And thanks a lot to the audience for staying for this talk, it's been a long day.
So this work, this talk draws from joint work with David Benzvi and Andy Nitsky and with
Kevin Costello.
And it's very closely related to work with Chris Beam, Matt Ballimore, Davide Gyoto and
Justin Hilburn.
I am especially thankful to Justin Hilburn for sitting down with me three years ago and
telling me what causality was and helping me work out the simplest example.
This is sort of a part of a series of investigations that are trying to take a lot of very sophisticated
results coming out of the mathematics of TQFTs in recent years and trying to take these
sophisticated results and extract what I think of as basic physics out of them.
In particular, trying to see what these results mean in supersymmetric quantum field theories
and their topological twists and the sorts of theories that I grew up playing with.
So one of these very fancy concepts that is really fundamental in many parts of mathematics
is causal duality.
So roughly speaking, its causality relates pairs of algebras.
The algebras themselves are not equivalent, but they have many similar properties.
In particular, what I'll be using today are examples where the categories of modules for
A and the dual algebra, A-Schweig, are suitably equivalent.
As I said, this has been a really fundamental concept in many parts of mathematics.
Initially the idea rose in the 60s and 70s in rational homotopy theory, in the work of
Cohen and Sullivan, and was developed algebraically by Priddy.
One of the fundamental examples, which I spent part of the talk on, appeared in work of Wernstein,
Gelfand and Gelfand.
Then it was taken up and blown up and generalized in lots of amazing ways.
So fundamental in mathematics, I want to explain today at least one way in which causality
is actually fundamental in quantum field theory, and quite prevalent.
My focus is going to be causality of associative algebras, more generally DG algebras, and
how they're related to boundary conditions in two-dimensional TQFTs.
Specifically, not just TQFTs on the nose, but things like twists of supersymmetric theories
whose cohomology is topological, but have more interesting structure there.
There are two major applications that have come out of this.
One of them is some more formal understanding of how gauging and ungauging works in topological
theories and perhaps in more general theories.
The other one has to do with gluing and ungluing.
I'll talk a little bit about that at the end.
So very often, one wants to take a quantum field theory and maybe manipulate it on a
half space, and it's very useful to slice it, do some manipulation, then glue it back
together.
In fact, causality governs precisely how that works.
Even though the talk is going to be largely in two dimensions to keep things concrete
and mathematically well-defined, I hope that by the end of the talk, the generalization
to more than two dimensions will be clear, at least in principle.
David Bensvie's talk on his work with Sem Gunningham talked about a version of causal
duality engaging in four dimensions.
Konstantin Telemann, the last years, has been developing this concept in Rosenzky-Witten
theory, and Kevin had some recent work saying that causal duality is actually fundamental
in ADSCFT.
Let me just start with some of the basic mathematical concepts.
Going to take A to B a DG algebra, and for many of the examples in the talk, just think
of it as an associative algebra, a graded associative algebra.
There's a bit of extra structure that's needed to define the dual mathematically, and that's
a homomorphism from A to the trivial DG algebra.
Now, a homomorphism from A to C is an algebra in particular means that A acts on C as a
module, and the causal dual of A is defined as endomorphisms of this trivial module that
commutes with the action of A.
There is a picture for that that I will show you in a second, but let me just take it as
a definition for the moment.
In physics, we usually see a little bit more structure.
In the ideal nicest possible situation, which is what I'll discuss, one has some category
and we'll think of it as a category of boundary conditions, and a few special objects in it.
Now, given any object in a category, one can define a functor from the category to modules
over the endomorphism algebra of that object called B above.
This is a sandwich functor.
In terms of boundary conditions, to apply this functor to any other object X, you form
a sandwich with B on one side and X on the other.
If these are boundary conditions in a two-dimensional theory, you get a Hilbert space on the strip
or generally on a sandwich.
Any local operators sitting on the B boundary condition are going to act in this Hilbert
space.
That Hilbert space forms a representation of the algebra of operators on B, and that's
what I mean by end B, the algebra of operators on the B boundary condition.
I'm also going to be sketchy in the talk about homology versus chains.
There are a few places where I will point out important differences.
I will espouse Ben's philosophy and say that everything is derived unless otherwise stated.
Suppose I wanted a bit more structure in physics.
Suppose that our category C has a pair of objects that are both very large, very large
in the sense that they see the entire category, mathematically you'd want to say that they
generate the entire category, specifically such that these maps, obtained by sandwiching
with either B or B dual, are actually isomorphisms.
Our category is completely equivalent to the category viewed as a module for B operators
or B shriek operators.
I'll call the two endomorphism algebras A and A shriek.
The second condition that I'll ask for is that B and B shriek are transverse boundary
conditions, in the sense that if we put a theory on a sandwich between these two special
boundary conditions, we get something trivial.
In a topological theory, the width of the sandwich doesn't matter, and by trivial I
mean in two dimensions that the Hilbert spaces see.
There are certain aspects of causal duality that also seem to apply to physical theories,
and I'll discuss later what could morally be causal dual pairs of boundary conditions
in a physical theory, and their transversality makes perfectly good sense.
You put the theory on a sandwich and flow to the IR, or shrink the sandwich to zero
size and you want a trivial theory resulting.
Given these conditions, it's fairly easy to show that the endomorphism algebras of these
two special boundary conditions are causal dual algebras given the previous definition.
Maybe the next slide will explain why I want transversality.
In terms of the definition of causal duality, there is this augmentation, this homomorphism
to C. Let me just put the slide up.
Transversality is what gives me C as a module over both of these.
I'll explain in applications why transversality is nice.
So, this C is going to be responsible for defining the functor, for defining an equivalence
functor between modules for one and the other.
Let me come back to that question in half an hour and see if it's still an issue.
There may also be ways to generalise this, but I do want to take C for now.
With the setup, we have C simultaneously as a module for local operators on one side and
local operators on the other side.
The two operator algebras have to commute because they're on different sides of the
strip.
The assumption that these boundary conditions are big enough, these objects are big enough,
guarantees that the stuff on the right is everything that commutes with the left and
vice versa, and that's where the duality comes from.
So, there are several sort of canonical examples of causal duality, every single one of which
shows up in physics and plays some fundamental role.
I will talk a little bit about sort of the, or example, the duality between the symmetric
algebra and vector space and the exterior algebra on the dual of that vector space.
That is easy to implement physically in a B model whose target is that vector space.
Another example which I'll go through might seem a little bit construed.
Then if you actually want to see something more natural, you move on to gauge theory.
Another really canonical example of causal duality relates the equivariance cohomology
of a point as a symmetric algebra, and the homology of G, which turns out to have the
structure of an exterior algebra.
This duality governs G actions in quantum mechanics.
This is some work I've been doing with David Bensky and Andy Nitsky.
In mathematics, this showed up, or this was applied in an essential way in the work of
Gorsky, Kotvitz, and McPherson.
There are a few others that I won't have much time to say anything about.
One I learned from work of Kevin Costellos, and it shows up as an important example in
a paper of Dylan Buttson and Filsang Yeoh.
This is a duality between the enveloping algebra of a Lie algebra and Schiavalli co-chains.
In other words, ghosts for that Lie algebra.
This shows up again in engaging and un-gaging in a different kind of two-dimensional theory.
The first time I ever learned about causal duality was actually in the study of 3D unequal
x4 theories and something called symplectic duality, which is sort of a fancier version
of this entire story, but still has to do ultimately with categories of boundary conditions
in 2D.
I won't, unfortunately, say anything about that.
Let me go through the simplest possible example.
I want to consider the B model with target C.
I thought of as a supersymmetric theory.
This is just a free chiral multiplet in two dimensions.
The local operators that are in the cohomology of the B-type supercharge consist of a complex
boson and a complex fermion.
The algebra of operators in the bulk is just a graded commutative algebra.
It's polynomials in a boson and a fermion.
If you're thinking about that B model language, you want to identify that with polyvector
fields on C.
I want to consider boundary conditions that preserve some symmetry, in particular the
U1 symmetries that rotate phi and psi.
There are two canonical ones.
One of them preserves the field phi at the boundary, but kills the fermion.
I'll call this a Neumann boundary condition.
The local operators that survive on the Neumann boundary condition are just insertions of
the boson.
We end up with a symmetric algebra as Nn.
The other boundary condition is Dirichlet, in the sense that it kills the boson.
It preserves the fermion, and we get an exterior algebra generated by this fermionic operator
psi as Nd.
There's only one component in Q cohomology, and that's the one that I'm killing.
There's a psi plus and a psi minus.
You're absolutely right.
I could describe these boundary conditions as one of them killing psi plus, the other
one killing psi minus, but only one of those two is in Q cohomology.
I'm simplifying the story.
These are transverse.
The way I've described it as having a boson and a fermion, it's clear that they're transverse.
Nothing survives when I collide them together.
You can say that nothing survives either in a Hilbert space language by putting the theory
on a strip or just asking what are the local operators at the juncture of Neumann and Dirichlet
boundary conditions?
You just get, well, just get basically multiples of the identity coming in from the bulk.
Now, I should say secretly, this copy of C is coming about as the cohomology of a complex
that's built out of operators on N and operators on D with a suitable differential.
So, transversality holds somewhat less trivially.
These boundary conditions are both big enough, at least if you're careful about how you define
the category of boundary conditions, these boundary conditions are big enough.
Any boundary condition can be expressed as a module over the symmetric algebra or the
exterior algebra.
Let me give you an example of how that comes about.
A typical boundary condition in such a theory can be obtained by taking Neumann, the one
that leaves the boson to survive at the boundary, and deforming it a bit so that, say, some
power of that boson vanishes.
Using a sandwich with Neumann produces a Hilbert space that, sort of by definition,
is given by polynomials in phi modulo the constraint that phi to the n vanishes.
That's automatically expressed as a module over the polynomial algebra.
What about the other one?
Given a suitable definition of this deformation, we could just compute.
We could form a sandwich and figure out what that Hilbert space is.
But since I'm trying to emphasize causal duality, let me explain how you get the exterior algebra
module from the symmetric algebra module.
You build a little strip like this.
The fact that the boundary conditions are transverse, again, means that there's a unique
thing I can insert at a junction between them.
Then the strip gives me an operation that goes from the Hilbert space on a Neumann sandwich
to the Hilbert space on a Dirichlet sandwich and back.
That can be translated into a more precise mathematical prescription.
We start with some symmetric algebra module, like C of phi mod phi to the n, and then
tensor with C. By tensor with C, I secretly mean tensor with that big complex, whose cohomology
is C. Identify the phis and apply the differential from that complex.
One works through that, one gets something interesting.
The basic vector space of this complex is polynomials in both phi and psi with phi to
the n set to zero and a differential phi psi.
That can be split up according to powers of phi, there are n different ones that show
up and a differential that goes from one to the other.
Every term in this complex is a two-dimensional vector space, exterior algebra on one variable.
The action of the exterior algebra, we're trying to build an exterior algebra module,
is sort of obvious, it's just multiplication by psi that obviously commutes with the differential.
I could compute the cohomology.
That would just give me two copies of C, one at the beginning and one at the end, everything
else cancels out pairwise.
This is the place where it's important to say DG module rather than module.
I will actually explain a little bit later in the case of gauge theory why you can't
just pass to cohomology.
These are not quasi-isomorphic, sorry, the complex and its cohomology are not quasi-isomorphic
as DG modules.
Another way to say it is that there's a higher infinity operation that ties these two copies
of C together.
If you want to include that in to the cohomology, you can just pass to two copies of C.
You need to somehow remember that there's this action of psi around.
I'll try to make that clear a little bit later.
What did we have?
We had a very simple-looking module for the symmetric algebra and a giant complex that
was a module for the exterior algebra.
These are both ways of looking at the exact same boundary condition in the B-model.
This embedded the basic causal duality in physics.
It might have seemed a little bit contrived.
It is not.
If you do more general B-models, there are generalizations of this picture.
To me, the most natural setting by far for causal dualities and gauge theories.
Let's consider a pure gauge theory.
I'll do it in two dimensions, but let me say some general words first.
Why is this a useful concept in general?
If we have pure gauge theory in any dimensions, there are always two transverse boundary
conditions, two canonical transverse boundary conditions.
One of them, Neimann, preserves gauge symmetry at the boundary.
In terms of fields, it sets the mixed components of the curvature to zero.
In other words, f with a component along the boundary and one component perpendicular
to the boundary.
The other one, Dirichlet, breaks gauge symmetry at the boundary to a flavor symmetry.
In terms of fields, it sets the connection along the boundary to zero.
If we start forming sandwiches out of Neimann and Dirichlet boundary conditions in gauge
theory, we end up with effectively d-dimensional quantum field theories that are versions
of the same d-dimensional theory, where a symmetry has been either gauged or left as
a global symmetry.
To make that a little bit concrete, let me observe again that the Dirichlet boundary
condition, by definition, breaks the gauge symmetry to a global symmetry.
You can even see operators that act effectively as currents for this global symmetry.
They come from the mixed components of f, which survive on the Dirichlet boundary condition.
Forming a sandwich with any other boundary condition leaves me with a d-dimensional theory
that has a global g-symmetry, and these f's precisely map to the currents in the d-dimensional
theory.
Given that the Dirichlet sandwich gives me something with global symmetry, the Neimann
sandwich gives me the same thing, but with the symmetry gauged.
In that sense, these two transverse boundary conditions control gauging and un-gauging
of general quantum field theories with g-symmetry.
Of course, the processes in a general quantum field theory are not reversible.
Once I form the sandwich, I can't unform the sandwich, but if I know how to take some theory
with g-symmetry and lift it to a boundary condition, then I can immediately form both
gauged and un-gauged versions in d-dimensions.
In a topological quantum field theory, one can do much better.
In particular, one can undo the sandwich.
If Neimann and Dirichlet are these special kinds of boundary conditions that generate
the entire category, in particular, if they're causal dual, then we can start passing freely
back and forth between gauged and un-gauged versions of the d-dimensional QFT.
At least that is the picture in principle.
I should say that actually implementing this in practice in dimensions greater than two
is very subtle.
Let me get to the main example of the talk.
This is work with David Benzvy and Andy Knightsky.
We wanted to consider the A-twist of pure gauge theory, pure supersymmetric gauge theory
in two dimensions.
If G is abelian, then this is basically the previous B-model example, because N equals
2, 2, U1 gauge theory can be dualized almost to a free-twisted chiral multiplet.
We're going to get a symmetric exterior algebra duality.
For general groups, the story is a little more interesting.
The boundary conditions for this theory come from supersymmetric quantum mechanics and
specifically the type of supersymmetric quantum mechanics that Witten considered in his famous
paper on Moore's theory.
This is Durand-type quantum mechanics with a global G-symmetry.
In the simplest case, one can consider a category whose objects are just labeled by smooth compact
Riemannian manifolds with a Riemannian G isometry, which will become this G-flavor symmetry.
A quantum mechanics whose target space is such a Riemannian manifold can be used to
define a boundary condition for 2D gauge theory just by minimal coupling, by using the bulk
gauge fields to gauge the boundary symmetry.
Moreover, the morphisms in this category are extremely easy to calculate.
If we put Riemannian manifolds X and Y on opposite sides of a strip and squeeze the
strip down to zero size, well, we just get quantum mechanics with G gauge symmetry and
target X cross Y.
The Hilbert space of that quantum mechanics is G-equivariant cohomology of X cross Y.
Those are the morphisms.
Let me look at the two canonical boundary conditions.
Neumann in this language is described by taking the Riemannian manifold X to just be a point
with a trivial G action.
The trivial G action means that G gauge symmetry is preserved at this boundary, and we can
check that this actually makes sense.
If we form a sandwich with a point on one side and some other Riemannian manifold X
on the other side and compute the Hilbert space, by the previous prescription we'll
get G-equivariant cohomology of point cross X, which is just G-equivariant cohomology
of X, which is the correct Hilbert space and gauged quantum mechanics with target X.
Moreover, we get the Hilbert space of gauge quantum mechanics as a module for operators
on the Neumann boundary conditions.
Now you can compute these operators in many different ways.
From the point of view of supersymmetric gauge theory, and just look at the supersymmetric
extension of Neumann and see that there's a complex Lie algebra-valued scalar, sigma,
that survives on the Neumann boundary condition.
This boundary condition should have gauge invariant polynomials in the scalar sigma.
In the sort of language I've been using here, we can formally compute the endomorphism algebra
as G-equivariant cohomology of point cross point.
It's operators between Neumann and Neumann, which is just G-equivariant cohomology of
a point, which is gauge invariant polynomials in the Lie algebra.
This acts on equivariant cohomology in a standard and extremely well-known way.
Let me just explain how that works in examples, in case you have not seen it before.
I want to go through three very basic examples.
Let me start with the Riemannian manifold just being S1 with a circle action.
Our sigma is just a complex scalar.
It's valued in the complexified Lie algebra of U1.
The Hilbert space engaged quantum mechanics on X is just C.
Once the action is free, we can just quotient it down and we'll get cohomology of a point.
But as a module for the symmetric algebra, it's C of sigma mod sigma.
A very different sort of example occurs when the G action has fixed points.
If we take the two-sphere and a U1 symmetry that rotates it with fixed points at the north
and south poles, there's a fixed point localization theorem in equivariant cohomology that says
the corresponding Hilbert space engaged quantum mechanics is just ordinary cohomology C2
times arbitrary polynomials in sigma.
This is a free module over the symmetric algebra.
A slightly more interesting example of a space with a free G action is S3.
If we view S3 as the Hopf vibration with U1 rotating the Hopf fibers,
and compute equivariant cohomology there, well, again, we can quotient down.
So as a vector space, the equivariant cohomology of S3 is just the ordinary cohomology of the quotient,
S2, which is C2.
But as a module for the polynomial algebra, it's C of sigma mod sigma squared.
Okay, what about the other guy, the Dirichlet boundary condition?
In the same sort of language, Dirichlet is realized as coupling to the Ramanian manifold G.
If we were in a higher dimensional quantum field theory, we would have said that coupling
to G on the boundary spontaneously breaks the symmetry, which is what we want.
We want to turn the gauge symmetry into global symmetry.
We can check, right, sorry.
So I'm saying that the stabilizer of any point on G under G multiplication is trivial.
That's the symmetry breaking.
We can check that explicitly by forming sandwiches and computing.
So colliding some manifold X with Dirichlet gives us a Hilbert space.
That's the equivariant cohomology of G times X.
We can divide by G and just get ordinary cohomology of X,
which is the Hilbert space in supersymmetric quantum mechanics, to an ungaged target X.
What about the operators?
Well, this same sort of analysis tells us how to find them.
We compute endomorphisms as G equivariant cohomology of G times G.
In other words, the ordinary cohomology of G, which may now seem a little bit surprising.
It implies that the Hilbert space of supersymmetric quantum mechanics,
to any manifold with a G symmetry, has the cohomology or dually the homology of G acting on it.
This is the analog of the exterior algebra in the previous examples.
It leads to what is called a homological G action in supersymmetric quantum mechanics.
It actually exists, though, as far as I know, it has not really been appreciated before.
I'm going to take a few minutes just to describe this thing.
Let me recall what Witten said.
If we have supersymmetric quantum mechanics with a Ramanian target X,
the full Hilbert space is differential forms on X.
The supercharges act as dnd dagger, the Hamiltonian, acts as the form Laplacian,
and we're in a situation where there's a Fermion number, which corresponds to form degree.
The space of supersymmetric ground states is identified with Q cohomology
of the full Hilbert space, in other words, the Ram cohomology of X.
So what happens in quantum mechanics when there's a flavor symmetry?
Well, the infinitesimal action of this group
geometrically is generated by Lie derivatives.
In the physics, it's generated by neuter charges.
These neuter charges, Lie derivatives, commute with supersymmetry,
which immediately implies that there should be an action of the Lie algebra
on the supersymmetric Hilbert space, but there's one other formula around.
The Lie derivative is actually Q exact.
It's an anti-commutator of dniV, contraction with a vector field V,
which directly implies that the supersymmetric Hilbert space
is a trivial representation of the Lie algebra.
You can check that the statement actually makes sense.
If you take a closed form and consider its Lie derivative, you get something exact.
So Lie derivative acts as zero.
That is the reason why we don't usually hear
a lot about flavor symmetry in supersymmetric quantum mechanics.
The point is that even though the infinitesimal action of the group is trivial,
the actual action is not.
There is something coming from the global structure of the group.
To understand it, one way to proceed is to sort of generalize the notion of topological descent.
The Lie derivative of a closed form is d of something.
We can remove a d and think of that something, namely IV omega, as a one form on the group.
By taking a Lie derivative, we're deforming on the group.
So we get something that looks like a one form.
It's, of course, still valued in differential forms on the manifold X.
Then we can just pick a one cycle and integrate.
The formula here is not exactly right.
What I mean by that integral is first to promote this one form that's actually living in the
tangent space of the identity to a form on the entire group and then integrate that.
This operation of applying IV and smearing over a cycle on the group
takes any P form on our manifold X to a P minus one form.
You could say it has Fermion number minus one and it commutes with the supercharge.
That is an easy calculation.
It's a standard sort of calculation that one does in topological descent.
And then one finds that this action of a one cycle on the group descends to the
supersymmetric Hilbert space.
In a very similar way, one can apply contraction with the vector fields multiple times and
integrate over a K cycle on the group to get an action of the entire homology of G
on the supersymmetric Hilbert space.
And that, at least very naively, is the homological G action.
There's a little bit more, which is where the letters D and G come in.
I'll explain in the next slide.
One thing to say at this point is that this action automatically endows the homology of G
with the structure of an algebra.
You can also think of that structure as coming just because G is not just a manifold.
It's a group.
There's a multiplication on the group that induces a product in homology.
And it ends up being an exterior algebra.
So examples.
I'm going to go through this a bit quickly.
So first, if the group is U1, the homology is two dimensional.
The point class is going to correspond to the identity in this algebra.
And the one form, the top class, is going to be a non-trivial operation that squares to zero.
So we get an exterior algebra on one variable.
Now, consider a space like S1 with rotations.
This action of gamma takes a one form and really just integrates it over S1.
On forms, it takes one form to zero forms.
And the operation descends to be something non-trivial on cohomology.
It takes the volume form to one.
If we instead consider something like S2 where there were fixed points,
the setup is totally different.
In cohomology, we still have this gamma, sorry, on forms, we still have this gamma operation.
But once we pass to cohomology, there's nothing in between.
So gamma has to be zero.
And so cohomology in this case really is just two copies of C as a trivial module
for the exterior algebra.
If we want to get something more interesting, we can consider other spaces with free actions
like S3.
So in this case, there are forms in degrees zero, one, two, three with d's raising degrees,
gamma's lowering degrees.
Again, by the same argument as above, if we pass to cohomology, we lose the gamma action.
Gamma has for me on number one.
There's no way it's going to relate the top copy of C to the bottom copy of C.
On the other hand, we can do something more interesting here and apply gamma.
And then notice that d is invertible because the cohomology in degree two vanishes.
So apply d inverse and apply gamma again and actually get a yet higher operation
that connects the top copy of C to the bottom copy of C.
This is the sort of A infinity thing that I hinted at before.
And sort of the reason that the full structure of this homological G action
is described not as ordinary exterior algebra modules, but as exterior algebra DG modules.
And so the precise statement is that forms on S3 is a DG module for the exterior algebra.
So that exists in quantum mechanics.
To summarize, if we have gauged quantum mechanics, there's a very old story.
Its Hilbert space is equivariant cohomology and a symmetric algebra acts on it.
If we consider ungauged quantum mechanics, we get durham cohomology and an exterior algebra acts on it.
In order to pass back and forth between the two, not only to gauge a G symmetry,
but to ungauge a G symmetry, you just have to keep track of these module actions.
They're like little handles that you can plug into to manipulate the structure of the symmetry.
And that manipulation is the duality picture.
So you can go between gauged and ungauged by tensoring with the current version of C.
So in this case, what is C? It's hums between Riemann and Dirichlet.
In other words, it's the G equivariant cohomology of G, which is, and cohomology is certainly just C.
The way the functor works is by tensoring with the entire complex of equivariant co-chains
and applying a differential. Really exactly the same way it worked in the B model.
As a physical note, I think from what I've said and what I've had time to say, this picture is
still very abstract. Almost every detail can be worked out very, very explicitly.
The gamma operations are disorder operators. They're fermionic. They're built with these IVs,
which are super partners of the neuter charges. They're fermionic operators decorated by a
disorder operator. And in the U1 case, which is just like the B model, they come from bulk
operators. And in the bulk, they're gaugeinos decorated by vortices. You know that they have
to be disorder operators of some sort because they're acting very nonlocally on the target
space. They're smearing along a G orbit. Let me just very quickly mention the last application.
This is going to sound very, very different. And certainly in the time I have to speak about it,
it's going to look very different. But the point is that this causal duality coming from
transverse big enough boundary conditions plays a dual role in slicing theories up and gluing
them back together. So the basic physical motivation is that we often want to do something
to half of a quantum field theory. For example, to create a duality interface.
For D, if you want to create an S duality interface, you apply S duality to half of
N equals 4 super young mills and get degrees of freedom trapped at the boundary. To analyze
those degrees of freedom, you want to split the theory up, dualize half along with the boundary
condition, and then glue it back together and just canonically see what you're left with on
those boundary conditions. So it's incredibly useful to be able to cut and glue like this.
So the gluing property of two boundary conditions that we can slice on is that when we couple them
back together, we end up with a trivial interface. So we can certainly take any pair of boundary
conditions we want for two copies of the theory on a half space. But we want them to have the
property that there's some sort of canonical coupling between them that leads to a trivial
interface when they're put back together. So engaged theory, this sort of property is usually
enjoyed by the Neumann and Dirichlet boundary conditions that I mentioned before. And you can
see fairly easily why that's the case. Dirichlet has a global g-symmetry.
Neumann can couple to global g-symmetries. And so if you cut along Dirichlet and Neumann,
you can glue them back together just by gauging the global symmetry on one side with the other
side. And you just fairly trivially end up with the original gauge theory on an entire space.
Right. And as I've been saying, the fact that Neumann and Dirichlet have this property
engaged theory that you can slice and glue along them is very far from a coincidence.
And in general, if you list the properties that I just described in the last slide in algebraic
language, you end up seeing that causal dual boundary conditions are exactly what you need.
So, yeah. In any sort of theory, once you have a pair of these special boundary conditions,
you can start manipulating in any way you want. Okay. Thank you.