**Abbreviations**

The following abbreviations are used in this manuscript:

CS Chern–Simons

ECS Extended Chern–Simons

### **Appendix A. Uniqueness of Consistent Interaction Vertex**

In this Appendix, we demonstrate the fact of the uniqueness of the interaction vertex in Equation (16) in the class of Poincare-covariant couplings that are polynomial in the invariants <sup>F</sup>*μ*, G*μ* without higher derivatives. We apply the method of the inclusion of not necessarily Lagrangian consistent interactions in [54].

Let us first explain the concept of interaction consistency in the class of non-Lagrangian field theories. The dynamics of the theory are determined by a system of partial differential equations (equations of motion) imposed onto the dynamical fields *ϕ<sup>I</sup>*(*x*):

$$\mathbb{T}\_a(\varphi^l(\mathbf{x}), \partial \varphi^l(\mathbf{x}), \partial^2 \varphi^l(\mathbf{x}), \dots) = 0 \,. \tag{A1}$$

The equations of motion do not necessarily follow from the least-action principle for any functional *<sup>S</sup>*[*ϕ*(*x*)], so *I* and *a* may run over different sets. For the free model, the left-hand side of Equation (A1) is supposed to be linear in the fields. The interactions are associated with the deformations of system Equation (A1) by non-linear terms. The equations of motion for the theory with coupling are polynomial in the fields

T*a* = <sup>T</sup>(0)*a* + <sup>T</sup>(1)*a* + <sup>T</sup>(2)*a* + ... = 0 . (A2)

where <sup>T</sup>(0), T(1) and T(2) are linear, quadratic and cubic in the dynamical variables. Throughout the section, the system Equation (A1) (or, equivalently, Equation (A2)) is supposed to be involutive. The concept of involution implies that Equation (A1) has no differential consequences of lower order (hidden integrability conditions). The ECS Equation (16) is involutive.

The defining relations for gauge symmetries and gauge identifies in the system of partial derivative Equation (A1) read

$$
\delta\_{\mathfrak{C}} \mathfrak{q}^{l} = \mathcal{R}^{l} \mathfrak{e}^{\mathfrak{a}}{}\_{\prime} \qquad \delta\_{\mathfrak{C}} \mathbb{T}\_{\mathfrak{d}} \vert\_{\mathbb{T}=0} = 0 \; ; \tag{A3}
$$

$$L^a{}\_A \mathbb{T}\_a \equiv 0 \,. \tag{A4}$$

where *<sup>R</sup>Iα*, *LaA* are certain differential operators. For non-Lagrangian equations, the gauge symmetries and gauge identities are not related to each other, so the multi-indices *A*, *α* are different. In the perturbative setting, the gauge symmetry and gauge identity generators are supposed to be polynomial in fields

$$R^I{}\_{a} = R^{(0)I}{}\_{a} + R^{(1)I}{}\_{a} + R^{(2)I}{}\_{a} + \dots, \qquad L^{a}{}\_{A} = L^{(0)a}{}\_{A} + L^{(1)a}{}\_{A} + L^{(2)a}{}\_{A} + \dots \tag{A5}$$

where *<sup>R</sup>*(0), *L*(0, *<sup>R</sup>*(1), *L*(1) and *<sup>R</sup>*(2), *L*(2) are field-independent, linear and quadratic in the dynamical variables. The interaction is consistent if all the gauge symmetries and gauge identities of the free model are preserved by the deformation of equations of motion. Equation (8) of paper [54] provides a simple formula for the computation of the number of physical degrees of freedom based on the orders of derivatives involved in the equations

of models, gauge symmetries and gauge identities. The free and non-linear theory must have the same number of physical degrees of freedom.

Relations in Equations (A3) and (A4) imply the following consistency conditions for <sup>T</sup>(*k*), *<sup>R</sup>*(*k*), *L*(*k*) and *k* = 0, 1, 2, . . .:

$$\mathcal{R}^{(0)I}{}\_{a}\partial I \mathbb{T}^{(0)}{}\_{a} = 0 ; \tag{A6}$$

$$\mathcal{R}^{(0)I}{}\_{a}\partial\_{I}\mathbb{T}^{(1)}{}\_{a} + \mathcal{R}^{(1)I}{}\_{a}\partial\_{I}\mathbb{T}^{(0)}{}\_{a} = 0;\tag{A7}$$

$$R^{(0)I}{}\_a \partial\_I \mathbb{T}^{(2)}{}\_a + R^{(1)I}{}\_a \partial\_I \mathbb{T}^{(1)}{}\_a + R^{(2)I}{}\_a \partial\_I \mathbb{T}^{(0)}{}\_a = 0 \tag{A8}$$

(where the symbol *∂I* denotes a variational derivative with respect to the field *ϕ<sup>I</sup>*);

$$L^{(0)a}{}\_A \mathbb{T}^{(0)}{}\_a = 0 \; ; \tag{A9}$$

$$\mathbb{E}^{(0)a}{}\_{A}\mathbb{T}^{(1)}{}\_{a} + L^{(1)a}{}\_{A}\mathbb{T}^{(0)}{}\_{a} = 0 ; \tag{A10}$$

$$L^{(0)a}{}\_A \mathbb{T}^{(2)}{}\_a + L^{(1)a}{}\_A \mathbb{T}^{(1)}{}\_a + L^{(2)a}{}\_A \mathbb{T}^{(0)}{}\_a = 0 \,. \tag{A11}$$

Equations (A6) and (A9) determine the gauge symmetry and gauge identity generators *<sup>R</sup>*(0), *L*(0) of the free theory. These quantities are usually given from the outset. Relations in Equations (A7) and (A10) determine the first-order corrections to the equations of motion *<sup>T</sup>*(1), gauge symmetry generators *R*(1) and gauge identity generators *L*(1). Relations of Equations (A7) and (A10) determine the second-order corrections to the equations of motion *<sup>T</sup>*(2), gauge symmetry generators *R*(2) and gauge identity generators *L*(2). The procedure of interaction construction can be extended to the third and higher orders. Once the most general covariant ansatz is applied for *<sup>T</sup>*(*k*), *k* = 1, 2, ... the procedure of Equations (A6)–(A11), ... allows a complete classification of consistent interactions in a given field theory. An important subtlety of this procedure is that some lower-order couplings can be inconsistent at the higher orders of perturbation theory. The first critical step is the extension of the first-order (quadratic) interaction vertex to the second order of perturbation theory.

Equation (3) determines the left-hand side of the free ECS equations T(0):

$$
\mathbb{T}^{(0)a}{}\_{\mu} = a\_1 \mathbb{F}^a{}\_{\mu} + a\_2 \mathbb{G}^a{}\_{\mu} + a\_3 \mathbb{K}^a{}\_{\mu} = 0 \,. \tag{A12}
$$

The gauge symmetries and gauge identities are defined by the gradient and divergence operators:

$$R^{(0)}\!\_{\mu} = \partial\_{\mu} \!\_{\prime} \qquad L^{(0)\mu} = \partial^{\mu} \!\_{\cdot} \tag{A13}$$

The free gauge identity of Equation (A6) and free gauge transformation read

$$
\partial\_{\mu} \mathbb{T}^{(0) \mu} \equiv 0 \,, \qquad \delta\_{\mathfrak{c}} A^{a}{}\_{\mu} = \partial\_{\mu} \mathbb{J}^{a}{}\_{\nu} \tag{A14}
$$

where *ζ* values are gauge transformation parameters. We consider the Poincare-covariant interactions that are expressed in terms of gauge covariants <sup>F</sup>*μ*, G*<sup>μ</sup>*, <sup>K</sup>*μ* Equation (14) with no higher-derivative terms being included in the coupling. In this case, the equations of motion are automatically preserved by the Yang–Mills gauge symmetry (Equations (A6)–(A8) are satisfied). Consistent interaction vertices of first and second orders are selected by the conditions in Equations (A10)–(A11). We elaborate on this problem below.

We assume that the equations of motion are polynomial in gauge covariants <sup>F</sup>*μ*, G*<sup>μ</sup>*, <sup>K</sup>*μ*. The linear term is given by the covariantization of the free Equation (3). The most general covariant first-order interaction vertex without higher-derivatives reads

$$\begin{aligned} \mathbb{T}^{(0)}\!\_{\mu} + \mathbb{T}^{(1)}\!\_{\mu} &= \\ \mathbb{T}^{(0)}\!\_{\mu} + a\_2 \mathcal{G}\_{\mu} + a\_3 \mathcal{K}\_{\mu} + \epsilon\_{\mu\nu\rho} \Big( \frac{1}{2} k\_1 [\mathcal{F}^{\nu}, \mathcal{F}^{\rho}] + k\_2 [\mathcal{F}^{\nu}, \mathcal{G}^{\rho}] + \frac{1}{2} k\_3 [\mathcal{G}^{\nu}, \mathcal{G}^{\rho}] \Big) \Big) , \end{aligned} \tag{A15}$$

where *kl*, *l* = 1, 2, 3 are constants. The covariant divergence of equations of motion reads

$$\begin{split} D\_{\mu} \mathbb{T}^{\mu} &= -\frac{1}{a\_{3}} [k\_{2} \mathcal{F}\_{\mu} + k\_{3} \mathcal{G}\_{\mu}, \mathbb{T}^{\mu}] + \\ + \frac{1}{a\_{3}} (a\_{3} \,^{2} - a\_{3} k\_{1} + a\_{2} k\_{2} - a\_{1} k\_{3}) [\mathcal{F}\_{\mu}, \mathcal{G}^{\mu}] + (k\_{2}^{2} - k\_{3} k\_{1}) \varepsilon\_{\mu\nu\rho} [\mathcal{F}^{\mu}, [\mathcal{F}^{\upsilon}, \mathcal{G}^{\rho}]] \,. \end{split} \tag{A16}$$

The gauge identity is satisfied in the first-order approximation in Equation (A10) if the coefficients *kl*, *l* = 1, 2, 3 satisfy the relation

$$
\alpha\_3 \, ^2 - \mathfrak{a}\_3 k\_1 + \mathfrak{a}\_2 k\_2 - \mathfrak{a}\_1 k\_3 = 0 \, . \tag{A17}
$$

The general solution to this equations reads

$$k\_1 = -\frac{\beta\_1 \,^2 a\_3}{\mathbb{C}(\beta; a)} + a\_1 \beta\_3, \qquad k\_2 = -\frac{\beta\_2 \beta\_1 a\_3}{\mathbb{C}(\beta; a)} \qquad k\_3 = -\frac{\beta\_2 \,^2 a\_3}{\mathbb{C}(\beta; a)} + a\_3 \beta\_1. \tag{A18}$$

where *β*1, *β*2, *β*3 are coupling parameters. The parameters *β*1, *β*2 determine the coupling vertex of Equation (16) (two constants determine a single coupling because the ratio *β*1/*β*2 is relevant). The constant *β*3 is responsible for another interaction vertex, which is consistent at the first order of perturbation theory. The interaction vertex of Equation (16) is self-consistent, with no higher-order corrections required for the equations of motion. The other coupling needs cubic corrections to the equations of motion in the fields. To prove the uniqueness of the interaction of Equation (16), we should demonstrate that the ansatz of Equations (A15) and (A18) is inconsistent at the second order of perturbation theory for *β*3 = 0.

The most general second-order covariant interaction vertex reads

$$\begin{aligned} \text{Tr}^{(2)}{}\_{\mu} &= l\_1 [\mathcal{F}\_{\nu\nu} [\mathcal{F}\_{\mu\nu} \mathcal{F}^{\nu}]] + l\_2 [\mathcal{F}\_{\nu\nu} [\mathcal{G}\_{\mu\nu} \mathcal{F}^{\nu}]] + l\_3 [\mathcal{F}\_{\nu\nu} [\mathcal{F}\_{\mu\nu} \mathcal{G}^{\nu}]] + \\ &+ l\_4 [\mathcal{G}\_{\nu\nu} [\mathcal{F}\_{\mu\nu} \mathcal{G}^{\nu}]] + l\_5 [\mathcal{G}\_{\nu\nu} [\mathcal{G}\_{\mu\nu} \mathcal{F}^{\nu}]] + l\_6 [\mathcal{G}\_{\nu\nu} [\mathcal{G}\_{\mu\nu} \mathcal{G}^{\nu}]] , \end{aligned} \tag{A19}$$

where *lp*, *p* = 1, 6 are constants. The covariant divergence of equations of motion reads (only cubic terms are written out)

*<sup>D</sup>μ*T*<sup>μ</sup>* = 1*α*3 *μνρ*[S*μν*,T*<sup>ρ</sup>*] + 12*<sup>C</sup>*1(*k*; *<sup>l</sup>*)*εμνρ*[F*μ*, [F*<sup>ν</sup>*, G*ρ*]] + 12*<sup>C</sup>*2(*k*; *<sup>l</sup>*)*εμνρ*[G*<sup>μ</sup>*, [G*<sup>ν</sup>*, F*ρ*]]+ + *l*1 2 [F*μ*, [F*<sup>ν</sup>*, *Dμ*F*<sup>ν</sup>* + *D<sup>ν</sup>*F*<sup>μ</sup>*]] + *l*3 − *l*2 2 [G*<sup>μ</sup>*, [F*<sup>ν</sup>*, *Dμ*F*<sup>ν</sup>* + *D<sup>ν</sup>*F*<sup>μ</sup>*]] + 2*l*2 − *l*3 2 [F*μ*, [G*<sup>ν</sup>*, *Dμ*F*<sup>ν</sup>* + *D<sup>ν</sup>*F*<sup>μ</sup>*]]+ + *l*5 2 [G*<sup>μ</sup>*, [G*<sup>ν</sup>*, *Dμ*F*<sup>ν</sup>* + *D<sup>ν</sup>*F*<sup>μ</sup>*]] + *l*32 [F*μ*, [F*<sup>ν</sup>*, *D<sup>μ</sup>*G*<sup>ν</sup>* + *D<sup>ν</sup>*G*<sup>μ</sup>*]] + *l*5 − *l*4 2 [F*μ*, [G*<sup>ν</sup>*, *D<sup>μ</sup>*G*<sup>ν</sup>*+ +*D<sup>ν</sup>*G*<sup>μ</sup>*]] + 2*l*4 − *l*5 2 [G*<sup>μ</sup>*, [F*<sup>ν</sup>*, *D<sup>μ</sup>*G*<sup>ν</sup>* + *D<sup>ν</sup>*G*<sup>μ</sup>*]] + *l*62 [G*<sup>μ</sup>*, [G*<sup>ν</sup>*, *D<sup>μ</sup>*G*<sup>ν</sup>* + *D<sup>ν</sup>*G*<sup>μ</sup>*]] . (A20)

Here, the following notation is used:

$$\mathbb{C}\_{1}(k;l) = k\_{2}^{2} - k\_{1}k\_{3} - 3l\_{1} - \frac{a\_{1}}{a\_{3}}l\_{4} - \frac{a\_{1}}{a\_{3}}l\_{5} - \frac{a\_{1}}{a\_{3}}l\_{6}, \qquad \mathbb{C}\_{2}(k;l) = 3l\_{2} + l\_{3} - \frac{a\_{2}}{a\_{3}}l\_{5} - \frac{a\_{1}}{a\_{3}}l\_{6}; \tag{A21}$$

$$\mathcal{S}^{\mu\nu} = l\_3[\mathcal{F}^{\mu}, [\mathcal{F}^{\nu}, \cdot]] + l\_4[\mathcal{G}^{\mu}, [\mathcal{F}^{\nu}, \cdot]] + l\_6[\mathcal{G}^{\mu}, [\mathcal{G}^{\nu}, \cdot]] - l\_4[[\mathcal{F}^{\mu}, \mathcal{F}^{\nu}], \cdot] - 1$$

<sup>−</sup>*l*5[[F*μ*, G*<sup>ν</sup>*], ·] − *l*6[[G*<sup>μ</sup>*, G*<sup>ν</sup>*], ·] .

The interaction is consistent at the second order of perturbation theory if the right-hand side of this expression causes the modulo free Equation (3) to vanish. The critical observation is that the expressions of the form [<sup>X</sup>*μ* , [Y*ν* , *Dμ*Z*ν* + *<sup>D</sup><sup>ν</sup>*Z*<sup>μ</sup>*]], where Y*<sup>μ</sup>*, <sup>Z</sup>*μ*, Y*μ* = <sup>F</sup>*μ* or G*μ* represent on-shell independent combinations of fields and their derivatives. Once they are made to vanish, we conclude

$$k\_2 \,^2 - k\_3 k\_1 = 0 \, , \qquad l\_p = 0 \, , \qquad p = \overline{1, 6} \, . \tag{A23}$$

(A22)

The general solution to these equations has the form (A18) with *β*3 = 0. Taking account of this fact, the interaction (16) is unique in the class of covariant couplings without higher derivatives.
