6D fractional quantum Hall effect
Received: March
6D fractional quantum Hall e ect
Jonathan J. Heckman 0 1 3
Luigi Tizzano 0 1 2
0 Box 516 , SE75120 Uppsala , Sweden
1 Philadelphia , PA 19104 , U.S.A
2 Department of Physics and Astronomy, Uppsala University
3 Department of Physics and Astronomy, University of Pennsylvania , USA
We present a 6D generalization of the fractional quantum Hall e ect involving membranes coupled to a threeform potential in the presence of a large background ux. The low energy physics is governed by a bulk 7D topological eld theory of abelian threeform potentials with a single derivative ChernSimonslike action coupled to a 6D antichiral theory of Euclidean e ective strings. We derive the fractional conductivity, and explain how continued fractions which gure prominently in the classi cation of 6D superconformal eld theories correspond to a hierarchy of excited states. Using methods from conformal eld theory we also compute the analog of the Laughlin wavefunction. Compacti cation of the 7D theory provides a uniform perspective on various lowerdimensional gapped systems coupled to boundary degrees of freedom. We also show that a supersymmetric version of the 7D theory embeds in Mtheory, and can be decoupled from gravity. Encouraged by this, we present a conjecture in which IIB string theory is an edge mode of a 10 + 2dimensional bulk topological theory, thus placing all twelve dimensions of Ftheory on a physical footing.
ChernSimons Theories; MTheory; Topological Field Theories; Topological

fourform
1 Introduction
2 7D bulk 3
Many body wavefunction
3.1
3.2
3.3
3.4
Landau wavefunction
Zero slope limit of the membrane
Large rigid limit
Point particle limit
Quasibranes and quasidranes
Compacti cation
5.1
Examples
Embedding in Mtheory
7 IIB as an edge mode
4
5
6
8
1
Conclusions
Introduction
the condensed matter and string theory literature, see e.g. [19{25]. There is by now a vast
literature on various ways that bulk gapped systems produce novel edge mode dynamics.1
1The list of such references is wellknown to experts. For a review of some aspects of this and other
spondence [27{29], and also as an interesting topological eld theory in its own right [30{35]:
with cI a collection of threeform potentials with gauge redundancy cI
subject to the imaginaryselfdual boundary condition (see e.g. [31]):
! cI + dbI and
S7D =
IJ Z
4 i
M7
cI ^ dcJ ;
(1.1)
(1.2)
(1.3)
Here, our sign convention is such that if we analytically continue to Lorentzian signature,
the resulting boundary condition will yield an antiselfdual threeform, in accord with the
convention adopted for 6D SCFTs in references [36{38]. The pairing
IJ is an integral,
positive de nite symmetric matrix which is the analog of the \Kmatrix" in the context
of the standard fractional quantum Hall e ect. The resulting 7D theory describes the low
energy dynamics of 2 + 1dimensional membranes. In the boundary, these will appear as
2D Euclidean strings coupled to antichiral twoform potentials bI such that:
cI = dbI :
In this context, the matrix
IJ 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 selfduality condition and quantization condition for threeform
uxes [31].2
To realize both the integer and fractional quantum Hall e ect, we activate a background
magnetic fourform eld strength, namely one in which all four legs thread the 6D boundary.
2The reason for this is that there is no integral basis of imaginaryselfdual threeforms on a generic
sixmanifold with compact threecycles. This is simply because the imaginaryselfduality condition depends
on a choice of metric, and as such, continuous variation of the metric destroys the chance to have an
correlation functions of nonlocal operators of the form:
0
Z
1
bI A ;
(1.4)
which from a 7D perspective involves replacing the integral of bI over a Riemann surface
by an integral of cI over a threechain 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.
nd that in the case of a free theory of antichiral twoforms, the
evaluation is relatively straightforward, and can be derived from general properties of
conformal invariance. Said di erently, 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 nd 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 eld 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 Ftheory), the full list of
IJ has actually been classi ed [36{39]. From a
geometric perspective, these matrices are nothing but the intersection pairings obtained from
the resolution of orbifold singularities of the form C2= U(
2
) for
U(
2
) a particular choice
of discrete subgroup of U(
2
). The structure of this singularity is in turn controlled by
continued fractions [40{42]:
p
q
= x1
x2
1
1
fractions for excitations above the ground state [2{6].
The 7D starting point also provides a unifying perspective on a variety of
lowerdimensional bulk topological systems coupled to dynamical edge modes. In these
systems, the analog of the Kmatrix is dictated by a tensor product
(7D)
(intersect), with
(intersect) the intersection pairing on the compacti cation manifold. Indeed, even our 7D
system can be viewed as the compacti cation of an 11D theory of veforms placed on a
background of the form M7
M4.
The higherdimensional uni cation in terms of this 11D theory suggests an irresistible
further extension to twelve dimensions, especially in the context of Ftheory, the
nonperturbative 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 10 + 2 dimensions. To avoid
pathologies with having two temporal directions (such as moving along closed timelike
(1.5)
{ 3 {
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 9 + 1dimensional spacetime.
The rest of this paper is organized as follows. First, in section 2, we discuss in more
detail the bulk 7D topological eld 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 antichiral
tensors. In section 4 we discuss the spectrum of quasiexcitations, and its interpretation
in terms of an 11D topological eld theory of veforms. In section 5 we brie y discuss
compacti cations of the 7D (and 11D) bulk topological eld theory to lower dimensions.
Section 6 discusses the embedding of the 7D theory in Mtheory, and section 7 presents
our conjecture on Ftheory as a 10 + 2dimensional 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 higherdimensional
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 rst develop the candidate
e ective eld theory which would govern the low energy physics.
Our starting point is the 7D action:3
(2.1)
(2.2)
S7D[cI ] =
cI ^ dcJ ;
IJ Z
4 i
M7
M7 = Rtime
M6 :
in which we further assume that the sevenmanifold is given by a product of the form:
Said di erently, when we proceed to quantize the theory, a wavefunction will involve
coordinates de ned on M6, which we then evolve from one time slice to another.
There are various subtleties in quantizing such ChernSimonslike theories, and we
refer the interested reader to the careful discussion presented in references [31, 32, 44, 45].
3Following [29], in order to properly de ne (2.1), we need to specify the global nature of the elds c's
and their gauge transformations. In this paper we assume that cI
cI + Z
p
2 Z(M7), where Z
2p Z(M7) is the
therefore need to specify a Lagrangian splitting of the phase space [31].
0
Z
S
1
for S 2 H3(M6; Z), and mI a vector of charges. The pairing IJ de nes an integral lattice
, and the mI take values in its dual
.
The bulk correlation function of such observables is:
h m1 (S1) m2 (S2) : : : mN (SN )i7D = exp42 i mI
2
1 IJ
mJ
X
where L(Si; Sj ) is the integral linking number of Si and Sj . Given two bulk operators
m (S) and
n (T ) for mI ; nJ 2
and S; T 2 H3(M6; Z), we get the braid relations [27]:
m (S) n (T ) =
n (T ) m (S)
exp 2 i mI
1 IJ nJ (S T ) ;
(2.7)
where S T is the intersection pairing for threecycles in M6.
The braiding relation of equation (2.7) tells us that in the quantum theory, we cannot
simultaneously specify the periods of the threeform for all threecycles in M6. Rather, we
must take a maximal sublattice of commuting periods and use this to specify the ground
state(s). Calling this lattice of threecycles L
H3(M6; Z), the degeneracy of the ground
state is:4
dGND = (det )dim L :
(2.8)
The threeform potential couples to 2 + 1dimensional membranes via integration over
its worldvolume. These degrees of freedom are the analog of the electrons present in the
fractional quantum Hall e ect. In contrast to that case, however, there is no guarantee
4A more formal way to understand this relation is to note that states of the 7D bulk theory must
assemble into representations of the Heisenberg group H 3(M6;
= ). The maximal set of commuting
uxes L
H3(M6;
the case of a 3D theory on R
= ) leads to a ground state degeneracy of dimension j
= j
dim L. Note also that in
for
a genus g Riemann surface, the degeneracy would be (det )g. Here
e ects of this (which do not alter the degeneracy of the ground state) can be found in reference [35].
M7
(2.9)
(2.10)
(2.11)
(2.12)
HJEP05(218)
the 7D bulk by M7, the 6D boundary by @M7 and time running into the boundary by t.
that these membranes are the genuine microscopic degrees of freedom for the 7D system.5
Indeed, from the perspective of Mtheory, it is widely expected that M2branes are also
simply a collective excitation.
Suppose now that we consider our bulk theory on the spacetime
M7 = R 0
M6 ;
so that at t = 0, we realize the boundary @M7 = M6, namely we evolve a state from
t =
1 to t = 0. See gure 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:
which if we analytically continue to M6 a Lorentzian signature manifold (with mostly +'s
in the metric) would de ne an antiselfduality relation.
From the perspective of the 6D boundary, we interpret the cI 's as threeform
eld
strengths for twoforms bI :
cI = dbI :
The twoform couples to a Euclidean e ective string, namely the spatial section of our
2 + 1dimensional membrane. Including a membrane of charge mI at a particular time can
therefore be described by inserting into the path integral the operator:
0
Z
1
5Moreover, there is a problem with formulating a theory of rst quantized membranes as fundamental
degrees of freedom as this generically introduces instabilities [46] (see also the review [47]). Here, however,
there is not much of an issue since we are adopting an e ective eld theory perspective. Indeed, even if the
membrane disintegrates, the macroscopic behavior we are considering should remain sensible since we are
only interested in collective features.
{ 6 {
M7
S7D[cI ; j] =
cI ^ dcJ
i jI ^ cI :
IJ Z
4 i
M7
jI =
2
IJ dcJ :
is a threechain with boundary
a Riemann surface wrapped by the
where
= R 0
membrane. See gure 2.
The choice of charge vector mI for the membrane depends on whether we view it as
a dynamic or static object in the 6D boundary theory. Indeed, though the mI take values
in the dual lattice
, we can also entertain a special class of charges in the sublattice
by instead working with mI =
IJ nJ for n
J 2
. 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 jI to the action as follows:
The equations of motion in the presence of a background source are:
S7D[cI ; C] =
cI ^ dcJ +
IJ Z
twopoint function for these threeindex objects de nes a sixindex conductivity
with all indices di erent.
Suppose now that we have a background threeform C which couples to the membranes.
In this interpretation, the cI simply correspond to various emergent gauge elds at low
energies. The coupling between the two is:
(2.13)
(2.14)
abcdef ,
(2.15)
HJEP05(218)
Physically, the membranes are all charged under C, which has eld strength G. In our
conventions, the G ux is quantized in units of 2 :6
1
2
Z
1
2
G 2 Z:
Much as in the lowerdimensional setting, we then obtain a fractional conductivity given by:
abcdef =
1 IJ
I J .
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 fourform
ux 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 threechain
with boundary a Riemann surface , and some
choice of charge vector mI in the dual lattice of charges
. We label each such insertion
in the path integral by an operator
(i) for i =; 1 : : : ; N . The operator associated with a
background fourform
ux follows from line (2.15):
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
0
I Z
D7
1
0
1
bI ^ GA ;
I Z
D6
Laughlin =
D (
1
) : : : (N) bkgndE6D
;
where the correlation function is evaluated in the boundary 6D theory.
(i) wrapped by the membranes.
for N such membranes to take the form:
where D7 = R 0
D6 is a sevendimensional domain, which at t = 0 is given by a
sixdimensional domain D6 which has no overlap with the locations of the Riemann surfaces
Putting all of this together, we expect the (unnormalized) many body wavefunction
Assuming that our 6D boundary theory is actually free, everything reduces to the
calculation of appropriate twopoint functions:
where we have introduced the unnormalized Landau wavefunction for a single membrane
moving in a background fourform
ux:
(i)
Landau = h bkgnd (i)i6D:
Our goal will be to estimate
Laughlin in various limits.
6Strictly speaking, what is really required is that the integration over a di erence of two fourcycles is
integral. There can still be a halfintegral shift, and this plays an important role in compacti cations of
Mtheory [48].
{ 8 {
In the previous section we presented a bulk perspective on the 6D fractional quantum Hall
e ect. 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
at space, namely R6. Note that our result also generalizes to other conformally
at
spaces such as S6, or a 6D ball (namely the hyperbolic space H6). For ease of exposition,
we shall focus on the theory of a single emergent threeform potential c, which couples to
a uniform background magnetic fourform
ux G = dC (locally), so we consider:
S7D[c; C] =
c ^ dc +
C ^ dc:
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 antichiral twoforms. In the case of a 2D chiral boson in
at space, there is a
wellknown (noncovariant) action given in reference [49]. In principle one could construct
a similar action for the 6D antichiral twoform.
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 twopoint function for nonlocal operators
such as:
m( ) m0 ( 0) 6D =
*
0
Z
1
0
for some choice of charges m and m0, and Riemann surfaces
and 0 wrapped by the
membranes. Here, the prime on b0 serves to remind us that the potential is supported on 0. For
now, we treat these Riemann surfaces as xed, though when we turn to the analysis of the
Landau wavefunction, we will show how to x their mean eld values. Since we are dealing
with a free eld theory, we apply Wick's theorem to such correlation functions to write:
Z
0
1+
6D
;
Z Z
0
bb0
+
6D
:
We therefore need to extract the integrated twopoint function for the b elds 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 di erently, our operators and correlation
functions are wellde ned.
As we have already remarked, we do not have a covariant action for our antichiral
twoform. Indeed, this is not even an operator in the 6D CFT. Rather, we know that the
threeform
eld strength h given locally by db is a wellde ned operator. Our strategy will
therefore be to compute the twopoint function for h, and to then integrate this twopoint
function over a pair of threechains with boundaries
and 0, respectively.
{ 9 {
where we have introduced the relative separation:
ra = x
a
ya:
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 antichiral twoform theory, we apply a projection to
imaginaryselfdual eld strengths, namely we set:
h =
(h(nc)
i 6D h(nc)):
The answer in this case follows directly from that given for a theory of twoforms with
no selfduality constraint imposed. We follow the procedure outlined in references [50{53].
Denoting the eld strength for this nonchiral twoform by hnc, the twopoint function is:
Dh(an1ca)2a3 (x)h(bn1bc2)b3 (y)
E
6D
=
18
3
2
1
m( ) m0 ( 0) 6D = exp
Z Z
0
hh0
+
6D
;
Laughlin =
Y
(3.4)
(3.5)
The twopoint function for the chiral theory is then:
hha1a2a3 (x)hb1b2b3 (y)i =
r[a1 r[b1 ab22 ab33]] +
r "ca1a2a3
which is a somewhat more involved expression than its counterpart in two dimensions.
To obtain the integrated twopoint function for the chiral twoforms, we now formally
integrate this result over a pair of threechains
and 0 inside of R6:
=
and @ 0 = 0. Returning to equation (2.20), we see that our expression for
the many body wavefunction:
reduces to the calculation of these integrated twopoint functions for the threeform
uxes.
So far, we have followed a quite similar plan to what is typically done in the 2D
fractional quantum Hall e ect; 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:
Depending on the strength of the fourform
ux, the membrane may either pu up to a
large rigid object, or may instead be more accurately approximated by a point particle.
This will in turn a ect 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 [51, 54]. 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 \pu ng up" for the membrane in
all six spatial directions. This in turn depends on the particular pro le for the fourform
ux. 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 fourform
ux in which the membranes are
wellapproximated 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:
*
0
2 i
1
0
Z
1+
bA
Z
D6
6D
=
*
0
Z
1+
bA
6D
:
(3.11)
One of the implicit assumptions we have made up to this point is that the Riemann surface
is held
xed. We now show that the presence of the fourform
ux actually causes the
membrane to pu up. As mentioned at the beginning of this section, we take the fourform
ux to be uniform, and write:
G =
1
4! Gabcddxa ^ dxb ^ dxc ^ dxd
where xa are local coordinates on R6 and we have taken some choice of constant Gabcd.
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 ya. Applying the Laplacian in the y variable to the correlation
function yields:
(y)
*Z G
D6
2
Z
+
To reach the righthand side, we have used the fact that the membrane is  by de nition 
localized along
, so we know that it has a delta function support for its source. Observe
that in our integral, the only legs of G which actually participate are those which are
normal to the Riemann surface . We denote these local coordinates by y .
?
1=pG.
On the other hand, we also know that if we simply consider the action of the Laplacian
in these four directions, then we have:
?y?2 = 8:
Integrating equation (3.13), we conclude that our integral is, in this limit, given by a
quadratic form in the normal directions:
where we have introduced the localized ux density:
* Z G
D6
2
? =
Z
At this point, we can now see why the presence of a background fourform
ux
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:
where G
? denotes the component of the fourform
ux with all legs transverse to
.
Continuing in the same vein, we see that provided we have at least three independent
contributions to the fourform
ux, we always get a pu ed up membrane. Said di erently,
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
e ect. 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). See
also
gure 3. In contrast to that case, however, we see that since the fourform
ux
always picks a preferred set of directions in the 6D space, we actually get a tensor of such
characteristic lengths.
1
2`2
?
G
16
?
(3.14)
(3.15)
(3.16)
(3.17)
1=pF ,
In the limit of a large magnetic ux, we also expect the worldvolume theory of the
membrane to simplify. Here we present a brief sketch of how we expect such a simpli cation 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 threeform potential and the membrane:
SM2
3
Z
M3
i C(3);
where 3 is an overall constant of proportionality set by the tension of the membrane, i :
M3 ! M7 is the embedding of the worldvolume of the membrane in the 7D spacetime, and
i C(3) denotes the pullback of the bulk threeform onto the worldvolume. The embedding
of the membrane is captured by
elds Xa( 1; 2; t). So, we can alternatively write the
form of this coupling in components, where we absorb various numerical prefactors into
the overall coupling:
SM2
CabcdXa
^ dXb ^ dXc:
We are in particular interested in the special case of a large, uniform magnetic G ux. In
this limit, we anticipate that the action is actually dominated by this topological term. We
refer to this as the zeroslope limit action:
will decouple.
uctuations, we can write:
Szeroslope = 3
GabcdXadXb ^ dXc ^ dXd:
The general algebraic structure associated with C
at and large has been considered quite
extensively in the literature (see e.g. [55{57] and references therein). As far as we are aware,
however, the case of large G ux 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 2bracket [58]. The quantum theory contains many technical complications,
including the appearance of nonassociative and noncommutative 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 e ective theory.
This is accomplished most cleanly in the limit where we have a large background G ux,
since in this case, the membranes are polarized, and most uctuations of the worldvolume
Consider, for example, the expansion of the elds Xa( 1; 2; t). In the limit of slow
(3.18)
(3.19)
(3.20)
3
Z
M3
Z
M3
Xa( 1; 2; t) = Xcam(t) + Pcam( ) + : : : ;
(3.21)
where to leading order, the Xcm's have no position dependence, and the Pcm's have no time
dependence. In this limit, then, we see that in the action of equation (3.20), each Xa eld
can support at most one worldvolume derivative. Consequently, we can integrate by parts
and present the action in the suggestive form:
Z
M3
Szeroslope = 3
Gabcd"
(3.22)
Observe that the composite operators X[a@ Xb] de ne a collection of abelian gauge elds
on the threedimensional space:
Interpreted in this way, we see that the G ux can be split according to pairs of indices
ab and cd, and with respect to these indices, it is a symmetric matrix. Combining the
parameter 3 with that from G, we can now present the canonical form of the zero slope
limit in terms of the gauge elds Y ab and a dimensionless matrix of couplings Kab;cd:
Szeroslope =
Kab;cd Z
Consider now the quantization of this worldvolume theory. In the gauge where Y0ab = 0,
the canonical commutation relation is:
Kab;cd[Y ab( ); Y cd( 0)] =
2 i"
2
(
0):
(3.25)
The above commutation relation is the direct analog of what one nds for electrons
moving in a large magnetic
eld. 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 NeveuSchwarz B eld [59]. It is quite tempting to also consider
the limit of a large number of M2branes, say in Mtheory. The backreaction of the
fourform
ux on an individual membrane should then give, in a suitable limit, a ChernSimons
action. It would be interesting to connect these observations to the constructions presented
in [60, 61].
3.3
Large rigid limit
Now that we have determined the impact of the fourform
ux 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 0, as well as a pair of charges m and
m0. In the limit where the background magnetic ux is very large, we see that the size of
the membrane in the transverse directions is quite small, going roughly as `e
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 R6, writing
C3 with local coordinates u; v; w such that the two Riemann surfaces are locally de ned by
1=p
G. In
the equations:
= fu = u0g \ fw = 0g
and
0 =
u = u00 \ fv = 0g ,
so that the common normal coordinate for both Riemann surfaces is parameterized by the
uplane, and the holomorphic coordinate:
speci es the separation between the two Riemann surfaces.
= u0
u0
0
(3.23)
(3.24)
(3.26)
(3.27)
To evaluate the correlation function of ( ) and
( 0) in this limit, we observe that
in the twopoint function for the b
eld, we are integrating (in momentum space) over
the w and v 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 uplane. With this in mind, we see that our problem reduces to a
twodimensional system.
In the related context where we compactify our 6D CFT on the space R
2
low energy theory on R2 is governed by a collection of chiral and antichiral bosons which
all descend from the antichiral twoform. Indeed, we can decompose the b eld on shell as:
HJEP05(218)
b = i( )!i + e ei( )! i;
e
where !i is a basis of harmonic antiselfdual twoforms on
dual twoforms on
0. Here, the index i = 1; : : : ; b2 (
e
0) and ei = 1; : : : ; b2+(
0, and ! i is a basis of
self0)
so that the i( ) are chiral bosons and the e ei( ) are antichiral bosons. Integrating over
the Riemann surfaces, we nd that the correlation function reduces to:
*
0
Z
1
0
=
e;
Z
0
1+
2D
and e
where in general the values of the
depend on integrating the basis of selfdual and
antiselfdual 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. [62{65] ):
e
=
(m
1m0)(
0):
e
=
(n n0)(
0):
(3.28)
(3.29)
(3.30)
(3.31)
Note that singlevaluedness of the associated OPE requires us to work in terms of charges
m and m0 which scale in appropriate units of , as per our discussion below equation (2.12).
The simplest possibility is to take m = m0 =
, though more generally we can contemplate
m = n
and m0 = n0 so that the di erence becomes:
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 di erence 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
= 0.
Consequently, we see that we can essentially carry over unchanged the calculation in the
2D limit. The precise values of the exponents
explicit choice of metric as well as the dynamics of the membranes moving in a background
and e also require information about the
charge density.
With this in place, we now generalize to other con gurations of a ne planes. It is
helpful to introduce holomorphic homogeneous coordinates Z
with Z4 = 1.7 The Riemann
surfaces can then be speci ed as:
= ff Z
0 = f 0 Z
= 0g \ fg Z
= 0 \ g0 Z
In this case, the analog of the holomorphic separation between the two Riemann surfaces
is now given by:
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:
in the obvious notation.
Point particle limit
(3.32)
(3.33)
(3.35)
(3.34)
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 wellapproximated by point particles moving in six spatial dimensions, and interacting
via exchange of the bulk 7D threeform. We can therefore proceed in two complementary
ways. On the one hand, we can simply calculate the scattering amplitude between two
nonrelativistic M2branes. Alternatively, we can work directly in terms of the 6D CFT,
and compute the long distance limit of the twopoint function for the antichiral twoforms,
suitably integrated over the small Riemann surfaces.
In either case, the problem reduces to that of a scattering amplitude as in gure 4.
We calculate the spin averaged value, neglecting issues of
ne structure. Indeed, if we
were to treat the Riemann surface as xed, we would get a fourform source given by the
corresponding delta function
(4), which dualizes to a twoform "cd and couples to the
beld via "cdbcd. At this point, it is convenient to work in Feynman gauge for the twopoint
function of the nonchiral twoform:
hb(anbc)(x)b(cndc)(y)i =
1
The spin averaged amplitude is then given by:
A(nc) =
m(i)
1m(j)
Vol( (i))
Vol( (j))
1
1
3 (x(i)
x(j))4
:
7A curious feature of this formulation is the appearance of homogeneous coordinates, and therefore
a CP3.
Additionally, by specifying our Riemann surfaces by a pair of points in CP3, we also obtain
the standard correspondence between twistor space and the complexi cation of conformally compacti ed
fourdimensional Minkowski space [66]. The presence of a background fourform
ux also suggests a
noncommutative (possibly covariant) deformation of this space (see e.g. [67{69]). It would be interesting to
develop a fourdimensional interpretation for the results of this paper.
Now, the amplitude receives two equal contributions, one from a basis of chiral
twoforms and another from antichiral twoforms, with no crossterms 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:
h
(i) (j)i6D = exp
m(i)
1m(j)
Vol( (i)) Vol( (j))
1
1
2 3 (x(i) x(j))4
: (3.38)
Observe that as we separate the particles, this correlator tends to one. In the opposite
limit, the apparent divergence is cut o
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
Quasibranes and quasidranes
As mentioned at the beginning of section 3, we can extend this analysis to more general
pairings IJ . Indeed, there is a wellknown interpretation of this in the standard fractional
quantum Hall e ect in terms of quasiparticle excitations/holes and their associated
emergent gauge elds. To better understand this in our system, it is actually helpful to view the
7D bulk topological eld theory as obtained from the compacti cation of an 11D theory of
a single abelian
veform with action:
Compactifying on a fourmanifold, we assume the 11D spacetime takes the form of a
Cartesian product M11 = M7
M4. There is a decoupled sector given by a theory of
threeforms. To study the structure of this subsystem, we can consider a basis of twocycles I
with pairing IJ =
I \ J . Dual to these cycles are harmonic twoforms !I . To perform
the reduction and maintain an integral basis of elds, it is actually most convenient to work
in terms of the related basis of twoforms !I =
IJ !J . Note that we also have:
Expanding the veform in terms of a basis of harmonic twoforms !I on M4 yields:
Z
M4
IJ =
!I ^ !J :
C(5) = c(I3) ^ !I + : : : ;
where the other terms in \. . . " refer to decoupled sectors. In this 11D theory, the degrees
of freedom in the boundary are Euclidean D3branes. These are wrapped over twocycles of
M4, and this gives rise to the membranes discussed in the previous section. In this geometric
construction, we take D3branes wrapped over collapsing twocycles. This leads to e ective
strings in six dimensions, which at the conformal xed point have vanishing tension.
Indeed, in Ftheory, the construction of 6D SCFTs involves compacti cation on a
singular base B = C2= U(
2
) with
U(2) a discrete subgroup of U(2).
Not all discrete
subgroups realize a 6D SCFT, in part because they are incompatible with the existence of
an elliptically
bered CalabiYau threefold with base B. 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 selfintersection di erent from
2. For additional details on the construction of
6D SCFTs in Ftheory, see references [36, 38] as well as [70].
An intriguing feature of 6D SCFTs with an Ftheory origin is the appearance of
continued fractions such as:
p
q
= x1
x2
1
1
As discussed in the introduction, the fractional quantum Hall e ect also exhibits a sequence
of continued fractions, and these numbers specify lling fractions for the spectrum of
physical excitations above the ground state [2{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 Atype base, this is a collection of curves of
selfintersection
xi with pairing:
2 x1
6
6
6
6
4
1
1 x2
1
= 6
1 : : :
1
1 xk 1 1 75
1 xk
3
7
7
7 :
7
(4.2)
(4.3)
HJEP05(218)
(4.4)
(4.5)
(4.6)
The orbifold singularity is then given by the group action on C2 with local coordinates u
and v as:
for
a primitive pth root of unity (for example
= exp(2 i=p)).
(u; v) 7! ( u; qv) ;
Here we see that the spectrum of \quasiparticles" are actually Euclidean e ective
strings dictated by the continued fraction of line (4.4)!
Moreover, we also know that
at least for 6D SCFTs realized in Ftheory, the space of possible pairings
constrained. For example, the selfintersection numbers must always obey 1
IJ is tightly
xi
Additionally, further blowups of the base do not shift the value of p or q, but do introduce
additional quasibranes [34, 71].
Turning the discussion around, it is natural to ask whether there is a top down
interpretation of \quasidranes" (the negation of a brane). Physically, this would appear to
descend from antiEuclidean D3branes. We leave an analysis of this issue for future work.
5
Compacti cation
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 eld theory. Motivated by these considerations, in this section we use
our higherdimensional 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 compacti cations of our 7D bulk theory, and the corresponding edge modes. For
simplicity, we focus on the case where there bulk
ux is switched o .
In fact, following up on our discussion in section 4, it is helpful to actually begin with
an 11D bulk theory of veforms with action:
4 i
M11
Some aspects of this theory have been studied in [31, 32]. Suppose now that we restrict
the form of the 11D spacetime to be a product of the form:
in which M11 p is a Lorentzian signature spacetime and Mp is a Euclidean signature space.
From the perspective of the boundary theory, we can take a limit in the space of metrics
where Mp is relatively small compared with @M11 p. In this sense, we can \compactify"
and reach a lowerdimensional theory de ned solely on M11 p.
To better understand the resulting theory, we decompose C(5) into a basis of harmonic
di erential forms de ned on Mp:
C(5) =
p bicpct(Mp)
X
X
i=0 ki=1
ki
^ !k(ii);
namely, we sum over all compact harmonic i forms with degeneracy label ki, and also sum
over all choices of i. Here, bicpct(Mp) = dim HDiR,cpct(Mp).8 There is a canonical pairing on
8Note that since we do not assume Mp is compact, we cannot assert a relation between bicpct(Mp) and
bcppcit(Mp).
(5.1)
(5.2)
(5.3)
HJEP05(218)
Mp between an iform !k and a (p i)form l given by:
(i);(p i) = h!k; li =
k;l
!k ^ l
;
Z
Mp
which de nes a matrix of integers. The compacti cation of our 11D theory therefore reduces
to a theory of abelian di erential forms:
S(11 p)D =
X
X
Observe that this action breaks up into di erent noninteracting sectors. We always have
a theory of (5
i)forms coupled to (5
p + i)forms, but these do not interact with the
other sectors.9 For this reason, we can treat these contributions independently.
Note that we do not assume Mp 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
det
6= 1, the boundary theory is most appropriately viewed as a relative quantum
eld
theory in the sense of reference [33] (see also [35]).
Let us give a few examples to show how we recover various topological eld theories from
this point of view. Isolating the contributions from the middle degree forms, we get the
following bulk theories:
(5.4)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
S7D =
S6D =
S5D =
S4D =
S3D =
S2D =
S1D =
IJ
(7D) Z
4 i
IJ
(6D) Z
2 i
IJ
(5D) Z
4 i
IJ
(4D) Z
2 i
IJ
(3D) Z
4 i
IJ
(2D) Z
2 i
IJ
(1D) Z
4 i
C(3) ^ dC(J3)
I
C(3) ^ dB(J2)
I
B(
2
) ^ dB(J2)
I
B(
2
) ^ dA(J1)
I
A(
1
) ^ dA(J1)
I
A(
1
) ^ d (J0)
I
(0) ^ d (J0);
I
9In a theory with additional bulk matter elds, there can be oneloop induced mixing terms upon
reduction to lower dimensions.
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 eld theories fall into two general subclasses, namely
ChernSimonslike theories with potentials of the same degree and either symmetric or
antisymmetric pairings, and BFlike theories with forms of di erent degree. In all of these
cases, we expect to realize interesting edge mode dynamics, which in many cases can be
understood from the compacti cation of a chiral fourform in ten dimensions to the
lowerdimensional setting. Supersymmetry provides an additional extension of these results and
leads to an even broader class of lowerdimensional theories.
One of the other lessons from string theory is that additional light degrees of freedom
are expected to emerge in limits where the compacti cation manifold develops singularities.
From this perspective, we can see that in many cases, we should expect to realize both
additional nonabelian structure and higher spin currents in the boundary theory.
We begin our discussion on e ective topological eld theories focusing rst on dimensions
6, 4 and 2. Here, in general, we expect a BFlike 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 fourdimensional action similar to (5.9) was instrumental in the
description of novel bosonic symmetry protected topological phases [72].
The 6D BFlike 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 [29]. It would be very interesting to determine the resulting theory of edge modes. It
would also likely shed further light on the compacti cation of 6D SCFTs to ve dimensions
(see e.g. [73{75]).
5.1.2
CSlike theories
Consider next the ChernSimonslike theories in which the di erential 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 ChernSimons
theory (5.10), which as we have already remarked is helpful in the study of the fractional
quantum Hall e ect [9, 13{15]. Our interest in this paper has of course been the 7D
generalization of this to threeforms.
Note that the theories in dimensions 7 and 3 have a symmetric matrix of couplings,
as dictated by the intersection pairing on the internal space. By contrast, the theories in
dimensions 5 and 1 have an antisymmetric matrix of couplings, again in accord with the
structure of the internal intersection pairing.
Finally, the 5D abelian ChernSimons theory of twoforms in line (5.8) has appeared
both in the condensed matter and high energy theory literature. For example, it
appears in the low energy e ective action of type IIB string theory on the background
AdS5
S5 [27{29]. Alternative applications of line (5.8) concern the study of both gapped
phases of matter [76] and discrete symmetries of gauge [77].
In our discussion up to this point, we have deliberately phrased our entire discussion
in terms of a 7D topological
eld theory which is wellde ned in its own right. It is
nevertheless of interest to see how the 7D ChernSimonslike theory we have been studying
arises in compacti cations of Mtheory. An added bene t 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 eld theory. At present, the only
way to construct examples of such theories are supersymmetric and involve embedding in
string/Mtheory/Ftheory, the rst examples of this type being found in references [78{80].
A helpful example to keep in mind is the special case of M5branes lling the rst
factor of R5;1
R?
C2= for
point when the M5branes all sit at the orbifold singularity of C2= and a common point
of the R? factor. This realizes the socalled class S
conformal eld theories studied in
references [37, 81, 82]. We pass to the partial tensor branch of the theory where e ective
strings have a tension by moving the M5branes apart from one another in the R? direction.
a discrete subgroup of SU(
2
). We realize a conformal xed
Observe that the geometry R5;1
R? is sevendimensional, so it is natural to expect a bulk
7D theory to reside here which couples to the 6D boundary de ned by the M5branes.
Let us now turn to the construction and study of this putative 7D massive
supermultiplet which contains a threeform 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 N = 1 massless supermultiplets (see
e.g. [83, 84]):
2
2
where the notation [0], [ 1 ], [1], [ 3 ], [2] respectively refers to an 8D scalar, Weyl fermion,
vector boson, gravitino and graviton. The notation [tp] refers to a pform potential. We
also have massive supermultiplets, where we denote massive elds by an overline:
3
2
3
2
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
8D Massless Supermultiplets:
G8D
N =1 = 1 [2] + 1
S(83D=2) = 1
V(81D) = 1 [1] + 1
+ 2 [0]
1
2
+ 2 [1] + 1
+ 1 [0] + 1 [t2]
8D Massive Supermultiplets:
V 8D(
1
) = 1 [1] + 1
In terms of the eld content, we have the following relations:
G8D
N =2 = G8D
N =1 + S(83D=2) + 2 V(81D)
S8D(3=2) = S(83D=2)
V 8D(
1
) = V(81D):
The 8D N = 2 multiplet G8D
N =2 arises from Mtheory compacti ed on a T 3.
Proceeding now to seven dimensions, we obtain the following irreducible 7D
supermultiplets:10
HJEP05(218)
G7D
So in other words, everything properly assembles into the following 6D multiplets:
8D ! 7D ! 6D
S(83D=2) ! S(73D=2) ! 2 S(+3=2) + 2 S(3=2):
10We thank D.S. Park for helpful discussions.
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
S(73D=2). 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 S(73D=2) further into 6D
elds:
8D ! 7D ! 6D
S(83D=2) ! S(73D=2) ! 2
3 +
2
+2
3
2
+10
+10
+8 [1]+8 [0]+4 [t2]:
And in six dimensions, we observe that we have the following N
= (1; 1) gravitino
multiplets:
6D Gravitino Supermultiplets:
S(+3=2) =
S(3=2) =
3 +
2
3
2
+ 2 [1] + 4
+ 2 [1] + 4
+ 2 [0] + 2 [t2]+
+ 2 [0] + 2 [t2] :
1
2
1
2
1
2
1
2
2
1 +
With this caveat dealt with, we now see that we have correctly identi ed a 7D
supermultiplet decoupled from the graviton:
One might now ask whether it is actually consistent to consider such a multiplet
decoupled from gravity. Indeed, the presence of the RaritaSchwinger elds would seem to
suggest that in any theory with a nontrivial Smatrix, gravity must indeed be included.
But if we have a trivial Smatrix (though a nontrivial theory!), we can consistently
decouple gravity and simultaneously retain our gravitino multiplet.11 To illustrate, observe
that when we compactify Mtheory on a fourmanifold, the action contains the following
terms with the threeform potential:
HJEP05(218)
Z
7D
M4
7D
Vol (M4)
dC(3) ^ dC(3) +
C(3) ^ dC(3):
(6.19)
In the case where we decouple gravity, Vol(M4) is quite large, so in this sense the threeform
is also \decoupled". Additionally, however, we see that if there is a nonzero fourform
ux,
as can happen if we have M5branes present, then there is a ChernSimonslike coupling in
seven dimensions. This places the threeform potential in a gapped phase and in particular
means that the resulting supermultiplet is massive.
The fermions of the threeform multiplet decouple in a similar fashion. We note that in
the 11D supergravity action, there is a coupling of the schematic form
(3=2) G(4)
(3=2),
so this ux also generates a mass for the gravitinos and Weyl fermions of S(73D=2). The four
zeroform and four oneform potentials all develop masses through the standard Stuckelberg
mechanism, and the kinetic terms:
d (0) + A(
1
)
2 + dA(
1
) 2 ;
with mass set by the compacti cation scale, just as for the threeform potential. The
nal set of terms involving the twoform potentials proceed in analogous fashion. Observe
that the magnetic dual of the twoform is a threeform, so we can alternatively just write
additional ChernSimons like couplings to decouple these modes as well. All told, then,
we see that the local excitations decouple, the only remnant in the 7D theory being a
topological sector.
One can also consider coupling the 7D gravitino multiplet to the 7D vector multiplet.
This proceeds, for example, through the topological term:
Z
7D
7D
C(3) ^ Tr(F ^ F );
where F is the twoform
eld strength and
7D is the \lift" of the 6D GreenSchwarz
coupling [70, 85{87] to seven dimensions.
It is also possible to couple our supersymmetric theory to defects carrying propagating
massless degrees of freedom. Observe that in this case, we need to impose suitable boundary
conditions which will break half of the supersymmetry, but allow us to couple to appropriate
supersymmetric edge modes.
11We thank M. Del Zotto for discussions on this point.
(6.20)
(6.21)
In the above we have only sketched the main elements of the 7D supersymmetric
theory. It would of course be most instructive to ll in these details in future work.
7
IIB as an edge mode
Motivated by the success of embedding our 7D topological theory in a supersymmetric
compacti cation of Mtheory, it is natural to ask whether a similar structure also holds
for IIB string theory. Indeed, IIB supergravity contains a chiral fourform potential with
selfdual eld strength and there are notorious subtleties in writing an o shell 10D action.
From our present perspective, the issue is in some sense forced once we demand proper
uxes in the quantum theory. In the special case where we restrict to
level one for the chiral fourform (as happens for the standard IIB supergravity action)
then we have an invertible theory in the sense of [33] so we expect to be able to decouple
the bulk and boundary. Nevertheless, such an answer is rather unsatisfying because one
could ask what happens at other \levels" of the chiral fourform, namely if we choose
a di erent normalization for the twopoint function. It is also disturbing that certain
quantum questions cannot be easily accessed due to the absence of an o shell action.
With this in mind, it is tempting to extend the standard 10D spacetime by an additional
spatial direction to write an 11D topological theory of
veforms [31, 32], much as we
already did in section 5. We have already seen the utility of this perspective in our geometric
uni cation of lowerdimensional topological theories.
Having gone to eleven dimensions, it is now irresistible to try and connect this with
the 12D geometric formulation of Ftheory [88]. Our aim in this section will be to present
some suggestive  though conjectural  aspects of how to view all of IIB superstring
theory as an edge mode of a bulk 10 + 2dimensional topological theory.12 In some sense
this resurrects the original formulation of Ftheory as genuinely living in 10 + 2 dimensions
rather than \merely" as a formal device for constructing nonperturbative 10D vacua. It
also points the way to the construction of new selfconsistent string vacua.
To accomplish this, we shall abandon from the start the notion of having propagating
degrees of freedom in all twelve dimensions. Indeed, propagating degrees of freedom with
two time directions leads to severe pathologies with causality. Rather, we shall immediately
restrict attention to edge modes localized on a 9 + 1dimensional subspace. Our goal will
be to see whether this can be t together selfconsistently.
To motivate our conjectural formulation of IIB as an edge mode, we shall proceed in a
\bottom up" fashion, piecing together various consistency conditions. These will include:
(
1
) We attempt to quantize the IIB selfdual veform ux by viewing the chiral fourform
potential as an edge mode coupled to a bulk topological theory.
(
2
) We require the bulk theory to preserve supersymmetry, namely we must be able to
assemble bulk supermultiplets.
(3) We must maintain the known SL(2; Z) covariance central to the formulation of IIB
and its geometrization in Ftheory.
12We thank C. Vafa for an inspirational discussion.
Let us now proceed to enforce conditions (
1
){(3). Along these lines, we brie y recall
that the RamondRamond sector of IIB string theory consists of a chiral fourform c(4), a
twoform c(
2
), and a zeroform c(0). Continuing with our general philosophy, we shall at
rst attempt to lift this to an 11D space with 10D boundary. In our 11D space we introduce
bulk higherform potentials C(5), C(3) and C(
1
) so that the edge modes amount to degrees
of freedom which cannot be gauged away, namely C(p) = dc(p 1) on the boundary.
We immediately face an obstacle because the 11D supergravity spectrum in 10 + 1
dimensions does not admit such degrees of freedom! There is, however, a loophole if we
proceed to 10 + 2 dimensions, namely ten spatial and two temporal dimensions.
Supergravity multiplets in a 12D spacetime with this exotic signature have been considered
previously, for example in references [89{96]. The key point is that in 10 + 2 dimensions,
one can impose a MajoranaWeyl condition on spinors which in turn allows for a match
between bosonic and fermionic degrees of freedom. The supersymmetry algebra in 10 + 2
dimensions has been studied in references [92, 95], and a proposed chiral supergravity
theory has been constructed in [91, 94, 95] (for a di erent perspective, see e.g. [97, 98]). The
general conclusion is that one ought to consider a theory with a fourform and twoforms,
as well as a pair of null oneforms, + and
.
d 1;1
d 1;1
Before proceeding to how this matches with the form content we have, let us brie y
explain how this earlier work makes contact with the more geometric formulation of
Ftheory on an elliptically
bered CalabiYau with section which has become the de facto
standard in modern practice. Along these lines, let us suppose that we have a standard
Ftheory background of the form R
CYn, where CYn is an elliptically bered
CalabiYau nfold with base Bn 1 a Kahler manifold of complex dimension (n
1) so that the total
spacetime dimension is 10 = d + 2(n
1). The existence of a section means that there is an
embedding of the 10D spacetime R
Bn 1 in the 12D geometry. As such, we also have
a local normal coordinate z 2 O( KBn 1 ), and corresponding in nitessimal oneform dz.
This dz can in turn be decomposed into real and imaginary parts, and thus we get a pair of
real oneforms, namely the Euclidean analogs of + and
presented above. Extending to
the 12D total space, we can also construct a connection which parameterizes the breaking
of SO(10; 2) to SO(9; 1):
(7.1)
(7.2)
(
1
) =
+ d
:
Here, the \ " refers to the contraction of oneform indices.13 This relation appears (in a
slightly di erent form) in reference [99] (see also [95]), for example.14
Continuing in Euclidean signature for our two extra dimensions, the local geometry
near the 10D spacetime is the Cartesian product of a cylinder C with the 10D spacetime.
Topologically, C = S1
R. The local pro le of the metric is in the conformal class:
ds2Euc = d 2 + dr2;
13For the perhaps more familiar case of breaking patterns of a compact Lie group, we would write
(
1
) =
igydg, where g is a generator of the group. Note that for a U(
1
) gauge theory with g = exp(i ),
the formal oneform reduces to d .
14In reference [95], the SL(2; Z) duality relation of IIB supergravity is imposed \by hand" at the classical
12D level, rather than as an emergent property from compacti cation on a T 2.
where the complex structure of the elliptic curve is identi ed with the combination of IIB
supergravity modes:
theory.
= c(0) + ie :
This elliptic bration is central to the geometrization of SL(2; Z) duality in IIB string
HJEP05(218)
so that is a periodic local coordinate, and r parameterizes the radial pro le of the cylinder.
Globally, we of course complete this local presentation of the metric to construct an elliptic
bration over the 10D spacetime, with metric in the conformal class:
ds2E = (dx + (x10D)dy) (dx + (x10D)dy) ;
(7.3)
(7.4)
(7.5)
Let us now turn to the Lorentzian signature version, as dictated by supersymmetry in
10 + 2 dimensions. We have already presented evidence that to properly quantize the 10D
chiral fourform, we ought to extend our spacetime by one additional spatial direction, i.e.,
the r coordinate of line (7.2). This means that if we insist on a supersymmetric theory in
(10; 2) signature, we need to analytically continue 7!
the metric in line (7.2) is instead:
i . So for us, the local pro le of
ds2Lor =
d 2 + dr2:
The presence of an additional timelike direction means that a reduction along a null
direction takes us from a 10 + 2dimensional spacetime to a 9 + 1dimensional spacetime.
At this point, something awkward appears to have happened. On the one hand, we
have argued based on ux quantization considerations and supersymmetry that we need to
add Lorentzian directions of signature (1; 1). On the other hand, the SL(2; Z) duality group
of IIB is most naturally formulated in terms of a Euclidean T 2, and simply analytically
continuing to Lorentzian signature fails to retain this structure.15 Again taking our cue
from the original paper [88], we note that the Narain lattice for strings on a Euclidean T 2
has signature (2; 2) so it is not inconceivable that a (1; 1) signature spacetime could still
replicate these features.
To put this on a rmer footing, we now observe that by construction, we do not allow
any localized excitations in the two extra directions, this being the point of dealing with
a bulk 12D topological theory in the
rst place. Nothing, however, disallows extended
objects to wrap around compact cycles of the geometry. In particular, returning to our
discussion of the Lorentzian cylinder geometry of line (7.5), we see that we can wrap an
extended object along the timelike circle parameterized by the
coordinate. The winding
number along this circle has a Fourier transform which yields a dual periodic coordinate e.
From this perspective, there is again a Euclidean signature torus lurking, but in comparison
with line (7.3), it will involve a dual coordinate x:
e
ds2E = (dxe + (x10D)dy) (dxe + (x10D)dy) :
e
(7.6)
15The main point is that it is not possible to de ne a family of Lorentzian signature torii in which the
analog of smoothly varies. We thank D.R. Morrison for helpful comments on this point.
the special case where the axiodilaton is constant. Here, we have passed from the (10; 2) signature
spacetime to one in which a dual winding coordinate e appears. This takes us to a dualized spacetime
of signature (11; 1) and in which the standard elliptic ber of Ftheory appears geometrically.
This sort of mixing of momentum and winding is a feature of geometric approaches to
Tduality such as double eld theory [100{102]. In some sense our discussion is less ambitious
since we restrict to a topological sector from the start. We thus have in our extra 1 +
1dimensions a pair of null directions along which we can compactify on lightlike circles.
Even though nothing propagates along them, extended objects can wrap the circles, and
the consistent description of such degrees of freedom leads to a Euclidean signature torus,
with complex structure the axiodilaton of IIB strings. See gure 5 for a depiction of the
geometry in the special case in which we have a constant elliptic
bration.
A satisfying byproduct of this proposal is that it helps to clarify the role of the elliptic
ber in Mtheory/Ftheory duality. Recall that in the standard picture, Mtheory on the
background CYn is also described by Ftheory on S1
CYn, where CYn denotes the \same"
elliptically
bered CalabiYau nfold. In this process, one is supposed to collapse the
Mtheory elliptic
ber to zero size to reach the Ftheory limit. In the lowerdimensional
context of IIA/IIB Tduality, one can consider similar limits, modulo the caveat that
we interchange momentum and winding degrees of freedom. Here, we see that a similar
consideration seems to hold, and in some sense is required to simultaneously satisfy our
earlier conditions (
1
){(3) mentioned at the beginning of this section. Precisely because
we can interchangeably work in terms of either a 10 + 2dimensional spacetime, or one in
which we trade a position coordinate for a winding coordinate to reach a dualized 11 +
1dimensional spacetime, we see that the procedure of dimensional reduction also carries
over. In particular, a reduction of di erential forms along the circle with coordinate e
proceeds as in a standard KaluzaKlein compacti cation.
With the geometry of the spacetime dealt with, let us now return to our proposed
mode content, now lifted to 10 + 2 dimensions. We have so far identi ed a veform,
threeform and oneform, namely C(5), C(3), and C(
1
). Additionally, taking our cue from earlier
work on supersymmetry in 10 + 2 dimensions, we impose an antiselfduality condition on
the sixform
eld strength F(6) =
12D F(6). Given that our motivation was to properly
quantize the uxes of the IIB theory, and we have now introduced a selfduality constraint
in 12D, one might ask whether it is necessary to proceed up to 13 dimensions, perhaps
along the lines of Stheory [103]. There is no reason to do so, at least from quantization
considerations. The reason is that because of the presence of a distinguished null
oneform, the way we read o the physical brane spectrum is di erent. Indeed, as we shortly
explain, the physical degrees of freedom associated with our
veform potential require
either contraction or wedging our
veform to a fourform or sixform, respectively. As
such, the physical degrees of freedom are associated with a 2 + 2brane and 4 + 2brane,
so we need not directly quantize the corresponding sixform
ux units (since we have no
3 + 2branes to speak of). Indeed, turning the discussion around, we see that the absence
of integral sixform
uxes actually eliminates the possibility of a 3 + 2brane, leaving us
only with objects such as a 2 + 2brane or 4 + 2brane!
Comparing with the earlier supersymmetry literature in 10 + 2 dimensions [89{96], we
now ask how we can recover a fourform potential and a pair of twoform potentials, as
well as the axiodilaton. This all falls into place upon contracting our pforms with the
oneforms + and
, namely we write:16
C(p)
C(p 1)
( ) :
(7.7)
Note that this automatically generates covariant transformation rules for the twoform
potentials and the scalars in the 10D IIB theory, and the antiselfduality relation projects
us onto an SL(2; Z) singlet for the chiral fourform potential.
A closely related point is that (as is wellknown from M/Ftheory duality) the periods
of our potentials along circles produces the expected IIB mode content. For example, the
threeform reduces to the NS and RR twoforms, and transform as a doublet of SL(2; Z).
The periods of the oneform potential C(
1
) along the two onecycles produce a pair of scalars,
and ratios of appropriate linear combinations reduces to the axiodilaton . Finally, the
periods of the veform are related by duality of its eld strength, namely F(6) =
12D F(6),
with F(6) = dC(5), so the boundary fourform transforms as an SL(2; Z) singlet.
So far, we have focused on the RR sector, and have already seen a partial uni cation
with some parts of the NS sector of the 10D boundary. To round out our discussion, we now
turn to the 12D origin for the remaining NS degrees of freedom, namely the 10D metric.
Since we have introduced explicit oneforms
+ and
, we expect the graviton to always
have a formally in nite mass in two of the twelve directions. The gauge redundancy for a
massless graviton:
gAB ! gAB + rACB + rBCA;
also means that we can alternatively package our mode content in terms of a 10D graviton,
and a oneform C(
1
), the one introduced earlier from a bottom up perspective. By the same
token, we expect a partially massive gravitino in 10 + 2 dimensions to ll out the necessary
Let us now turn to the sense in which our system is actually in a gapped phase. Starting
fermionic degrees of freedom.
from the 11D action:
duality manifest.
16Here, we are working in the (11; 1) signature geometry with a winding coordinate so as to make SL(2; Z)
1 Z
4 i
11D
C(5) ^ dC(5),
(7.8)
(7.9)
(7.10)
(7.11)
12D
12D
12D
(
1
) C(5) ^ G(4) ^ G(4);
we can extend to 12D using our privileged oneform (
1
) of line (7.1). By the same token, we
can also introduce a related BFlike term to produce a mass gap for the oneform C(
1
) and
its pairing to the threeform C(3). Writing out all proposed couplings to the higherform
S12D;top =
(
1
) ^ C(5) ^ dC(5) +
(
1
) C(3) ^ dC(9)
where G(4) is the fourform
eld strength for the threeform potential, and we have used
the magnetic dual of C(
1
), namely the nineform C(9). The coupling of the second line has
appeared in the Ftheory literature before [97, 98, 104, 105]. Observe that we have dropped
prefactors of \i" because we have two Lorentzian directions. Finally, note that the only
appearance of the metric occurs in the privileged null direction via contraction with
(
1
).
This dependence is su ciently mild that we again obtain a trivial stress energy tensor, as
required for a topological theory of this type. As far as we are aware, the couplings of
line (7.10) have not been considered before.
We can also include a partially massive graviton, along the lines of reference [106]. Such
theories have appeared in string theory and related holographic setups, see e.g. [107{109].
Let us note that in massive gravity, there are various concerns about superluminal
propagation. Here, however, we are taking a rather di erent limit from what is typically discussed
in phenomenological applications: for us, the mass in the decoupled directions is formally
in nite, rather than small. An additional comment is that in the related context of warped
compacti cation, a graviton zero mode can be localized on a lowerdimensional
spacetime [110]. This is rather suggestive considering that in Ftheory models, the base of the
model has positive curvature, and thus the curvature is negative in the directions normal
to the base inside the CalabiYau.
Returning to the topological couplings of our 12D action, note that in the 10D
boundary, the rst term reduces to a kinetic term for a chiral fourform, and the second term
provides a Stuckelberg mass for the RR and NS twoforms. Indeed, in Ftheory
backgrounds with nontrivial SL(2; Z) monodromy, these twoforms are generically projected
out, receiving a mass due to such interactions. Line (7.11) reduces to the wellknown IIB
coupling c(4) ^h(R3R) ^h(N3)S, which could in principle receive additional oneloop contributions
from integrating out components of the massive gravitino.
It is also tempting to extend each stringy excitation of the 10D boundary by a 1 +
1dimensional direction in 12D, as for example in [92{96]. Following our topological route, we
extend the F1 and D1branes to a topological 2 + 2brane which couples to the fourform
obtained through the contraction (
1
) C(5). Some aspects of 2 + 2dimensional branes have
recently been studied for example in [111], and for earlier work, see e.g. [112{116]. We can
also extend a 3 + 1dimensional D3brane to a 4 + 2dimensional topological brane which
couples to the sixform
(
1
) ^ C(5).
This also points the way to constructing new string vacua. For example, though we
have for the most part concentrated on the case of unit coe cient in line (7.9), it is quite
tempting to broaden our horizons to more general couplings of the form:
N Z
4 i
11D
C(5) ^ dC(5).
(7.12)
There is a simple interpretation of this as winding modes along a timelike circle of the 12D
geometry. Indeed, in an early speculative attempt to mimic the Mtheory interpretation of
D0branes as momentum along a circle, reference [117] sought to interpret IIB Dinstantons
(namely D 1branes) in terms of \momentum" along a circle of a 12D spacetime. Here, we
see that a background condensate of Dinstantons produces a corresponding shift in the level
of the veform theory. In this formulation, we identify N
1 with a winding number for the
metric, since the case of no Dinstantons ought to correspond to the case of the \usual" IIB
background with canonically normalized twopoint function for the chiral fourform in ten
dimensions. This provides a somewhat complementary motivation for IIB matrix models,
and it would be interesting to revisit earlier proposals such as [118{120]. Departing one
step further, it is tempting to consider 12D geometries without an elliptic bration.
Apparently, then, the natural setting for many of our considerations resides in a 10 +
2dimensional bulk topological theory. This appears to be compatible with the conditions of
supersymmetry, SL(2; Z) duality, and also evades the pathologies of twotime physics and
potential issues with massive gravity. We leave more detailed checks of this proposal for
future work.
Summarizing, the 12D topological interpretation of superstrings is intriguing.
8
In this paper we have proposed a sixdimensional generalization of the fractional quantum
Hall e ect which makes use of the correspondence between a bulk 7D topological eld
theory and a 6D theory of edge modes. The bulk is given by a ChernSimons like 7D
theory of threeforms, and the edge mode theory is that of free antichiral twoforms.
In the presence of a large background magnetic fourform
ux, there is a direct analog
with the usual fractional quantum Hall e ect. We determined the leading order behavior
of the analogous Laughlin wavefunction in various limits, and have also explained how
this higherdimensional starting point provides a unifying perspective on several
lowerdimensional systems involving a bulk topological eld theory coupled to edge modes. This
7D theory embeds in a limit of Mtheory decoupled from gravity, and this in turn motivates
a speculative conjecture on the interpretation of Ftheory as a topological theory in 10 + 2
dimensions coupled to 9 + 1dimensional edge modes associated with IIB strings. In the
remainder of this section we discuss some avenues for future investigation.
There is by now a nearly complete list of supersymmetric 6D CFTs. One of the
original motivations for this work was to better understand the topological structure of
these fascinating theories. It is therefore quite tempting to ask whether we can develop a
supersymmetric version of our analysis. For some discussion of a supersymmetric version
of the fractional quantum Hall e ect in two dimensions, see for example [121, 122].
In this work focused on the simplest situation where there is a single connected
component to the boundary. In lower dimensions, it is often quite fruitful to consider additional
disconnected components to the boundary. This in turn is intimately connected with the
spectrum of nonlocal operators in the theory.
Another thread of our analysis is the unifying framework it provides for generating a
rich class of lowerdimensional phenomena. We can already anticipate that various integer
couplings can in many cases be recast as topological properties of a compacti cation
manifold. It would be exciting to use this perspective to develop a systematic analysis of these
One of the original motivations for this work was to better understand the topological
sector of little string theories (for early constructions see e.g. [78, 123{128]), in which the
pairing
IJ is no longer invertible since it has a zero eigenvalue [39, 129]. Such little string
theories are nonlocal, and thus not controlled by the standard axioms of local quantum
eld theory. It is natural to ask whether a suitable 7D theory exists with edge modes given
by a little string theory.
We have also seen that our 7D topological theory admits a supersymmetric extension
which embeds as a decoupling limit of the physical Mtheory. Topological Mtheory
provides a uni ed framework for understanding various aspects of topological strings [130] (see
also [131, 132]). This theory is formulated in terms of an abelian threeform, but instead
involves compactifying on a manifold of G2 metric holonomy with action the integral of C(3) ^
7DC(3), where the metric is itself constructed from the associative threeform [133, 134].
Adding such a term to a theory of a threeform gauge potential also leads to a mass gap.
It would thus be interesting to compare the IR behavior of these two 7D theories.
Proceeding up to 10 + 2 dimensions, we presented some tantalizing hints of an
interpretation of Ftheory as a bulk topological theory. Though quite conjectural, it already
provides a novel starting point for realizing new classes of string vacua. It would clearly
be very interesting to provide further evidence for this proposal. A particularly
fascinating aspect of such a correspondence is the emergence of the IIB graviton as an edge
mode excitation.
Acknowledgments
We thank F. Apruzzi, M. Del Zotto, T.T. Dumitrescu, O.J. Ganor, D.R. Morrison,
D.S. Park and C. Vafa for helpful discussion. We also thank O.J. Ganor and D.R. Morrison
for comments on an earlier draft. JJH thanks the 2016 and 2017 Summer workshops at the
Simons Center for Geometry and Physics as well as the Aspen Center for Physics Winter
Conference in 2017 on Superconformal Field Theories in d
4, NSF grant PHY1066293,
for hospitality during part of this work. The work of JJH is supported by NSF CAREER
grant PHY1452037. LT thanks UNC Chapel Hill and the ITS at CUNY for hospitality
during this work. The work of LT is supported by VR grant #20145517 and by the
\Geometry and Physics" grant from Knut and Alice Wallenberg Foundation.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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