The 12th annual graduate student topology and geometry conference will be at UT Austin this April. The registration deadline is coming up on February 1. The website for registration and more information is http://ma.utexas.edu/conferences/gstgc14/. Many of the talks are given by graduate student participants, and we have lots of interesting faculty speakers in a range of topology/geometry topics. I encourage all grad students to register and submit a talk proposal (expository or original research).
I just want to draw your attention to a survey that appeared on the arXiv on January 1: A symplectic prolegomenon, by Ivan Smith. The main point is to motivate and illustrate the Fukaya category, and to show how its algebraic structures amplify the power of Floer cohomology. Smith uses the running examples of the nearby Lagrangian conjecture and the symplectic mapping class group to demonstrate these algebraic structures (the Oh spectral sequence, the exact triangle associated to a Dehn twist, …) in action. There are lots of applications throughout, and one nice feature is that section 5 consists of explicit descriptions of the Fukaya categories of six (families of) symplectic manifolds.
So it’s great winter break reading, check it out! It collects together a lot of information that had previously been scattered over a bunch of different papers. And it includes the take-home messages of a number of rather intimidating papers.
This is my last post on defining the Seiberg-Witten equations and invariants for closed 4-manifolds based on a learning seminar at UT. Maybe I’ll post about some applications later on.
The Seiberg-Witten Configuration Space
We start with a Riemannian 4-manifold (M,g) and a Spinc structure on M. As we have seen, this data gives rise to the associated bundle and the determinant line bundle .
Let be the set of all Hermitian connections on . We have seen that such a connection gives rise to a connection on which is compatible with the Clifford multiplication.
The Seiberg-Witten configuration space is defined as
The Seiberg-Witten equations
The Seiberg-Witten equations take an element as their input. We are now prepared to define these equations.
As discussed in an earlier post, a connection A on gives rise to a connection on . Note that for , so . Therefore we have a Clifford structure
which composes with the connection
to get a Dirac operator
Denote the curvature of the connection A by . Then the curvature is a matrix of 2-forms on M, so we can consider its self-dual and anti-self dual parts and .
Let denote the traceless part of the endomorphism .
Now we can define the (perturbed) Seiberg-Witten equations. Fix a closed 2-form (the pertubation parameter). Then the Seiberg-Witten equations are:
The input to these equations is an element . The elements of which are solutions to these equations are called (-)monopoles.
The Gauge Action
The gauge group is . It acts on by
While it seems natural enough to act on the section by multiplication, why do we define the action ? Specifically where is the 2 coming from?
A is the connection of the determinant line bundle L of . We would really like to think of the gauge group as acting on . If acts on by multiplication , then the induced action on is multiplication by . (This goes back to the fact that in coordinate charts, the spinc structure is obtained by tensoring the spin structure with the square root of the determinant line bundle L.) Now we can look at how this acts on the covariant differentiation induced by the connection A on L. Here the natural action is conjugation
For we can consider its stabilizer in . If the stabilizer of C is trivial, we say C is irreducible, otherwise we say C is reducible. It is easy to show that the reducible elements are exactly those with , and that their stabilizers are the constant maps into .
The Seiberg-Witten moduli space
The Seiberg-Witten solution space is the space of elements for which the Seiberg-Witten equations are satisfied. To obtain the moduli space from this, we want to mod out by the gauge action. In order for this to be well defined, we first need to check that the space is invariant under the gauge action.
For the first equation, we can prove that because and because we can think locally that so taking its exterior derivative gives 0. Furthermore , so the first equation is invariant under the gauge action.
For the second equation, can be understood by breaking up the dirac operator into the composition of the Clifford multiplication and the connection on .
The discussion above about why the gauge group acts as it does on A is related to the fact that . Applying the Clifford multiplication to this connection acting on and using the Leibniz rule for connections eventually simplifies to show that so the solutions to are invariant under the gauge action.
Therefore we can mod out the Seiberg Witten solution space by the gauge action to get a well-defined space.
Properties of the Seiberg Witten moduli space
The reason the Seiberg-Witten equations are so useful is that the moduli space is actually a compact smooth manifold in many cases. When there are no reducible solutions to the equations, the moduli space defined by a generic perturbation is a smooth manifold (one needs to show that the linearization of a map defined by the Seiberg Witten equations and the gauge action is Fredholm and then use Sard-Smale to show that generic perturbations correspond to regular values).
Compactness of the manifold requires some analytic estimates. The Weitzenbock forumla is the main tool in obtaining bounds on solutions to the Seiberg-Witten equations.
After going through hard work to show these properties, which I am avoiding here, one just needs to worry about reducible solutions. Notice that if there are reducible solutions then they satisfy for our chosen perturbation. Since both of these forms are closed, they represent cohomology classes. The cohomology class of the curvature is independent of A, so we only have reducible solutions when . When the dimension of the positive second homology is at least 1, then a generic perturbation will avoid this phenomenon.
The Seiberg-Witten invariant of a 4-manifold is given by the homology class of the moduli space of solutions in the configuration space. This configuration space is homotopy equivalent to so its cohomology has a canonical generator in even degrees. By evaluating this generator against the homology class of the Seiberg-Witten moduli space we obtain an integer .
A priori this integer depends on the metric and perturbation, but when , the subspace of perturbations which allows for reducible solutions (bad perturbations) is codimension 2. Since the space of metrics on a manifold is convex, we can find a path through the space of metrics and good perturbations connecting any two pairs which lifts to a cobordism between the moduli space at and the moduli space at . Therefore SW gives a diffeomorphism invariant of the 4-manifold, and it has been used very effectively to distinguish many homeomorphic but not diffeomorphic 4-manifolds (exotic pairs).
When , there is a codimension 1 space of bad perturbations which forms a wall between two chambers. Within each chamber stays constant, and there is a well-understood wall-crossing formula describing the difference of SW in the two different chambers. By keeping track of a little more information, it is still possible to use information from the Seiberg-Witten invariants to distinguish exotic pairs (this has been used a lot for finding exotic ).
This is my third post on the set-up for the Seiberg-Witten invariants of 4-manifolds. The next post will finally define the Seiberg-Witten equations and invariants, so this is the last bit of background.
Symbols, generalized Laplacians, and Dirac operators
In order to define the Seiberg-Witten equations, we need to understand certain partial differential operators called Dirac operators. If you don’t know the formal definition of partial differential operators and their symbols, here is a link with some definitions and examples.
The class of all second order partial differential operator with the same symbol as the usual Laplacian: are called generalized Laplacians. Note that the symbol of a generalized Laplacian is an isomorphism on each fiber for , which means generalized Laplacians are elliptic operators. An elliptic operator L is good because there are estimates on the norms of solutions to equations of the form . This allows us to use Fredholm theory to describe the space of solutions to equations using elliptic operators. (In particular the linearization of an elliptic operator is Fredholm, i.e. has finite dimensional kernel and cokernel).
Dirac operators are 1st order partial differential operators which square to a generalized Laplacian. Dirac operators inherit many of the nice properties of Laplacians, specifically they are also elliptic (though in a weaker sense than the Laplacian–my vague understanding is that the bounds we get from ellipticity of the Laplacian are uniform, whereas the bounds we get from ellipticity of a Dirac operator depend on the point in the manifold; in the case of compact manifolds these coincide).
Dirac Operators and Clifford multiplication
We mentioned above that the symbol of a generalized Laplacian, (which is the square of a Dirac operator) is , for . Additionally, one can show that the symbol of a Dirac operator (which squares to a generalized Laplacian), is the square root of the symbol of the generalized Laplacian. Therefore so gives us a Clifford multiplication. In conclusion, a Dirac operator give rise to a Clifford structures by taking its symbol.
Conversely, given a Clifford structure, (equivalently ) and a connection we can compose them
and the resulting operator is a Dirac operator.
A Riemannian manifold M has a distinguished connection, the Levi-Civita connection , which has nice properties namely it preserves the metric g (this can be phrased either as or ), and it is torsion free meaning . Basically, this is a natural connection on TM when a Riemannian metric g is given.
Using the metric and orientation on M, the structure bundle of TM reduces to an -bundle. Namely, we can find gluing maps defining the tangent bundle that map into : which define a principal -bundle . The Levi-Civita connection on TM induces a principal -connection on specified locally by
We have the double cover map , which induces, by differentiating at 1, an isomorphism .
If we have a Spin structure on M, this means there are lifts such that . These define a principal Spin(n) bundle . In this case, the Levi-Civita connection on induces a connection on which is locally defined by
So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.
The representations , and give rise to an associated bundle . The spin connection on M induces a connection on whose local matrix valued 1-forms are defined by
Recall that acts on by the Clifford multiplication . The composition of the Clifford multiplication with the induced connection on yields a Dirac operators .
Remember, a -bundle is specified by gluing data
satisfying the cocycle condition
We want to understand structures for M and their connections. Let be a structure on M given by the bundle .
Letting , the associated spinor bundle to is , which splits into . A connection on the bundle will induce a connection on . Also note that has a Clifford structure, inherited from the map .
In the case that M has a spin structure, and .
In the general case, we will construct connections on the associated bundles using the Levi-Civita connection on M, and a choice of connection on the determinant line bundle of .
In the case that TM is the trivial bundle, the determinant line bundle has a square root, and and . We have the natural lift of the Levi-Civita connection to . This induces a natural connection on the associated bundle , which we can tensor with any connection on the line bundle to get a connection on .
Remember that had a Clifford structure c as well as a natural connection which together give rise to a Dirac operator. We obtain a similar structure on by twisting the triple with a line bundle with connection to obtain a triple where
Therefore over trivial charts, a choice of connection A on gives rise to a Dirac triple .
In general the determinant line bundle does not have a global square root, though over any trivial chart it does. When the determinant line bundle has a square root, the connections on are related to the connection on as follows. If the connection on is defined by
then the induced connection on is defined by
We can always choose a connection on . This induces a connection over each trivial chart on . Then we can twist this in to the locally defined Dirac triples , to obtain on each trivial chart . Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple .
Here is the second post on setting up the Seiberg-Witten equations on a 4-manifold, based on our learning seminar at UT Austin. The first post is here.
Clifford Algebras and Structures
For a vector space V with inner product g, its Clifford Algebra is defined as the tensor algebra of V modded out by all relations generated by setting ,
For a vector bundle , any map satisfying for all (equivalently satisfying for all v) extends to a representation
Such a map is called a Clifford structure.
There are two reasons we are interested in Clifford algebras and Clifford structures for Seiberg-Witten theory. The first is their relation to Spin and Spinc structures. The second is their relation to Dirac operators. In this post we will focus on their relation to Spin and Spinc structures, and discuss Dirac operators next.
Clifford algebras and Spin
Let denote the Clifford algebra of with its standard inner product. Let denote the standard orthonormal basis for . Consider the multiplicative subgroup of generated by unit vectors of . This is called .
There is a natural grading on induced by a bijection identifying . The integer grading on the exterior power reduces to a grading (even/odd) on the Clifford algebra. This yields a splitting . Define to be the intersection of with the even summand .
Before, we defined to be the universal double cover of . We can show this new definition of Spin agrees with the old definition, by explicitly constructing a double cover map from this subset of to .
There is an action of on given by signed conjugation (using the multiplicative structure of the Clifford algebra). If is a unit vector (i.e. a generator of ) then for any
Here we have used the fact that for unit vectors so , and the relation in the Clifford algebra. This can be interpreted geometrically: the action is the reflection of x over the hyperplane orthogonal to v.
The group of orthogonal transformations is generated by reflections over hyperplanes, so we have a representation called the twisted adjoint representation:
defined by where (extend linearly). Restricting this to this is just usual conjugation, which corresponds to an even number of reflections so the image lies in :
This map is a surjective group homomorphism, and by studying the elements of which lie in the center of , we see that the kernel of is two elements . Because these are nice smooth compact Lie groups, this implies that is a covering map. To check it is not the trivial double cover, we can find a path in between -1 and 1 given by
for [observe this path is a product of two unit vectors at each t and is thus in ].
Therefore this definition of agrees with the previous one.
The spinor representation
We have already seen that so and naturally admit two complex rank two representations coming from the projections onto the two factors of . However, it is useful to understand these representations from the Clifford algebra perspective so that the representations carry the additional information of a Clifford structure. In fact, there is a complex representation of the entire (complexified) Clifford algebra which splits into a direct sum of two complex rank two representations, which behave nicely with respect to the grading on the Clifford algebra. More specifically:
Theorem: There is a complex vector space with , and an -linear isomorphism
such that and .
To prove this, we have to define , , and the map c, and then verify that c is an algebra isomorphism satisfying the specified properties. There are a lot of things to check so I will define everything, and say a few things about how the map c works which hopefully make it more believable that c is an algebra isomorphism.
Let with standard coordinates and standard almost complex structure J. This almost complex structure gives rise to a splitting of , where is the i-eigenspace of J and is the -i-eigenspace of J. We have orthonormal bases for these pieces given by:
Define , and its splitting by and .
Now we need to define with the properties specified in the theorem. We will define c on elements of and then extend this to a map on the Clifford algebra by setting and extending complex linearly. To specify c on , it suffices to say what c does to vectors in and .
For , is the endomorphism of obtained by wedging with v:
For is contraction with :
One needs to check that this respects the Clifford algebra structure, and is an isomorphism. Initially, this may look wrong because for example when
and it seems like we should have . However, the algebra structure we want to preserve is complex linear on and has the Clifford structure only on the piece. Therefore, for example when ,
For basis elements, the map c is a sum of the exterior and interior products. To compute for example, we split this into the and parts, so
If you want to be slightly more convinced without completing the proof that for real elements of it is fairly easy at this point to check that at least on the part of (since any map that starts with contraction vanishes and for a complex number, and ).
We get the last property in the theorem easily from the definition of c. For , either raises or lowers by 1, wedge power of an element of . Therefore sends to and vice versa. Extending this over the entire Clifford algebra, we see that the endomorphisms in preserve and (since they switch between an even number of times) and sends to .
Note: We can rewrite the isomorphism as a map
This will be useful when we use this representation to form associated bundles and consider sections of those bundles and maps between the spaces of sections.
This theorem generalizes for , producing a complex vector space which splits where , whose endomorphisms are isomorphic to , where preserves the splitting and switches the components. In the odd dimensional case, the situation is slightly different, but reduces to the even case by showing that . For the purposes of Seiberg-Witten Floer homology, it will be useful to know which implies .
Now that we have this representation of the complexification of the Clifford algebra, we can restrict to get a representation of Spin. Because , and preserves the splitting , we get two representations
Note the image of lands in automorphisms instead of only endomorphisms because elements of are invertible in . These two representations correspond to the same ones we obtain by identifying and projecting onto one component.
We can extend these maps to by defining
by for , .
Note this is well defined since .
Given a Spin or Spinc structure on a manifold, these representations give rise to associated bundles . These bundles show up in the set-up for the Seiberg-Witten configuration space, which I will get to in another post.
Seiberg-Witten theory has been one of the most useful tools for understanding exotic 4-manifolds. It has been around for awhile now, but it involves a lot of geometric definitions and analytic proofs so it is difficult to approach as a grad student. Ciprian Manolescu’s recent disproof of the triangulation conjecture has brought Seiberg-Witten Floer homology into the spotlight again, which has convinced a group of us at UT Austin to go through and learn this stuff carefully from the beginning. Since this seems potentially useful to a wider audience, I’m posting some of what we have gone through. The first 2 or 3 posts will be loosely based on talks in our learning seminar given by Cagri, Richard, and me, and the written reference we have relied most on is a book by Nicolaescu called Notes on Seiberg-Witten Theory.
We started from the beginning defining curvature and connections on vector bundles and principal bundles. That part is a bit definitional/computational and not too blog friendly, so I’ll just include a link: ConnectionsCurvature. Here I’ll start with spin and spinc structures from a fairly topological perspective. In the next post I’ll talk about their relations to Clifford structures and Dirac operators, and eventually get to connections on these bundles and various associated bundles, so we can define the Seiberg Witten invariants.
Spin and Spinc structures on 3 and 4-manifolds
We can specify any vector bundle , (specifically we will be concerned with the tangent bundle) over a manifold M with fiber uniquely up to isomorphism by cocycle data , where is an open cover of M, and satisfy the “cocycle condition”
The vector bundle is formed by taking the disjoint union over all of the trivial bundles and quotienting out by identifications of the fibers and above a point by the isomorphism . The cocycle condition ensures that everything glues up coherently to a global vector bundle. Similarly a principal G-bundle can be specified by cocycle data where now and for . It is frequently useful for computations to think about vector bundles and principal bundles through these local trivializations, keeping track of the gluing maps.
Given an orientable n-manifold, we consider its tangent bundle described by gluing maps . By choosing a metric and orientation, we can reduce its structure group to , meaning we can assume the maps have image in . We can use these gluing maps to construct a principal bundle (the bundle of orthonormal frames) whose associated bundle is the tangent bundle. Spin and Spinc structures are types of lifts of this principal bundle.
For all , so has a double cover, which conveniently, is also a Lie group called .
One can show explicitly that
by constructing 2-fold covers and using the following idea. Identify with the quaternions. Observe that can be identified with the unit quaternions, and that the unit quaternions act by conjugation on the imaginary quaternions in a norm-preserving way. This action by conjugation induces a map from to whose kernel is . Similarly an action of on the quaternions can be defined by . Again one can check this action is orthogonal so there is an induced map whose kernel is two points.
A spin structure on an n-manifold M is a bundle over M which lifts the principal bundle associated to .
We can also define , and similarly define structures on a manifold.
On a 4-manifold, a spin structure gives rise to two rank 2 complex associated bundles as follows. has two natural projection maps onto , . These can be viewed as representations, so if is a structure on , we obtain two complex rank 2 associated bundles
We will see these representations again in the context of Clifford structures, when we discuss how sits inside a Clifford algebra.
If admits a structure, we also have two projections:
These similarly admit two complex rank two associated bundles . In this case .
Obstructions to Spin and Spinc structures:
The obstruction to a structure is the Stiefel-Whitney class , which can be viewed as a Cech cohomology class as follows. If are the gluing maps for the bundle defining the structure group for , each map has exactly two lifts to maps . Then
This collection is a Cech 2-cycle and so it represents a Cech cohomology class, called the second Stiefel-Whitney class . When satisfy the cocycle condition: , this cohomology class vanishes and the define a spin structure.
Because , a structure can be specified by cocycle data relating to the gluing maps for the structure bundle of the tangent bundle. This cocycle data is given by a collection of maps
satisfying two requirements
(1) (the structure is a cover of the bundle)
(2) (cocycle condition)
Focusing on the maps , we almost get a bundle except that instead of the cocycle condition we have that . Because is abelian, by looking instead at , we obtain gluing maps satisfying the cocycle condition, so they form a genuine bundle, or equivalently a complex line bundle L. We can calculate as follows. Write . Then
is an integer (since ), and the define the cocycle representing .
There is a relationship between given by requirement (2) above. Namely,
(where where the group structure is multiplication, and where the group structure is addition.) Since represents , and represents , the existence of a structure implies .
For any structure the associated line bundle L constructed above is called . It is not obvious from this definition that this is the determinant of any vector bundle, but in fact it will be the determinant of the spinor bundle associated to the Spinc structure via the spinor representation we will discuss in the next post.
Note that the set of line bundles over M acts on the set of structures as follows. For a line bundle L defined by gluing maps , and a structure defined by gluing maps , is defined by . Observe that so . One can prove that the action of the line bundles on the set of structures is free and transitive.
For any manifold with a spin structure, there is a canonical structure , obtained by composing the maps with the obvious map sending to where denotes the equivalence class by modding out by . Given this canonical structure, any other is represented by gluing cocycles where satisfy the cocycle condition (since do). In other words the define a complex line bundle L. Any structure is given by , and the associated line bundle is . Therefore any structure canonically determines a square root of for any structure .
Now we’ll work on going from rulings to augmentations. Fuchs does this using what he calls “splashes” in diagrams, but I find it easier to see this using Sabloff’s method of defining an augmentation for the dipped diagram as one can then get an augmentation for the original diagram.
Given a ruling for the original diagram in plat position, we will define an augmentation for the dipped diagram. First, augment if the ruling is switched at , augment , the crossing of strands and in the -lattice, if strands and are paired between and , (this is what we called Property (R) in part II), and augment , the crossing of strands and in the -lattice, if one of the following holds:
- ruling looks like (a) at the previous crossing and strands and are crossing strands,
- ruling looks like (b) or (c) at the previous crossing and strands and are crossing or companion strands,
- ruling looks like (e) or (f) at the previous crossing and strands and are companion strands.
Recall the various crossing configurations.
Let’s check for a couple of these cases that this gives an augmentation of the dipped diagram. In other words, check that for each crossing in the dipped diagram there are an even number of totally augmented disks in the diagram with positive corner at that crossing.
First, check the left end of the diagram. Since, in the ruling strands and are paired at the left, we know the crossing in the first -lattice of strands and is augmented for .
Most of the crossings in the dips I will leave for you to check, but we will check the dip after a crossing of configuration (c). Thus the ruling is switched at that crossing. Suppose strands and cross at the crossing and that strand is paired with and strand is paired with .
Since the ruling is switched at the crossing, we know the crossing is augmented. We also see that the following other crossings are augmented as well, from the pairing of the strands in the ruling.
To check whether on the crossings in the dip after the original crossing, look for totally augmented disks.
Clearly there aren’t any totally augmented disks with positive corner at , so .
We see that there are two totally augmented disks contributing to of the crossing in the -lattice of strands and and so . (Recall that we are working mod 2.)
We see that there are two totally augmented disks contributing to of the crossing in the -lattice of strands and and so .
Similarly, we have disks for crossings and in the -lattice of, respectively, strands and and strands and .
None of the remaining crossings in the - or -lattice have totally augmented disks, so we have checked that on this region of the dipped diagram, is an augmentation.
Now, let’s look at the right end of the dipped diagram. Since we have the ruling of the original diagram, we know that at the right end of the diagram, strands and are paired in the ruling for . Following our algorithm, this means that the crossings in the -lattice of strands and are augmented for .
Thus we have the following totally augmented disks for .
Again, we see that there are two totally augmented disks with positive corner at , so .
Thus, with some checking of the remaining cases, we have shown that the augmentation of the dipped diagram we defined, is in fact an augmentation and so, given a way to define an augmentation of the dipped diagram of a knot from a ruling of the knot.