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