6D Fractional Quantum Hall Effect

6D Fractional Quantum Hall Effect

Abstract

We present a 6D generalization of the fractional quantum Hall effect involving membranes coupled to a three-form potential in the presence of a large background four-form flux. The low energy physics is governed by a bulk 7D topological field theory of abelian three-form potentials with a single derivative Chern-Simons-like action coupled to a 6D anti-chiral theory of Euclidean effective strings. We derive the fractional conductivity, and explain how continued fractions which figure prominently in the classification of 6D superconformal field theories correspond to a hierarchy of excited states. Using methods from conformal field theory we also compute the analog of the Laughlin wavefunction. Compactification of the 7D theory provides a uniform perspective on various lower-dimensional gapped systems coupled to boundary degrees of freedom. We also show that a supersymmetric version of the 7D theory embeds in M-theory, and can be decoupled from gravity. Encouraged by this, we present a conjecture in which IIB string theory is an edge mode of a -dimensional bulk topological theory, thus placing all twelve dimensions of F-theory on a physical footing.


PENN

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA


Uppsala

Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

\authors

Jonathan J. Heckman\worksat\PENN1 and Luigi Tizzano\worksat\Uppsala2

1 Introduction

Extra dimensions provide a unifying perspective on a variety of lower-dimensional phenomena. This is by now quite commonplace in developing connections between the higher-dimensional world of string theory and various low energy effective field theories. It also figures prominently in the analysis of many condensed matter systems, especially in the context of a bulk theory in a gapped phase coupled to boundary modes.

A classic example is the 2D fractional quantum Hall effect (FQHE) [1, 2, 3, 4, 5, 6] and its connection with a bulk 2+1-dimensional Chern-Simons theory [7, 8, 9, 10, 11, 12, 13, 14, 15]. This involves an interplay between the theory of 2D chiral conformal field theories (CFTs), and a bulk topological field theory [16]. For a recent introduction to the subject see [17]. Various generalizations of this phenomena are by now available, including a higher-dimensional (integer) quantum Hall effect [18]. For additional discussion of the higher-dimensional quantum Hall effect in the condensed matter and string theory literature, see e.g. [19, 20, 21, 22, 23]. There is by now a vast literature on various ways that bulk gapped systems produce novel edge mode dynamics.3

From this perspective, it is natural to seek out additional examples of chiral conformal field theories as potential generalizations of the fractional quantum Hall effect to higher dimensions. Along these lines, there has recently been significant progress in understanding the construction of 6D supersymmetric conformal field theories (6D SCFTs). An important ingredient in these theories is that in the deformation away from the conformal fixed point, there are always anti-chiral two-form potentials with an anti-self-dual three-form field strength. This is a higher-dimensional analog of a chiral boson, and as such, there is no known way to write a Lorentz covariant action.

Figure 1: A representation of the bulk-boundary geometry considered in this paper. We denote the 7D bulk by , the 6D boundary by and time running into the boundary by .

Our generalization of the fractional quantum Hall effect will involve a 6D anti-chiral theory of edge modes coupled to a -dimensional bulk with Euclidean boundary . The 7D action we propose to study has been considered before in the high energy theory literature, both in the context of the topological sector of the AdS/CFT correspondence [25, 26, 27], and also as an interesting topological field theory in its own right [28, 29, 30, 31, 32, 33]:

(1.1)

with a collection of three-form potentials with gauge redundancy and subject to the imaginary-self-dual boundary condition (see e.g. [29]):

(1.2)

Here, our sign convention is such that if we analytically continue to Lorentzian signature, the resulting boundary condition will yield an anti-self-dual three-form, in accord with the convention adopted for 6D SCFTs in references [34, 35, 36]. The pairing is an integral, positive definite symmetric matrix which is the analog of the “-matrix” in the context of the standard fractional quantum Hall effect. The resulting 7D theory describes the low energy dynamics of -dimensional membranes. In the boundary, these will appear as 2D Euclidean strings coupled to anti-chiral two-form potentials such that:

(1.3)

In this context, the matrix governs the braiding statistics for Euclidean strings in the boundary 6D system. The existence of this bulk action is in some sense necessary to properly quantize the 6D theory since it is otherwise impossible to simultaneously impose a self-duality condition and quantization condition for three-form fluxes [29].4

To realize both the integer and fractional quantum Hall effect, we activate a background magnetic four-form field strength, namely one in which all four legs thread the 6D boundary. Using the 7D perspective, we argue that the many body wavefunction is constructed from correlation functions of non-local operators of the form:

(1.4)

which from a 7D perspective involves replacing the integral of over a Riemann surface by an integral of over a three-chain with boundary . Much as in the 2D case, our many body wavefunction factorizes as the product of a piece controlled by correlation functions involving the ’s, and a Landau wavefunction which dictates the overall size of droplets. Perhaps not surprisingly, this latter wavefunction is in turn controlled by the background four-form flux.

Though this correlation function is likely to be quite difficult to compute for an interacting 6D CFT, we find that in the case of a free theory of anti-chiral two-forms, the evaluation is relatively straightforward, and can be derived from general properties of conformal invariance. Said differently, the absence of a Lagrangian formulation for 6D chiral CFTs does not impede our analysis. The end result is somewhat more involved than that of the 2D Laughlin wavefunction, but even so, we find that it reduces to a quite similar form in a zero slope limit where the membranes are large and rigid. Other limits dictated by the relative energy scales set by the field strength and membrane tension lead to deviations from this simple behavior.

In the context of 6D CFTs with supersymmetry (which can be realized in a geometric phase of F-theory), the full list of has actually been classified [34, 35, 36, 37]. From a geometric perspective, these matrices are nothing but the intersection pairings obtained from the resolution of orbifold singularities of the form for a particular choice of discrete subgroup of . The structure of this singularity is in turn controlled by continued fractions [38, 39, 40]:

(1.5)

From the perspective of the fractional quantum Hall effect, these are interpreted as filling fractions for excitations above the ground state [2, 3, 4, 5, 6].

The 7D starting point also provides a unifying perspective on a variety of lower-dimensional bulk topological systems coupled to dynamical edge modes. In these systems, the analog of the -matrix is dictated by a tensor product , with the intersection pairing on the compactification manifold. Indeed, even our 7D system can be viewed as the compactification of an 11D theory of five-forms placed on a background of the form .

The higher-dimensional unification in terms of this 11D theory suggests an irresistible further extension to twelve dimensions, especially in the context of F-theory, the non-perturbative formulation of IIB string theory. With this in mind, we present a speculative conjecture on what such a 12D theory ought to look like, showing that many elements can indeed be realized, albeit for a supersymmetric theory in dimensions. To avoid pathologies with having two temporal directions (such as moving along closed timelike curves), we demand from the start that our theory is purely topological in the bulk, namely the only propagating degrees of freedom are localized along a -dimensional spacetime.

The rest of this paper is organized as follows. First, in section 2, we discuss in more detail the bulk 7D topological field theory which will form the starting point for our analysis. Using this perspective, we determine the associated fractional conductivities for membranes, and also present a formal answer for the analog of the Laughlin wavefunction. Next, in section 3 we evaluate the Laughlin wavefunction for a free theory of anti-chiral tensors. In section 4 we discuss the spectrum of quasi-excitations, and its interpretation in terms of an 11D topological field theory of five-forms. In section 5 we briefly discuss compactifications of the 7D (and 11D) bulk topological field theory to lower dimensions. Section 6 discusses the embedding of the 7D theory in M-theory, and section 7 presents our conjecture on F-theory as a -dimensional topological theory. Sections 6 and 7 are likely to be of more interest to a high energy theory audience, and have been written so that they can be read independently of the other sections. We present our conclusions in section 8.

2 7D Bulk

As mentioned in the introduction, our interest is in developing a higher-dimensional generalization of the fractional quantum Hall system. In this section, we lay out the main ingredients we use, focusing here on the 7D bulk description. Indeed, because we at present lack a microscopic description of 6D CFTs, it seems most fruitful to first develop the candidate effective field theory which would govern the low energy physics.

Our starting point is the 7D action:5

(2.1)

in which we further assume that the seven-manifold is given by a product of the form:

(2.2)

Said differently, when we proceed to quantize the theory, a wavefunction will involve coordinates defined on , which we then evolve from one time slice to another.

There are various subtleties in quantizing such Chern-Simons-like theories, and we refer the interested reader to the careful discussion presented in references [29, 30, 41, 42]. Following the discussion in reference [25], we see that in a gauge where we set:

(2.3)

the canonical commutation relation for three-forms with all legs in spatial directions reads:

(2.4)

where and are coordinates on . Note that to properly quantize the theory, we therefore need to specify a Lagrangian splitting of the phase space [29].

Much as in the case of 3D Chern-Simons theory, there is a class of observables of the form:

(2.5)

for , and a vector of charges. The pairing defines an integral lattice , and the take values in its dual .

The bulk correlation function of such observables is:

(2.6)

where is the integral linking number of and . Given two bulk operators and for and , we get the braid relations [25]:

(2.7)

where is the intersection pairing for three-cycles in .

The braiding relation of equation (2.7) tells us that in the quantum theory, we cannot simultaneously specify the periods of the three-form for all three-cycles in . Rather, we must take a maximal sublattice of commuting periods and use this to specify the ground state(s). Calling this lattice of three-cycles , the degeneracy of the ground state is:6

(2.8)

The three-form potential couples to -dimensional membranes via integration over its worldvolume. These degrees of freedom are the analog of the electrons present in the fractional quantum Hall effect. In contrast to that case, however, there is no guarantee that these membranes are the genuine microscopic degrees of freedom for the 7D system.7 Indeed, from the perspective of M-theory, it is widely expected that M2-branes are also simply a collective excitation.

Suppose now that we consider our bulk theory on the spacetime

(2.9)

so that at , we realize the boundary , namely we evolve a state from to . See figure 1 for a depiction of this geometry. In this case, we need to impose suitable boundary conditions for our system. Since we are assuming a Euclidean signature boundary manifold, we take:

(2.10)

which if we analytically continue to a Lorentzian signature manifold (with mostly in the metric) would define an anti-self-duality relation.

From the perspective of the 6D boundary, we interpret the ’s as three-form field strengths for two-forms :

(2.11)

The two-form couples to a Euclidean effective string, namely the spatial section of our -dimensional membrane. Including a membrane of charge at a particular time can therefore be described by inserting into the path integral the operator:

(2.12)

where is a three-chain with boundary a Riemann surface wrapped by the membrane.

Figure 2: State insertion of a membrane wrapping a three-cycle with boundary .

The choice of charge vector for the membrane depends on whether we view it as a dynamic or static object in the 6D boundary theory. Indeed, though the take values in the dual lattice , we can also entertain a special class of charges in the sublattice by instead working with for . This distinction will prove important when we come to the structure of the Laughlin wavefunction where we will need to further restrict our choice of charges in this way.

A background collection of membranes is conveniently described by adding a source term to the action as follows:

(2.13)

The equations of motion in the presence of a background source are:

(2.14)

Adhering to the standard condensed matter terminology we dualize to a three-form. The two-point function for these three-index objects defines a six-index conductivity , with all indices different.

Suppose now that we have a background three-form which couples to the membranes. In this interpretation, the simply correspond to various emergent gauge fields at low energies. The coupling between the two is:

(2.15)

Physically, the membranes are all charged under , which has field strength . In our conventions, the -flux is quantized in units of :8

(2.16)

Much as in the lower-dimensional setting, we then obtain a fractional conductivity given by:

(2.17)

Our discussion so far has focused on the bulk gapped system. Of course, it is also important to understand the structure of the boundary theory. We expect the many body wavefunction given by inserting a large number of membrane states in the presence of a background four-form flux to be in the same universality class as that of the genuine ground state. Each membrane corresponds to the insertion of an operator of the form given by line (2.12), for some choice of three-chain with boundary a Riemann surface , and some choice of charge vector in the dual lattice of charges . We label each such insertion in the path integral by an operator for . The operator associated with a background four-form flux follows from line (2.15):

(2.18)

where is a seven-dimensional domain, which at is given by a six-dimensional domain which has no overlap with the locations of the Riemann surfaces wrapped by the membranes.

Putting all of this together, we expect the (unnormalized) many body wavefunction for such membranes to take the form:

(2.19)

where the correlation function is evaluated in the boundary 6D theory.

Assuming that our 6D boundary theory is actually free, everything reduces to the calculation of appropriate two-point functions:

(2.20)

where we have introduced the unnormalized Landau wavefunction for a single membrane moving in a background four-form flux:

(2.21)

Our goal will be to estimate in various limits.

3 Many Body Wavefunction

In the previous section we presented a bulk perspective on the 6D fractional quantum Hall effect. Our aim in this section will be to extract additional details on the structure of the ground state wavefunction. To obtain concrete formulas, we focus on the case of a 6D CFT in flat space, namely . Note that our result also generalizes to other conformally flat spaces such as , or a 6D ball (namely the hyperbolic space ). For ease of exposition, we shall focus on the theory of a single emergent three-form potential , which couples to a uniform background magnetic four-form flux (locally), so we consider:

(3.1)

The generalization to the action of line (2.15) follows a similar line of argument.

The starting point for our analysis is the observation that on the 6D boundary, we have a theory of anti-chiral two-forms. In the case of a 2D chiral boson in flat space, there is a well-known (non-covariant) action given in reference [46]. In principle one could construct a similar action for the 6D anti-chiral two-form.

We shall not follow this route, but will instead simply appeal to the fact that we have a 6D CFT, and use the resulting structure of correlation functions for local operators. Indeed, we anticipate that our analysis will generalize to more involved interacting theories.

With this in mind, we need to extract the two-point function for non-local operators such as:

(3.2)

for some choice of charges and , and Riemann surfaces and wrapped by the membranes. Here, the prime on serves to remind us that the potential is supported on . For now, we treat these Riemann surfaces as fixed, though when we turn to the analysis of the Landau wavefunction, we will show how to fix their mean field values. Since we are dealing with a free field theory, we apply Wick’s theorem to such correlation functions to write:

(3.3)

We therefore need to extract the integrated two-point function for the -fields of our boundary theory. Strictly speaking, such a correlation function is not gauge invariant. Note, however, that by integrating over a closed Riemann surface, we should expect to obtain an answer independent of a particular gauge. Said differently, our operators and correlation functions are well-defined.

As we have already remarked, we do not have a covariant action for our anti-chiral two-form. Indeed, this is not even an operator in the 6D CFT. Rather, we know that the three-form field strength given locally by is a well-defined operator. Our strategy will therefore be to compute the two-point function for , and to then integrate this two-point function over a pair of three-chains with boundaries and , respectively.

The answer in this case follows directly from that given for a theory of two-forms with no self-duality constraint imposed. We follow the procedure outlined in references [47, 48, 49, 50]. Denoting the field strength for this non-chiral two-form by , the two-point function is:

(3.4)

where we have introduced the relative separation:

(3.5)

Let us note that in this expression, there is in principle also a Dirac delta function contact term. This involves details about the microscopic theory, and can be removed by a suitable counterterm consistent with 6D conformal invariance. We note that this expression holds both in Minkowski and Euclidean spacetimes (by suitable choice of metric).

To reach the expression for the anti-chiral two-form theory, we apply a projection to imaginary-self-dual field strengths, namely we set:

(3.6)

The two-point function for the chiral theory is then:

(3.7)

which is a somewhat more involved expression than its counterpart in two dimensions.

To obtain the integrated two-point function for the chiral two-forms, we now formally integrate this result over a pair of three-chains and inside of :

(3.8)

where and . Returning to equation (2.20), we see that our expression for the many body wavefunction:

(3.9)

reduces to the calculation of these integrated two-point functions for the three-form fluxes.

So far, we have followed a quite similar plan to what is typically done in the 2D fractional quantum Hall effect; We have expressed the Laughlin wave function as a correlation function in a boundary CFT, and have also presented a formal expression for its structure (see e.g. [16] as well as the review in [17]).

But in contrast to the case of the 2D system which involves point particles, we are now faced with extended objects which carry an intrinsic tension:

(3.10)

Depending on the strength of the four-form flux, the membrane may either puff up to a large rigid object, or may instead be more accurately approximated by a point particle. This will in turn affect the shapes of the Riemann surfaces wrapped by the membranes.

We expect that a complete analysis will involve a generalization of the loop equations in QCD to the case of membranes, perhaps along the lines of references [48, 51]. Even so, we can still use the structure of the correlation functions just extracted to approximate these dynamics. Our goal in the remainder of this section will be to characterize the typical size of a membrane, and to then use this to extract the behavior of the Laughlin wave function in various regimes. To this end, in the next subsection, we show that the Landau wavefunction factor always leads to a certain amount of “puffing up” for the membrane in all six spatial directions. This in turn depends on the particular profile for the four-form flux. After this, we turn to two special limits. In the limit where the membranes are very large compared to the intrinsic length scale , we show that the Laughlin wavefunction actually reduces to a form quite close to that of the standard 2D Laughlin wavefunction. We also consider the opposite regime of dilute four-form flux in which the membranes are well-approximated by point particles.

3.1 Landau Wavefunction

Our aim in this subsection will be to extract additional details on the structure of the Landau wavefunction factor appearing in equation (2.19), namely, the correlation function:

(3.11)

One of the implicit assumptions we have made up to this point is that the Riemann surface is held fixed. We now show that the presence of the four-form flux actually causes the membrane to puff up. As mentioned at the beginning of this section, we take the four-form flux to be uniform, and write:

(3.12)

where are local coordinates on and we have taken some choice of constant . Suppose that we work in the limit where the Riemann surface is small. Then, we can parameterize its location, to leading order, by the position of the center of mass, which we denote by coordinates . Applying the Laplacian in the variable to the correlation function yields:

(3.13)

To reach the righthand side, we have used the fact that the membrane is –by definition– localized along , so we know that it has a delta function support for its source. Observe that in our integral, the only legs of which actually participate are those which are normal to the Riemann surface . We denote these local coordinates by .

Figure 3: Depiction of a quantum Hall droplet of characteristic radius .

On the other hand, we also know that if we simply consider the action of the Laplacian in these four directions, then we have:

(3.14)

Integrating equation (3.13), we conclude that our integral is, in this limit, given by a quadratic form in the normal directions:

(3.15)

where we have introduced the localized flux density:

(3.16)

At this point, we can now see why the presence of a background four-form flux prevents the membrane from collapsing to zero size. Indeed, from the above analysis, we see that if we consider the width of the membrane in the directions normal to , there is a characteristic size indicating the spread of the associated Gaussian:

(3.17)

where denotes the component of the four-form flux with all legs transverse to . Continuing in the same vein, we see that provided we have at least three independent contributions to the four-form flux, we always get a puffed up membrane. Said differently, there is no direction we can place a membrane so that it is not polarized.

It is helpful at this point to compare with the standard case of the 2D quantum Hall effect. There, we also have a droplet size, with characteristic length set by , in appropriate units. Here, we see the analog of this formula in equation (3.17). In contrast to that case, however, we see that since the four-form flux always picks a preferred set of directions in the 6D space, we actually get a tensor of such characteristic lengths.

3.2 Zero Slope Limit of the Membrane

In the limit of a large magnetic flux, we also expect the worldvolume theory of the membrane to simplify. Here we present a brief sketch of how we expect such a simplification to occur. Some elements of our discussion are necessarily schematic, but we anticipate it will be useful for further investigations.

Consider the topological coupling between the three-form potential and the membrane:

(3.18)

where is an overall constant of proportionality set by the tension of the membrane, is the embedding of the worldvolume of the membrane in the 7D spacetime, and denotes the pullback of the bulk three-form onto the worldvolume. The embedding of the membrane is captured by fields . So, we can alternatively write the form of this coupling in components, where we absorb various numerical pre-factors into the overall coupling:

(3.19)

We are in particular interested in the special case of a large, uniform magnetic G-flux. In this limit, we anticipate that the action is actually dominated by this topological term. We refer to this as the zero-slope limit action:

(3.20)

The general algebraic structure associated with flat and large has been considered quite extensively in the literature (see e.g. [52, 53, 54] and references therein). As far as we are aware, however, the case of large G-flux has not been considered in much detail. Part of the complication with this sort of coupling is that it leads to a generalization of the Poisson bracket to a Nambu 2-bracket [55]. The quantum theory contains many technical complications, including the appearance of non-associative and non-commutative algebraic structures.

We bypass these complications (interesting though they may be) by focusing exclusively on the low energy limit, i.e. leading derivative contributions to the effective theory. This is accomplished most cleanly in the limit where we have a large background G-flux, since in this case, the membranes are polarized, and most fluctuations of the worldvolume will decouple.

Consider, for example, the expansion of the fields . In the limit of slow fluctuations, we can write:

(3.21)

where to leading order, the ’s have no position dependence, and the ’s have no time dependence. In this limit, then, we see that in the action of equation (3.20), each field can support at most one worldvolume derivative. Consequently, we can integrate by parts and present the action in the suggestive form:

(3.22)

Observe that the composite operators define a collection of abelian gauge fields on the three-dimensional space:

(3.23)

Interpreted in this way, we see that the G-flux can be split according to pairs of indices and , and with respect to these indices, it is a symmetric matrix. Combining the parameter with that from , we can now present the canonical form of the zero slope limit in terms of the gauge fields and a dimensionless matrix of couplings :

(3.24)

Consider now the quantization of this worldvolume theory. In the gauge where , the canonical commutation relation is:

(3.25)

The above commutation relation is the direct analog of what one finds for electrons moving in a large magnetic field. Indeed, in that context, the resulting 1D topological quantum mechanics is the starting point for the zero slope limit of open string theory in the presence of a large Neveu-Schwarz B-field [56]. It is quite tempting to also consider the limit of a large number of M2-branes, say in M-theory. The backreaction of the four-form flux on an individual membrane should then give, in a suitable limit, a Chern-Simons action. It would be interesting to connect these observations to the constructions presented in [57, 58].

3.3 Large Rigid Limit

Now that we have determined the impact of the four-form flux on the size of the droplets, we turn to the evaluation of the rest of the Laughlin wave function in line (2.20). This requires us to specify a pair of Riemann surfaces and , as well as a pair of charges and . In the limit where the background magnetic flux is very large, we see that the size of the membrane in the transverse directions is quite small, going roughly as . In this limit, then, we approximate the membranes as wrapping very large Riemann surfaces (in units of the membrane tension length), and very thin in the transverse directions. To perform explicit computations, it is helpful to introduce a complex structure for , writing with local coordinates such that the two Riemann surfaces are locally defined by the equations:

(3.26)

so that the common normal coordinate for both Riemann surfaces is parameterized by the -plane, and the holomorphic coordinate:

(3.27)

specifies the separation between the two Riemann surfaces.

To evaluate the correlation function of and in this limit, we observe that in the two-point function for the field, we are integrating (in momentum space) over the and directions. Consequently, we are actually working in the limit of low momentum in these two directions. The correlation function will therefore be dominated by momenta in the -plane. With this in mind, we see that our problem reduces to a two-dimensional system.

In the related context where we compactify our 6D CFT on the space , the low energy theory on is governed by a collection of chiral and anti-chiral bosons which all descend from the anti-chiral two-form. Indeed, we can decompose the -field on shell as:

(3.28)

where is a basis of harmonic anti-self-dual two-forms on , and is a basis of self-dual two-forms on . Here, the index and so that the are chiral bosons and the are anti-chiral bosons. Integrating over the Riemann surfaces, we find that the correlation function reduces to:

(3.29)

where in general the values of the and depend on integrating the basis of self-dual and anti-self-dual forms over the Riemann surfaces. This in turn depends on the details of the metric. There is, however, an important aspect of this correlator which is protected by topology (see e.g. [59, 60, 61, 62]):

(3.30)

Note that single-valuedness of the associated OPE requires us to work in terms of charges and which scale in appropriate units of , as per our discussion below equation (2.12). The simplest possibility is to take , though more generally we can contemplate and so that the difference becomes:

(3.31)

Returning to the evaluation of our correlation function, we see that there is a rather close similarity to the case of line (3.29). The main difference is that in the limit we have just taken, we have discarded various global data such as the topology of the Riemann surfaces. By construction, however, we have assumed that the only intersections occur when . Consequently, we see that we can essentially carry over unchanged the calculation in the 2D limit. The precise values of the exponents and also require information about the explicit choice of metric as well as the dynamics of the membranes moving in a background charge density.

With this in place, we now generalize to other configurations of affine planes. It is helpful to introduce holomorphic homogeneous coordinates with .9 The Riemann surfaces can then be specified as:

(3.32)
(3.33)

In this case, the analog of the holomorphic separation between the two Riemann surfaces is now given by:

(3.34)

Assuming we remain in the rigid limit for all surfaces, we see that the resulting contribution to the Laughlin wavefunction in line (2.20) takes the form:

(3.35)

in the obvious notation.

3.4 Point Particle Limit

We can also evaluate the form of the many body wavefunction in the limit where the relative separation between a pair of Riemann surfaces is quite large. In this case, the membranes are well-approximated by point particles moving in six spatial dimensions, and interacting via exchange of the bulk 7D three-form. We can therefore proceed in two complementary ways. On the one hand, we can simply calculate the scattering amplitude between two non-relativistic M2-branes. Alternatively, we can work directly in terms of the 6D CFT, and compute the long distance limit of the two-point function for the anti-chiral two-forms, suitably integrated over the small Riemann surfaces.

Figure 4: Exchange of an anti-chiral two-form between two membranes wrapped on Riemann surfaces and .

In either case, the problem reduces to that of a scattering amplitude. We calculate the spin averaged value, neglecting issues of fine structure. Indeed, if we were to treat the Riemann surface as fixed, we would get a four-form source given by the corresponding delta function , which dualizes to a two-form and couples to the -field via . At this point, it is convenient to work in Feynman gauge for the two-point function of the non-chiral two-form:

(3.36)

The spin averaged amplitude is then given by:

(3.37)

Now, the amplitude receives two equal contributions, one from a basis of chiral two-forms and another from anti-chiral two-forms, with no cross-terms between the two. Because of this, the chiral case is half as large. Putting this together, we reach our estimate for the correlation function in this limit:

(3.38)

Observe that as we separate the particles, this correlator tends to one. In the opposite limit, the apparent divergence is cut off by the short distance behavior already described in the previous subsection. The crossover between these two regimes occurs precisely when the characteristic size of the Riemann surface becomes comparable to the separation.

4 Quasi-Branes and Quasi-Dranes

As mentioned at the beginning of section 3, we can extend this analysis to more general pairings . Indeed, there is a well-known interpretation of this in the standard fractional quantum Hall effect in terms of quasi-particle excitations / holes and their associated emergent gauge fields. To better understand this in our system, it is actually helpful to view the 7D bulk topological field theory as obtained from the compactification of an 11D theory of a single abelian five-form with action:

(4.1)

Compactifying on a four-manifold, we assume the 11D spacetime takes the form of a Cartesian product . There is a decoupled sector given by a theory of three-forms. To study the structure of this subsystem, we can consider a basis of two-cycles with pairing . Dual to these cycles are harmonic two-forms . To perform the reduction and maintain an integral basis of fields, it is actually most convenient to work in terms of the related basis of two-forms . Note that we also have:

(4.2)

Expanding the five-form in terms of a basis of harmonic two-forms on yields:

(4.3)

where the other terms in “…” refer to decoupled sectors. In this 11D theory, the degrees of freedom in the boundary are Euclidean D3-branes. These are wrapped over two-cycles of , and this gives rise to the membranes discussed in the previous section. In this geometric construction, we take D3-branes wrapped over collapsing two-cycles. This leads to effective strings in six dimensions, which at the conformal fixed point have vanishing tension.

Indeed, in F-theory, the construction of 6D SCFTs involves compactification on a singular base with a discrete subgroup of . Not all discrete subgroups realize a 6D SCFT, in part because they are incompatible with the existence of an elliptically fibered Calabi-Yau threefold with base . In the resolved phase, we have a generalization of Dynkin diagrams of ADE type. In fact, only the A- and D- series can have curves of self-intersection different from . For additional details on the construction of 6D SCFTs in F-theory, see references [34, 36] as well as [67].

An intriguing feature of 6D SCFTs with an F-theory origin is the appearance of continued fractions such as:

(4.4)

As discussed in the introduction, the fractional quantum Hall effect also exhibits a sequence of continued fractions, and these numbers specify filling fractions for the spectrum of physical excitations above the ground state [2, 3, 4, 5, 6]. From an 11D perspective we can explain the appearance of this structure in terms of geometrical properties of the internal directions.

For example, in the generalization of an A-type base, this is a collection of curves of self-intersection with pairing:

(4.5)

The orbifold singularity is then given by the group action on with local coordinates and as:

(4.6)

for a primitive root of unity (for example ).

Here we see that the spectrum of “quasi-particles” are actually Euclidean effective strings dictated by the continued fraction of line (4.4)! Moreover, we also know that at least for 6D SCFTs realized in F-theory, the space of possible pairings is tightly constrained. For example, the self-intersection numbers must always obey . Additionally, further blowups of the base do not shift the value of or , but do introduce additional quasi-branes [32, 68].

Turning the discussion around, it is natural to ask whether there is a top down interpretation of “quasi-dranes” (the negation of a brane). Physically, this would appear to descend from anti-Euclidean D3-branes. We leave an analysis of this issue for future work.

5 Compactification

One of the general paradigms of many condensed matter systems is the presence of a gapped bulk coupled to edge modes. It is also widely believed that this gapped phase is described by a topological field theory. Motivated by these considerations, in this section we use our higher-dimensional starting point as a tool in generating consistent examples of such phenomena. Our aim in this section will be to understand the class of theories generated from compactifications of our 7D bulk theory, and the corresponding edge modes.

In fact, following up on our discussion in section 4, it is helpful to actually begin with an 11D bulk theory of five-forms with action:

(5.1)

Some aspects of this theory have been studied in [29, 30]. Suppose now that we restrict the form of the 11D spacetime to be a product of the form:

(5.2)

in which is a Lorentzian signature spacetime and is a Euclidean signature space. From the perspective of the boundary theory, we can take a limit in the space of metrics where is relatively small compared with . In this sense, we can “compactify” and reach a lower-dimensional theory defined solely on .

To better understand the resulting theory, we decompose into a basis of harmonic differential forms defined on :

(5.3)

namely, we sum over all compact harmonic forms with degeneracy label , and also sum over all choices of . Here, .10 There is a canonical pairing on between an -form and a -form given by:

(5.4)

which defines a matrix of integers. The compactification of our 11D theory therefore reduces to a theory of abelian differential forms:

(5.5)

Observe that this action breaks up into different non-interacting sectors. We always have a theory of -forms coupled to -forms, but these do not interact with the other sectors.11 For this reason, we can treat these contributions independently.

Note that we do not assume is compact. This means that the intersection pairings we generate, and thus the resulting matrix of couplings need not be square matrices, and when they are square, they need not have determinant one. For square matrices with , the boundary theory is most appropriately viewed as a relative quantum field theory in the sense of reference [31] (see also [33]).

5.1 Examples

Let us give a few examples to show how we recover various topological field theories from this point of view.  Isolating the contributions from the middle degree forms, we get the following bulk theories:

(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)

in the obvious notation. There is a vast literature on nearly all of these theories, and so we shall limit our discussion to a few general comments.

These bulk topological field theories fall into two general subclasses, namely Chern-Simons-like theories with potentials of the same degree and either symmetric or anti-symmetric pairings, and BF-like theories with forms of different degree. In all of these cases, we expect to realize interesting edge mode dynamics, which in many cases can be understood from the compactification of a chiral four-form in ten dimensions to the lower-dimensional setting. Supersymmetry provides an additional extension of these results and leads to an even broader class of lower-dimensional theories.

One of the other lessons from string theory is that additional light degrees of freedom are expected to emerge in limits where the compactification manifold develops singularities. From this perspective, we can see that in many cases, we should expect to realize both additional non-abelian structure and higher spin currents in the boundary theory.

BF-like theories

We begin our discussion on effective topological field theories focusing first on dimensions , and . Here, in general, we expect a BF-like theory as in equations (5.7), (5.9), (5.11).

Such BF theories feature prominently in long distance limits of various high energy physics systems and also play an important role in the description of gapped phases of matter. For example, a four-dimensional action similar to (5.9) was instrumental in the description of novel bosonic symmetry protected topological phases [69].

The 6D BF-like theory appears to have not received as much attention. Some details about this theory (and about BF theories of various dimensions) can be found in Appendix A of [27]. It would be very interesting to determine the resulting theory of edge modes. It would also likely shed further light on the compactification of 6D SCFTs to five dimensions (see e.g. [70, 71, 72]).

CS-like theories

Consider next the Chern-Simons-like theories in which the differential forms all have the same degree. This occurs when the number of spacetime dimensions is odd.

The most familiar example in this class is given by abelian 3D Chern-Simons theory (5.10), which as we have already remarked is helpful in the study of the fractional quantum Hall effect [9, 13, 14, 15]. Our interest in this paper has of course been the 7D generalization of this to three-forms.

Note that the theories in dimensions and have a symmetric matrix of couplings, as dictated by the intersection pairing on the internal space. By contrast, the theories in dimensions and have an anti-symmetric matrix of couplings, again in accord with the structure of the internal intersection pairing.

Finally, the 5D abelian Chern-Simons theory of two-forms in line (5.8) has appeared both in the condensed matter and high energy theory literature. For example, it appears in the low energy effective action of type IIB string theory on the background [25, 26, 27]. Alternative applications of line (5.8) concern the study of both gapped phases of matter [73] and discrete symmetries of gauge [74].

6 Embedding in M-theory

In our discussion up to this point, we have deliberately phrased our entire discussion in terms of a 7D topological field theory which is well-defined in its own right. It is nevertheless of interest to see how the 7D Chern-Simons-like theory we have been studying arises in compactifications of M-theory. An added benefit of this approach is that we will automatically show that there is a supersymmetric extension of our 7D theory.

Along these lines, we now take the 6D boundary theory to be a Lorentzian signature manfold, so that the 7D bulk coordinate is an additional spatial direction. Recall that we are interested in physical systems with interacting degrees of freedom localized along the 6D boundary which realize a chiral conformal field theory. At present, the only way to construct examples of such theories are supersymmetric and involve embedding in string / M-theory / F-theory, the first examples of this type being found in references [75, 76, 77].

A helpful example to keep in mind is the special case of M5-branes filling the first factor of for a discrete subgroup of . We realize a conformal fixed point when the M5-branes all sit at the orbifold singularity of and a common point of the factor. This realizes the so-called class conformal field theories studied in references [35, 78, 79]. We pass to the partial tensor branch of the theory where effective strings have a tension by moving the M5-branes apart from one another in the direction. Observe that the geometry is seven-dimensional, so it is natural to expect a bulk 7D theory to reside here which couples to the 6D boundary defined by the M5-branes.

Let us now turn to the construction and study of this putative 7D massive supermultiplet which contains a three-form potential and is decoupled from gravity. To see how this comes about, it is actually helpful to proceed up to eight dimensions, where supersymmetry has a chiral structure. Here, we have the following 8D massless supermultiplets (see e.g. [80, 81]):

8D Massless Supermultiplets: (6.1)
(6.2)
(6.3)
(6.4)

where the notation , , , , respectively refers to an 8D scalar, Weyl fermion, vector boson, gravitino and graviton. The notation refers to a -form potential. We also have massive supermultiplets, where we denote massive fields by an overline:

8D Massive Supermultiplets: (6.5)
(6.6)
(6.7)

In terms of the field content, we have the following relations:

(6.8)
(6.9)
(6.10)

The 8D multiplet arises from M-theory compactified on a .

Proceeding now to seven dimensions, we obtain the following irreducible 7D supermultiplets:12

(6.11)
(6.12)
(6.13)

The presence of the additional factor of two in the gravitino and Weyl spinors has to do with the way we count degrees of freedom for our spinors; In 8D we have a Weyl spinor, whereas in 7D we cannot have a Weyl spinor, and instead impose an appropriate reality condition.

Note that we have not “removed a vector multiplet” from the gravitino multiplet . The reason this is not correct to do can be seen either by directly constructing the appropriate supermultiplet, or indirectly, by considering a further reduction to 6D, where removing such a multiplet would make it impossible to construct appropriate 6D supermultiplets. To see this explicitly, we decompose further into fields:

(6.14)
(6.15)

And in six dimensions, we observe that we have the following gravitino multiplets:

(6.16)