Download Topological Quantum Field Theories in Topological Recursion

Topological Quantum Field Theories in
Topological Recursion
Campbell Wheeler
Paul Norbury
The University of Melbourne
Outline
We aim to find an elementary construction of a class of 2-dimensional Topological
Quantum Field Theories that arise out of Topological Recursion. We know that there
is some 2-dimensional Topological Quantum Field Theory contained in the output
of some Topological Recursion due to results in Cohomological Field Theory [6].
We consider a graphical approach to both 2-dimensional Topological Quantum Field
Theory and Topological Recursion and use these to try and find this construction.
Topological Quantum Field Theories
Firstly we will discuss some of the physical motivation behind Topological Quantum
Field Theories then describe there axioms and how these can be interpreted using
category theory. Then we’ll see how they are motivated by the physics. Many of
these ideas are attributed to M. Atiyah and E. Witten and the following papers and
book go into much more depth than we will here [1] [7]. We will however try to keep
the discussion simpler.
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Motivation
Not surprisingly the motivation behind the axioms of a Topological Quantum Field
Theory comes from Quantum Field Theory. To define a Quantum Field Theory on
some manifold we must specify some action, S(φ), which is a function of the fields,
φ, on the manifold. This action completely determines the physics on this manifold.
When we say physics we mean expectation values and correlation functions between
physical observables which is the output of a Quantum Field Theory. One way
calculations are carried out is through Feynman integrals. They are of the following
form
Z
hW i = W eiS(φ) dφ
where the right is an integral over all fields of the theory. In analogy with the
partition function of statistical mechanics we can define what is called the partition
function of our theory which is like the expectation value of 1. It is given by the
following
Z
Z = eiS(φ) dφ
Now for particularity nice actions, that are independent of the metric given to the
manifold the Quantum Field Theory is on, the partition function Z will be an invariant of the manifold. An example of such an action is the Chern-Simons action [1] [4].
These integrals aren’t well defined so we try to formulate axioms that will not rely on
these ill defined integrals but capture the information they hold. The direct link between the axioms is complex and needs more discussion than we will go into. See [4]
for more. The axioms given next are based on axioms given in [4] which are based
on axioms given in [1].
Axioms
A (n+1)-dimensional Topological Quantum Field Theory associates to every n-dimensional
(smooth compact orientable) manifold, Σ, a vector space, Z(Σ), and to every (n+1)dimensional (smooth compact orientable) manifold with boundary, M , to a linear
functional, Z(M ), with the following properties. (A visual aid is given below)
• (Vacuum) Z(∅) = C
• (Duality) Z(Σ∗ ) = Z(Σ)∗
• (Multiplicity) Z(Σ1 t Σ2 ) = Z(Σ1 ) ⊗ Z(Σ2 )
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These aren’t the only rules but they are the main ones and give a feel for how it
works. We wish to simplify the discussion so we don’t need to list an extensive list
of axioms. It is important to note that gluing is achieved through contracting, the
natural map from the tensor product of a vector space and it’s dual to the field. This
will require us gluing boundary components with opposite orientation. See [1] [4] for
more.
Functorality
The notation used above to denote the vector space, Z(Σ), associated to the ndimensional (smooth compact orientable) manifold, Σ, is for a reason. The reason
is that if we list the way the association works with all the rules and we know a
little category theory we can see that this association is actually a functor from two
categories. It is a functor from the category of (n+1)-dimensional cobordisms to
the category of vector spaces (both with some extra structure). Noting this we can
define a (n+1)-dimensional Topological Quantum Field Theory as follows.
An (n+1)-dimensional Topological Quantum Field Theory is a functor, Z, from the
category of (d+1)-dimensional cobordisms to vector spaces.
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This definition is compact but using the rules directly is often clearer and in what
follows we won’t really touch on the fact that a Topological Qunatum Field Theory,
Z, is a functor.
2-dimensional Topological Quantum Field Theories
We will build up to the statement of a classification theorem of 2-dimensional Topological Quantum Field Theories. We will use results from the classification of surfaces
to do this. We will then also describe some algebra structure every 2-dimensional
Topological Quantum Field Theory will have and use this to show that every 2dimensional Topological Quantum Field Theory is equivalent to some Frobenious
Algebra and visa versa.
Classification of Surfaces
Theorem - Classification of surfaces with boundary A (smooth connected
compact orientable) surface with boundary is completely determined by its number
of connected boundary components and its genus (i.e. the number of handles). [6]
This means we will write a given surface with boundary as Mg,n where g represents
the genus and n represents the number of connected boundary components.
Boundary Components Note also that the each connected boundary component
is isomorphic to S 1 , the circle. This means that from the axiom of multiplicity we can
get the vector space of any boundary component from the vector space associated to
the circle. We do this by applying the axiom which will give us some tensor product
of the vector space Z(S 1 ).
Constructing Surfaces It is important to note that every connected surface (besides the sphere, two holed sphere/cylinder and the torus) can be built out of gluing
three holes spheres (pairs of pants in the language of [6] or trinions in the language
of [4]). This is easy to see after we have the classification theorem and a pen and
paper. From this we can see that every surface can be built using one holed spheres,
two holed spheres and there holed spheres.
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Classification
From the construction of surfaces and the possible boundary components we can
see that the following association completely determines a 2-dimensional Topological
Quantum Field Theory.
|
Z(S 1 )
1
η
5
c
|
Every vector space and surface can be gotten by taking disjoint unions or by pasting
one holed spheres and three holed spheres. We use the reversed orientation of the
cylinder to paste things together. Note that in this way the cylinder acts like a semimetric (i.e. can be negative) as we can use it as a non-degenerate inner product.
Pasting and disjoint union are governed by the axioms so once these are specified
everything is determined.
Algebra Structure
We can view the three holed sphere as giving us some kind of multiplication when we
contract (glue) it to a cylinder. The following is a visual proof that this multiplication
is associative.
In fact this multiplication is unital, commutative and associative. Also note that
with the cylinder we have a non-degenerate inner product that satisfies the following
ha, b · ci = ha · b, ci
the proof is again given visually below.
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This collection of facts describes a Frobenious algebra and in fact every Frobenious
algebra is determined by the same four things we listed above. A vector space, commutative and associative multiplication, a unit and a non-degenerate inner product
that satisfies the above statement.
Semi-Simple
If a Frobenious algebra has a basis {e1 , ..., en } that satisfies the relation below we
call this Frobenious algebra semi-simple.
ei · ej = δij ei
Note δij is the Kronecker delta. From this definition one can see that we have
1=
n
X
ei
i
Calculations are then very trivial if we are given such as basis. In fact given such a
basis the following numbers will determine our Frobenious algebra.
{h1, e1 i, ..., h1, en i}
Using some properties of the Frobenious algebra this is the same as the following.
{he1 , e1 i, ..., hen , en i}
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These numbers are then what we need to describe a semi-simple 2-dimensional Topological Quantum Field Theories. This is interesting as if we have a semi simple Topological Quantum Field Theory there is a list of numbers that completely determine
the theory.
Problem
Finding the idempotent basis is not necessarily easy. To find the basis in terms of
a general basis we need to solve multiple quadratic equations in multiple variables.
This becomes computationally difficult when we have large numbers of variables and
equations. An example of the kind equations needed to be solved is given here
α2 + 3β 2 + 2γ 2 = α
2αβ + 4βγ = β
γ 2 + 2αγ + 3β 2 = γ
(This example came from the center of the group algebra generated by S3 and is only
a subset of the list of equations needed to be solved to find the basis) There are a
few solutions to these equations that are interesting when we consider the full set of
equations. They are given here.
α=β=γ=
1
6
α = −β = γ =
1
6
2
α= ,
3
β = 0 and γ =
−1
3
They will actually be the coefficients needed to switch the standard basis for the
center of the group algebra to the idempotent basis.
Graphical Approach
We are going to describe a graphical method that can be used to compute various
linear functionals from the four things that classify our surface. This method is
reminiscent of that used when discussing Feynman Diagrams.
Pair of Pants Decomposition to a Graph
As mentioned in the classification of surfaces every surface can be constructed out
of pairs of pants, that is three holed spheres. Instead of considering a possible
construction (note that in general there will be many possible ways to construct a
given surface out of pairs of pants) we will now consider a possible decomposition
given some surface. Once given this decomposition we will associate to this a trivalent
graph (each internal vertex having three edges going into it). The way we will do
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this is by letting each pair pants be a vertex and each boundary component or glued
boundary be represented by an edge. This process is illustrated here. Note that the
number of edges and vertices (3g − 3 + n and 2g − 2 + n respectively) is independent
of our decomposition.
Colourings of Graphs
Once we have graph we can start to determine our linear functional. The way we do
this is by colouring the external vertices with the vectors we wish to input into our
functional. A colouring is basically just some labelling. So we label the end of an
edge by some vector. We can only consider basis elements as the input as anything
else will be some linear combination of these and the linear functionality means we
can calculate that using the linearity. Once we have an input we wish to compute, we
colour the ends of each edge (i.e. each edges has two colours) by the basis vectors and
consider all possible colourings. We associate a weight to each colouring and then
sum over all possible colourings. Say we consider a 2-dimensional Frobenious algebra
with basis {e0 , e1 }. The following is an example of a colouring for the Topological
Quantum Field Theory described below in Benefits.
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Weight of a Colouring
The weight associated to a colouring comes from associating a weight to each edge
and vertex then multiplying these together. These weights come from the reciprocal
of the cylinder of the two coloured vectors of an edge (the cylinder = edge = η) and
the value of the pair of pants of the three coloured vectors of a vertex (the pair of
pants = vertex = c) respectively. Note that one needs to check that when we sum
over all colourings we get the the same result no matter what decomposition we use.
This is of course true.
Benefits
For a graph with 6 internal edges we already have 212 possible colourings for a two
dimensional vector space so this doesn’t seem to help in any calculation. However
if we choose a nice basis most of the weights may go to zero and then we can rule
out all but few colourings which greatly simplifies the kind of calculation we’d have
to do otherwise which would involve large strings of tensor products. An especially
nice example is with the idempotent basis which gives either zero or one possible
non-zero colouring for each input. For a slightly more complex example let {e0 , e1 }
be a basis for Z(S 1 ). Consider the following outputs for the cylinder and the pair of
pants.
1 if i 6= j
η(ei , ej ) =
0 if i = j
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c(ei , ej , ek ) =
1 if i + j + k is odd
0 if i + j + k is even
This is an example that arises in Cohomological Field Theories. Notice only colourings with opposite vectors on each edge will be non-zero. In fact with this example
there will be either 0 or 2g non-zero colourings and each colouring will have weigh 1
in this way we can actually determine the following result.
P
g
2
if g + n − nj=1 ij is odd
P
Z(Mg,n )(ei1 , ..., ein ) =
0 if g + n − nj=1 ij is even
The idea is that either we have no colourings or that each handle gives us two possible
colourings. A picture is given here to give an intuitive idea of the argument one would
use to prove this.
Topological Recursion
We will give a brief introduction to Topological Recursion and how we can describe
it through a similar graphical approach to that used in the topological quantum field
theory. We will talk about some key differences and similarities.
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Plane Curves and Symplectic Invariants
A plane curve is a subset of C2 defined by a polynomial equation in the two variables.
The set is given by the expression {(x, y) ∈ C2 | p(x, y) = 0} for some polynomial, p
in x and y. This set will be some surface or a one dimensional complex manifold
immersed in C2 . This will not necessarily be an embedding and we may have self
intersection (e.g. the Klein Bottle can be immersed in R3 but can’t be embedded).
Eynard and Orantin developed a sequence of meromorphic differentials ωng to recursively define symplectic invariants F g = ω0g on a genus zero immersed plane curve
where the branch points in the coordinate x are simple (have only one twist). These
invariants are invariant under automorphisms of C2 that preserve the symplectic
form dx ∧ dy. See the introduction in [5] and also [2] and [3].
The Recursion
The recursion that they defined has the following base and kernel which are defined
in terms of the immersion of the plane curve, (x, y). Note this kernel will need only
be defined close to the branch points of x in it’s second variable and that when z is
near a branch zb denotes the unique point such that x(z) = x(b
z ).
ω10 (z1 ) = y(z1 )dx(z1 )
ω20 (z1 , z2 ) = B(z1 , z2 ) =
dz1 dz2
(z1 − z2 )2
Rz
K(z1 , z) =
B(z1 , z 0 )
2(y(z) − y(b
z ))dx(z)
zb
From these kernels the recursion is defined as follows.
X g
X
g−1
g2
g
1
ωn+1
Res K(z1 , z) ωn+2
(z, zb, zS ) +
ω|I|+1
(z1 , zS ) =
(z, zI )ω|J|+1
(b
z , zJ )
a∈A
z=a
ItJ=S
g1 +g2 =g
where S = {2, ..., n + 1}, I 6= ∅ and J 6= ∅ and A is the set of branch points.
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Diagrammatic Representation
This expression looks a little messy but there is a nice way to think about it that
makes it much clearer. We view the ωng as connected surfaces with n boundary points
and genus g. Then we can picture the recursion as follows.
• View K as a pair of pants
• Consider all ways to glue a disjoint union of connected surfaces to two of the
holes in the pair of pants to get n boundary components and genus g.
• Weight each possible gluing by the product of the ωng ’s associated to each
connected surface being glued.
• Sum over all gluings considered and then sum over the residue at every branch
point.
The process is given visually here.
Graphical Representation
We can represent these diagrams graphically like we did for the 2-dimensional Topological Quantum Field Theories but there is much more structure that needs to be
added. Representing them graphically we can actually ignore the fact we are using a
recursion and go straight to the expression we’re interested in. Doing this we don’t
just consider one pair of pants decomposition we consider every possible decomposition. There is even more structure on top of that. A full description of this process
can be found in section 4.5 of [2].
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Research
A large portion of this project is actually trying to find the Topological Quantum
Field Theory in this Topological Recursion. We have came to some kind of result
but this still needs more work. We’ll briefly highlight where we expect to see the
Topological Quantum Field Theory. We’ll also briefly describe some ideas of how
the recursion could lead to a Topological Quantum Field Theory.
Where is the Topological Quantum Field Theory?
Here are some examples of the ωng for the plane curve y 2 −yx+1 = 0 with x(z) = z + z1
and y(z) = z. Note zb = z1 .
ω30 (z1 , z2 , z3 )
ω11 (z1 )
=
=
!
1
1
1
1
−
dz1 dz2 dz3
2 (1 − z1 )2 (1 − z2 )2 (1 − z1 )3 2 (1 + z1 )2 (1 + z2 )2 (1 + z1 )3
!
1
1
1
1
1
1
1
1
1
1
1
1
−
−
−
+
+
dz1
16 (1 − z1 )4 16 (1 − z1 )3 32 (1 + z1 )2 16 (1 + z1 )4 16 (1 + z1 )3 32 (1 + z1 )2
We expect to see a Topological Quantum Field Theory contained in the coefficient
of the highest order pole. We think of the poles as representing the input of various
basis vectors. The dimension of our vector space will in fact be the number of poles.
This is an interesting problem as there are many different coefficients in font of these
poles including intersection numbers on the moduli space of curves which are quite
complex mathematical objects.
A closer look at K
It is interesting to look at the behaviour of K when we consider close to the branch
points of the second variable. We can write K as follows.
K(z1 , z) =
1
dz1
1 z − zb
2 y(z) − y(b
z ) (z − z1 )(b
z − z1 ) dx(z)
If we consider z → a where a ∈ A and assume dy
(a) 6= 0 then we have the following.
dz
1
z − zb
1
dz1
lim
K(z1 , z) →
2
2 z→a y(z) − y(b
z ) (a − z1 ) dx(z)
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It seems that the limit in this formula may play an important role in this problem. It seems similar to a derivative of y this may possibly be important. Taking
this limiting case of K makes it seem very similar to the B (note the second last
term). So maybe this contains the information of the edge as well as the vertex
which may be interesting. The other aspect that may be important is the terms of
z
to
the form db
z . This will add another derivative into the terms of the recursion ( db
dz
be exact) so this will play an interesting role. The aim of this direction is try and see
possible coefficients that can be pulled out of the recursion to get back a Topological
Quantum Field Theory.
Acknowledgements
I would like to thank AMSI and the University of Melbourne for giving me the
opportunity to take part in this research project. I’d also really like to thank Paul
Norbury for the countless number of conversations and for advice. This has been a
really enjoyable experience and I think that this is largely due to the person I did
a lot of the hands on work with. So lastly I’d like to thank Anupama Pilbrow for
working on the project with me and for the many valued conversations.
References
[1] M. Atiyah. The geometry and physics of knots. Cambridge University Press,
1990.
[2] N. Orantin B. Eynard. Invarients of algebraic curves and topological expansion,
2007.
[3] N. Orantin B. Eynard. Topological recursion in enumerative geometry and random matrices. Journal of Physics A: Mathematical and Theoretical, 42, 2009.
[4] R. Lawrence. An introduction to topological field theory, 1994.
[5] P. Norbury. Counting lattice points in the moduli space of curves, 2008.
[6] Paul Norbury. Various conversations, 2014/2015.
[7] E. Witten. Quantum field theory and the jones polynomial. Comunications in
Mathematical Physics, 1989.
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