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Electric-Magnetic Duality On A Half-Space
Edward Witten
Rutgers University
May 12, 2008
(work with D. Gaiotto)
This will be a talk about electric-magnetic duality in 4-dimensional N=4 super Yang-Mills theory. That is not a new story.
What is new is that we will consider duality on a half-space.
But I will start with motivation from 2 dimensions.
Classically, the simplest form of duality is the relation between two scalar fields
and that obey (in two dimensions)
or in more detail the
Cauchy-Riemann equations
This condition implies that
and both obey the Laplace
equation (= the massless wave equation)
or
When interpreted quantum mechanically, this
simple relation leads to many adventures
including mirror symmetry.
If we do all this on a
manifold with boundary
then the relation
implies that if obeys Dirichlet boundary
conditions (i.e., it is zero, or constant, on the boundary) then obeys Neumann boundary conditions (i.e., its normal derivative vanishes on the boundary) and vice-versa.
This has an analog in the nonlinear case that the scalar field φ is replaced by a map Φ:D→X, where D is a two-manifold and X is some target space.
Possible boundary conditions in this case, and their transformation under duality, have been much-studied, because of their importance in string theory and their role in mirror symmetry. (Kontsevich has formulated this as“Homological mirror symmetry.”) What we will be discussing is the four-dimensional analog.
This has an analog in statistical mechanics
For the Ising model, Kramers-Wannier duality reverses order and disorder so it
exchanges Dirichlet boundary conditions on the spins (ordered) with Neumann
(disordered)
The relation between scalar fields in two dimensions has an analog for gauge fields in four dimensions. Classically, one considers abelian gauge theory with field strength and one observes that Maxwell’s equations in vacuum
are invariant under the exchange which exchanges
electric and magnetic fields
This turns out to have a very deep analog in the quantum theory for nonabelian gauge group. The analog is called S-duality.
There are actually quite a few important details in setting this up, such as the role of supersymmetry (maximal supersymmetry, or N=4, to be exact)
but I will try to hide all these details today. We are just going to consider a four-manifold with boundary and ask what electric-magnetic duality does in this
context.
We consider the N=4 theory, with some
gauge group G, just on the half plane
to the left of this line
Of course, we need a boundary
condition. There are two obvious
choices, Dirichlet and Neumann
and
has a simple physical interpretation: It is the boundary condition for an interface between vacuum and a superconductor
There is no equally accessible realization of the other boundary condition … which should be related to electric confinement rather than the magnetic Meissner effect
These look like they might be dual under the transformation
since this transformation in a sense exchanges and
The two boundary conditions (when suitably extended to the other fields) are both supersymmetric, but they preserve different supersymmetries. However, S does transform the supersymmetries properly.
It is actually true in abelian gauge theory that Dirichlet and Neumann boundary conditions are dual to each other. Any of the usual derivations of S-duality in abelian gauge theory can be used to show this.
However, it can hardly be true in nonabelian gauge theory. The reason is that the symmetries are wrong.
When we impose Dirichlet boundary conditions, we require that on the boundary. Then we divide only by gauge transformations that equal 1 on the boundary. We are left with G acting as a group of global symmetries via gauge transformations that are constant, but not 1, on the boundary. There are (in nonabelian theory) local fields at the boundary that transform nontrivially under the global symmetry
This doesn’t have any analog in the
Neumann case. With Neumann boundary
conditions, we are dividing by all gauge
transformations on the half-space,
including those that are non-trivial at the
boundary. We are not left with any global
symmetry. So Neumann cannot be dual
to Dirichlet.
One might ask why there is no contradiction
in the abelian case in claiming that
Neumann is dual to Dirichlet. The
answer is twofold. First, in abelian
gauge theory, the Bianchi identity
remains valid, of course, if we restrict to the
boundary. But the boundary is three-dimensional and in three dimensions
the equation is an equation for a conserved current. That is, we define
on the boundary a conserved current
So this operation gives us a conserved current that is defined only on the boundary and only for Neumann boundary conditions (with Dirichlet boundary conditions this current vanishes)
The other argument (in which we derived a global symmetry from gauge transformations that are constant, but not 1, on the boundary) only made sense in the Dirichlet case.
So for either Dirichlet or Neumann boundary conditions, but for rather different reasons, there is, for G=U(1), a U(1) global symmetry with a conserved current that is nonzero only on the boundary.
On the Neumann side, since the current is , the conserved charge
density is the component of the magnetic field normal to the boundary … So the conserved charge is the first Chern class, integrated over a spatial section of the boundary … we note that this spatial section is a two-manifold.
This means that the conserved charge is not carried by any local operators, but that is also true on the Dirichlet side since U(1) is abelian
The conclusion is that for abelian gauge theory, Dirichlet is dual to Neumann, but that this cannot be true for nonabelian gauge theory.
The dual of Neumann is something else that preserves the same supersymmetry as Dirichlet, and the dual of Dirichlet is something else that preserves the same supersymmetry as Neumann.
There is a fairly direct route to generalize Neumann. In the case of Neumann
boundary conditions, the gauge fields are non-trivial on the boundary and therefore, they can be coupled to additional degrees of freedom that are only defined on the boundary. The
additional degrees of freedom may make up any superconformal field theory with G
symmetry, the superconformal group being
Three-dimensional superconformal field theories with this amount of supersymmetry can have “mirror symmetry” (Intriligator and Seiberg, 1996) exchanging the Higgs and Coulomb branches, and this turns out to be an important ingredient.
This is how to generalize Neumann boundary conditions.
Gauge theory with Dirichlet boundary conditions cannot be sensibly coupled to new degrees of freedom at the boundary,
but something else happens. N=4 super Yang-Mills contains six scalar fields in the adjoint representation of G. When we reduce the symmetry in the presence of the boundary, these split into two groups of three, say and
For Neumann boundary conditions, supersymmetry requires that and
should be constant. But with Dirichlet boundary conditions, one learns that
should be constant, but need only obey “Nahm’s equations”
where measures the distance from the boundary.
Nahm’s equations have superconformally
invariant solutions that, however, are singular at
Pick any set of generators in the Lie algebra , that is any three elements of that Lie algebra that obey
and cyclic permutations, and set
Because this solution is singular at
it isn’t allowed as a solution of the theory with ordinary Dirichlet boundary conditions.
But this gives us an opportunity: we modify the boundary conditions by requiring this kind of singularity. So, for every su(2) embedding in the Lie algebra of G, we get a new boundary condition with the same SUSY as Dirichlet boundary conditions.
So we’ve generalized Dirichlet and we’ve generalized Neumann, and although this doesn’t yet give the most general half-BPS boundary condition, it turns out that it is general enough to contain the dual of standard Dirichlet and standard Neumann.
First of all, let us try to guess the dual of standard Neumann. This should be a Dirichlet-like boundary condition with no global symmetry.
The Dirichlet-like boundary condition defined by an embedding with generators
has the global symmetry group G broken to the subgroup H that commutes
with … so we can get a Dirichlet-like
boundary condition with no global symmetry by picking the
embedding to be irreducible.
This actually is the dual of naïve Neumann boundary conditions
For one can show this via a brane construction … one needs to use facts about Nahm’s equations that are standard but whose relevance in this problem is a little surprising
To find the dual of naïve Dirichlet boundary conditions, we go back to Nahm’s equations
but now we look at them in another way. With naïve Dirichlet, is supposed to
be nonsingular at y=0, so we are not going to find a non-zero conformally invariant solution of Nahm’s equations.
However, the equations have many solutions for which for
… these solutions form a moduli
space of supersymmetric (but not
conformally invariant) vacua
for the gauge theory
on a half space
is acted on by the global symmetry
group G
What can be the dual of that?
With Neumann boundary conditions, Nahm’s equations don’t come in. However, the
gauge theory on the half-space can be
coupled to a boundary CFT with G symmetry, and then the combined system
4d gauge theory + 3d CFT can have a moduli space of vacua, which must coincide with what we get from Nahm’s
equations on the other side.
Let us formulate this a little more thoroughly.
We call the gauge group on the Dirichlet side . And we call the dual gauge group on the Neumann side
The global symmetry group of the Dirichlet problem is also . The CFT that we couple to on the Neumann side must have
symmetry.
The symmetry of the CFT is gauged – that is how we couple it to the 4d gauge theory on the half-space – and the
symmetry remains as a global symmetry that matches the global symmetry on the Dirichlet side.
Therefore, we are looking for a 3d CFT that has symmetry. Moreover, we know something about its vacua.
The 3d CFT can have both a Higgs branch and a Coulomb branch of vacua (and possibly mixed branches). In coupling to the bulk gauge theory, acts on one of the branches, say the Higgs branch. This one is then “killed” and what survives is the moduli space of vacua of the combined system (gauge theory on a half space plus 3d CFT) is the Coulomb branch of the CFT. This one must be acted on by .
The Coulomb branch of the CFT must therefore match the moduli space of
vacua that we find on the Dirichlet side by
solving Nahm’s equations.
This information determines what the CFT should be.
To avoid going into too much detail, I’ll just state the answer for
The theory is a self-mirror theory that was one of the original examples of Intriligator and Seiberg. It can be constructed as the IR limit of a U(1) theory coupled to two
charged hypermultiplets, and appears many ways in string theory.
Coulomb branchOne SU(2)acts here
Higgs branchThe other SU(2) acts here
For there is a generalization of this
involving a certain quiver.
We have basically found three ways to study the CFT – which we call -
that is needed to describe the dual of Dirichlet boundary conditions.
One method involves
brane constructions
This is most simple for
but has analogs for other classical
groups.
The second method is to consider M-theory
on
where and are finite subgroups of SU(2).
This leads to many 3d CFT’s, depending on the membrane charges and related data, and the theories we want are among them
at least for many G.
There is actually a third method which gives the desired CFT for any G … this uses a surprising construction known as the Janus solution
(Bak, Gutperle and Hirano 2003, Clark,Freedman, Karch, and Schnabl 2005, Clark and Karch 2005, D’Hoker, Estes, and Gutperle 2006a,2006b, 2007, Bak, Gutperle, and Hirano 2007, Kim, Koh, and Lee 2008)
The Janus solution was originally described
in supergravity, but it can also be seen in weakly coupled field theory
(Clark et al, D’Hoker et al, Kim et al, DG+EW)
It describes a supersymmetric situation in which the gauge coupling varies as a function of one coordinate, say y
In general the coupling varies as an arbitrary function
Very weak coupling to the left, very strong coupling to the right
Now consider this situation:
To the right, it is better to use a dual description with gauge fields of the dual
group In the limit that the coupling
is very weak on both sides, we get something, living at the interface, that has
global symmetry and is weakly coupled to
gauge fields of that live on the left and
to gauge fields of that live on the right.
I claim that the something is a 3d superconformal field theory with
symmetry. We can determine
its Higgs and Coulomb branches by using
Nahm’s equations
And all of the evidence is that this gives the same superconformal field theory with is needed to understand the dual of Dirichlet boundary conditions.
I hope we can understand duality a little better by studying it.
Electric-Magnetic Duality On A Half-Space
Edward Witten
Rutgers University
May 12, 2008