On Fractional Quantum Hall Solitons and Chern-Simons Quiver Gauge Theories
aa r X i v : . [ h e p - t h ] M a r On Fractional Quantum Hall Solitons andChern-Simons Quiver Gauge Theories
Adil Belhaj ∗ Centre of Physics and Mathematics, CPM-CNESTEN, Rabat, MoroccoGroupement National de Physique des Hautes Energies, Rabat, Morocco
October 26, 2018
Abstract
We investigate a class of hierarchical multiple layers of fractional quantum Hall soli-ton (FQHS) systems from Chern-Simons quivers embedded in M-theory on the cotangenton a 2-dimensional complex toric variety V , which is dual to type IIA superstring ona 3-dimensional complex manifold CP × V fibered over a real line R . Based on M-theory/Type IIA duality, FQHS systems can be derived from wrapped D4-branes on2-cycles in CP × V type IIA geometry. In this realization, the magnetic source can beidentified with gauge fields obtained from the decomposition of the R-R 3-form on ageneric combination of 2-cycles. Using type IIA D-brane flux data, we compute the fillingfactors for models relying on CP and the zeroth Hirzebruch surface. Keywords : Quantum Hall Solitons, M-theory, Type IIA string, Chern-Simons Quivers. ∗ [email protected] Introduction
Three dimensional Quantum Hall Systems (QHS) of condensed matter physics can be mod-eled using D-branes of type II superstrings [1, 2, 3]. The first ten dimensional superstringpicture of Quantum Hall Effect (QHE) in (1+2)-dimensions has been given in terms of a stackof K D6-branes, a spherical D2-brane as well as dissolved D0-branes in D2 and a stack of N F-strings stretching between D2 and D6-branes [1]. The F-strings ending on the D2-branehave an interpretation in terms of the fractional quantum Hall particles and they are chargedunder the U ( ) world-volume gauge field associated with the D0-branes. These solitonicobjects behave as magnetic flux quanta disolved in the D2-brane world-volume on which theQHS system reside. This string picture of QHE in (1+2)-dimensions has been extended to thecompactification of type IIA superstring theory on the K3 surface with singularities classifiedby Dynkin diagrams[4, 5, 6].Alternatively, some efforts have been devoted to study fractional quantum Hall solitons(FQHS) in connections with (2+1)-dimensional Chern-Simons (CS) theory constructed byAharony, Bergman, Jafferis and Maldacena. Recall that, ABJM theory is a 3-dimensional N = k × U(N) − k gauge symmetry proposed to be dual to M-theorypropagating on AdS × S / Z k , with an appropriate amount of fluxes, or type IIA superstringon AdS × CP for large k and N with k ≥ N in the weakly interacting regime [7]. Inthe decoupling limit, the corresponding three dimensional conformal field theory (CFT ) isobtained from the physics of the multiple M2-branes placed at the orbifold space C / Z k . Ithas been shown that QHE can be realized on the world-volume action of the M5-brane fillingAdS inside AdS [8]. The model has been derived from d = ( − ) . FQHE systems can be also embedded in ABJM theory by adding fractionaltype IIA D-branes [9].Recently, a possible extension of FQHS in ABJM theory has been given using type IIAdual geometry considered as the blown up of CP by four-cycles which are isomorphic to CP . Based on D6-branes wrapped 4-cycles and interacting with the R-R gauge fields inABJM-like geometry, hierarchical multiple layers of FQHS systems have been given in [10].Exended constructions can be characterized by a vector q i and a real, symmetric and in-vertible symmetric matrix K ij . These parameters classify various fractional quantum Hall(FQH) states. The choice of K ij play important role in the embedding of CS in type II super-strings and M-theory compactified on deformed singular Calabi-Yau manifolds. In particular,the matrix K ij can be identified with the intersection numbers of compact cycles in the inter-nal space. These numbers are, up to some details, the opposite of the Cartan matrices of Liealgebras. This link may lead to a nice correspondence between FQHS models and 2-cyclesinvolved in the deformation of toric singularities. More specifically, to each simple 2-cycle,2e associate a FQHS model.In this work we discuss such CS realizations of QHS in 1+2 dimensions from M-theoryon a real eight dimensional manifold. The manifold is realized explicitly as the cotangentbundle over a 2-dimensional complex toric variety V . Up some details, the obtained modelscan be compraed to Gaiotto-Witten theory describing N = q i and K ij are connected via the filling factor, the embedding of CS in the string theorygives a toric geometric calculation for the filling factor. More precisely, starting from giventwo toric varieties describing quantum Hall systems and using CS gauge modeling a laSusskind, we determine the filling factors. In particular, based on M-theory/Type IIA duality,we give first Chern-Simons type theories describing 3-dimensional FQHS systems using D4-branes wrapping 2-cycles and interacting with the R-R gauge fields living on CP × V typeIIA dual geometry. This class of QHS can be considerd as a possible extension of the worksgiven discusing the QHE in ABJM and its generalization [8, 9, 10]. This allows a geometricinterpretation of filling factors in terms of the intersections between 2-cycles in H ( V , Z ) .The matrix K ij can be identified with the intersection matrix for curves embedded in CP × V . Then, we present explicit examples in order to illustrate the general idea. In particular,we discuss the case of CP and the Hirzebruch surfaces.The organization of this work is as follows. In the next section, we study FQHS systems inABJM-like theory using D6-branes wrraping 4-cycles in CP . In section 3, we derive a class ofFQHS in (1+2)-dimensions using M-theory/Type IIA duality. In particular, we discuss FQHSmodel based on CP , then we extend the analysis to the Hirzebruch surfaces in section 4.The last section is devoted to our conclusion. We start this section by recalling that the quantum Hall states are characterized by the fillingfactor ν describing the ratio between the electronic density and the flux density. When ν is a fractional value, it is called fractional quantum Hall effect (FQHE) for interactingelectron systems. The first proposed series of the fractional quantum states was given byLaughlin and they are characterized by the filling factor ν L = k where k is an even integerfor a boson electron and an odd integer for a fermionic electron [12]. At low energy, thismodel can be described by a 3-dimensional U(1) Chern-Simons theory coupled to an externalelectromagnetic field ˜ A with the following effective action S CS = − k π Z R A ∧ dA + Z R q π ˜ A ∧ dA , (2.1)3here A is the dynamical gauge field, ˜ A is an external electromagnetic field, and where q is the charge of the electron [13, 14]. Extended models are characterized by a vector q i anda real, symmetric and invertible symmetric matrix K ij . These parameters play an importantrole in quiver construction of FQHE embedded in type II superstrings [4, 6] and M-theorycompactified on 8-dimensional manifolds [5]. Following the Susskind approach for abelianfield theory, these models can be described by the following action S ∼ π R R K ij A i ∧ dA j + R R q i ˜ A ∧ dA i . (2.2)The external gauge field ˜ A couples now to each current ⋆ dA i with charge strengths eq i .The K ij matrix and the q i charge vector in this effective field action suggests some physicalconcepts. Following the Wen-Zee model [14], K ij and q i are interpreted as order parametersand classify the various QHS states. Integrating over the all abelian gauge fields A i , one getsthe formulae for the filling factor of the system ν = q i K − ij q j . (2.3)In what follows, we see that such CS quivers describing FQHS can be embedded either inABJM theory with U ( N ) k × U ( N ) − k gauge symmetry or more generally in toric CS quivergauge theories. For simplicity, we first consider the case of the U(1) k CS gauge theory inABJM theory. Indeed, this model can be obtained from a D6-brane wrapping a four-cycleclass [ C ] in H ( CP , Z ) which is one-dimensional. On the gauge theory side of ABJM, thegauge symmetry U ( N ) k × U ( N ) − k becomes U ( N + M ) k × U ( N ) − k . To derive the first part ofthe action (2.1), we take just the U(1) abelian part of U ( M ) k and assume that the remaininggauge factors are spectators. Indeed, on the D6-brane one can write down the following S WZ action S WZ ∼ T Z R F ∧ F ∧ A RR , (2.4)where T is the D6-brane tension and where the gauge field A RR is the R-R 3-form coupledto the D2-brane of type IIA superstring. Integrating by parts and integrating the result overthe 4-cycle C , one gets − k π Z R A ∧ F (2.5)where k = π R C ( dA RR ) is produced now by k D4-flux. To couple the system to an exter-nal gauge field, we need to turn on the RR 5-form A RR which is coupled to the D4-brane.Decomposing this gauge field as follows A RR → ˜ A ∧ ω (2.6)4here ω is a harmonic 4-form dual to the four-cycle C , the WZ action R R A RR ∧ F on aD6-brane gives the second term of the action (2.1), namely, q Z R ˜ A ∧ F (2.7)where q = R C w and ˜ A is an U(1) gauge field which serves as the external gauge field thatcouples to the gauge fields living on the D6-brane world-volume. The above effective actionon the D6-brane world-volume reproduces the following filling factor ν = q k . (2.8)From this equation, it follows that the filling factor depends on the D4-branes and the har-monic 4-forms defined on CP . It turns out that known values could be obtained by takingparticular choices of such parameters. Moreover, these FQHS in AdS /CFT have been ex-tended to models based on IIA dual geometry realized as the blown up of CP by four-cycleswhich are isomorphic to CP . In particular, we proposed a stringy hierarchical descriptionof multi-layer systems in terms of wrapped D6-branes on the blown up four-cycles [10].In what follows, we investigate FQHS system embedded in CS quiver gauge theories aris-ing from M-theory on the cotangent bundle on a two dimensional complex toric variety V .More precisely, we discuss the case of two dimensional complex projective space and thezeroth Hirzebruch surface. Many models have been given to extend ABJM theory. They are conjectured to be gauge fieldduals of
AdS background in type IIA superstring and M-theory compactifications on eightdimensional manifolds. In particular, these kinds of CS quiver theories can be obtained interms of M2-branes placed at hyper toric singularities of eight-dimensional manifolds. Recallthat, the simple model is described by abelian gauge factors U ( ) k × U ( ) k × . . . × U ( ) k n ,where k i denote the Chern-Simons levels for each abelian factor U ( ) . Geometrically, thesemodels can be encoded in a quiver formed by n nodes where each factor U ( ) is associatedwith a node while the matter is represented by the link between nodes [15, 16]. For thesemodels, it has been shown that Chern-Simons levels k i are subject to ∑ i k i =
0. (3.1)5or the ∏ i U ( N i ) k i non abelian gauge group, the Chern-Simons levels k i and the set of theranks of the gauge groups N i are constrained by ∑ i k i N i =
0. (3.2)The CS quivers we consider here are obtained from M-theory on the cotangent bundle overa two dimensional toric variety V . This internal space is built in terms of a bi-dimensionalU(1) r sigma model with eight supercharges ( N =
4) and r + r + ∑ i = Q ai [ φ α i ¯ φ i β + φ β i ¯ φ i α ] = ~ ξ a ~ σ αβ , a =
1, . . . , r (3.3)where Q ai is a matrix charge specified later on. φ α i ’s ( α =
1, 2 ) denote the component fielddoublets of each hypermultiplets ( i = r + ) . ~ ξ a are the Fayet-Illiopoulos (FI) 3-vectorcouplings rotated by the SU(2) symmetry, and ~ σ αβ are the traceless 2 × ξ a = ξ a = ξ a >
0, (3.3) describes the cotangent bundle over acomplex two-dimensional toric variety V defined by + r ∑ i = Q ai | φ i | = ξ a , a =
1, . . . , r . (3.4)Using M-theory/type IIA duality, it follows that M theory compactified on such geometriesis dual to type IIA superstring on 3-dimensional complex manifolds X fibered over a realline R with D6-branes. In fact, one can show that X can be described as a CP fibrationover the base V . Instead of being general, let us consider first a toy model where V is CP associated with r = Q i = (
1, 1, 1 ) . The corresponding quiver has3 nodes, where each node is associated with an U( N ) gauge symmetry factor. For this CSquiver, we have a level vector ( k , k , k ) such that k + k + k =
0. (3.5)It turns our that this model could be related to ABJM theory by adding D6-branes wrapping4-cycles of the blown up CP by CP at singular toric points. In this way of thinking andsolving the constraints (3.2), the gauge symmetry U ( N ) k × U ( N ) − k of ABJM theory canchange into U ( N ) k × U ( N ) k × U ( N ) − k . The IIA/M-theory duality predicts that this gaugetheory should be dual to type IIA superstring on AdS × CP × CP . Roughly speaking,the CS gauge theory describing FQHS model can be obtained from D4-branes moving on6 P × CP . In type IIA geometry, D4-branes can wrap CP and a particular complex curveclass [ C ] in H ( CP , Z ) . On the gauge theory side, the gauge symmetry becomes U ( N + M ) k × U ( N ) k × U ( N + M ′ ) − k . The constraint (3.2) requires that M = N M ′ = N . (3.6)As before, the CS quiver theory describing our FQHS system will be in the U ( N ) k × U ( N ) − k part of U ( N ) k × U ( N ) k × U ( N ) − k . Extracting an U ( ) × U ( ) , we can obtain the FQH fieldtheory from D4-branes wrapping individually CP and C . To see that let us consider first thecase of the abelain part U(1) corresponding to C . Indeed, on the 5-dimensional world-volumeof each D4-brane we have U(1) gauge symmetry. The corresponding effective action can takethe following form S D ∼ T Z R × C d σ e − φ q − det ( G + π F ) + T Z R × C F ∧ F ∧ A RR (3.7)where T is the brane tension and where the gauge field A RR is the R-R 1-form coupled tothe D0-brane of type IIA superstring. Ignoring the first term and integrating by part, the WZaction on the D4-brane world-volume becomes Z R × C F ∧ F ∧ A RR = − Z R × C A ∧ F ∧ ( dA RR ) (3.8)Then, we get the well known Chern-Simons term S WZ ∼ Z R A ∧ F . (3.9)The presence of the R-R gauge field sourced by the anti-D6-flux leads to − k = π R C ( dA RR ) .To couple the system to an external gauge field, one needs an extra D4-brane wrapping thecycle C in the presence of the RR 3-form A RR which is sourced by a D2-brane dual to aD4-brane. After the compactification, this gauge field decomposes into A RR → ˜ A ∧ ω (3.10)where ω is a harmonic 2-form on C . In this way, the WZ action R A RR ∧ F on a D4-branewrapping C gives q Z R ˜ A ∧ F (3.11)where ˜ A is the U(1) gauge field which can be obtained from the dimensional reduction ofthe RR 3-form A RR . This U(1) gauge field can be interpreted as a magnetic external gaugefield that couples to our QHS. We can follow the same steps to construct a non abelian7ffective Chern-Simons gauge theory with U(2) gauge fields. Roughly speaking, thanks to ν = ν ( U(2) ) + ν ( U(1) ) , and putting the charge like q i = (
1, 1, 1 ) , the corresponding fillingfactor reads as ν = k − k = k . (3.12) Let us extend the result obtained in previous sections to quivers with more than three nodes.There are various ways of doing that. One possibility can be realized by replacing twodimensional projective spaces either by the Hirzebruch surfaces or the del Pezzo surfaces.The corresponding CS quivers involve more than three U ( N i ) gauge symmetry factors. Thegeneral study is beyond the scope of the present work, though we will consider an explicitexample corresponding to the zeroth Hirzebruch surfaces F . Other examples may be dealtwith in a similar manner. We will briefly comment on various simple extension in the con-clusion.Recall from literature that F is a two-dimensional toric surface defined by a trivial fibra-tion of CP over CP . In N = F is realized as the target spaceU(1) × U(1) gauge theory with four chiral fields with charges (
1, 1, 0, 0 ) and (
0, 0, 1, 1 ) . Asproposed in [16], the corresponding CS quiver is given by U ( N ) × U ( N ) × U ( N ) × U ( N ) where the CS levels are ( k , − k , k , − k ) . This model corresponds to a circular quiver with fournodes. The FQHS system we are interested in appears as a CS quiver model obtained fromgauge theories living on the world-volume of the wrapped D4-branes on 2-cycles in typeIIA geometry which is given by the trivial fibration of CP over F . Indeed, there are threecomplex curves on which the D4-branes can be wrapped. D4-branes can wrap CP and twocomplex curves class [ C i ] in H ( F , Z ) . In this realization of FQHS, the fractional D4-braneswill give three gauge factors. Indeed, consider three stacks ( M , M , M ) of D4-branes. Onthe IIA gauge theory side, the gauge symmetry becomes U ( N + M ) k × U ( N ) − k × U ( N + M ) − k × U ( N + M ) k . A particular solution of (3.2) requires that M = N , M = N , M = N . (4.1)Our CS quiver theory describing FQHS systems will be in the U ( N ) k × U ( N ) − k × U ( N ) k part. As before, thanks to ν = ν ( U(1) × U(1) ) + ν ( U(2) ) , we can find the the filling factorof the model. Indeed, consider first the bi-layer system corresponding to U ( ) × U ( ) quiver8auge theory. The corresponding matrix K ij takes the form K ij ( U ( ) × U ( )) = k k where the integer k can be interpreted geometrically in terms of the self intersection of com-plex curves with genus g >
0. From type IIA string point of view, this model can be obtainedfrom two D4-branes wrapping two identical complex curves. Evaluating (2.3) for the charges q i = ( q , q ) yields ν = q + q k . (4.2)This is a general expression depending on the vector charge. Particular choice of the D-braneflux and the charge vector may recover some observed values of the filling factor. Addingnow the non abelian contributions, the total filling factor can be written as ν = q + q − q k . (4.3)It is worth noting that the vanishing filling factor behavior has to do with the condition q + q = q . This can be fixed by the flux on the world-volume D6-branes. The Hallconductivity is quantized in terms of the CS level being identified with the D6-flux and thecharges given in terms of the integral of harmonic 2-forms over dual 2-cycles embedded intype IIA geometry. In this letter, we have studied a class of FQHS in toric CS quivers from M-theory on thecontangent bundle over a two dimensional toric variety V . The dual type IIA geometriesare realized as AdS × CP × V . Using M-theory/Type IIA duality, we have presentedhierarchical stringy descriptions using quiver gauge models living on stacks of D4-braneswrapping 2-cycles in CP × V and interacting with R-R gauge fields. In particular, wehave given two simple examples of FQHS systems. The first model is based on the CSquiver relying on CP × CP , while the second one is associated with quiver gauge theorydescribing system based on D4-branes which wrap 2-cycles inside CP × F . This analysiscan be adapted to other multilayered systems by considering more general two dimensionaltoric manifolds. The manifolds are given by C r + / C ∗ r , where the C ∗ r actions are given by z i → λ q ai z i , i =
1, . . . , r + a =
1, . . . , r . Clearly, type IIA geometry will have h = r + r + r + V arelocal ALE spaces with ADE singularities.On the other hand, motivated by supersymmetric part of ABJM theory and supersym-metric QHE, it should be interesting to study such realizations in terms of M-theory on eightdimensional spaces. We hope, all these open questions will be addressed in future works. Acknowledgments : The author would like to thank his mother for patience. He thanksalso his collaborators for discussions on related topics.
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