7.2 M2-branes
In this section, I would like to briefly mention the main results involving the amount of progress recently
achieved in the description of
parallel M2-branes, referring to the relevant literature when appropriate.
This will be done taking the different available perspectives on the subject: a purely kinematic approach,
based on supersymmetry and leading to 3-algebras, a purely field theory approach leading to three
dimensional CFTs involving Chern–Simons terms, a brane construction approach, in which one infers the
low energy effective description in terms of an intersection of branes and the connection between all these
different approaches.
The main conclusion is that the effective theory describing
M2-branes is a
,
gauge theory with four complex scalar fields
(
) in the
representation, their complex conjugate fields in the
representation and their fermionic
partners [12
]. The theory includes non dynamical gauge fields with a Chern–Simons action with levels
and
for the two gauge groups. This gauge theory is weakly coupled in the large
limit (
)
and strongly coupled in the opposite regime
, for which a weakly coupled gravitational
description will be available if
.
Supersymmetry approach:
Inspection of the
SYM supersymmetry transformations
and the geometrical intuition coming from M2-branes suggest that one look for
a supersymmetric field theory with field content involving eight scalar fields
and their fermionic partners
, and being invariant under a set of supersymmetry
transformations whose most general form is
This was the original approach followed in [26
], based on a real vector space with basis
,
, endowed with a triple product
where the set of
are real, fully antisymmetric in
and satisfy the fundamental identity
Closure of the supersymmetry algebra requires Eq. (501), but also shows the appearance of an extra gauge
symmetry [26]. To deal properly with the latter, one must introduce an additional (non-dynamical) gauge
field
requiring one to consider a more general set of supersymmetry transformations [27, 274]
Here
is a covariant derivative, whereas
and
define triple products on the
algebra.
Closure of the supersymmetry algebra determines a set of equations of motion that can
be derived, which form a Lagrangian. It was soon realised that under the assumptions of a
real vector space, essentially the only 3-algebra is the one defined by
, with
defining an inner product, and satisfying
[399, 412, 226].
Interestingly, it was pointed out in [488] that such supersymmetric field theory could be rewritten as a
Chern–Simons theory. The latter provided a link between a purely kinematic approach, based on
supersymmetry considerations, and purely field theoric results that had independently been
developed.
Field theory considerations:
Conformal field theories have many applications. In the particular context of
Chern–Simons matter theories in
, they can describe interesting IR fixed points in condensed matter
systems. Here I am interested in their supersymmetric versions to explore the AdS4/CFT3
conjecture.
Let me start this overview with
theories.
Chern–Simons theories coupled to
matter
include a vector multiplet
, the dimensional reduction of the four dimensional
vector multiplet,
in the adjoint representation of the gauge group
, and chiral multiplets
in representations
of
the latter. Integrating out the D-term equation and the gaugino, one is left with the action
where
and
are the bosonic and fermionic components of the chiral superfield
and the gauge
field
is non-dynamical.
There are
generalisations, but since their construction is more easily argued for
starting with the field content of an
theory, let me review the latter first. The field
content of the
theories adds an auxiliary (non-dynamical) chiral multiplet
in the
adjoint representation of
and pairs chiral multiplets
into a set of hypermultiplets
by requiring them to transform in conjugate representations, as the notation suggests. The
theory does not contain Chern–Simons terms, but a superpotential
for each pair.
theories are constructed by the addition of Chern–Simons terms, as in Eq. (503), and
the extra superpotential
. Integrating out
leads to a superpotential
The resulting
theory has the same action as Eq. (503) with the addition of the above
superpotential.
In [12
], an
theory based on the gauge group
was constructed. Its field content
includes two hypermultiplets in the bifundamental and the Chern–Simons levels of the two
gauge groups were chosen to be equal but opposite in sign. Denoting the bifundamental chiral
superfields by
and their anti-bifundamental by
, the superpotential then equals
After integrating out the auxiliary fields
,
As discussed in [12
], the four bosonic fields
transform in the
of
,
matching the generic
R-symmetry in
super-CFTs. For a more thorough discussion of
global symmetries and gauge invariant observables, see [12
].
It was argued in [12
] that the
theory constructed above was dual to
M2-branes on
for
. Below, I briefly review the brane construction in which their argument is based. This
will provide a nice example of the notion of geometrisation (or engineering) of supersymmetric field theories
provided by brane configurations.
Brane construction:
Following the seminal work of [282
], one can associate low energy effective field theories
with the dynamics of brane configurations stretching between branes. Consider a set of
D3-branes wrapping the
direction and ending on different NS5-branes according to the array
This gives rise to an
gauge theory in
dimensions, along the
directions, whose field content includes a vector multiplet in the adjoint representation and 2
complex bifundamental hypermultiplets, describing the transverse excitations to both D3-branes and
NS5-branes [282
].
Adding
D5-branes, as illustrated in the array below,
breaks supersymmetry to
and adds
massless chiral multiplets in the
and
representation of each of the
gauge group factors. Field theoretically, this
construction allows a set of mass deformations that can be mapped to different geometrical
notions [282, 72
, 12
]:
- Moving the D5-branes along the 78-directions generates a complex mass parameter.
- Moving the D5-branes along the 5-directions generates a real mass, of positive sign for the fields
in the fundamental representation and of negative sign for the ones in the anti-fundamental.
- Breaking the
D5-branes and NS5-branes along the 01234 directions and merging them
into an intermediate
5-brane bound state generates a real mass of the same sign for
both
and
representations. This mechanism is a web deformation [72]. The merging
is characterised by the angle
relative to the original NS5-brane subtended by the bound
state in the 59-plane. The final brane configuration is made of a single NS5-brane in the
012345 directions and a
5-brane in the
, where
stands for the
direction.
is fixed by supersymmetry [14].
After the web deformation and at low energies, one is left with an
Yang–Mills–Chern–Simons theory with four massless bi-fundamental matter multiplets (and their complex
conjugates), and two massless adjoint matter multiplets corresponding to the motion of the D3-branes in
the directions 34 common to the two 5-branes.
The enhancement to an
theory described in the purely field theoretical context is realised in
the brane construction by rotating the
5-brane in the 37 and 48-planes by the same amount as in
the original deformation. Thus, one ends with a single NS5-brane in the 012345 and a
5-brane along
.This particular mass deformation ensures all massive adjoint
fields acquire the same mass, enhancing the symmetry to
. Equivalently, there must
exist an
R-symmetry corresponding to the possibility of having the same
rotations in the 345 and 789 subspaces. Thus, the
supersymmetric field theory must be
.
The connection to
is obtained by flowing the
theory to the IR [12
]. Indeed, by
integrating out all the massive fields, we recover the field content and interactions described in the field
theoretical
construction. The enhancement to
for
was properly discussed
in [276].
It was realised in [12
] that under T-duality in the
direction and uplifting the configuration to
M-theory, the brane construction gets mapped to
M2-branes probing some configuration of
KK-monopoles. These have a supergravity description in terms of hyper-Kähler geometries [224]. Flowing
to the IR in the dual gravitational picture is equivalent to probing the near horizon of these geometries,
which includes the expected AdS4 factor times a quotient of the 7-sphere.
The Chern–Simons theory has a
coupling constant. Thus, large
has a weakly coupled
description. At large
, it is natural to consider the ’t Hooft limit:
fixed. The gauge theory is
weakly coupled for
and strongly coupled for
. In the latter situation, the supergravity
description becomes reliable and weakly coupled for
[12
].
Matching field theory, branes and 3-algebra constructions:
The brane derivation of the supersymmetric field
theory relevant to describe multiple M2-branes raised the natural question for what the connection was, if
any, with the 3-algebra formulation that stimulated all these investigations. The answer was found in [28
].
The main idea was to consider a 3-algebra based on a complex vector space endowed with a triple product
and an inner product
The change in the notation points out antisymmetry only occurs in the first two indices. Furthermore, the
constants
satisfy the following fundamental identity,
It was proven in [28] that this set-up manages to close the algebra on the different fields giving rise to some
set of equations of motion. In particular, the
conformal field theories described in [12] could be
rederived for the particular choices
Thus, the 3-algebra approach based on complex vector spaces is also suitable to describe these string theory
models. Furthermore, it provides us with a mathematical formalism capable of describing more general
set-ups.