Extended Nappi-Witten Geometry for the Fractional Quantum Hall Effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Extended Nappi-Witten Geometry for the Fractional Quantum Hall Effect
Patricio Salgado-Rebolledo and Giandomenico Palumbo Universit´e Libre de Bruxelles and International Solvay Institutes,ULB-Campus Plaine CP231, B-1050 Brussels, Belgium School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland (Dated: February 9, 2021)Motivated by the recent progresses in the formulation of geometric theories for the fractional quan-tum Hall states, we propose a novel non-relativistic geometric model for the Laughlin states based onan extension of the Nappi-Witten geometry. We show that the U(1) gauge sector responsible for thefractional Hall conductance, the gravitational Chern-Simons action and Wen-Zee term associated tothe Hall viscosity can be derived from a single Chern-Simons theory with a gauge connection thattakes values in the extended Nappi-Witten algebra. We then provide a new derivation of the chiralboson associated to the gapless edge states from the Wess-Zumino-Witten model that is induced bythe Chern-Simons theory on the boundary.
INTRODUCTION
The Fractional Quantum Hall Effect (FQHE) is recog-nized as one of the most important physical phenomenain condensed matter physics [1–5]. Being its microscopicorigin a central research topic in topological phases ofmatter, it has given rise to an enormous amount of devel-opments in quantum mesoscopic physics. In the low en-ergy regime, the fractional quantum Hall states (FQHs)can be described by Abelian and non-Abelian Chern-Simons theories, while the corresponding chiral edgestates by rational conformal field theories (CFTs) [5–9].In the case of Laughlin states, the topological field theorydepends on an emergent U(1) gauge field and the electro-magnetic field, while the edge states are described by chi-ral bosons. Recently, there have been an intense researchon the geometric aspects of the FQHE. On one hand,the incompressibility of the FQHs is due to the presenceof the strong external magnetic field and many-body in-teractions, which is encoded in the Girvin-Macdonald-Platzman (GMP) mode [8, 10, 11]. This mode can benaturally understood as a propagating non-relativisticspin-2 boson related to an emergent quantum geometry[12–14]. On the other hand, the background (ambient)geometry plays a central role in the Hall viscosity, whichis a linear response effect of the Hall fluid in the bulk[15–19]. For all these reasons, several geometric mod-els for the FQHE have been recently proposed [20–37].These 2+1-dimensional effective field theories are basedon non-relativistic geometry. In other words, the modelsare spatially covariant, and their formulation rely one theNewton-Cartan geometry [33]. This non-relativistic the-ory is a geometric reformulation of this Newton’s gravity,that imitates the geometric formulation of general rela-tivity. It has been shown that Newton-Cartan theory isbased on the gauging of the Bargmann algebra (namely,centrally extended Galilei algebra) [38–40]. Importantly,there have been many developments and generalizationsof this theory by replacing the Bargmann algebra withthe Newton-Hooke algebra, the Maxwell algebra, etc. Be- hind these extensions of the Newton-Cartan theory, thereappears the Nappi-Witten algebra [41]. This special alge-bra plays also an important role in certain Wess-Zumino-Witten (WZW) models and pp-wave spacetime [42], andstring-inspired 1+1-dimensional gravity [43]. Thus, gen-eralized non-relativistic geometries can provide a novelscenario where to characterize the geometric features ofFQHs. Notice, this research line follows in spirit the re-cent studies of the geometric aspects of topological insu-lators and topological superconductors where relativisticnon-Riemannian geometries have been employed [44–48].In this work, we will present a novel geometric modelfor the Laughlin states given by a Chern-Simons (CS)theory with the gauge connection that takes values inan extended Nappi-Witten algebra. We will show thatthis CS action naturally contains not only the Wen-Zee[49] and non-relativistic gravitational CS [20] terms butalso the U(1) topological terms responsible for the frac-tional Hall conductance [5]. From this topological effec-tive field theory we will derive the chiral WZW model onthe boundary of the system. This CFT will allow us todescribe the gapless edge states in terms of a chiral bosonthat takes contribution from both the charge and gravi-tational sectors. Finally, we will show that the extendedNappi-Witten geometry can be naturally embedded ina full non-relativistic space-time AdS-Lorentz geometry.Our work paves the way for the characterization of FQHEfrom the the point of view of generalized non-relativisticgeometries, where both the charge and gravitational sec-tors are encoded into a unified geometric formalism.
EXTENDED NAPPI-WITTEN ALGEBRA
The Nappi-Witten algebra is the central extension ofthe Euclidean algebra in two dimensions [42, 50][ P a , P b ] = − ǫ ab T , [ J, P a ] = ǫ ba P b , (1)where a = { , } , P a stands for the translations in thetwo-dimensional plane, J is the generator of rotations,and T is central. Its Lorentzian version is isomorphic tothe Newton-Hook algebra [51] as well as to the Maxwellalgebra in two dimensions [52]. The Maxwell algebradescribes particle systems in the presence of a constantelectromagnetic field [53, 54]. Thus, the Nappi-Wittenalgebra has been found in the description of the integerQHE [55, 56] because it naturally contains the magnetictranslation algebra in two dimensions. Furthermore ithas been shown that the Nappi-Witten geometry prop-erly describes the momentum-space cigar geometry of acertain kind of two-dimensional topological phases [45].Based in the previous discussion, it seems natural toexpect the Nappi-Witten algebra to be relevant in thedescription of two-dimensional interacting systems wherea constant external electromagnetic field plays a centralrole. The quantum field theory of the Laughlin statesis characterized by an extra field content given by anemergent gauge field a [5]. Thus, as a first attempt todescribe these quantum states, we add an Abelian gener-ator Y to Eq.(1) and consider the direct product Nappi-Witten × u (1). The central extension of the translations T will be associated to the external electromagnetic field A and the Abelian generator Y , to the emergent gaugefield a . The Nappi-Witten × u (1) algebra admits a non-degenerate invariant bilinear form given by h JJ i = µ , h P a P b i = µ δ ab , h JT i = − µ , h Y J i = ρ , h Y Y i = ρ . (2)The relevant gauge fields of the system are to encoded ina connection one-form A = A Aµ T A dx µ where µ = 0 , , T A = { J, P a , Z, Y } collectivelydenotes the generators of the Nappi-Witten × u (1) alge-bra. Explicitly, the gauge connection has the form A = ωJ + 1 ℓ e a P a + AT + aY. (3)where ω is the connection of rotations (spin connection), e a the spatial dreibein and we have introduced a parame-ter ℓ with dimensions of length in such a way that the Liealgebra generators are dimensionless. We now introducethe following shift in the connection one-form in Eq.(3) ω → ω + βa, (4)which can be translated into the definition of a new Liealgebra generator Z = Y + βJ (5)This leads to the following commutation relations[ P a , P b ] = − ǫ ab T , [ J, P a ] = ǫ ba P b = 1 β [ Z, P a ] , (6)which we will refer to as extended Nappi-Witten algebra .This algebra admits a non-degenerate invariant bilinear form given by h JJ i = µ , h P a P b i = µ δ ab , h JT i = − µ . h ZJ i = µ , h ZT i = − βµ , h ZZ i = µ , (7)where µ i are real constants. One can show that imple-menting the shift in Eq.(4) in the connection in Eq.(3)and using the Nappi-Witten × u (1) algebra leads to thegauge connection for the extended Nappi-Witten algebra A = ωJ + 1 ℓ e a P a + AT + aZ. (8)The corresponding curvature F = (1 / F Aµν T A dx µ ∧ dx ν has the form F = dωJ + 1 ℓ R a P a + daZ + RT, (9)where we have defined the one-forms R a = de a + ǫ ab e b ( ω + βa ) ,R = dA + 12 ǫ ab e a e b . (10)A gauge transformation δ ξ A = dξ + [ A , ξ ] leads to thefollowing transformation laws for the gauge fields δ ξ ω = dξ J , δ ξ a = dξ Z , δ ξ A = dξ T + 1 ℓ ǫ ab ξ P a e b ,δ ξ e a = dξ P a + ǫ ab (cid:16) ξ J e b − ωξ P b + β (cid:0) ξ Z e b − aξ P b (cid:1) (cid:17) . (11)Diffeomorphisms with parameter χ µ act on the connec-tion as L χ A µ = F µν χ ν + δ ξ a A a A µ (12)and thus the are on-shell equivalent to gauge transforma-tions with parameter ξ a A aµ . TOPOLOGICAL HALL RESPONSE FROM THECHERN-SIMONS THEORY
Because we are willing to derive the topological re-sponse of the fractional Hall states, in this section weconsider the Chern-Simons action for the connection inEq.(8), given by S = − k π ˆ (cid:28) A ∧ d A + 23 A ∧ A ∧ A (cid:29) , (13)with k an integer level. Importantly, this effective actioncan be derived from a microscopic theory by integrat-ing out the fermionic fields associated to gapped spinfulmatter minimally coupled to the extended Nappi-Wittengeometry (notice, spinful matter is compatible with atorsionful geometric background). By employing Eq.(7),the above action can be rewritten as follows (from nowon the wedge product between forms will be omitted forsimplicity) S = − k π ˆ (cid:20) µ ωdω + µ ℓ e a R a − µ Adω − µ β Ada + 2 µ adω + µ ada (cid:21) . (14)Variation with respect to the spatial dreibein e a leadsto the field equation R a = 0, which in turn yields thefollowing equation for torsion T a ≡ de a + ǫ ab e b ω = − βǫ ab e b a. (15)This equation allows us to formally express the dreibeinin terms of the spin connection and the field a . Now, byvarying the action with respect to the field a , we obtain a = βµ µ A − µ µ ω. (16)By replacing this expression in Eq.(14) and by taking thefollowing identifications of the parameters k = µ = 1 , µ = 2¯ sν, µ = c ,µ = ¯ s √ ν, β = 12¯ s √ ν , (17)we finally obtain S = ˆ (cid:20) (cid:18) ν ¯ s π − c π (cid:19) ωdω + ν π AdA + ν ¯ s π Adω (cid:21) , (18)which is compatible with the effective geometric actionfor the FQHE analyzed in Refs.[20–28] although it is im-portant to bear in mind that our spin connection is tor-sionful. Here, c is the chiral central charge, ν is the frac-tional filling and ¯ s is the average orbital spin. For theLaughlin states we have: c = 1, ν = 1 / (2 p + 1) and¯ s = (2 p + 1) /
2, with p an integer. The first and thirdterms in the action are usually referred as the gravita-tional Chern-Simons [20] and Wen-Zee term [49], respec-tively. Here, the coefficient in front of the U(1) CS termis associated to the Hall conductance, while the coeffi-cient in front of the Wen-Zee term is related to the Hallviscosity [16–19, 57]. These are the two main topologicalresponses in the Abelian FQH states on manifolds withgenus 0 and 1 [31]. The electron Hall density and currentderived from the above action are respectively given by ρ = ν π B + ν ¯ s π R,J i = ν π ǫ ij E j + ν ¯ s π ǫ ij E j , (19)where B and E j are the magnetic and electric fields, re-spectively while R = (2 / p | e | ) ǫ ij ∂ i ω j and E j = ∂ j ω − ∂ ω j are the Abelian Ricci scalar and the gravi-electric field, respectively. The Hall viscosity comes from the re-sponse of the system to shear or strain [15]. Becauseour spin connection is torsionful, we follow here the ap-proach developed in Refs.[22, 58] that allows us to definethe symmetric Cauchy stress-mass tensor from the spincurrent s µ . This current is given by the variation of ourtopological action with respect to ωs µ = ν ¯ s π ǫ µνλ ∂ ν A λ + (cid:18) ν ¯ s π − c π (cid:19) ǫ µνλ ∂ ν ω λ , (20)where the first term can be rewritten as follows ν ¯ s π ǫ µνλ ∂ ν A λ = η H u µ , (21)with u µ = (1 , (1 /B ) ǫ ij E j ) the covariant drift velocityand η H = ν ¯ sB/ π the Hall viscosity [58]. Importantly,on a manifold with boundary both the Abelian and thegravitational CS terms contribute to the boundary gap-less modes. These edge states are usually identified bychiral bosons. In the next section, we will derive the ac-tion for the chiral boson for the Laughlin states startingfrom the WZW action induced on the boundary by ourCS theory in Eq.(13). CHIRAL BOSON FROM THE WZW MODEL
It is well known that Chern-Simons theories are gaugeinvariant only on compact manifolds. For this reason, inthis section we consider a manifold with boundary suchthat the CS action gives rise to a chiral WZW model onthe boundary [59]. This CFT represents the natural ef-fective theory for the gapless edge excitations of the FQHstates. In the specific case of Laughlin states, the edgestates are given by a chiral boson [5–9] that is usually de-scribed by the Floreanini-Jackiw action in flat space [60].In our framework, we now derive the chiral boson froma WZW model. In fact, the CS action can be reducedto a boundary theory by solving the corresponding fieldequations for the spatial connection F ij = 0 , (22)and then plugging the solution back in the action. Alocal solution is given by A i = g − ∂ i g. (23)Imposing the condition A t + v A φ = 0 (24)at the boundary and taking care of the surface integralsin (13), the action reduces to the following chiral WZWmodel [61] S WZW = k π ˆ π dφ ˆ dt (cid:10) g − ˙ gg − g ′ + v ( g − g ′ ) (cid:11) + k π ˆ M D(cid:0) g − d g (cid:1) E , (25)where ˙ g = ∂ t g , g ′ = ∂ φ g and we have chosen for simplic-ity a 2+1-dimensional manifold M = D × R , where D is topologically equivalent to a disk and φ is the angularvariable associated to the unitary circle that representsthe spatial boundary S ≡ ∂D . Because we are mainlyinterested in the kinematics of the chiral boson, we lookfor the left-invariant Maurer-Cartan form and neglect theexplicit derivation of the topological winding number ofthe WZW action. One way to find the left-invariantMaurer-Cartan form [59] is to solve the correspondingMaurer-Cartan equation d Ω + Ω ∧ Ω = 0 , (26)whereΩ = g − dg = Ω J J + Ω Z Z + Ω aP P a + Ω T T. (27)By employing the commutations of the extended Nappi-Witten algebra, the Maurer-Cartan equations can beshown to be equivalent to the following systems of equa-tions d Ω J = 0 , d Ω Z = 0 ,d Ω aP + ǫ ab Ω bP (Ω J + β Ω Z ) = 0 ,d Ω T − ǫ ab Ω aP Ω bP = 0 . (28)The first two equations implyΩ J = dθ , Ω Z = dϕ, (29)while for the others we haveΩ aP = dσ a − ǫ ab σ b ( dθ + βdϕ ) , Ω T = dϑ + 12 ǫ ab σ a dσ b + 12 σ a σ a ( dθ + βdϕ ) , (30)with θ , ϕ and ϑ three real scalar fields and σ a a vectorfield. By defining the following coordinates x ± = 12 (cid:18) t + 1 v φ (cid:19) , ∂ ± = ∂ t + v∂ φ , (31)with v the Fermi velocity, we find the explicit form of theWZW action S W ZW = ˆ dtdφ (cid:20) µ ∂ + θθ ′ + 2 µ ∂ + θϕ ′ + µ ∂ + ϕϕ ′ + µ (cid:16) ∂ + σ a σ ′ a − (2 ∂ + ϑ + ǫ ab σ a ∂ + σ b )( θ ′ + βϕ ′ ) (cid:17)(cid:21) . (32)We see that ϑ is a Lagrange multiplier which gives riseto the following constraint ∂ + θ ′ + β∂ + ϕ ′ = 0 , (33)which implies θ = − βϕ + a ( t ) + b ( x − ) , (34) with a ( t ) and b ( x − ) arbitrary functions of their argu-ments. On the other hand the field equation for σ a isgiven by ∂ + σ ′ a + ǫ ab ∂ + σ b ( θ ′ + βϕ ′ ) = 0 . (35)By replacing these expressions back in the action lead toan action for a single chiral boson, given by S W ZW = c π (cid:18) ν ¯ s (cid:19) ˆ dφ dt ( ˙ ϕϕ ′ + vϕ ′ ϕ ′ ) , (36)which agrees with the boundary theory previously de-rived in literature [26, 27] for the Laughlin states where1 / ( ν ¯ s ) = 4 ν and c = 1. Importantly, the above chiralboson action can be naturally employed to describe theedge states of spin- j Laughlin states [30–32] where¯ s = 12 ν − j. (37) ADS-LORENTZ ALGEBRA AND EXTENDEDNEWTON-CARTAN GEOMETRY
The extended Nappi-Witten algebra can be naturallyembedded in a full non-relativistic space-time algebra.This allows to interpret the Nappi-Witten geometry asa sub-manifold of a particular extended Newton-Cartangeometry [62] . In 2+1 dimensions, the Galilei algebraadmits two central extensions. This leads to the extendedBargmann symmetry [63–66][
J, G a ] = ǫ ba G b , [ J, P a ] = [ H, G a ] = ǫ ba P b , [ G a , G b ] = − ǫ ab S , [ G a , P b ] = − ǫ ab M . (38)One can consider the following an extension of theBargmann algebra by including a new set of generators { Z, Z a , T } and the commutation relations[ J, Z a ] = [ H, P a ] = [ Z, G a ] = ǫ ba Z b , [ P a , P b ] = [ G a , Z b ] = − ǫ ab T . (39)This is the Maxwellian Exotic Bargmann algebra [67]and defines a central extension of the “electric” non-relativistic Maxwell algebra [68, 69]. One can furtherextend the Bargmann algebra by introducing a param-eter ℓ with dimension of length and the commutationrelations [41][ Z, Z a ] = ǫ ba Z b , [ H, Z a ] = [ Z, P a ] = ǫ ba P b , [ P a , Z b ] = − ǫ ab M , [ Z a , Z b ] = − ǫ ab T . (40)This algebra can be obtained as a non-relativistic limit ofthe AdS-Lorentz algebra in 2+1 dimensions [70], whichin turn is a semi-simple extension of the Maxwell algebra[71]. As it happens in the relativistic case, the Maxwelland the AdS-Lorentz symmetries are related by an In¨on¨u-Wigner contraction. Indeed, one can use a length param-eter ℓ to reinsert dimensions in the Lie algebra generatorsas H → ℓH, P a → ℓP a , M → ℓM,Z → ℓ Z, Z a → ℓ Z a , T → ℓ T, (41)Thus, it is clear that in the limit ℓ → ∞ the non-relativistic AdS-Lorentz algebra reduces to the electricMaxwell symmetry. One can see that the extendedNappi-Witten algebra (6) is the sub-algebra of Eqs.(38)-(40) spanned by the generators { J, P a , Z, T } , where onehas to use the redefinition Z → βZ .At the relativistic level, the Maxwell algebra has beenpreviously used to construct a geometric model for thegapped boundary of three-dimensional topological insu-lators [44]. However, in order to go beyond the integerQHE, a Chern-Simons term for the emergent U(1) gaugefield is necessary. This term can be obtained by eitherextending the Maxwell to include its dual space, or by go-ing to the AdS-Lorentz extension [48]. Since the FQHEis intrinsically non-relativistic, it is natural to expectthat the non-relativistic AdS-Lorentz is a good candidateto construct an effective geometric description. Here,we have shown that this is indeed the case. A Chern-Simons action (13) invariant under the non-relativisticAdS-Lorentz algebra is constructed by means of the con-nection one-form A = ωJ + τ H + aZ + ω a G a + e a P a + k a Z a + mM + sS + AT. (42)We can chose an absolute time and fix the reference frameby imposing the conditions τ µ = δ µ , ω aµ = 0 (43)Furthermore, we consider the particular case where k aµ = 0 , m µ = 0 = s µ . (44)One can show that in this case the Chern-Simons actioninvariant under the non-relativistic AdS-Lorentz algebrareduces to the effective model in Eq.(14). It is importantto note, however, that non-degenerate invariant bilinearform for the non-relativistic AdS-Lorentz symmetry thatgeneralized (7) is given by h G a G b i = λ δ ab , h G a P b i = h P a Z b i = λ δ ab h P a P b i = h G a Z b i = h Z a Z b i = µ δ ab h JM i = h HS i = h HT i = h ZM i = − λ , h JT i = h ZS i = h HM i = h ZT i = − µ h JS i = − λ , h JJ i = µ , h JH i = λ , h JZ i = µ , h HH i = λ , h HZ i = λ , h ZZ i = µ . (45) This expression is more general tan the invariant bilin-ear form derived in [41], which is the one that comesfrom the relativistic AdS-Lorentz symmetry upon con-traction. This indicates that the model here presentedis purely non-relativistic and does not have a relativisticcounterpart.It is known that the gauging of the Bargmann alge-bra leads to Newton-Cartan geometry [38–40]. Similarly,since the AdS-Lorentz algebra is an extension of theBargmann symmetry, its gauging leads to an extendedNewton-Cartan geometry. This is in complete analogywith the extended relativistic geometry that follows fromMaxwell algebra [72] (see also [73]), which is an exten-sion of the Poincar´e symmetry that underlies Minkowskispace. CONCLUSIONS AND OUTLOOK
In this paper, we have proposed a novel geometricmodel for the Laughlin states. We have shown thatthe U(1) and gravitational CS terms together with theWen-Zee term can be derived from a single CS actionwhere the gauge connection takes values in the extendedNappi-Witten algebra. Besides the topological responsein the bulk given by the Hall conductance and Hall vis-cosity, we have provided a novel way to derive the ef-fective field theory for the chiral boson that lives on theedge of the system. We have shown that the extendedNappi-Witten symmetry can be naturally embedded inthe non-relativistic AdS-Lorentz algebra, which is a par-ticular extension of the Bargmann algebra in 2+1 dimen-sions. In this way, the geometry behind our model canbe thought as part of a generalized Newton-Cartan ge-ometry. Several directions will be considered in futurework. In particular, we will extend our formalism by in-cluding multi charged emergent gauge fields to describeFQH states beyond the Laughlin states [5, 74] and a sec-ond emergent metric to properly encode the GMP modeand nematic states [35, 36]. Finally, we will define anovel non-relativistic higher-spin Nappi-Witten algebrato properly describe the higher-spin modes in the FQHE[75, 76]. Our work paves the way for the descriptionof the geometric and topological features of interactingtopological fluids through generalized non-relativistic ge-ometries where both the charge and gravitational sectorsare dealt in a unified framework.
Acknowledgments:
The authors are pleased to ac-knowledge discussions with Barry Bradlyn. This workwas partially supported by FNRS-Belgium (conventionsFRFC PDRT.1025.14 and IISN 4.4503.15), as well as byfunds from the Solvay Family. [1] R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[2] F. D. M. Haldane, Phys. Rev. Lett. , 605 (1983).[3] J. K. Jain, Phys. Rev. Lett. , 199 (1989).[4] A. Lopez and E. Fradkin, Phys. Rev. B , 5246 (1991).[5] X.-G. Wen, International Journal of Modern Physics B , 1711 (1992).[6] X. G. Wen, Phys. Rev. B , 12838 (1990).[7] M. Stone, Annals of Physics , 38 (1991).[8] S. Iso, D. Karabali, and B. Sakita,Physics Letters B , 143 (1992).[9] A. Cappelli, G. V. Dunne, C. A. Trugenberger, and G. R.Zemba, Nuclear Physics B , 531 (1993).[10] S. M. Girvin, A. H. MacDonald, and P. M. Platzman,Phys. Rev. B , 2481 (1986).[11] A. Cappelli, C. A. Trugenberger, and G. R. Zemba,Nuclear Physics B , 465 (1993).[12] F. D. M. Haldane, Phys. Rev. Lett. , 116801 (2011).[13] B. Yang, Z.-X. Hu, Z. Papi´c, and F. D. M. Haldane,Phys. Rev. Lett. , 256807 (2012).[14] S. Golkar, D. X. Nguyen, and D. T. Son,Journal of High Energy Physics , 21 (2016).[15] J. E. Avron, R. Seiler, and P. G. Zograf,Phys. Rev. Lett. , 697 (1995).[16] N. Read, Phys. Rev. B , 045308 (2009).[17] N. Read and E. H. Rezayi,Phys. Rev. B , 085316 (2011).[18] C. Hoyos and D. T. Son,Phys. Rev. Lett. , 066805 (2012).[19] B. Bradlyn, M. Goldstein, and N. Read,Phys. Rev. B , 245309 (2012).[20] A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, andE. Fradkin, Phys. Rev. Lett. , 016805 (2015).[21] G. Y. Cho, Y. You, and E. Fradkin,Phys. Rev. B , 115139 (2014).[22] B. Bradlyn and N. Read,Phys. Rev. B , 125303 (2015).[23] B. Bradlyn and N. Read,Phys. Rev. B , 165306 (2015).[24] A. G. Abanov and A. Gromov,Phys. Rev. B , 014435 (2014).[25] A. Gromov and A. G. Abanov,Phys. Rev. Lett. , 266802 (2014).[26] A. Gromov, K. Jensen, and A. G. Abanov,Phys. Rev. Lett. , 126802 (2016).[27] S. Moroz, C. Hoyos, and L. Radzihovsky,Phys. Rev. B , 195409 (2015).[28] A. Cappelli and E. Randellini,Journal of High Energy Physics , 105 (2016).[29] M. Geracie, D. T. Son, C. Wu, and S.-F. Wu,Phys. Rev. D , 045030 (2015).[30] T. Can, M. Laskin, and P. Wiegmann,Phys. Rev. Lett. , 046803 (2014).[31] S. Klevtsov and P. Wiegmann,Phys. Rev. Lett. , 086801 (2015).[32] F. Ferrari and S. Klevtsov,Journal of High Energy Physics , 86 (2014).[33] D. T. Son, arXiv:1306.0638.[34] C. Wu and S.-F. Wu,Journal of High Energy Physics , 120 (2015).[35] A. Gromov and D. T. Son,Phys. Rev. X , 041032 (2017). [36] A. Gromov, S. D. Geraedts, and B. Bradlyn,Phys. Rev. Lett. , 146602 (2017).[37] V. Dwivedi and S. Klevtsov,Phys. Rev. B , 205158 (2019).[38] R. Andringa, E. Bergshoeff, S. Panda, andM. de Roo, Class. Quant. Grav. , 105011 (2011),arXiv:1011.1145 [hep-th].[39] R. Banerjee, A. Mitra, and P. Mukherjee,Phys. Lett. B , 369 (2014), arXiv:1404.4491 [gr-qc].[40] E. Bergshoeff, A. Chatzistavrakidis, L. Ro-mano, and J. Rosseel, JHEP , 194 (2017),arXiv:1708.05414 [hep-th].[41] D. M. Pe˜nafiel and P. Salgado-Rebolledo, Phys. Lett. B , 135005 (2019),arXiv:1906.02161 [hep-th].[42] C. R. Nappi and E. Witten,Phys. Rev. Lett. , 3751 (1993).[43] D. Cangemi and R. Jackiw,Phys. Rev. Lett. , 233 (1992).[44] G. Palumbo, Annals of Physics , 15 (2017).[45] G. Palumbo, The European Physical Journal Plus , 23 (2018).[46] G. Palumbo and J. K. Pachos,Annals of Physics , 175 (2016).[47] P. Maraner, J. K. Pachos, and G. Palumbo,Scientific Reports , 17308 (2019).[48] R. Durka and J. Kowalski-Glikman,Physics Letters B , 516 (2019).[49] X. G. Wen and A. Zee, Phys. Rev. Lett. , 953 (1992).[50] J. M. Figueroa-O’Farrill and S. Stanciu,Phys. Lett. B , 40 (1994), arXiv:hep-th/9402035.[51] P. D. Alvarez, J. Gomis, K. Kamimura, andM. S. Plyushchay, Phys. Lett. B , 906 (2008),arXiv:0711.2644 [hep-th].[52] H. Afshar, H. A. Gonz´alez, D. Grumiller, andD. Vassilevich, Phys. Rev. D , 086024 (2020),arXiv:1911.05739 [hep-th].[53] R. Schrader, Fortsch. Phys. , 701 (1972).[54] J. Gomis and A. Kleinschmidt, JHEP , 085 (2017),arXiv:1705.05854 [hep-th].[55] C. Duval and P. A. Horvathy,Phys. Lett. B , 284 (2000), arXiv:hep-th/0002233.[56] P. A. Horvathy, L. Martina, and P. C. Stichel,Phys. Lett. B , 87 (2005), arXiv:hep-th/0412090.[57] O. Golan, C. Hoyos, and S. Moroz,Phys. Rev. B , 104512 (2019).[58] M. Geracie, K. Prabhu, and M. M. Roberts,Journal of High Energy Physics , 89 (2017).[59] R. Dijkgraaf and E. Witten,Communications in Mathematical Physics , 393 (1990).[60] R. Floreanini and R. Jackiw,Phys. Rev. Lett. , 1873 (1987).[61] O. Coussaert, M. Henneaux, and P. vanDriel, Class. Quant. Grav. , 2961 (1995),arXiv:gr-qc/9506019.[62] E. Cartan, Annales Sci. Ecole Norm. Sup. , 325 (1923).[63] J.-M. L´evy-Leblond, in Group theory and its applications (Elsevier, 1971) pp. 221–299.[64] D. R. Grigore, J. Math. Phys. , 460 (1996),arXiv:hep-th/9312048.[65] S. K. Bose, Commun. Math. Phys. , 385 (1995).[66] E. A. Bergshoeff and J. Rosseel,Phys. Rev. Lett. , 251601 (2016),arXiv:1604.08042 [hep-th]. [67] L. Avil´es, E. Frodden, J. Gomis, D. Hi-dalgo, and J. Zanelli, JHEP , 047 (2018),arXiv:1802.08453 [hep-th].[68] N. Gonz´alez, G. Rubio, P. Salgado, andS. Salgado, Phys. Lett. B , 433 (2016),arXiv:1604.06313 [hep-th].[69] J. Gomis, A. Kleinschmidt, and J. Palmkvist,JHEP , 109 (2019), arXiv:1907.00410 [hep-th].[70] P. Concha and E. Rodr´ıguez, JHEP , 085 (2019),arXiv:1906.00086 [hep-th].[71] D. V. Soroka and V. A. Soroka,Adv. High Energy Phys. , 234147 (2009), arXiv:hep-th/0605251.[72] G. W. Gibbons, J. Gomis, and C. N.Pope, Phys. Rev. D , 065002 (2010),arXiv:0910.3220 [hep-th].[73] P. Salgado-Rebolledo, JHEP , 039,arXiv:1905.09421 [hep-th].[74] A. Cappelli and L. Maffi,Journal of Physics A: Mathematical and Theoretical , 365401 (2018).[75] S. Golkar, D. X. Nguyen, M. M. Roberts, and D. T. Son,Phys. Rev. Lett. , 216403 (2016).[76] Z. Liu, A. Gromov, and Z. Papi´c,Phys. Rev. B98