphic and antiholomorphic weights $(h,\bar h)$ transforms under a finite conformal map

$$z\mapsto f(z), \qquad \bar z\mapsto \bar f(\bar z),$$

as

$$\boxed{ O(z,\bar z) \mapsto \left({df\over dz}\right)^h \left({d\bar f\over d\bar z}\right)^{\bar h} O(f(z),\bar f(\bar z)). }$$

For an infinitesimal transformation

$$f(z)=z+\epsilon(z), \qquad \bar f(\bar z)=\bar z+\bar\epsilon(\bar z),$$

we expand

$$\left({df\over dz}\right)^h=1+h\partial\epsilon+O(\epsilon^2),$$

and

$$O(f,\bar f) =O+\epsilon\partial O+\bar\epsilon\bar\partial O+O(\epsilon^2).$$

Therefore

$$\boxed{ \delta O = \epsilon\partial O+\bar\epsilon\bar\partial O +h(\partial\epsilon)O+\bar h(\bar\partial\bar\epsilon)O. }$$

For a purely holomorphic field $O_n(z)$ of weight $h_n$, this reduces to

$$\boxed{ \delta O_n(z)=\epsilon(z)\partial O_n(z)+h_n(\partial\epsilon)(z)O_n(z). }$$

This formula is the conformal analogue of a Ward identity. Instead of a $U(1)$ current generating phase rotations, the stress tensor generates local conformal transformations. The next page derives this statement as an OPE.

A primary field transforms by moving its insertion to $f(z)$ and multiplying by the local scale factor $(df/dz)^h$. Expanding $f(z)=z+\epsilon(z)$ gives $\delta O=\epsilon\partial O+h(\partial\epsilon)O$ in the holomorphic sector.

Example: charges from the current OPE

Let $J(z)$ be the holomorphic current of a free complex fermion,

$$J(z)=:\!\psi^\dagger\psi\!:(z).$$

The OPE

$$J(z)\psi(w)\sim -{\psi(w)\over z-w}$$

means that the contour charge

$$Q={1\over2\pi i}\oint_w dz\,J(z)$$

acts as

$$Q\psi(w)=-\psi(w).$$

For $\psi^\dagger$,

$$J(z)\psi^\dagger(w)\sim +{\psi^\dagger(w)\over z-w},$$

so

$$Q\psi^\dagger(w)=+\psi^\dagger(w).$$

Now take a neutral mass bilinear,

$$M(w,\bar w)=\psi_+^\dagger(w)\psi_-(\bar w).$$

The vector charge of $M$ is zero: $\psi_+^\dagger$ and $\psi_-$ carry opposite vector charges in the bilinear. But its axial charge is not zero, because the two chiral sectors transform oppositely under $U(1)_A$. This is the operator-level reason why mass terms break axial symmetry while preserving vector symmetry:

$$m\bar\Psi\Psi \sim m\left(\psi_+^\dagger\psi_-+\psi_-^\dagger\psi_+\right).$$

The massless theory has chiral current algebra; the mass term couples the two chiralities and destroys separate chiral conservation.

Summary

Two-dimensional massless fermions make the current viewpoint unusually sharp. The vector current couples to a background gauge field. Localizing the global $U(1)$ symmetry gives the Ward identity, while making the gauge field dynamical turns the current equation into a gauge constraint.

The same current-coupled field $A$ gives two important theories depending on its quadratic kernel. An algebraic $A_+A_-$ kernel makes $A$ auxiliary and produces the local Thirring interaction $J_+J_-$. A Maxwell $F^2$ kernel gives QED₂, whose gauge field has no ordinary photon polarization but generates nonlocal current interactions and, with massless fermions, a gauge-invariant mass scale through the anomaly.

The massless theory has two chiral currents. Classically they are separately conserved. Quantum mechanically, a background electric field preserves the vector current but produces an axial anomaly. In holomorphic language, the current OPE $J(z)J(w)\sim1/(z-w)^2$ controls charge assignments and Ward identities.

Finally, the page ends with the finite and infinitesimal transformation law for primary fields. This prepares the stress-tensor OPE: the stress tensor is to conformal transformations what $J(z)$ is to $U(1)$ phase rotations.

Common pitfalls

The Thirring model and QED₂ are related but not identical. Both can be described using a field $A$ coupled to the fermion current, but the quadratic action for $A$ is different. Algebraic $A$ gives a local four-fermion interaction; Maxwell $A$ gives a nonlocal interaction after elimination.

The matter current is not the gauge-field equation of motion. In a dynamical gauge theory, $e^{-2}\partial_\mu F^{\mu\nu}-J^\nu$ is an Euler–Lagrange operator: it vanishes on shell and has contact terms in quantum correlators. Calling it $J_{\mathrm{tot}}^\nu$ does not make it an independent Noether current, and it does not say that $J^\nu$ vanishes.

The vector and axial symmetries behave differently in the quantum theory. A regulator that preserves vector gauge invariance produces an axial anomaly. Reversing this choice would break gauge invariance, which is not acceptable if $A_\mu$ is a gauge field.

A marginal operator need not be trivial. The operator $J\bar J$ has dimension two in two dimensions, so it is marginal. In the Abelian Thirring case it is exactly marginal and changes scaling dimensions continuously.

Exercises

Exercise 1: Integrating out the Thirring auxiliary field

Integrate out the auxiliary field in

$$S_{\mathrm{aux}}[A,J] = \int d^2x\, \left( {1\over g}A_+A_- - A_+J_- - A_-J_+ \right).$$

Show that the resulting interaction is local and proportional to $J_+J_-$.

Solution

The equations of motion for $A_+$ and $A_-$ are

$${1\over g}A_- - J_-=0, \qquad {1\over g}A_+ - J_+=0.$$

Thus

$$A_-=gJ_-, \qquad A_+=gJ_+.$$

Substitute this back into the action:

$$S_{\mathrm{aux}}[A_\star,J] = \int d^2x\, \left( {1\over g}(gJ_+)(gJ_-) -(gJ_+)J_- -(gJ_-)J_+ \right).$$

The three terms are

$$gJ_+J_- - gJ_+J_- - gJ_+J_-=-gJ_+J_-.$$

Therefore

$$S_{\mathrm{aux}}[A_\star,J] =-g\int d^2x\,J_+J_-.$$

The interaction is local because the auxiliary-field kernel had no derivatives. A Maxwell kernel would instead have to be inverted, producing an inverse Laplacian and hence a nonlocal current-current interaction.

Exercise 2: Splitting the vector and axial Ward identities

Assume the vector current is conserved and the axial current has anomaly

$$\partial_\mu J_V^\mu=0, \qquad \partial_\mu J_A^\mu=\mathcal A, \qquad \mathcal A\equiv -{1\over\pi}F_{01}.$$

Using the light-cone identifications

$$\partial_\mu J_V^\mu=\partial_-J_+ + \partial_+J_-, \qquad \partial_\mu J_A^\mu=\partial_-J_+ - \partial_+J_-,$$

derive the separate equations for $\partial_-J_+$ and $\partial_+J_-$.

Solution

Let

$$X=\partial_-J_+, \qquad Y=\partial_+J_-.$$

The two Ward identities become

$$X+Y=0, \qquad X-Y=\mathcal A.$$

Adding the equations gives

$$2X=\mathcal A,$$

so

$$X={1\over2}\mathcal A.$$

Then $Y=-X$, hence

$$Y=-{1\over2}\mathcal A.$$

Therefore

$$\partial_-J_+={1\over2}\mathcal A, \qquad \partial_+J_-=-{1\over2}\mathcal A.$$

These signs follow from the stated convention for $J_A$. A different convention for $\epsilon^{\mu\nu}$ flips the right-hand sides but leaves the distinction between vector conservation and axial anomaly intact.

Exercise 3: Infinitesimal primary-field transformation

Let $O(z,\bar z)$ be a primary field of weights $(h,\bar h)$, transforming as

$$O(z,\bar z) \mapsto (f'(z))^h(\bar f'(\bar z))^{\bar h}O(f(z),\bar f(\bar z)).$$

For

$$f(z)=z+\epsilon(z), \qquad \bar f(\bar z)=\bar z+\bar\epsilon(\bar z),$$

derive the infinitesimal variation of $O$ to first order in $\epsilon$ and $\bar\epsilon$.

Solution

To first order,

$$f'(z)=1+\partial\epsilon(z), \qquad \bar f'(\bar z)=1+\bar\partial\bar\epsilon(\bar z).$$

Therefore

$$(f')^h=1+h\partial\epsilon+O(\epsilon^2), \qquad (\bar f')^{\bar h}=1+\bar h\bar\partial\bar\epsilon+O(\bar\epsilon^2).$$

Also,

$$O(f,\bar f) =O(z,\bar z)+\epsilon\partial O+\bar\epsilon\bar\partial O+O(\epsilon^2).$$

Multiplying these expansions and keeping only first-order terms gives

$$\delta O = \epsilon\partial O+ \bar\epsilon\bar\partial O+ h(\partial\epsilon)O+ \bar h(\bar\partial\bar\epsilon)O.$$

For a holomorphic field with $\bar h=0$ and no $\bar z$ dependence, this reduces to

$$\delta O(z)=\epsilon(z)\partial O(z)+h(\partial\epsilon)(z)O(z).$$

Exercise 4: Charge from a current contour

Use the OPE

$$J(z)O_q(w)\sim {q\over z-w}O_q(w)$$

to show that the contour charge $Q_C=(2\pi i)^{-1}\oint_C dz\,J(z)$ acts on $O_q(w)$ as multiplication by $q$, when $w$ lies inside $C$.

Solution

Only the singular part of the OPE contributes to the contour integral. Therefore

$$Q_C O_q(w) ={1\over2\pi i}\oint_C dz\,{q\over z-w}O_q(w).$$

Since $O_q(w)$ is independent of the integration variable $z$, and since $C$ encloses $w$,

$${1\over2\pi i}\oint_C {dz\over z-w}=1.$$

Thus

$$Q_C O_q(w)=qO_q(w).$$

If $w$ lies outside $C$, the contour encloses no pole and the result is zero. This is the contour form of the Ward identity.

References

• S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model,” Physical Review D 11 (1975), 2088–2097.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer (1997), chapters 5–7.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers (1987), chapters 1 and 9.
• J. Schwinger, “Gauge Invariance and Mass. II,” Physical Review 128 (1962), 2425–2429.
• W. E. Thirring, “A Soluble Relativistic Field Theory,” Annals of Physics 3 (1958), 91–112.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press (2021), chapters 29–31.