JHEP05(2018)120 Springer May 7, 2018 May 18, 2018 : : March 21, 2018 : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP05(2018)120 b luigi.tizzano@physics.uu.se , and Luigi Tizzano . a 3 1708.02250 The Authors. Chern-Simons Theories, M-Theory, Topological Field Theories, Topological c

We present a 6D generalization of the fractional quantum Hall effect involv- , [email protected] E-mail: Department of Physics andPhiladelphia, Astronomy, PA University 19104, of U.S.A. Pennsylvania, Department of Physics andBox Astronomy, 516, Uppsala SE-75120 University, Uppsala, Sweden b a Open Access Article funded by SCOAP ArXiv ePrint: on a physical footing. Keywords: States of Matter conformal field theory wepactification also of the compute 7D the theorygapped analog provides systems a of uniform coupled the perspective to on Laughlinmetric boundary various version wavefunction. degrees lower-dimensional of of Com- the freedom.Encouraged 7D We by theory also this, embeds show we in that presenta a a M-theory, 10+2-dimensional conjecture and supersym- in bulk can topological which be theory, IIB thus decoupled string placing from theory all is gravity. twelve dimensions an of edge F-theory mode of four-form flux. The low energyabelian physics three-form is potentials governed with by a aa bulk 6D single anti-chiral 7D derivative theory topological Chern-Simons-like of field action Euclideanand theory coupled effective explain strings. of to how continued We fractions derive the whichperconformal figure fractional field prominently conductivity, theories in correspond the to classification a of hierarchy 6D of su- excited states. Using methods from Abstract: ing membranes coupled to a three-form potential in the presence of a large background Jonathan J. Heckman 6D fractional quantum Hall effect JHEP05(2018)120 1 4 ] and its connec- 6 – 1 ]. There is by now a vast ]. Various generalizations 25 17 ]. This involves an interplay – 15 19 – 7 – 1 – 13 17 9 22 11 16 25 14 19 ]. 20 1 31 26 ]. For a recent introduction to the subject see [ ]. For additional discussion of the higher-dimensional quantum Hall effect in 16 18 A classic example is the 2D fractional quantum Hall effect (FQHE) [ The list of such references is well-known to experts. For a review of some aspects of this and other 5.1 Examples 3.1 Landau wavefunction 3.2 Zero slope3.3 limit of the Large membrane rigid3.4 limit Point particle limit 1 the condensed matter and stringliterature theory on literature, various see ways e.g. that [ bulk gapped systems produce novelcondensed edge matter systems mode (with dynamics. ansee eye e.g. towards holography) the geared review towards [ a high energy theory audience, tion with a bulk 2+1-dimensionalbetween Chern-Simons the theory theory [ offield 2D theory chiral [ conformalof field this theories phenomena (CFTs), are by andHall now a effect available, [ including bulk a topological higher-dimensional (integer) quantum Extra dimensions provide anomena. unifying This is perspective by on nowdimensional quite a world commonplace of in variety developing string of connectionsfigures theory lower-dimensional between prominently and the phe- in higher- various low thecontext energy analysis of effective of a field bulk many theories. theory condensed in matter It a also systems, gapped especially phase coupled in to the boundary modes. 1 Introduction 6 Embedding in M-theory 7 IIB as an edge8 mode Conclusions 4 Quasi-branes and quasi-dranes 5 Compactification 3 Many body wavefunction Contents 1 Introduction 2 7D bulk JHEP05(2018)120 ]: 35 and (1.2) (1.3) (1.1) – I 30 db + I c 2 is an integral, ]. → 31 IJ I c such that: ]): -matrix” in the context I 31 with Euclidean boundary K b 7 M . ]. The pairing Ω , 7 J 38 – ∂M dc | I ∧ 36 ic . I I c 7 db Z M = + = 7 IJ – 2 – πi I 4 Ω c ∂M | I = c D D 7 6 governs the braiding statistics for Euclidean strings in S ∗ IJ ]. In such situations one expects to be able to decouple the bulk and boundary 33 ], and also as an interesting topological field theory in its own right [ 29 – 27 . The 7D action we propose to study has been considered before in the high 6 a collection of three-form potentials with gauge redundancy M I = c To realize both the integer and fractional quantum Hall effect, we activate a background Our generalization of the fractional quantum Hall effect will involve a 6D anti-chiral From this perspective, it is natural to seek out additional examples of chiral conformal 7 The reason for this is that there is no integral basis of imaginary-self-dual three-forms on a generic six- 2 manifold with compact three-cycles.on This a is choice simplyintegral of because basis metric, the of and imaginary-self-duality imaginary-self-dualthis condition three-forms. as condition depends such, in In the the continuous specialthe physical variation case sense context, of where of there det the reference Ω istheories = [ metric an since 1 additional the destroys since caveat boundary then the to theory we chance has have an a to invertible well-defined have quantum partition field an function. theory in properly quantize the 6D theorya since self-duality it condition is and otherwise quantization condition impossible for to three-form simultaneously fluxes impose magnetic [ four-form field strength, namely one in which all four legs thread the 6D boundary. 2D Euclidean strings coupled to anti-chiral two-form potentials In this context,the the boundary matrix 6D Ω system. The existence of this bulk action is in some sense necessary to the resulting boundary condition willconvention yield adopted an for anti-self-dual three-form, 6Dpositive in SCFTs accord definite in with symmetric the references matrix [ of which the is standard the fractional analog quantumenergy Hall of dynamics effect. the of The “ 2 + resulting 1-dimensional 7D membranes. theory describes In the the low boundary, these will appear as with subject to the imaginary-self-dual boundary condition (see e.g. [ Here, our sign convention is such that if we analytically continue to Lorentzian signature, spondence [ field strength. This isno a known higher-dimensional way analog to of write a a chiral Lorentz boson, covariant action. andtheory as of such, there edge is modes∂M coupled to aenergy 6 theory + literature, 1-dimensional both in bulk the context of the topological sector of the AdS/CFT corre- field theories as potentialdimensions. generalizations Along of these lines, the therethe has fractional recently construction quantum been of significant Hall progress 6D effect intant supersymmetric understanding ingredient to conformal in higher field these theories theoriespoint, is (6D there that SCFTs). are in the An always deformation impor- anti-chiral away two-form from potentials the conformal with fixed an anti-self-dual three-form JHEP05(2018)120 (1.5) (1.4) , with (intersect) ]. From a geo- Ω 39 ⊗ – a particular choice ) D 36 (7 U(2) over a Riemann surface I b for Γ , U(2) Γ .   / I 2 ... b ]. C 1 − 6 Σ 3 Z – x I 2 1 − im 2   x – 3 – − has actually been classified [ 1 x IJ = q p . 4 Φ(Σ) = exp M × 7 ]: -matrix is dictated by a tensor product Ω M K 42 – 40 over a three-chain with boundary Σ. Much as in the 2D case, our many I c the intersection pairing on the compactification manifold. Indeed, even our 7D The higher-dimensional unification in terms of this 11D theory suggests an irresistible The 7D starting point also provides a unifying perspective on a variety of lower- In the context of 6D CFTs with supersymmetry (which can be realized in a geometric Though this correlation function is likely to be quite difficult to compute for an in- (intersect) further extension toperturbative twelve formulation dimensions, of IIB especially stringconjecture in theory. on the With what this context in suchcan of mind, indeed a we F-theory, be present 12D the realized, a theory albeit speculative pathologies non- ought for with a to having supersymmetric look theory two like, in temporal 10 showing + directions 2 that (such dimensions. many as To elements avoid moving along closed timelike dimensional bulk topologicaltems, systems the coupled analog of to the Ω dynamical edge modes.system can In be these viewedbackground as sys- of the the compactification form of an 11D theory of five-forms placed on a From the perspective of thefractions fractional for quantum excitations Hall above effect, the these ground are state interpreted [ as filling phase of F-theory), themetric full perspective, list these matrices of arethe Ω nothing resolution but of the orbifold intersectionof pairings singularities discrete obtained of from subgroup the ofcontinued form fractions U(2). [ The structure of this singularity is in turn controlled by the 2D Laughlin wavefunction, butin even a so, zero we slope find limit thatrelative where it the energy reduces membranes scales are to set large afrom and quite by this rigid. similar the simple Other form behavior. field limits dictated strength by and the membrane tension lead to deviations teracting 6D CFT, weevaluation find is that relatively straightforward, in and theformal can case be invariance. derived of Said from a differently, generalCFTs the free properties does absence theory of not of impede con- of our a anti-chiral analysis. Lagrangian two-forms, The formulation the end for result 6D is somewhat chiral more involved than that of by an integral of body wavefunction factorizes as theinvolving the product Φ’s, of a andPerhaps piece a not controlled Landau surprisingly, by wavefunction this correlation which latter functions four-form dictates wavefunction flux. the is overall in size turn of controlled by droplets. the background correlation functions of non-local operators of the form: which from a 7D perspective involves replacing the integral of Using the 7D perspective, we argue that the many body wavefunction is constructed from JHEP05(2018)120 ]. ’s 7 c 45 (2.1) (2.2) , ) is the and 7 44 , 6 M presents ( Z 7 π 32 p 2 , Z 31 we briefly discuss , we discuss in more 5 ), where 7 2 M ( Z π p 2 Z + I c ]. We shall neglect these subtleties in , J ∼ 43 I , dc . c 31 6 ∧ I M c 7 × Z M IJ πi time ), we need to specify the global nature of the fields . To properly define this theory, it is helpful (much as in 4 Ω – 4 – 7 R 2.1 M 3 = ] = 7 I c [ M D 7 S , which we then evolve from one time slice to another. 6 we discuss the spectrum of quasi-excitations, and its interpretation M 4 we evaluate the Laughlin wavefunction for a free theory of anti-chiral 3 ], in order to properly define ( -forms with integral periods on p 29 . 8 discusses the embedding of the 7D theory in M-theory, and section 6 There are various subtleties in quantizing such Chern-Simons-like theories, and we Our starting point is the 7D action: The rest of this paper is organized as follows. First, in section Following [ 3 space of closed the ordinary Chern-Simons case) toone needs view the the analog 7-manifold of asWu spin structure. the structure boundary This on of a is anfurther three-manifold, necessary 8-manifold. which discussion to is More have on played a formally, in thiswhat properly seven and dimensions follows defined by related theory since a points, of 4-form mucheffect self-dual see of which forms references can our on [ all analysis the be boundary. will recast For focus in terms on of coarse boundary features correlators. of the 6D fractional quantum Hall dinates defined on refer the interested reader to the careful discussion presented inand references their [ gauge transformations. In this paper we assume that in which we further assume that the seven-manifold is given by a product of theSaid form: differently, when we proceed to quantize the theory, a wavefunction will involve coor- effective field theory which would govern the low energy physics. As mentioned in the introduction,alization our of interest the is fractional in quantum developingdients Hall a we system. higher-dimensional use, gener- focusing In this herea on section, the microscopic we 7D lay description bulk out of description. the 6D main CFTs, Indeed, ingre- it because seems we at most present fruitful lack to first develop the candidate are likely to beso of that more they interest can toin be a section read high independently energy of theory the audience, other and sections. have been We written present2 our conclusions 7D bulk Next, in section tensors. In section in terms of ancompactifications 11D of topological the field 7DSection theory (and of 11D) bulk five-forms. topologicalour In field conjecture section theory on to F-theory lower as dimensions. a 10 + 2-dimensional topological theory. Sections the only propagating degrees of freedom are localized along adetail 9 the + 1-dimensional bulk spacetime. 7D topologicalysis. field theory Using which willmembranes, this form and the perspective, also starting present we a point formal for determine answer our for the anal- the associated analog of fractional the Laughlin conductivities wavefunction. for curves), we demand from the start that our theory is purely topological in the bulk, namely JHEP05(2018)120 ]: 27 (2.6) (2.7) (2.8) (2.4) (2.5) . Here ]. g , 35   ) j ,  . Rather, we ,S i ) 6 S T ( M · . Note also that in L L S N ( ]. , dim ≤ |  ) 31 Λ J y / X idimension n ) / ) for 2 ), and ∗ T are coordinates on )Φ S Λ Z ( take values in its dual Λ ( , , ) is the integral linking number of S 6 2 y n j ( 6 I m M is the intersection pairing for three-cycles in m ( M ,S m 3 i ( Φ )Φ T 3 S and 1 H · ( H 4 S ⊂ x L ( S ) and Φ 1 ∈ L S The three-form potential couples to 2 + 1-dimensional membranes via integration over The braiding relation of equation ( The bulk correlation function of such observables is: Much as in the case of 3D Chern-Simons theory, there is a class of observables of m ( A more formal way to understand this relation is to note that states of the 7D bulk theory must S 4 Φ h m assemble into representations offluxes the Heisenberg group the case of a 3Dwe theory have on neglected subtleties havingeffects to of due this with (which the do presence not of alter torsion. the Some degeneracy additional of comments the on ground the state) can be found in reference [ its worldvolume. These degreesfractional of quantum freedom Hall are effect. the analog In of contrast the to electrons that present case, in however, the there is no guarantee simultaneously specify the periods ofmust the take three-form a for maximal allstate(s). sublattice three-cycles of in Calling commuting this periods latticestate and of is: use three-cycles this to specify the ground where Φ where for Λ, and the where therefore need to specify a Lagrangian splitting of thethe phase form: space [ the canonical commutation relation for three-forms with all legs in spatial directions reads: Following the discussion in reference [ JHEP05(2018)120 5 s (2.9) (2.11) (2.10) (2.12) . t ]). Here, however, 47 ’s as three-form field I 7 c at a particular time can I M m @ , , namely we evolve a state from , 6   7 I ] (see also the review [ M c , ∂M 46 6 | Γ Z I = I M ic . 7 I im × 0 db   ∂M = + ≤ = 7 and time running into the boundary by R – 6 – I 7 c ∂M a Lorentzian signature manifold (with mostly +’ = | 7 I 7 6 ∂M c M M D (Γ) = exp M 6 ∗ m Φ for a depiction of this geometry. In this case, we need to 1 t : I b , the 6D boundary by 7 M = 0. See figure = 0, we realize the boundary t t to . A representation of the bulk-boundary geometry considered in this paper. We denote From the perspective of the 6D boundary, we interpret the Suppose now that we consider our bulk theory on the spacetime −∞ Moreover, there is a problem with formulating a theory of first quantized membranes as fundamental 5 = degrees of freedom as thisthere is generically not introduces much instabilities of [ membrane an disintegrates, issue the since macroscopic weonly are behavior interested adopting we in an are collective effective considering field features. should theory remain perspective. sensible Indeed, since even we if are the The two-form couples to2 + 1-dimensional a membrane. Euclidean Including effective atherefore membrane string, be of described namely charge by the inserting spatial into the section path of integral the our operator: which if we analytically continuein to the metric) would define an anti-self-duality relation. strengths for two-forms t impose suitable boundary conditionssignature for boundary our manifold, system. we take: Since we are assuming a Euclidean simply a collective excitation. so that at Figure 1 the 7D bulk by that these membranes are theIndeed, genuine from microscopic the degrees of perspective freedom of for M-theory, the it 7D system. is widely expected that M2-branes are also JHEP05(2018)120 , (2.15) (2.14) (2.13) abcdef Γ. σ ∂ take values I m to a three-form. The I j . I . dc I c ∧ ∧ C I 7 j @ which couples to the membranes. 7 Z M Z M C I i πi ν 2 − . + J J J Λ. This distinction will prove important dc dc dc ∧ ∈ M2 π IJ I ∧ J 2 c Ω I 7 n – 7 – c Z 7 M = Z M I for IJ πi j 4 Ω IJ J πi for the membrane depends on whether we view it as 4 n Ω I IJ ] = m ] = , j simply correspond to various emergent gauge fields at low I = Ω c ,C [ I I I c D c [ 7 m S D 7 . S 2 , we can also entertain a special class of charges in the sublattice Λ ∗ Σ is a three-chain with boundary Σ a Riemann surface wrapped by the × 0 . State insertion of a membrane wrapping a three-cycle Γ with boundary ≤ R to the action as follows: I j Figure 2 Suppose now that we have a background three-form A background collection of membranes is conveniently described by adding a source The choice of charge vector In this interpretation, the energies. The coupling between the two is: Adhering to the standard condensed mattertwo-point terminology function we dualize for thesewith three-index all objects indices different. defines a six-index conductivity The equations of motion in the presence of a background source are: restrict our choice of charges in this way. term a dynamic or static objectin in the the dual 6D lattice boundaryby Λ theory. instead Indeed, working though with the when we come to the structure of the Laughlin wavefunction where we will need to further where Γ = membrane. See figure JHEP05(2018)120 (2.18) (2.19) (2.20) (2.21) (2.16) . In our G ,   = 0 is given by a six- G t ∧ ) i I ( Landau b , 6 Ψ D D Z 6 N . We label each such insertion I ∗ E ≤ πi . (2.17) i ν . Y 2 . The operator associated with a J ≤ D 1 ν 6   I i bkgnd ν × ) i , which has field strength Φ . ( ) D 6 IJ C ...,N Z : Φ 6
N i ): 1 π ( 1 ) = exp ∈ , j − Φ ( = G   Ω Φ bkgnd 2.15 i ) ... G i Φ ( h π – 8 – 1 Z ∧ Φ 2 (1) for h 4-cycle I = Φ c ) N π = i in various limits. 7 1 D ( ≤ 2 D Z = I Y ) i

4 ) 0 = , and Riemann surfaces Σ and Σ 0 (Σ D ] = m 0 6

m ) 0 serves to remind us that the potential is supported on Σ c, C [ and 0 (Σ b given locally by D 0 7 (Σ)Φ m m h . Note that our result also generalizes to other conformally flat S m 6 Φ R

(Σ)Φ m , or a 6D ball (namely the hyperbolic space Φ 6

S As we have already remarked, we do not have a covariant action for our anti-chiral We shall not follow this route, but will insteadWith simply this appeal in to mind, we the need fact to that extract the we two-point function for non-local operators The starting point for our analysis is the observation that on the 6D boundary, we have two-form. Indeed, this isthree-form not field even strength an operatortherefore in be the to 6D compute CFT. thefunction Rather, two-point over we function a know for pair that of the three-chains with boundaries Σ and Σ We therefore need to extract theary integrated two-point theory. function for the Strictlyhowever, speaking, that by such integrating a over aanswer correlation closed independent function Riemann of surface, is a we notfunctions should particular are expect gauge gauge. well-defined. to invariant. obtain Said an differently, Note, our operators and correlation Landau wavefunction, we will show howwith to a fix free their field mean theory, field we values. apply Since Wick’s we theorem are dealing to such correlation functions to write: for some choice of charges branes. Here, the prime on now, we treat these Riemann surfaces as fixed, though when we turn to the analysis of the have a 6D CFT, andIndeed, use we the anticipate resulting that structure our of analysis correlation will functions generalize for to more local involved operators. interactingsuch theories. as: The generalization to the action of line ( a theory of anti-chiral two-forms.well-known (non-covariant) In action the given case in ofa reference similar a [ action 2D for chiral boson the 6D in anti-chiral flat two-form. space, there is a we shall focus ona the uniform theory background of magnetic a four-form single flux emergent three-form potential In the previous section weeffect. presented a Our bulk aim perspective in onground this the state section 6D wavefunction. will fractional To obtain be quantumin concrete Hall to formulas, flat extract we additional focus space, on details namely spaces the on case such the of as structure a of 6D CFT the 3 Many body wavefunction JHEP05(2018)120 ]. 53 (3.7) (3.8) (3.9) (3.4) (3.5) (3.6) – (3.10) 50 ,   ] ]  3 3 ] b a 3 b δ 2 r 2 b a 2 b δ 1 2 1 3 b b : r [ , a [ 6 2 r a D 1 R 1 6 a [ ) ca i + r ( Landau 0 ε , the two-point function is: 6 c Ψ r nc hh − N i 0 h 6 ] Γ ≤ Z 3 3 i ), we see that our expression for . b a Y + inside of ) Γ ≤ δ Z ] ] ) 1 0 0 2 3 2 3 b b a a nc 2.20 × ( δ δ . 1 2 2 h 1 mm b a D a . 3 b [ 6 δ ) a D δ − i 1 ∗ 6 y ) b   [ 1 j ∗ * ( r π` 6 − i 1 1 r Φ a a (2 ) [ − i x r ( × ) = – 10 – Φ  = exp = 3 nc h 2 ( ]). 8 18 π a 1 N D h r M r 6 ( 17 ≤ 3 T

= 9 1 2 ) π 0 Y i

x Ψ Γ x ( ( ∂ 3 3 a a 2 ) 2 a a 1 nc 1 ( a a h h D ] as well as the review in [ h 16 Γ = Σ and ∂ But in contrast to the case of the 2D system which involves point particles, we are now So far, we have followed a quite similar plan to what is typically done in the 2D frac- To obtain the integrated two-point function for the chiral two-forms, we now formally To reach the expression for the anti-chiral two-form theory, we apply a projection to The answer in this case follows directly from that given for a theory of two-forms with (see e.g. [ faced with extended objects which carry an intrinsic tension: reduces to the calculation of these integrated two-point functions fortional the quantum three-form Hall fluxes. effect; wefunction have in expressed a the boundary Laughlin CFT, wave and function have as also a presented correlation a formal expression for its structure where the many body wavefunction: which is a somewhat more involved expression than itsintegrate counterpart this in result two over dimensions. a pair of three-chains Γ and Γ The two-point function for the chiral theory is then: counterterm consistent with 6D conformalboth invariance. in Minkowski We and note Euclidean that spacetimes this (by expression suitable holds choiceimaginary-self-dual of field metric). strengths, namely we set: where we have introduced the relative separation: Let us note that interm. this This expression, involves there details is about in the principle microscopic theory, also and a can Dirac be delta removed function by a contact suitable no self-duality constraint imposed. WeDenoting follow the the field procedure strength outlined for in this references non-chiral [ two-form by JHEP05(2018)120 . abcd (3.12) (3.13) (3.11) G . D 6 ]. Even so, we . + ⊥ 54 y   , b 51 Σ Z im .   (4) Σ variable to the correlation δ exp y D d 6 dx ∧ ∗ bkgnd ∧ ), namely, the correlation function: Φ π c G 2 * 6 dx D 2.19 Z = ∧ − D b 6 = , we show that the Laughlin wavefunction dx + ∗ ` D   ∧ 6 b a + which actually participate are those which are – 11 – Σ 0 Z b dx G and we have taken some choice of constant Σ im Z b 6   abcd R ∧ G exp 1 π 4! G 2   6 b = D Z ∧ . Applying the Laplacian in the G * a G ) y 6 y ( D Z ∆ πi ν 2   exp * = are local coordinates on a x Landau We expect that a complete analysis will involve a generalization of the loop equations Ψ To reach the righthand side, welocalized have used along the Σ, fact so thatthat the we membrane know in is that — our it by definition hasnormal integral, — a to the delta the only function Riemann support surface legs Σ. for of its We source. denote these Observe local coordinates by Suppose that we workparameterize in its the location, limit where towe the leading denote Riemann by order, coordinates surface byfunction Σ the yields: is position small. of Then, the we center can of mass, which flux to be uniform, and write: where One of the implicit assumptionsΣ we have is made held up to fixed. thismembrane point We to is now puff that up. show the that As Riemann mentioned the surface at presence the of beginning the of this four-form section, flux we actually take the causes four-form the 3.1 Landau wavefunction Our aim in thisLandau wavefunction subsection factor appearing will in be equation ( to extract additional details on the structure of the large compared to theactually intrinsic reduces length to scale aWe form also quite consider close the to oppositewell-approximated that regime by of of point the dilute particles. four-form standard flux 2D in Laughlin which wavefunction. the membranes are dynamics. Our goalsize in of the a remainder membrane,function of in and this various to regimes. section then Towavefunction will this factor use end, be always this leads in to to to theall characterize a next extract six certain subsection, the the amount spatial we typical of behavior show directions. “puffing thatflux. of This up” the After the for Landau in this, the Laughlin turn we membrane depends wave turn in on to the two special particular limits. profile for In the the limit four-form where the membranes are very large rigid object, orThis may will in instead turn be affect more the accurately shapes approximated of the byin QCD Riemann a to surfaces the point wrapped case particle. of bycan membranes, the still perhaps membranes. along use the the lines structure of references of [ the correlation functions just extracted to approximate these Depending on the strength of the four-form flux, the membrane may either puff up to a JHEP05(2018)120 , F √ / (3.17) (3.14) (3.15) (3.16) 1 ). See . ∼ G 3.17 ` √ / 1 ∼ polarized. , not ⊥ ρ 2 ⊥ . 8 y (4) Σ − δ D = . 6 π ⊥ D 6 G 16 ∧ ∗ = 8 + 0 π 2 ⊥ ∼ b G 2 y 6 Σ – 12 – 2 ⊥ ⊥ Z 1 D ` Z b 2 ∆ ν ∧ = π G 2 ⊥ 6 ρ D Z ν * ), we conclude that our integral is, in this limit, given by a 3.13 indicating the spread of the associated Gaussian: ⊥ ` . Depiction of a quantum Hall droplet of characteristic radius . In contrast to that case, however, we see that since the four-form flux denotes the component of the four-form flux with all legs transverse to Σ. 3 ⊥ Figure 3 G It is helpful at this point to compare with the standard case of the 2D quantum Hall At this point, we can now see why the presence of a background four-form flux pre- On the other hand, we also know that if we simply consider the action of the Laplacian effect. There, wein also appropriate have units. aalso droplet Here, figure size, we withalways see characteristic picks length the a set preferred analog set by characteristic of of lengths. this directions in formula the in 6D space, equation we ( actually get a tensor of such where Continuing in the samecontributions to vein, the we four-form flux, seethere we is that always no get provided direction a we we puffed can have up place membrane. at a Said least membrane differently, so three that independent it is vents the membrane from collapsingthat to if zero we size. considercharacteristic Indeed, the size from width the of above the analysis, membrane we in see the directions normal to Σ, there is a where we have introduced the localized flux density: Integrating equation ( quadratic form in the normal directions: in these four directions, then we have: JHEP05(2018)120 : i field (3.18) (3.19) (3.22) (3.20) (3.21) a X ’s have no time ), each . cm P  ] 3.20 d X . ). In the limit of slow ρ d , t ∂ c 2 [ dX . , σ X c 1 ∧ ...,  σ c ν ( dX ∂ a ) + ∧ dX  X ] σ b b ( , ∧ b X a cm (3) dX ). So, we can alternatively write the µ P C flat and large has been considered quite ∂ , t ∧ dX ∗ a 2 [ i a a C 3 ) + X X , σ t M 1 (  dX Z σ ( a 3 abcd cm abc ] and references therein). As far as we are aware, a µνρ – 13 – µ G X C ε X 57 3 3 ⊃ – M M abcd ) = Z 55 Z M2 G 3 , t 3 S 2 3 µ µ M , σ = ’s have no position dependence, and the ⊃ Z 1 3 2 σ cm ( µ M a X S = ]. The quantum theory contains many technical complications, X 58 zero-slope S zero-slope S is the embedding of the worldvolume of the membrane in the 7D spacetime, and is an overall constant of proportionality set by the tension of the membrane, 7 3 M µ denotes the pullback of the bulk three-form onto the worldvolume. The embedding Consider, for example, the expansion of the fields We bypass these complications (interesting though they may be) by focusing exclu- Consider the topological coupling between the three-form potential and the membrane: → (3) 3 C ∗ dependence. In this limit,can then, support we at see that most in oneand the worldvolume present derivative. action the Consequently, of action we equation in can ( the integrate suggestive by form: parts will decouple. fluctuations, we can write: where to leading order, the including the appearance of non-associative and non-commutative algebraic structures. sively on the low energyThis limit, is i.e. accomplished leading mostsince derivative cleanly contributions in in to this the the case, limit the effective where membranes theory. we are polarized, have a and most large fluctuations background of G-flux, the worldvolume The general algebraic structure associatedextensively with in the literature (see e.g.however, [ the case of largeplication G-flux with has this not sort been of considered couplingto is in a that much Nambu it detail. leads 2-bracket Part to [ of a the generalization of com- the Poisson bracket We are in particular interestedthis in limit, the we anticipate special that case therefer of action to is a this actually large, as dominated uniform the by magnetic this zero-slope topological G-flux. limit term. action: In We of the membrane isform captured of this by coupling fields the in overall components, coupling: where we absorb various numerical pre-factors into where M i In the limit ofbrane a to large simplify. magnetic Here flux, weoccur. we present also Some a expect elements brief of sketch thebe of our worldvolume how discussion useful theory we are for of expect necessarily further the such schematic, investigations. mem- a but simplification we to anticipate it will 3.2 Zero slope limit of the membrane JHEP05(2018)120 and . In = 0, : G (3.23) (3.24) (3.27) (3.25) m ab √ 0 ). This , writing / Y 6 ab,cd 1 R K 2.20 ∼ eff ` , (3.26) } = 0 v . ) 0 σ ∩ { .

− 0 0 cd u σ ( 2 = dY δ , as well as a pair of charges 0 u ∧ µν  . ab ] = b πiε Y ]. It is quite tempting to also consider 0 2 0 0 3 X define a collection of abelian gauge fields u − µ 59 M ] ∂ b Z − a [ X 0 )] = µ u X 0 and Σ ∂ πi – 14 – ab,cd σ a 4 = = [ ( K ξ cd } X ν ab µ and a dimensionless matrix of couplings = Y ,Y ab ) = 0 µ σ Y ( w such that the two Riemann surfaces are locally defined by , we can now present the canonical form of the zero slope ab µ G Y zero-slope [ } ∩ { S 0 u, v, w u ab,cd = K u { Σ = with that from 3 µ , and with respect to these indices, it is a symmetric matrix. Combining the ]. cd 61 , The above commutation relation is the direct analog of what one finds for electrons with local coordinates 60 and . In the limit where the background magnetic flux is very large, we see that the size of 0 3 -plane, and the holomorphic coordinate: so that the common normalu coordinate for both Riemann surfaces is parameterized by the specifies the separation between the two Riemann surfaces. perform explicit computations, it is helpfulC to introduce a complex structurethe for equations: requires us to specify a pairm of Riemann surfaces Σ and Σ the membrane in the transversethis directions limit, is then, quite we small, approximate going(in the roughly units membranes as of as the wrapping membrane very tension large length), Riemann and surfaces very thin in the transverse directions. To in [ 3.3 Large rigidNow limit that we have determinedwe the turn impact to of the the four-form evaluation flux of on the the size rest of of the the droplets, Laughlin wave function in line ( quantum mechanics is thethe starting presence point of for the athe zero large limit slope Neveu-Schwarz of B-field limit a [ ofform large flux open number on of string an M2-branes, theory individualaction. say membrane in should in It then would M-theory. be give, The interesting in to backreaction a connect suitable of these limit, the observations a to four- Chern-Simons the constructions presented the canonical commutation relation is: moving in a large magnetic field. Indeed, in that context, the resulting 1D topological limit in terms of the gauge fields Consider now the quantization of this worldvolume theory. In the gauge where on the three-dimensional space: Interpreted in this way,ab we see that theparameter G-flux can be split according to pairs of indices Observe that the composite operators JHEP05(2018)120 ) 0 ). Σ = 0. , the 0 × 2.12 (3.29) (3.30) (3.31) (3.28) ξ Σ (Σ × + 2 Σ × 2 , . . . , b R is a basis of self- -field on shell as: b i e ω = 1 i e , e ρ , and 0 ¯ ξ ) and Σ ρ 0 ξ Σ × = ) in this limit, we observe that × . 0 ) 0 D . 2 ) , (Σ Σ also require information about the 0 i e · + − 2 Σ e ρ ω · )   ξ b 0 ( )(Σ 0 i e Σ Z and ) are anti-chiral bosons. Integrating over )(Σ 0 , . . . , b 0 e m ξ φ ρ ( 1 n i e im − + Ω = 1 e i φ Ω   n i ω ( depend on integrating the basis of self-dual and = Ω, though more generally we can contemplate – 15 – ) m 0 − ( e ξ ρ ( exp i − m = φ ). The main difference is that in the limit we have just   field, we are integrating (in momentum space) over e ρ = = b and = e b ρ Σ − ρ Z m b 3.29 ρ − im ρ   exp . Here, the index is governed by a collection of chiral and anti-chiral bosons which 0 ] ): * 2 Σ 65 -plane. With this in mind, we see that our problem reduces to a R – Ω so that the difference becomes: × u 0 n 62 = ) are chiral bosons and the 0 ξ ( directions. Consequently, we are actually working in the limit of low mo- m i φ v which scale in appropriate units of Ω, as per our discussion below equation ( is a basis of harmonic anti-self-dual two-forms on Σ 0 i and m ω Ω and n Returning to the evaluation of our correlation function, we see that there is a rather In the related context where we compactify our 6D CFT on the space To evaluate the correlation function of Φ(Σ) and Φ(Σ w = and By construction, however, we haveConsequently, assumed we that see the that only we2D intersections can limit. occur essentially when The carry preciseexplicit over values choice unchanged of of the metric the calculation as exponents charge well in as density. the the dynamics of the membranes moving in a background close similarity to the case of linetaken, ( we have discarded various global data such as the topology of the Riemann surfaces. Note that single-valuedness of them associated OPE requires usThe to simplest work possibility in is terms tom take of charges anti-self-dual forms over the Riemannmetric. surfaces. This There in is, turntopology however, depends (see an on e.g. the important [ details aspect of of the this correlator which is protected by where in general the values of the where dual two-forms on Σ so that the the Riemann surfaces, we find that the correlation function reduces to: low energy theory on all descend from the anti-chiral two-form. Indeed, we can decompose the in the two-point functionthe for the mentum in theseby two momenta directions. in the Thetwo-dimensional system. correlation function will therefore be dominated JHEP05(2018)120 . - b 4 (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) The Riemann . , we also obtain 4 3 7 ) ) j CP ( = 1. x 1 4 and couples to the − Z ) i cd ( ε ]). It would be interesting to x , , ( 69 ij with – bc 3 e ρ ij 1 . η π ¯ α ξ 67

4 ad } ij Z ) × ρ η ij y ξ
= 0 , ) − = 0 N − 0 δ ) j α ≤ g α ( x bd 0 γ ( Z η Z f Y 0 α i

} ∩ { D ) 6 ) αβγδ = i i ) takes the form: – 16 – ( ε ) = 0 = 0 i j ) ( α = α y Φ 2.20 0 ( Z ) Z i Σ 0 α ( , α Vol(Σ f f Σ (nc) cd Φ ξ {  h b , which dualizes to a two-form ) × ]. The presence of a background four-form flux also suggests a non- N x = (4)
Σ ≤ ( 66 0 ) δ j Σ = ( Σ Y ibrane). We Physically, leave this an would analysis appear of this to 5 issue for future work. Compactification strings dictated byat the least continued for fraction 6Dconstrained. of SCFTs line realized For example, in ( Additionally, the F-theory, further the self-intersection blowups numbers of space the must ofadditional base always possible quasi-branes do obey [ not pairings 1 shift Ω the value of JHEP05(2018)120 (5.6) (5.7) (5.8) (5.9) (5.4) (5.5) (5.10) (5.11) (5.12) . ) i + p − i (5 l dC ∧ ) i − i , (5 l k θ , C p J J (2) J (1) J J (1) (2) ∧ (3) J (0) J (0) − k 11 Z dB dA dA dφ dC dB ω dφ p M ∧ ∧ ∧ ∧ ∧ ∧ ) ∧ Z i M − I I I I (2) (2) I (1) I (1) (3) (3) p I (0) = ]). i ( given by: πi , C C B B A A φ ,l i )-forms, but these do not interact with the ) l 4 i l 35 i i ( k θ Z Z Z Z Z Z Z , θ Ω + k ) ) ) ) ) ) ) ) – 20 – p p ω D D D D D D D h πi πi πi πi πi πi πi M (7 IJ (6 IJ (5 IJ (4 IJ (3 IJ (2 IJ (1 IJ − ( 4 4 2 4 2 4 2 =1 i i = )-form Ω Ω Ω Ω Ω Ω Ω X l i − ) is compact. This means that the intersection pairings p cpct i b ======− p ) − ] (see also [ p p p ( D D D D D D D M , 33 M 3 2 1 7 6 5 4 ) ( =1 i S S S S S S S i ( k,l X k Ω i cpct b and a ( p =0 i X k ω = D ) p )-forms coupled to (5 − i -form i For this reason, we can treat these contributions independently. (11 − 9 S 6= 1, the boundary theory is most appropriately viewed as a relative quantum field Note that we do not assume between an In a theory with additional bulk matter fields, there can be one-loop induced mixing terms upon 9 p reduction to lower dimensions. Let us give a fewthis examples point to of show view. howfollowing we bulk recover theories: Isolating various the topological contributions field from theories the from middle degree forms, we get the det Ω theory in the sense of reference [ 5.1 Examples Observe that this action breaksa up theory into of different (5 non-interactingother sectors. sectors. We always have we generate, and thus thewhen resulting they matrix are of square, couplings need they not need be not square matrices, have and determinant one. For square matrices with which defines a matrix of integers.to The a compactification of theory our of 11D theory abelian therefore differential reduces forms: M JHEP05(2018)120 ). 5.11 ), ( 5.9 ), ( ) has appeared 5.7 5.8 ]. 72 ) was instrumental in the 5.9 ]. 77 ) concern the study of both gapped 5.8 – 21 – ]. Our interest in this paper has of course been the 7D 15 – 13 , 9 ] and discrete symmetries of gauge [ 76 ]. Alternative applications of line ( 29 ]). – 75 27 – [ 5 73 ), which as we have already remarked is helpful in the study of the fractional S × ]. It would be very interesting to determine the resulting theory of edge modes. It 5.10 Finally, the 5D abelian Chern-Simons theory of two-forms in line ( Note that the theories in dimensions 7 and 3 have a symmetric matrix of couplings, The most familiar example in this class is given by abelian 3D Chern-Simons the- Such BF theories feature prominently in long distance limits of various high energy The 6D BF-like theory appears to have not received as much attention. Some details One of the other lessons from string theory is that additional light degrees of freedom These bulk topological field theories fall into two general subclasses, namely Chern- 5 29 both in thepears condensed in matter the andAdS high low energy energy theory effectivephases literature. of action matter of [ For type example, IIB it string ap- theory on the background quantum Hall effectgeneralization [ of this to three-forms. as dictated by thedimensions intersection 5 pairing and on 1 thestructure have internal an of space. the anti-symmetric By matrix internal contrast, intersection of pairing. the couplings, theories again in in accord with the Consider next the Chern-Simons-likesame theories degree. in This which occurs the when differential the forms number all of haveory spacetime the dimensions ( is odd. about this theory (and about BFof theories of [ various dimensions) can bewould found in also appendix likely A shed further light(see on e.g. the [ compactification of 6D SCFTs to five5.1.2 dimensions CS-like theories physics systems and alsomatter. play For an example, importantdescription a role of four-dimensional in novel action bosonic the similar symmetry description to protected of topological ( phases gapped [ phases of additional non-abelian structure and higher spin currents5.1.1 in the boundary theory. BF-like theories We begin our discussion on6, effective 4 topological and 2. field theories Here, focusing in general, first we on expect dimensions a BF-like theory as in equations ( understood from the compactification ofdimensional a setting. chiral Supersymmetry four-form provides inleads an ten to additional dimensions an extension to even of the broader these lower- class results of and lower-dimensional theories. are expected to emerge in limitsFrom where this the compactification perspective, manifold develops we singularities. can see that in many cases, we should expect to realize both we shall limit our discussion to a few generalSimons-like comments. theories withsymmetric potentials pairings, of and the BF-likecases, same theories we degree expect with to forms and realize of either interesting different symmetric edge degree. mode or dynamics, anti- In which all in of many these cases can be in the obvious notation. There is a vast literature on nearly all of these theories, and so JHEP05(2018)120 ]. 80 (6.5) – 78 direction. ⊥ R ] (6.4) 3 ] (6.2) t ] (6.1) 3 [ 2 t · [ t [ · -form potential. We · p Γ and a common point ] + 1 / 2 ] + 1 2 t 2 [ t C · [0] + 1 [ · · + 1 = 1 massless supermultiplets (see  [0] + 1 ] refers to a conformal field theories studied in · [0] + 2 p 1 2 N t · Γ  · S + 1 + 2  .  1 2 1 2 [0]  · · [1] + 1  [0] (6.3) · · · – 22 – + 1 + 2 + 2    [1] + 2 1 2 is seven-dimensional, so it is natural to expect a bulk · [1] + 3 3 2 1 2  · ⊥ ·   ], [2] respectively refers to an 8D scalar, Weyl fermion, · · R 3 2 + 1 + 2 ×  1  3 2 , 5 3 2  [1] + 1 · · [2] + 1  [1] + 1 R · · · ], [1], [ 1 2 Γ for Γ a discrete subgroup of SU(2). We realize a conformal fixed / = 1 = 1 2 = 1 = 1 = 1 ]. We pass to the partial tensor branch of the theory where effective 2) C (1) / D 2) =1 8 / × D (3 (1) D D 82 8 8 8 N (3 , V ⊥ D 8D Massive Supermultiplets: 8 S V G R 8D Massless Supermultiplets: S 81 , × 1 , 37 factor. This realizes the so-called class ]): 5 R ⊥ 84 , R 83 Let us now turn to the construction and study of this putative 7D massive supermulti- A helpful example to keep in mind is the special case of M5-branes filling the first Along these lines, we now take the 6D boundary theory to be a Lorentzian signa- where the notation [0],vector [ boson, gravitino andalso graviton. have massive The supermultiplets, notation where [ we denote massive fields by an overline: plet which contains a three-formcomes about, potential it and is is actually decoupled helpfulhas to from a proceed chiral gravity. up structure. To to see eight Here,e.g. dimensions, how we [ where this have supersymmetry the following 8D references [ strings have a tension by moving theObserve M5-branes that apart the from geometry one another7D in theory the to reside here which couples to the 6D boundary defined by the M5-branes. string/M-theory/F-theory, the first examples of this type being found in referencesfactor [ of point when the M5-branesof all sit the at the orbifold singularity of ture manfold, so thatthat the we are 7D interested bulkalong in the coordinate 6D physical is boundary systems anway which with to additional realize interacting construct a spatial examples degrees chiral direction. of conformal of such field freedom Recall theories theory. are localized At supersymmetric present, and the involve only embedding in In our discussionin up terms to of thisnevertheless a point, of interest 7D we to topological see havearises how field the deliberately in 7D compactifications theory phrased Chern-Simons-like of theory which ourautomatically M-theory. we have is show An entire been that added well-defined studying discussion there benefit in is of a its this supersymmetric approach own extension is of right. that our we 7D will It theory. is 6 Embedding in M-theory JHEP05(2018)120 (6.6) (6.7) (6.8) (6.17) (6.12) (6.14) (6.15) (6.16) (6.11) (6.13) . ] 2 1) gravitino t , [ fields: . · . + − 3 ] ] ] (6.10) D ] (6.9) 2 2 3 T 2 t t t = (1 [ [ [ t [ · · [0]+4 · · · N ] + 1 2 [1]+8 . [0] + 2 [0] + 2 t · [0] + 1 [ · · 2) · · D / further into 6 8 +8 (1) − (3 V + 2 + 2 − S 2) + 4 ·  / · − + D  1 2 7 (3   [0] + 3 1 2  S 1 2 1 2 · · + 2  + 2   · · · 2) 2) / / D + 4 +10 8 (3 + (3  + . S + 1 + 1 S 1 2  · + − [1] + 4  [0] + 1 2 2   · · ·  . D · 1 2 1 2 2) =1 – 23 – 6 / D D   → D 8 8 N 8 · · (3 + 4 + 3 (1) 2) → G S V +10   / D [1] + 8 3 2 1 2 − 7 (3 = = = D ·    7 S · · 3 2 2) [1] + 4 [1] + 4 (1) =2 /  D arises from M-theory compactified on a → → · · + 4 · D 8 N (3 8  D 2) G D =2 8 / 3 2 V +2 8 D D + 2 + 2 8 [1] + 2 [2] + 2  S 8 N (3 + · · · − +  S G   3 2 3 2 3 2  · = 1 = 1 = 2   2 D 2) D = = =1 7 / 6 (1) D D → 7 7 N (3 2) 2) V Gravitino Supermultiplets: / / 2) S → G / − + (3 (3 D 7 D (3 D S S 6 7 S = 2 multiplet 10 → → N 2) D / 8 D . The reason this is not correct to do can be seen either by directly constructing 8 (3 Note that we have not “removed a vector multiplet” from the gravitino multiplet Proceeding now to seven dimensions, we obtain the following irreducible 7D super- 2) S We thank D.S. Park for helpful discussions. / D 10 7 (3 So in other words, everything properly assembles into the following 6D multiplets: And in sixmultiplets: dimensions, we observe that we have the following the appropriate supermultiplet, orwhere removing indirectly, such by a consideringsupermultiplets. multiplet a To would see further make this reduction explicitly, it we to impossible decompose 6D, to construct appropriate 6D with the way wewhereas count in degrees 7D of freedom wecondition. cannot for have our a spinors; Weyl in spinor, 8D and we instead have impose aS Weyl an spinor, appropriate reality The presence of the additional factor of two in the gravitino and Weyl spinors has to do multiplets: The 8D In terms of the field content, we have the following relations: JHEP05(2018)120 , 2) / (3 (6.18) (6.19) (6.20) (6.21) Ψ · (4) . The four 2) G / · D 7 (3 2) S / . (3 ] . 3 Ψ t To illustrate, observe (3) 11 dC ] + [ 2 ∧ t [ · (3) C D , Z 7 , 2 [0] + 3

) π (4) · 2 F (1) G 4 ∧ is the “lift” of the 6D Green-Schwarz + 4 Z dA M

F  D πi 1 7 1 2 + 4 µ Tr(  2 ) is quite large, so in this sense the three-form

· 4 + ∧ (1) M – 24 – (3) (3) A C + dC D [1] + 8 Z 7 · (0) D ∧ ∗ 7 µ dφ + 4

(3)  3 2 dC  D · Z 7 ) 4 = 2 M 2) / D ] to seven dimensions. 7 (3 Vol ( 87 S – 85 , 70 is the two-form field strength and F It is also possible to couple our supersymmetric theory to defects carrying propagating One can also consider coupling the 7D gravitino multiplet to the 7D vector multiplet. The fermions of the three-form multiplet decouple in a similar fashion. We note that in But if we have a trivial S-matrix (though a non-trivial theory!), we can consistently One might now ask whether it is actually consistent to consider such a multiplet We thank M. Del Zotto for discussions on this point. 11 coupling [ massless degrees of freedom. Observe thatconditions in which this will case, break we half need of tosupersymmetric the impose supersymmetry, edge but suitable allow boundary modes. us to couple to appropriate This proceeds, for example, through the topological term: where that the magnetic dualadditional of the Chern-Simons two-form like is couplings awe to three-form, see decouple so that we these the cantopological modes alternatively sector. local as just excitations well. write decouple, All the told, only then, remnant in the 7D theory being a zero-form and four one-form potentials allmechanism, develop masses and through the the standard kinetic St¨uckelberg terms: with mass setfinal by set the of compactification terms involving scale, the just two-form potentials as proceed for in analogous the fashion. three-form Observe potential. The seven dimensions. This places themeans three-form that potential the in a resulting gapped supermultiplet phase is and massive. in particular the 11D supergravity action, thereso is this a flux coupling also of generates the a schematic form mass for the gravitinos and Weyl fermions of In the case where we decoupleis gravity, Vol( also “decoupled”. Additionally, however, weas see can that happen if if there we is have a M5-branes non-zero present, four-form then flux, there is a Chern-Simons-like coupling in suggest that in any theory with a non-trivial S-matrix,decouple gravity gravity and must simultaneously indeed retain be ourthat included. gravitino when multiplet. we compactifyterms M-theory with on the a three-form four-manifold, potential: the action contains the following tiplet decoupled from the graviton: decoupled from gravity. Indeed, the presence of the Rarita-Schwinger fields would seem to With this caveat dealt with, we now see that we have correctly identified a 7D supermul- JHEP05(2018)120 In some sense ], much as we 12 32 , 31 ] so we expect to be able to decouple 33 ]. Our aim in this section will be to present 88 ) covariance central to the formulation of IIB Z – 25 – , . We have already seen the utility of this perspective in our geometric 5 assemble bulk supermultiplets. and its geometrization in F-theory. potential as an edge mode coupled to a bulk topological theory. We require the bulk theory to preserve supersymmetry, namely weWe must must be maintain able the to known SL(2 We attempt to quantize the IIB self-dual five-form flux by viewing the chiral four-form To motivate our conjectural formulation of IIB as an edge mode, we shall proceed in a To accomplish this, we shall abandon from the start the notion of having propagating With this in mind, it is tempting to extend the standard 10D spacetime by an additional Having gone to eleven dimensions, it is now irresistible to try and connect this with From our present perspective, the issue is in some sense forced once we demand proper In the above we have only sketched the main elements of the 7D supersymmetric We thank C. Vafa for an inspirational discussion. 12 (2) (3) (1) “bottom up” fashion, piecing together various consistency conditions. These will include: degrees of freedom intwo all time twelve directions dimensions. leads to Indeed, severerestrict pathologies propagating with attention degrees causality. to of Rather, edge we freedom shallbe modes with immediately to localized see on whether a this 9 can + 1-dimensional be subspace. fit together Our self-consistently. goal will some suggestive — thoughtheory conjectural as an — edge aspectsthis mode resurrects of of the how a original to formulation bulkrather of 10 view F-theory than + as 2-dimensional all “merely” genuinely topological of as livingalso theory. a in IIB points 10 formal + superstring the 2 device dimensions way for to constructing the non-perturbative construction of 10D new vacua. self-consistent It string vacua. spatial direction toalready write did in an section 11Dunification topological of theory lower-dimensional topological of theories. five-forms [ the 12D geometric formulation of F-theory [ then we have an invertiblethe theory bulk in and the boundary. sensecould of Nevertheless, ask [ such what ana happens answer is different at rather normalization other unsatisfyingquantum for “levels” because questions the of cannot one be the two-point easily function. chiral accessed four-form, due It to namely the is if absence also of we an disturbing choose off-shell that action. certain for IIB string theory.self-dual Indeed, field strength IIB and supergravity there contains are a notorious subtleties chiral in four-form writingquantization potential an with of off-shell 10D fluxes action. inlevel one the for quantum the theory. chiral In four-form (as the happens special for case the where standard we IIB restrict supergravity to action) theory. It would of course be most instructive7 to fill in these IIB details as in future anMotivated work. edge by mode the successcompactification of of embedding M-theory, it our is 7D natural topological to theory ask in whether a a supersymmetric similar structure also holds JHEP05(2018)120 . ), , a iθ dz (7.1) (7.2) (4) c ]). The = exp( 98 g , 97 1) so that the total − n 14 . with the 10D spacetime. 2 on the boundary. T C This relation appears (in a 1) − p ( presented above. Extending to 13 is an elliptically fibered Calabi- dc − n µ = CY , ) p 2 . ]), for example. ( and − dr C 95 + ], and a proposed chiral supergravity the- dµ µ + so that the edge modes amount to degrees · in the 12D geometry. As such, we also have 2 , where . 95 1 + , n − (1) dθ − µ ]. The key point is that in 10 + 2 dimensions, µ n C 92 – 26 – CY = ), and corresponding infinitessimal one-form 96 = 1 – × ×B − ] (for a different perspective, see e.g. [ and ] (see also [ 1 n (1) and 1 1). The existence of a section means that there is an , 89 2 Euc , B 1 µ 1 + 99 − 95 − ds − K . Continuing with our general philosophy, we shall at , (3) µ d d n − C R R (0) ( , 94 c , + 2( (5) 91 . d ∈ O C ) duality relation of IIB supergravity is imposed “by hand” at the classical dθ z a K¨ahlermanifold of complex dimension ( Z . The local profile of the metric is in the conformal class: , 1 R − n × 1): B is a generator of the group. Note that for a U(1) gauge theory with , 1 g S = ], the SL(2 C 95 , and a zero-form ” refers to the contraction of one-form indices. , where · (2) dg c 2) to SO(9 † , can in turn be decomposed into real and imaginary parts, and thus we get a pair of ig − -fold with base dz n Continuing in Euclidean signature for our two extra dimensions, the local geometry Before proceeding to how this matches with the form content we have, let us briefly We immediately face an obstacle because the 11D supergravity spectrum in 10 + 1 Let us now proceed to enforce conditions (1)–(3). Along these lines, we briefly recall = For the perhaps more familiar case of breakingIn patterns reference of [ a compact Lie group, we would write 13 14 (1) µ the formal one-form reduces to 12D level, rather than as an emergent property from compactification on a slightly different form) in reference [ near the 10D spacetimeTopologically, is the Cartesian product of a cylinder real one-forms, namely the Euclideanthe analogs 12D of total space, weof can SO(10 also construct a connection which parameterizes the breaking Here, the “ Yau spacetime dimension is 10 = embedding of the 10D spacetime a local normal coordinate This explain how this earliertheory work on makes an contact ellipticallystandard with fibered in the Calabi-Yau modern more with practice.F-theory geometric section background formulation which Along of the has of these form become F- lines, let the us de suppose facto that we have a standard between bosonic and fermionicdimensions degrees has of been freedom. studied The inory supersymmetry references has [ algebra been in constructed 10general in + conclusion [ 2 is that oneas ought well to as consider a a pair theory of with null a one-forms, four-form and two-forms, dimensions does not admitproceed such to degrees 10 of +gravity 2 freedom! dimensions, multiplets There namely in is, ten apreviously, however, spatial for a 12D example and loophole spacetime in two if referencesone temporal with we [ can dimensions. this impose exotic Super- a signature Majorana-Weyl condition have on been spinors considered which in turn allows for a match that the Ramond-Ramond sectortwo-form of IIB string theoryfirst consists attempt of to a lift this chiral tobulk four-form an higher-form 11D space potentials with 10Dof boundary. freedom In our which 11D cannot space be we introduce gauged away, namely JHEP05(2018)120 . 2 e θ T (7.6) (7.3) (7.4) (7.5) ) duality group Z , Again taking our cue ) duality in IIB string . , Z coordinate. The winding 15 , and simply analytically ) ) , 2 θ T dy dy ) ) D D . So for us, the local profile of ), we see that we can wrap an 10 10 iθ x x 1) signature spacetime could still 7.5 ( ( , . τ τ 2 7→ − . + + dr φ θ e x − : d dx + e x ie 2 )( )( + dθ dy dy parameterizes the radial profile of the cylinder. ) ) − (0) r – 27 – D D c 1). On the other hand, the SL(2 = 10 10 , = x x ( ( τ 2 Lor τ τ ds + + e x d dx = ( = ( ). This means that if we insist on a supersymmetric theory in ], we note that the Narain lattice for strings on a Euclidean e 2 E 2 E 7.2 88 ds ds ) is instead: 7.2 2) so it is not inconceivable that a (1 , ), it will involve a dual coordinate 7.3 smoothly varies. We thank D.R. Morrison for helpful comments on this point. is a periodic local coordinate, and τ θ coordinate of line ( To put this on a firmer footing, we now observe that by construction, we do not allow At this point, something awkward appears to have happened. On the one hand, we Let us now turn to the Lorentzian signature version, as dictated by supersymmetry in 2) signature, we need to analytically continue r The main point is that it is not possible to define a family of Lorentzian signature torii in which the , 15 with line ( analog of objects to wrap arounddiscussion compact of cycles the of Lorentzianextended cylinder the object geometry geometry. along of the In linenumber timelike particular, along ( circle this returning parameterized circle by to has the From a our this Fourier perspective, transform there which is yields again a a Euclidean dual signature periodic torus coordinate lurking, but in comparison from the original paper [ has signature (2 replicate these features. any localized excitations ina the bulk two extra 12D directions, topological this theory being in the the point of first dealing place. with Nothing, however, disallows extended have argued based on fluxadd quantization Lorentzian considerations directions and of supersymmetry signature that (1 of we need IIB to is mostcontinuing naturally to formulated Lorentzian in signature terms fails of to a retain Euclidean this structure. The presence of an additionaltion timelike takes direction us means from that a a 10 reduction + along 2-dimensional a spacetime null to direc- a 9 + 1-dimensional spacetime. 10 + 2 dimensions. Wechiral have four-form, already we presented ought evidence to thatthe extend to our properly spacetime quantize by the one(10 10D additional spatial direction, i.e., the metric in line ( supergravity modes: This elliptic fibrationtheory. is central to the geometrization of SL(2 Globally, we of course complete thisfibration local over presentation the of 10D the spacetime, metric to with construct metric an in elliptic the conformal class: where the complex structure of the elliptic curve is identified with the combination of IIB so that JHEP05(2018)120 e θ 2) signature , denotes the “same” n for a depiction of the 5 CY , where n CY × 1 S appears. This takes us to a dualized spacetime . Additionally, taking our cue from earlier e θ . Given that our motivation was to properly (1) C ]. In some sense our discussion is less ambitious (6) – 28 – F 102 D – , and 12 (3) 100 − ∗ C -fold. In this process, one is supposed to collapse the M- , = n (5) (6) C F is also described by F-theory on n 1) and in which the standard elliptic fiber of F-theory appears geometrically. , CY . Depiction of the 12D geometry associated with F-theory and IIB as an edge mode in With the geometry of the spacetime dealt with, let us now return to our proposed A satisfying byproduct of this proposal is that it helps to clarify the role of the elliptic form and one-form, namely work on supersymmetry in 10the + six-form 2 field dimensions, we strength imposequantize the an fluxes anti-self-duality of condition the on in IIB 12D, theory, and one we might have now ask introduced whether a self-duality it constraint is necessary to proceed up to 13 dimensions, perhaps which we trade a positiondimensional coordinate spacetime, for we a winding seeover. coordinate that to In the reach particular, a procedureproceeds dualized a as of 11 in reduction + dimensional a 1- of reduction standard differential Kaluza-Klein also compactification. forms carries alongmode the content, now circle lifted with to 10 coordinate + 2 dimensions. We have so far identified a five-form, three- context of IIA/IIBwe T-duality, interchange one momentum can andconsideration winding consider seems degrees similar to of limits, hold, freedom.earlier and modulo conditions Here, in (1)–(3) the we some mentioned seewe caveat sense can at that that is interchangeably the a required work similar beginning in to terms of simultaneously of this satisfy either our section. a 10 Precisely + 2-dimensional because spacetime, or one in fiber in M-theory/F-theory duality.background Recall that in theelliptically standard fibered picture, Calabi-Yau M-theory ontheory the elliptic fiber to zero size to reach the F-theory limit. In the lower-dimensional dimensions a pair ofEven though null nothing directions propagates alongthe along consistent which them, description we extended of can objects suchwith degrees can compactify complex of wrap structure on freedom the the leads lightlikegeometry axio-dilaton circles, to circles. in of and a the IIB Euclidean special signature strings. case torus, See in figure which we have a constant elliptic fibration. of signature (11 This sort of mixing ofduality such momentum as and double winding field theory issince [ a we feature restrict of to geometric a approaches topological to sector T- from the start. We thus have in our extra 1 + 1- Figure 5 the special case where thespacetime to axio-dilaton one is in constant. which a Here, dual winding we coordinate have passed from the (10 JHEP05(2018)120 , ) ). Z , (6) Z , F ], we (7.8) (7.7) D 96 12 – 89 −∗ = . Finally, the τ -forms with the (6) p F ) singlet. Z , , A C , we expect the graviton to always B − , (7.9) . ∇ µ 1) (5) ) + −  p ( ( dC B and C C ∧ the possibility of a 3 + 2-brane, leaving us + A ≡ µ (5) ∇ ) p C – 29 – ( + along the two one-cycles produce a pair of scalars, D C 16 Z · 11 (1) AB g  C πi 1 µ eliminates 4 → ]. There is no reason to do so, at least from quantization AB 1) signature geometry with a winding coordinate so as to make SL(2 g , 103 , namely we write: − , the one introduced earlier from a bottom up perspective. By the same µ ) singlet for the chiral four-form potential. , so the boundary four-form transforms as an SL(2 (1) Z , C (5) and dC + µ = (6) F Let us now turn to the sense in which our system is actually in a gapped phase. Starting So far, we have focused on the RR sector, and have already seen a partial unification A closely related point is that (as is well-known from M-/F-theory duality) the periods Comparing with the earlier supersymmetry literature in 10 + 2 dimensions [ Here, we are working in the (11 16 from the 11D action: duality manifest. also means that we canand alternatively a package our one-form mode content intoken, terms we of expect a a 10D partiallyfermionic graviton, massive degrees gravitino in of 10 freedom. + 2 dimensions to fill out the necessary turn to the 12DSince origin we for have the introduced remaining explicithave one-forms NS a degrees formally of infinite mass freedom,massless in graviton: namely two the of 10D the metric. twelve directions. The gauge redundancy for a The periods of the one-form potential and ratios of appropriateperiods linear of the combinations five-form reduces are related towith by duality the of its axio-dilaton field strength, namely with some parts of the NS sector of the 10D boundary. To round out our discussion, we now potentials and the scalars inus the onto 10D an IIB SL(2 theory, and the anti-self-duality relation projects of our potentials alongthree-form circles reduces produces to the the expected NS IIB and mode RR content. two-forms, For and example, transform the as a doublet of SL(2 well as the axio-dilaton.one-forms This all falls into place upon contracting our Note that this automatically generates covariant transformation rules for the two-form so we need not3 directly + 2-branes quantize to the speak correspondingof of). six-form integral flux Indeed, six-form turning units fluxes theonly (since actually discussion with we around, objects have we such no see as that a the 2 + absence 2-branenow or ask 4 + how 2-brane! we can recover a four-form potential and a pair of two-form potentials, as considerations. The reasonform, is the way that we because readexplain, of off the the the physical physical presence degreeseither brane of spectrum of contraction a is or freedom distinguished different. wedging associatedsuch, null Indeed, our the with one- as five-form physical our we degrees to five-form shortly of a potential freedom four-form require are or associated six-form, with respectively. a 2 As + 2-brane and 4 + 2-brane, along the lines of S-theory [ JHEP05(2018)120 . ]. (1) and µ 109 – (7.10) (7.11) (1) ]. Such C 107 ]. We can 106 116 – (9) 112 dC ∧
(3) ). By the same token, we C · 7.1 (1) µ ) reduces to the well-known IIB ]. Observe that we have dropped , D of line ( (4) Z 12 7.11 105 G ]. Following our topological route, we , . The coupling of the second line has π (1) 1 ∧ 4 µ 96 (9) – 104 C + (4) , 92 G (5) 98 , ∧
dC 97 . Some aspects of 2+2-dimensional branes have – 30 – ∧ ], and for earlier work, see e.g. [ (5) (5) C C (5) · · 111 C (1) (1) ∧ . µ µ . Writing out all proposed couplings to the higher-form ) monodromy, these two-forms are generically projected (5) (1) Z (3) D , µ C C Z 12 D ∧ Z π 12 1 4 , namely the nine-form π (1) 1 , which could in principle receive additional one-loop contributions 4 µ + (1) NS (3) = C h ∧ top ” because we have two Lorentzian directions. Finally, note that the only D, RR (3) i 12 h S ∧ is the four-form field strength for the three-form potential, and we have used (4) ]. This is rather suggestive considering that in F-theory models, the base of the ) have not been considered before. c (4) G 110 7.10 It is also tempting to extend each stringy excitation of the 10D boundary by a 1 + 1- Returning to the topological couplings of our 12D action, note that in the 10D bound- We can also include a partially massive graviton, along the lines of reference [ extend the F1- and D1-branesobtained to through the a contraction topological 2recently + been 2-brane studied which for couples example toalso in the extend [ four-form a 3 +couples 1-dimensional to D3-brane the to six-form a 4 + 2-dimensional topological brane which out, receiving a masscoupling due to such interactions.from Line integrating ( out components of the massive gravitino. dimensional direction in 12D, as for example in [ model has positive curvature,to and the thus base the inside curvature the is Calabi-Yau. negative in the directionsary, normal the first termprovides reduces a to mass St¨uckelberg a forgrounds kinetic the term with RR for non-trivial a SL(2 and chiral NS four-form, two-forms. and the Indeed, second in term F-theory back- tion. Here, however, we arein taking phenomenological a applications: rather for differentinfinite, limit us, rather from the than what mass small. is in An typicallycompactification, the additional discussed decoupled comment a is directions graviton that is intime zero formally the [ related mode context of can warped be localized on a lower-dimensional space- required for a topologicalline ( theory of this type. As fartheories as have appeared we in are stringLet aware, us theory note the and that related couplings in holographic massive of gravity, setups, there are see various e.g. concerns [ about superluminal propaga- the magnetic dual of appeared in the F-theory literature beforepre-factors [ of “ appearance of the metricThis occurs dependence in is the sufficiently privileged mild null that direction we via again contraction obtain with a trivial stress energy tensor, as where can also introduce a relatedits BF-like pairing term to to produce thepotentials, a three-form mass we gap have: for the one-form we can extend to 12D using our privileged one-form JHEP05(2018)120 ), it is quite 7.9 ]. Departing one 120 – 118 1 with a winding number for the − N . (7.12) ] sought to interpret IIB D-instantons (5) 117 dC ∧ (5) C – 31 – D Z 11 πi N 4 ) duality, and also evades the pathologies of two-time physics and Z , -branes) in terms of “momentum” along a circle of a 12D spacetime. Here, we 1 − D There is by now a nearly complete list of supersymmetric 6D CFTs. One of the Summarizing, the 12D topological interpretation of superstrings is intriguing. Apparently, then, the natural setting for many of our considerations resides in a 10+2- This also points the way to constructing new string vacua. For example, though we dimensions coupled to 9remainder + 1-dimensional of this edge section modes we associated discuss with some IIB avenues for strings.original future motivations investigation. In for the thisthese work fascinating was theories. to It better is therefore understand quite the tempting topological to structure ask whether of we can develop a with the usual fractionalof quantum Hall the effect. analogous We Laughlinthis determined wavefunction higher-dimensional the in leading starting order variousdimensional point behavior limits, systems provides involving and a a bulk have unifying7D topological theory also perspective field embeds explained theory in on a coupled how limita several to of speculative edge lower- M-theory conjecture modes. decoupled on from This the gravity, and interpretation this of in F-theory turn as motivates a topological theory in 10 + 2 In this paper we haveHall proposed effect a six-dimensional which generalization makestheory of the use and fractional of a quantum the 6Dtheory correspondence theory of of between three-forms, edge aIn and modes. bulk the the presence 7D edge The of topological mode bulk a field is theory large given is background by magnetic that a four-form of Chern-Simons flux, free like there anti-chiral 7D is two-forms. a direct analog future work. 8 Conclusions step further, it is tempting to consider 12D geometriesdimensional without bulk an topological elliptic theory. fibration. Thissupersymmetry, appears SL(2 to be compatiblepotential with issues the with conditions of massive gravity. We leave more detailed checks of this proposal for of the five-form theory. In thismetric, formulation, since we the identify case ofbackground no D-instantons with ought canonically to normalized correspond two-point todimensions. function the This for case of the provides the chiral a “usual”and four-form somewhat IIB in it complementary ten motivation would for be IIB interesting matrix to models, revisit earlier proposals such as [ There is a simple interpretationgeometry. of Indeed, this in as an winding early modesD0-branes speculative along as attempt a momentum along to timelike a mimic circle circle, the of(namely reference M-theory the [ interpretation 12D of see that a background condensate of D-instantons produces a corresponding shift in the level have for the most parttempting concentrated to on broaden the our case horizons of to unit more coefficient general in couplings line of ( the form: JHEP05(2018)120 ]. ∧ (3) ] (see 134 C , ]. 130 133 122 , ]), in which the 121 ]. Such little string 128 – 129 , 123 , 39 78 4, NSF grant PHY-1066293, ≥ d – 32 – metric holonomy with action the integral of 2 G ]). This theory is formulated in terms of an abelian three-form, but instead in- is no longer invertible since it has a zero eigenvalue [ IJ 132 , , where the metric is itself constructed from the associative three-form [ (3) 131 Proceeding up to 10 + 2 dimensions, we presented some tantalizing hints of an inter- We have also seen that our 7D topological theory admits a supersymmetric extension Another thread of our analysis is the unifying framework it provides for generating a One of the original motivations for this work was to better understand the topological In this work focused on the simplest situation where there is a single connected compo- C D 7 Conference in 2017 onfor Superconformal hospitality Field during Theories part in ofgrant this PHY-1452037. work. LT The thanksduring work UNC of this JJH Chapel work. is Hill“Geometry supported and and by The the Physics” NSF work ITS grant CAREER from of at Knut CUNY LT and for is Alice hospitality Wallenberg supported Foundation. by VR grant #2014-5517 and by the Acknowledgments We thank F.D.S. Apruzzi, Park and C. M. Vafa forfor Del helpful comments discussion. on Zotto, an We also earlier T.T. thankSimons draft. O.J. Center Dumitrescu, JJH Ganor for thanks and the D.R. Geometry O.J. Morrison 2016 and and Ganor, Physics 2017 as Summer D.R. well workshops at as Morrison, the the Aspen Center for Physics Winter provides a novel startingbe point for very realizing interesting newnating to classes aspect provide of of further string suchmode vacua. evidence a excitation. for It correspondence would this is clearly proposal. the emergence of A the particularly IIB fasci- graviton as an edge ∗ Adding such a termIt to would a thus theory be of interesting a to three-form compare gauge the potential IRpretation also behavior of leads of to F-theory these a two as mass 7D a gap. theories. bulk topological theory. Though quite conjectural, it already which embeds as avides decoupling a limit unified framework of for the understandingalso physical various [ aspects M-theory. of topological Topological strings M-theoryvolves [ compactifying pro- on a manifold of sector of little stringpairing theories Ω (for early constructionstheories see are e.g. [ non-local, andfield thus theory. not It is controlled natural byby to a the ask little standard whether string a axioms suitable theory. of 7D theory local exists quantum with edge modes given rich class of lower-dimensional phenomena.couplings We can can in already many anticipate cases thatifold. be various It recast integer would as be topologicalpossibilities. exciting properties to of use a this compactification perspective man- to develop a systematic analysis of these of the fractional quantum Hall effect in two dimensions,nent see to for the example boundary. [ disconnected In lower components dimensions, to it the isspectrum boundary. often of This quite non-local in fruitful operators turn to in is consider the intimately additional theory. connected with the supersymmetric version of our analysis. For some discussion of a supersymmetric version JHEP05(2018)120 B ]. ] B 364 B 41 ]. Phys. Rev. (1983) 75 SPIRE Nucl. Phys. , ]. , IN 56 [ ]. SPIRE ]. , Int. J. Mod. Phys. IN , Nucl. Phys. SPIRE Phys. Rev. , IN , SPIRE [ SPIRE IN ]. [ IN cond-mat/9501022 [ (1991) 2637 [ ]. ]. SPIRE (1984) 2390] [ Helv. Phys. Acta IN B 43 ]. , [ SPIRE SPIRE (1987) 1252 52 IN arXiv:1606.06687 IN (1995) 99 . [ [ 58 (1983) 1395 SPIRE IN 50 456 [ , 2016, Phys. Rev. Two-dimensional magnetotransport in the , (1982) 1559 (1991) 274 (1983) 605 (1990) 135 48 Erratum ibid. – 33 – [ 51 15 B 44 (1990) 8133 Off diagonal long range order, oblique confinement and Phys. Rev. Lett. , Phys. Rev. Lett. , Lect. Notes Phys. 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