## Seiberg Witten 4: Moduli spaces and invariants

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 $\sigma$ on M. As we have seen, this data gives rise to the associated bundle $S_\sigma=S_\sigma^+\oplus S_\sigma^-$ and the determinant line bundle $det(\sigma)$.

Let $\mathfrak{A}_\sigma(M)$ be the set of all Hermitian connections on $det(\sigma)$. We have seen that such a connection gives rise to a connection on $S_\sigma$ which is compatible with the Clifford multiplication.

The Seiberg-Witten configuration space is defined as
$\mathcal{C}_\sigma(M)=\mathfrak{A}_\sigma(M)\times \Gamma(S_\sigma^+)$

The Seiberg-Witten equations

The Seiberg-Witten equations take an element $(A,\psi)\in \mathcal{C}_\sigma(M)$ as their input. We are now prepared to define these equations.

As discussed in an earlier post, a connection A on $det(\sigma)$ gives rise to a connection on $S_\sigma^+$. Note that for $\xi\in T^*M$, $c(\xi)\in c(Cl^-(4))$ so $c(\xi):S_\sigma^+\to S_\sigma^-$. Therefore we have a Clifford structure
$c: \Gamma(T^*M\otimes S_\sigma^+)\to \Gamma(S_\sigma^-)$
which composes with the connection
$\nabla^A: \Gamma(S_\sigma^+)\to \Gamma(T^*M\otimes S_\sigma^+)$
to get a Dirac operator
$D_A: \Gamma(S_\sigma^+)\to \Gamma(S_\sigma^-).$

Denote the curvature of the connection A by $F_A$. Then the curvature is a matrix of 2-forms on M, so we can consider its self-dual and anti-self dual parts $F_A^+$ and $F_A^-$.

Let $(\psi\otimes \psi^*)_0$ denote the traceless part of the endomorphism $\psi\otimes \psi^*:S_\sigma^+\to S_\sigma^+$.

Now we can define the (perturbed) Seiberg-Witten equations. Fix a closed 2-form $\eta\in \Omega^2(M)$ (the pertubation parameter). Then the Seiberg-Witten equations are:
$SW_{(\sigma,\eta)}=\begin{cases} \frac{1}{2}c(F_A^++i\eta^+)-(\psi\otimes \psi^*)_0=0\\ {D}_A\psi = 0\end{cases}$

The input to these equations is an element $(A,\psi)\in \mathcal{C}_\sigma(M)$. The elements of $\mathcal{C}_\sigma(M)$ which are solutions to these equations are called ($(\sigma,\eta)$-)monopoles.

The Gauge Action

The gauge group is $\mathfrak{G}_\sigma(M)=\{\gamma: M\to U(1)| \text{ smooth}\}$. It acts on $\mathcal{C}_\sigma(M)$ by
$\gamma\cdot (A,\psi) = (A-2d\gamma/\gamma, \gamma\psi)$

While it seems natural enough to act on the section $\psi$ by multiplication, why do we define the action $\gamma\cdot A=A-2d\gamma/\gamma$? Specifically where is the 2 coming from?

A is the connection of the determinant line bundle L of $S_{\sigma}^+$. We would really like to think of the gauge group as acting on $S_{\sigma}^+$. If $g\in \mathfrak{G}$ acts on $s\in S^+$ by multiplication $s \mapsto gs$, then the induced action on $\sigma\in L=\wedge^2 S_{\sigma}^+$ is multiplication by $g^2$. (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 $\nabla_A$ induced by the connection A on L. Here the natural action is conjugation

$g^2\nabla_A(g^{-2}s)=g^2d(g^{-2})\otimes s +\nabla_As=-2g^{-1}dg\otimes s +\nabla_As$

For $C=(A,\psi)\in \mathcal{C}_\sigma(M)$ we can consider its stabilizer in $\mathfrak{G}_\sigma(M)$. 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 $\psi\equiv 0$, and that their stabilizers are the constant maps into $S^1$.

The Seiberg-Witten moduli space

The Seiberg-Witten solution space is the space of elements $(A,\phi)$ 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 $F_A^+=F_{A-2g^{-1}dg}^+$ because $F_{A-2g^{-1}dg}^+=(F_A-d(2g^{-1}dg))^+$ and $d(g^{-1}dg)=0$ because we can think locally that $g^{-1}dg=d(log(g))$ so taking its exterior derivative gives 0. Furthermore $(g\phi)\otimes (g\phi)^*=gg^{-1}\phi\otimes \phi^*=\phi\otimes \phi^*$, so the first equation is invariant under the gauge action.

For the second equation, $D_{A-2g^{-1}dg(g\phi)}$ can be understood by breaking up the dirac operator into the composition of the Clifford multiplication and the connection $\nabla_A$ on $S_\sigma^+$.

The discussion above about why the gauge group acts as it does on A is related to the fact that $\nabla_(A-2g^{-1}dg)=\nabla_A-g^{-1}dg\otimes I_{S^+}$. Applying the Clifford multiplication to this connection acting on $g\phi$ and using the Leibniz rule for connections eventually simplifies to show that $D_{A-2g^{-1}dg}(g\phi)=g D_{A}\phi$ so the solutions to $D_A\phi=0$ 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 $(A,0)$ then they satisfy $F_A^+=\eta$ for our chosen perturbation. Since both of these forms are closed, they represent cohomology classes. The cohomology class of the curvature $[F_A^+]=-2\pi ic_1(L)^+$ is independent of A, so we only have reducible solutions when $[\eta]=-2\pi ic_1(L)^+$. 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 $\mathbb{CP}^\infty$ 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 $SW_{M,g,\eta}(L)\in \mathbb{Z}$.

A priori this integer depends on the metric and perturbation, but when $b_2^+>1$, 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 $(g_1,\eta_1), (g_2,\eta_2)$ which lifts to a cobordism between the moduli space at $(g_1,\eta_1)$ and the moduli space at $(g_2,\eta_2)$. 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 $b_2^+=1$, there is a codimension 1 space of bad perturbations which forms a wall between two chambers. Within each chamber $SW_{M,g,\eta}(L)$ 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 $\mathbb{CP}^2\#N\overline{\mathbb{CP}^2}$).

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## Seiberg Witten 3: Dirac operators, Spin and Spinc connections

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: $\sigma_L(\xi)=-|\xi|^2I\in End(E,E)$ are called generalized Laplacians. Note that the symbol $\sigma_L(\xi): E_x\to E_x$ of a generalized Laplacian is an isomorphism on each fiber for $\xi\neq 0$, 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 $Lu=v$. 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 $\sigma_L(\xi)=-|\xi|^2I$, for $\xi\in \Gamma(T^*M)$. 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 $(\sigma_D(\xi))^2=-|\xi|^2I$ so $\sigma_D$ 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, $c: \Gamma(T^*M)\to \Gamma(End(E))$ (equivalently $c: \Gamma(T^*M\otimes E)\to \Gamma(E)$) and a connection $\nabla: \Gamma(E)\to \Gamma(T^*M\otimes E)$ we can compose them

$D:\Gamma(E)\xrightarrow{\nabla}\Gamma(T^*M\otimes E)\xrightarrow{c}\Gamma(E)$

and the resulting operator is a Dirac operator.

Spin connections

A Riemannian manifold M has a distinguished connection, the Levi-Civita connection $\nabla^M$, which has nice properties namely it preserves the metric g (this can be phrased either as $\nabla^Mg=0$ or $\nabla(g(X,Y))=g(\nabla X,Y)+g(X,\nabla(Y))$), and it is torsion free meaning $\nabla_XY-\nabla_YX-[X,Y]=0$. 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 $SO(n)$-bundle. Namely, we can find gluing maps defining the tangent bundle that map into $SO(n)$: $\{g_{\alpha\beta}: U_\alpha\cap U_\beta \to SO(n)\}$ which define a principal $SO(n)$-bundle $P_{SO(n)}\to M$. The Levi-Civita connection on TM induces a principal $SO(n)$-connection on $P_{SO(n)}$ specified locally by

$\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{so}(n).$

We have the double cover map $\tau: Spin(n)\to SO(n)$, which induces, by differentiating at 1, an isomorphism $\tau_*: \mathfrak{spin}(n)\to \mathfrak{so}(n)$.

If we have a Spin structure on M, this means there are lifts $\widetilde{g}_{\alpha\beta}: U_\alpha\cap U_\beta\to Spin(n)$ such that $\tau\circ \widetilde{g}_{\alpha\beta}=g_{\alpha\beta}$. These define a principal Spin(n) bundle $P_{Spin(n)}$. In this case, the Levi-Civita connection on $P_{SO(n)}$ induces a connection $\widetilde{\nabla}^M$ on $P_{Spin(n)}$ which is locally defined by

$\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{spin}(n).$

So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.

The representations $\rho_\pm: Spin(4)\to Aut(\mathbb{S}^\pm)$, and $\rho=(\rho_+,\rho_-)$ give rise to an associated bundle $S_0=P_{Spin}\times_\rho \mathbb{S}$. The spin connection on M induces a connection $\nabla^{S_0}$ on $S_0$ whose local matrix valued 1-forms are defined by

$\rho_*\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_\alpha)\otimes End(\mathbb{S}).$

Recall that $T^*M$ acts on $S_0$ by the Clifford multiplication $c: Cl(TM)\otimes \mathbb{C}\to End(S_0)$. The composition of the Clifford multiplication with the induced connection on $S_0$ yields a Dirac operators $D_0$.

$\mathbf{Spin^c}$ connections

Remember, a $Spin^c(n)$-bundle is specified by gluing data

$\{(h_{\alpha\beta}, z_{\alpha\beta}): U_{\alpha}\cap U_{\beta} \to Spin(n)\times U(1)\}$
satisfying the cocycle condition

$(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})=\pm (1,1).$
We want to understand $Spin^c$ structures for M and their connections. Let $\sigma$ be a $Spin^c$ structure on M given by the $Spin^c(4)$ bundle $P_{Spin^c}$.

Letting $\rho^c=(\rho^c_+,\rho^c_-)$, the associated spinor bundle to $\sigma$ is $S_\sigma=P_{Spin(4)}\times_{\rho^c} \mathbb{S}$, which splits into $S^\pm_\sigma = P_{Spin(4)}\times_{\rho^c_\pm}\mathbb{S}^{\pm}$. A connection on the $Spin^c$ bundle will induce a connection on $S_\sigma, S^+_\sigma,S^-_\sigma$. Also note that $S_\sigma$ has a Clifford structure, inherited from the map $c: Cl(V)\otimes \mathbb{C}\to End(\mathbb{S})$.

In the case that M has a spin structure, $P_{Spin^c}=P_{Spin}\otimes (det\sigma)^{1/2}$ and $S_\sigma = S_0\otimes (det\sigma)^{1/2}$.

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 $\sigma$.

In the case that TM is the trivial bundle, the determinant line bundle has a square root, and $P_{Spin^c}=P_{Spin}\otimes (det\sigma)$ and $S_\sigma=S_0\otimes (det\sigma)^{1/2}$. We have the natural lift $\widetilde{\nabla}^M$ of the Levi-Civita connection to $P_{Spin}$. This induces a natural connection $\nabla^{S_0}$ on the associated bundle $S_0$, which we can tensor with any connection on the line bundle $(det\sigma)^{1/2}$ to get a connection on $S_{\sigma}=S_0\otimes (det\sigma)^{1/2}$.

Remember that $S_0$ had a Clifford structure c as well as a natural connection $S_0$ which together give rise to a Dirac operator. We obtain a similar structure on $S_\sigma$ by twisting the triple $(S_0,\nabla^{S_0}, c)$ with a line bundle with connection $(L,\nabla^L)$ to obtain a triple $(S_0\otimes L, \nabla, c_L)$ where

$\nabla(s\otimes x) = \nabla^{S_0}s\otimes x +s\otimes \nabla^Lx$

and

$c_L: \Omega^*M \xrightarrow{c}End(S_0)\xrightarrow{\cdot \otimes I_L} End(S_0\otimes L)$

Therefore over trivial charts, a choice of connection A on $(det\sigma)^{1/2}$ gives rise to a Dirac triple $(S_\sigma,\nabla_A, c_\sigma)$.

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 $det(\sigma)$ are related to the connection on $(det(\sigma))^{1/2}$ as follows. If the connection on $det(\sigma)$ is defined by

$\{\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}$

then the induced connection on $(det(\sigma))^{1/2}$ is defined by

$\{\frac{1}{2}\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}.$

We can always choose a connection on $det(\sigma)$. This induces a connection over each trivial chart on $(det(\sigma))^{1/2}$. Then we can twist this in to the locally defined Dirac triples $(S_0,\nabla^{S_0},c)$, to obtain $(S_\sigma, \nabla, c)$ on each trivial chart $U_\alpha$. Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple $(S_\sigma,\nabla, c)$.

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## Seiberg-Witten Theory 2: Clifford Structures and Spinors

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 $v\otimes v=-g(v,v)1$,

$Cl(V)= \otimes V/\langle v\otimes v +g(v,v)1: v\in V\rangle.$

For a vector bundle $E\to M$, any map $c: T^*M\to End(E)$ satisfying $c(u)c(v)+c(v)c(u)=-2g(u,v)I_E$ for all $u,v\in \Gamma(T^*M)$ (equivalently satisfying $c(v)^2=-|v|I_E$ for all v) extends to a representation
$c:Cl(T^*M)\to End(E)$
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 $Cl(n)$ denote the Clifford algebra of $\mathbb{R}^n$ with its standard inner product. Let $(e_1,\cdots, e_n)$ denote the standard orthonormal basis for $\mathbb{R}^n$. Consider the multiplicative subgroup of $Cl(n)$ generated by unit vectors of $\mathbb{R}^n$. This is called $Pin(n)$.

There is a natural $\mathbb{Z}/2$ grading on $Cl(n)$ induced by a bijection $Cl(n)\leftrightarrow \bigwedge^* \mathbb{R}^n$ identifying $e_{i_1}\cdots e_{i_k} \leftrightarrow e_{i_1}\wedge \cdots \wedge e_{i_k}$. The integer grading on the exterior power reduces to a $\mathbb{Z}/2$ grading (even/odd) on the Clifford algebra. This yields a splitting $Cl(n)= Cl^+(n)\oplus Cl^-(n)$. Define $Spin(n)$ to be the intersection of $Pin(n)$ with the even summand $Cl^+(n)$.

Before, we defined $Spin(n)$ to be the universal double cover of $SO(n)$. We can show this new definition of Spin agrees with the old definition, by explicitly constructing a double cover map from this subset of $Cl(n)$ to $SO(n)$.

There is an action of $Cl(n)$ on $\mathbb{R}^n$ given by signed conjugation (using the multiplicative structure of the Clifford algebra). If $v\in \mathbb{R}^n$ is a unit vector (i.e. a generator of $Pin(n)$) then for any $x\in \mathbb{R}^n$
$-vxv^{-1} = vxv = x-2\langle x,v \rangle v$
Here we have used the fact that for unit vectors $-vv=1$ so $v^{-1}=-v$, and the relation $vx+xv=-2g(v,x)$ in the Clifford algebra. This can be interpreted geometrically: the action $(v,x)\mapsto -vxv^{-1}$ 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:
$\rho: Pin(k)\to O(k)$
defined by $\rho(y)x = yx\varepsilon(y^{-1})$ where $\varepsilon(Cl^\pm(n))=\pm 1$ (extend linearly). Restricting this to $Spin(k)$ this is just usual conjugation, which corresponds to an even number of reflections so the image lies in $SO(k)$:
$\rho: Spin(k)\to SO(k)$
This map is a surjective group homomorphism, and by studying the elements of $Spin(k)$ which lie in the center of $Cl(k)$, we see that the kernel of $\rho: Spin(k)\to SO(k)$ is two elements $\{\pm 1\}$. Because these are nice smooth compact Lie groups, this implies that $\rho$ is a covering map. To check it is not the trivial double cover, we can find a path in $Spin(k)$ between -1 and 1 given by
$\gamma(t)=\cos(t)+e_1e_2\sin(t)=-(e_1\cos(t/2)+e_2\sin(t/2))(e_1\cos(t/2)-e_2\sin(t/2))$
for $t\in [-\pi,\pi]$ [observe this path is a product of two unit vectors at each t and is thus in $Spin(n)$].
Therefore this definition of $Spin(n)$ agrees with the previous one.

The spinor representation

We have already seen that $Spin(4)\cong SU(2)\times SU(2)$ so $Spin(4)$ and $Spin^c(4)$ naturally admit two complex rank two representations coming from the projections onto the two factors of $SU(2)$. 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 $Cl(4)$ which splits into a direct sum of two complex rank two representations, which behave nicely with respect to the $\mathbb{Z}/2$ grading on the Clifford algebra. More specifically:

Theorem: There is a complex vector space $\mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-$ with $\dim_{\mathbb{C}}\mathbb{S}^+=\dim_{\mathbb{C}}\mathbb{S}^-=2$, and an $\mathbb{C}$-linear isomorphism
$c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})$
such that $c(Cl^+(4))\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-)$ and $c(Cl^-(4)\cong Hom(\mathbb{S}^+,\mathbb{S}^-)\oplus Hom(\mathbb{S}^-,\mathbb{S}^+)$.

To prove this, we have to define $\mathbb{S}$, $\mathbb{S}^\pm$, 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 $V=\mathbb{R}^4$ with standard coordinates and standard almost complex structure J. This almost complex structure gives rise to a splitting of $V\otimes \mathbb{C} = V^{1,0}\oplus V^{0,1}$, where $V^{1,0}$ is the i-eigenspace of J and $V^{0,1}$ is the -i-eigenspace of J. We have orthonormal bases for these pieces given by:
$V^{1,0}=span\left(\varepsilon_1 := \frac{1}{\sqrt{2}}(e_1-if_1), \varepsilon_2 := \frac{1}{\sqrt{2}}(e_2-if_2)\right)$
$V^{0,1}=span\left(\overline{\varepsilon}_1 := \frac{1}{\sqrt{2}}(e_1+if_1), \overline{\varepsilon}_2 := \frac{1}{\sqrt{2}}(e_2+if_2)\right)$

Define $\mathbb{S}:= \bigwedge^* V^{1,0}$, and its splitting by $\mathbb{S}^+ := \bigwedge^{even}V^{1,0}$ and $\mathbb{S}^- := \bigwedge^{odd}V^{1,0}$.

Now we need to define $c: Cl(V)\otimes \mathbb{C} \to End (\mathbb{S})$ with the properties specified in the theorem. We will define c on elements of $V\otimes \mathbb{C}$ and then extend this to a map on the Clifford algebra by setting $c(e_{i_1}\cdots e_{i_k})=c(e_{i_1})\cdot \cdots \cdot c(e_{i_k})$ and extending complex linearly. To specify c on $V\otimes \mathbb{C}$, it suffices to say what c does to vectors in $V^{1,0}$ and $V^{0,1}$.

For $v\in V^{1,0}$, $c(v)$ is the endomorphism of $\mathbb{S}$ obtained by wedging with v:
$c(v)(u_1\wedge \cdots u_k)=\sqrt{2}v\wedge u_1\wedge \cdots \wedge u_k$

For $\overline{v}\in V^{0,1}$ $c(\overline{v})$ is contraction with $\overline{v}$:
$c(\overline{v})(u_1\wedge \cdots u_k) = \sqrt{2}\sum_{j=1}^k (-1)^j g(v,u_j)u_1\wedge \cdots \wedge \widehat{u_j} \wedge \cdots u_k$

One needs to check that this respects the Clifford algebra structure, and is an isomorphism. Initially, this may look wrong because for example when $v\in V^{1,0}$
$c(v)^2(u_1\wedge \cdots \wedge u_k) = v\wedge v\wedge u_1\wedge \cdots \wedge u_k=0$
and it seems like we should have $c(v)^2=-|v|^2I$. However, the algebra structure we want to preserve is complex linear on $Cl(V)\otimes \mathbb{C}$ and has the Clifford structure only on the $Cl(V)$ piece. Therefore, for example when $v=e_j-if_j\in V^{1,0}$,
$0=c(e_j-if_j)^2 = (c(e_j)-ic(f_j))^2 = (c(e_j))^2-ic(e_j)c(f_j)-ic(f_j)c(e_j)-(c(f_j))^2 = |e_j|^2-i2g(e_j,f_j)-|f_j|^2$

For basis elements, the map c is a sum of the exterior and interior products. To compute for example, $c(e_j)$ we split this into the $V^{1,0}$ and $V^{0,1}$ parts, so
$c(e_j)=c\left(\frac{1}{2}(e_j-if_j)+\frac{1}{2}(e_j+if_j)\right)=\frac{\sqrt{2}}{2}\left((e_j-if_j)\wedge\cdot +\iota_{e_j-if_j} \right)$
If you want to be slightly more convinced without completing the proof that $c(v)^2=-|v|^2I$ for real elements of $Cl(V)$ it is fairly easy at this point to check that $c(e_j)^2=-I$ at least on the $\bigwedge^0V^{1,0}$ part of $\mathbb{S}=\bigwedge V^{1,0}$ (since any map that starts with contraction vanishes and $\iota_x(y\wedge f)=-fg(x,y)$ for $f\in \bigwedge^0V^{1,0}$ a complex number, and $x,y\in V\subset Cl(V)\otimes 1$).

We get the last property in the theorem easily from the definition of c. For $v\in V\otimes \mathbb{C}$, $c(v)$ either raises or lowers by 1, wedge power of an element of $\mathbb{S}=\bigwedge V^{1,0}$. Therefore $c(v)$ sends $\mathbb{S}^+$ to $\mathbb{S}^-$ and vice versa. Extending this over the entire Clifford algebra, we see that the endomorphisms in $c(Cl^+(4))$ preserve $\mathbb{S}^+$ and $\mathbb{S}^-$ (since they switch between $\mathbb{S}^\pm$ an even number of times) and $c(Cl^-(4))$ sends $\mathbb{S}^\pm$ to $\mathbb{S}^\mp$.

Note: We can rewrite the isomorphism $c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})$ as a map
$c: Cl(4)\otimes \mathbb{C}\otimes \mathbb{S}\to \mathbb{S}$.
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 $Cl(2n)$, producing a complex vector space $\bigwedge V^{1,0}$ which splits where $dim(V)=2n$, whose endomorphisms are isomorphic to $Cl(2n)\otimes \mathbb{C}$, where $Cl^+$ preserves the splitting and $Cl^-$ switches the components. In the odd dimensional case, the situation is slightly different, but reduces to the even case by showing that $Cl(2n-1)\cong Cl^+(2n)$. For the purposes of Seiberg-Witten Floer homology, it will be useful to know $Cl(3)\cong Cl^+(4)$ which implies $Cl(3)\otimes \mathbb{C}\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-)$.

Spinor bundles

Now that we have this representation of the complexification of the Clifford algebra, we can restrict to get a representation of Spin. Because $Spin(4)\subset Cl^+(4)$, and $c(Cl^+(4))$ preserves the splitting $\mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-$, we get two representations
$\rho_{\pm}: Spin(4)\to Aut(\mathbb{S}^\pm)$
Note the image of $Spin(4)$ lands in automorphisms instead of only endomorphisms because elements of $Spin(4)$ are invertible in $Cl(4)$. These two representations correspond to the same ones we obtain by identifying $Spin(4)\cong SU(2)\times SU(2)$ and projecting onto one component.

We can extend these maps to $Spin^c$ by defining
$\rho^c_{\pm}: Spin^c(4)\to Aut(\mathbb{S}^\pm)$
by $\rho^c_\pm((g,z))=z\rho_{\pm}(g)$ for $g\in Spin(4)$, $z\in U(1)$.

Note this is well defined since $\rho^c_\pm((-g,-z))=\rho^c_\pm((g,z))$.

Given a Spin or Spinc structure on a manifold, these representations give rise to associated bundles $S^\pm \to M$. These bundles show up in the set-up for the Seiberg-Witten configuration space, which I will get to in another post.

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## Seiberg Witten Theory 1: Spin, Spinc structures

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 $V^n$ uniquely up to isomorphism by cocycle data $(\{U_{\alpha}\}, \{g_{\beta \alpha}\})$, where $\{U_\alpha\}$ is an open cover of M, and $g_{\beta \alpha}: U_\alpha \cap U_\beta \to GL(V^n)$ satisfy the “cocycle condition”
$g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=I_V$

The vector bundle is formed by taking the disjoint union over all $\alpha$ of the trivial bundles $U_\alpha \times V$ and quotienting out by identifications of the fibers $x\times V\subset U_\alpha\times V$ and $x\times V \subset U_\beta \times V$ above a point $x\in U_\alpha \cap U_\beta$ by the isomorphism $g_{\beta \alpha}(x)$. 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 $(\{U_\alpha\}, \{g_{\beta \alpha}\})$ where now $g_{\beta \alpha}: U_\alpha\cap U_\beta \to G$ and $g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=1_G$ for $x\in U_\alpha\cap U_\beta$. 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 $\{g_{\alpha\beta}: U_\alpha\cap U_\beta\to GL(n,\mathbb{R})\}$. By choosing a metric and orientation, we can reduce its structure group to $SO(n)$, meaning we can assume the maps $g_{\alpha\beta}$ have image in $SO(n)$. We can use these gluing maps to construct a principal $SO(n)$ 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 $n\geq 3$, $\pi_1(SO(n))=\mathbb{Z}/2$ so $SO(n)$ has a double cover, which conveniently, is also a Lie group called $Spin(n)$.

One can show explicitly that
$Spin(3)=SU(2)$
$Spin(4)=SU(2)\times SU(2)$
by constructing 2-fold covers $SU(2)\to SO(3)$ and $SU(2)\times SU(2)\to SO(4)$ using the following idea. Identify $\mathbb{R}^4$ with the quaternions. Observe that $SU(2)\cong S^3$ 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 $SU(2)$ to $SO(3)$ whose kernel is $\pm 1$. Similarly an action of $SU(2)\times SU(2)$ on the quaternions can be defined by $(P,Q)\cdot X = PXQ^{-1}$. Again one can check this action is orthogonal so there is an induced map $SU(2)\times SU(2)\to SO(4)$ whose kernel is two points.

A spin structure on an n-manifold M is a $Spin(n)$ bundle over M which lifts the principal $SO(n)$ bundle associated to $TM$.

We can also define $Spin^c(n)=(U(1)\times Spin(n)/\{\pm 1\}\to SO(n)$, and similarly define $Spin^c$ structures on a manifold.

On a 4-manifold, a spin structure gives rise to two rank 2 complex associated bundles as follows. $Spin(4)=SU(2)\times SU(2)$ has two natural projection maps onto $SU(2)$, $pr_1,pr_2$. These can be viewed as representations, so if $\widetilde{P}$ is a $Spin$ structure on $M^4$, we obtain two complex rank 2 associated bundles
$S^\pm = \widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2$
We will see these representations again in the context of Clifford structures, when we discuss how $Spin(n)$ sits inside a Clifford algebra.

If $M^4$ admits a $spin^c$ structure, we also have two projections:
$pr_{1,2}: Spin^c(4)=U(1)\times SU(2)\times SU(2)/\{\pm 1\}\to U(2)$
These similarly admit two complex rank two associated bundles $W^{\pm}=\widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2$. In this case $\bigwedge^2W^+=\bigwedge^2W^-$.

Obstructions to Spin and Spinc structures:

The obstruction to a $Spin$ structure is the Stiefel-Whitney class $w_2(M)$, which can be viewed as a Cech cohomology class as follows. If $\{g_{\alpha\beta}: U_\alpha\cap U_\beta\to SO(n)\}$ are the gluing maps for the $SO(n)$ bundle defining the structure group for $TM$, each map $g_{\alpha\beta}$ has exactly two lifts to maps $h_{\alpha\beta}: U_{\alpha}\cap U_\beta\to Spin(n)$. Then
$w_{\alpha\beta\gamma}:=h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}\in \mathbb{Z}/2=\ker(Spin(n)\to SO(n))$
This collection $\{w_{\alpha\beta\gamma}\}$ is a Cech 2-cycle and so it represents a Cech cohomology class, called the second Stiefel-Whitney class $w_2(M)$. When $h_{\alpha\beta}$ satisfy the cocycle condition: $w_{\alpha\beta\gamma}=1$, this cohomology class vanishes and the $h_{\alpha\beta}$ define a spin structure.

Because $Spin^c(n)=(Spin(n)\times S^1)/\{\pm (1,1)\}$, a $Spin^c$ structure can be specified by cocycle data relating to the gluing maps $\{g_{\alpha\beta}\}$ for the $SO(n)$ structure bundle of the tangent bundle. This cocycle data is given by a collection of maps
$h_{\alpha\beta}: U_\alpha\cap U_\beta \to Spin(n)$
$z_{\alpha\beta}: U_\alpha\cap U_\beta \to S^1$
satisfying two requirements

(1) $\rho(h_{\alpha\beta})=g_{\alpha\beta}$ (the $Spin^c$ structure is a cover of the $SO(n)$ bundle)
(2) $(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})\in\{(1,1),(-1,-1)\}$ (cocycle condition)

Focusing on the maps $z_{\alpha\beta}$, we almost get a $U(1)$ bundle except that instead of the cocycle condition we have that $z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}=\pm 1$. Because $U(1)=S^1$ is abelian, by looking instead at $\lambda_{\alpha\beta}=z_{\alpha\beta}^2$, we obtain gluing maps satisfying the cocycle condition, so they form a genuine $U(1)$ bundle, or equivalently a complex line bundle L. We can calculate $c_1(L)$ as follows. Write $\lambda_{\alpha\beta}=e^{i\theta_{\alpha\beta}}$. Then
$n_{\alpha\beta\gamma}=\frac{1}{2\pi}(\theta_{\alpha\beta}+\theta_{\beta\gamma}+\theta_{\gamma\alpha})$
is an integer (since $e^{i(\theta_{\alpha\beta}+\theta_{\beta\gamma}+\theta_{\gamma\alpha})})=1$), and the $n_{\alpha\beta\gamma}$ define the cocycle representing $c_1(L)\in H^2(M;\mathbb{Z})$.

There is a relationship between $w_{\alpha\beta\gamma}:= h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}$ given by requirement (2) above. Namely,
$w_{\alpha\beta\gamma} = sign(z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}) = n_{\alpha\beta\gamma} \mod 2$
(where $w_{\alpha\beta\gamma}\in \{-1,1\}=\mathbb{Z}/2$ where the group structure is multiplication, and $n_{\alpha\beta\gamma}\in \{0,1\}=\mathbb{Z}/2$ where the group structure is addition.) Since $\{w_{\alpha\beta\gamma}\}$ represents $w_2(M)$, and $\{n_{\alpha\beta\gamma}\}$ represents $c_1(L)$, the existence of a $Spin^c$ structure implies $w_2(M)\equiv c_1(L) \mod 2$.

For any $Spin^c$ structure $\sigma$ the associated line bundle L constructed above is called $det(\sigma)$. 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 $Spin^c$ structures as follows. For a line bundle L defined by gluing maps $\zeta_{\alpha\beta}: U_{\alpha}\cap U_\beta \to S^1$, and a $Spin^c$ structure $\sigma$ defined by gluing maps $[h_{\alpha\beta}, z_{\alpha\beta}]$, $\sigma\otimes L$ is defined by $[h_{\alpha\beta}, z_{\alpha\beta}\zeta_{\alpha\beta}]$. Observe that $det(\sigma\otimes L)=det(\sigma)\otimes L^2$ so $c_1(\sigma\otimes L)=c_1(det(\sigma))+2c_1(L)$. One can prove that the action of the line bundles on the set of $Spin^c$ structures is free and transitive.

For any manifold with a spin structure, there is a canonical $Spin^c$ structure $\sigma_0$, obtained by composing the maps $h_{\alpha\beta}:U_{\alpha}\cap U_\beta \to Spin(n)$ with the obvious map $Spin(n)\to Spin(n)\times S^1/\{\pm 1\}$ sending $g\in Spin(n)$ to $[(g,1)]$ where $[\cdot]$ denotes the equivalence class by modding out by $\pm 1$. Given this canonical $Spin^c$ structure, any other is represented by gluing cocycles $\{(h_{\alpha\beta},z_{\alpha\beta})\}$ where $\{z_{\alpha\beta}\}$ satisfy the cocycle condition (since $\{h_{\alpha\beta}\}$ do). In other words the $\{z_{\alpha\beta}\}$ define a complex line bundle L. Any $Spin^c$ structure is given by $\sigma_0\otimes L$, and the associated line bundle is $det(\sigma_0\otimes L)=L^{\otimes 2}$. Therefore any $Spin$ structure canonically determines a square root of $det(\sigma)$ for any $Spin^c$ structure $\sigma$.

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## From Rulings to Augmentations

This is part III of a post on the relationship between augmentations and rulings. If you missed parts I and II, you can find them here and here.

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 $c_k$ if the ruling is switched at $c_k$, augment $a_{rs}$, the crossing of strands $r$ and $s$ in the $a$-lattice, if strands $r$ and $s$ are paired between $c_k$ and $c_{k+1}$, (this is what we called Property (R) in part II), and augment $b_{rs}$, the crossing of strands $r$ and $s$ in the $b$-lattice, if one of the following holds:

• ruling looks like (a) at the previous crossing $c_k$ and strands $r$ and $s$ are crossing strands,
• ruling looks like (b) or (c) at the previous crossing $c_k$ and strands $r$ and $s$ are crossing or companion strands,
• ruling looks like (e) or (f) at the previous crossing $c_k$ and strands $r$ and $s$ are companion strands.

Recall the various crossing configurations.

(From Sabloff’s paper.)

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 $2k$ and $2k-1$ are paired at the left, we know the crossing in the first $a$-lattice of strands $2k$ and $2k-1$ is augmented for $1\leq k\leq m$.

We then see that we have the totally augmented disks depicted.
So $\epsilon'\circ\partial=0$ on this portion of the diagram.

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 $i$ and $i+1$ cross at the crossing $c_k$  and that strand $i$ is paired with $L$ and strand $i+1$ is paired with $K$.
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 $\epsilon'\partial=0$ 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 $c_k$, so $\epsilon'\partial c_k=0$.

We see that there are two totally augmented disks contributing to $\epsilon'\circ\partial$ of the crossing $b_1$ in the $b$-lattice of strands $K$ and $i$ and so $\epsilon'\partial(b_1)=1+1=0$. (Recall that we are working mod 2.)
We see that there are two totally augmented disks contributing to $\epsilon'\circ\partial$ of the crossing $b_2$ in the $b$-lattice of strands $L$ and $i$ and so $\epsilon'\partial(b_1)=1+1=0$.
Similarly, we have disks for crossings $b_3$ and $b_4$ in the $b$-lattice of, respectively, strands $K$ and $i+1$ and strands $L$ and $i+1$.
None of the remaining crossings in the $a$- or $b$-lattice have totally augmented disks, so we have checked that on this region of the dipped diagram, $\epsilon'$ 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 $2k$ and $2k-1$ are paired in the ruling for $1\leq k\leq m$. Following our algorithm, this means that the crossings in the $a$-lattice of strands $2k$ and $2k-1$ are augmented for $1\leq k\leq m$.
Thus we have the following totally augmented disks for $q_k$.
Again, we see that there are two totally augmented disks with positive corner at $q_k$, so $\epsilon'\circ\partial(q_k)=1+1=0$.

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.

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## From Augmentations to Rulings

This is part II of a post on the relationship between augmentations and rulings. Here is part I if you missed it.

Given a front diagram of a Legendrian knot in plat position, we will use Sabloff’s method of dips to show that if there exists an augmentation of the Chekanov-Eliashberg DGA, then there exists a normal ruling of the diagram. We will extend the augmentation $\epsilon$ of the diagram to an augmentation $\epsilon'$ of the dipped diagram satisfying Property (R) which will simultaneously build a normal ruling of the original front diagram.

We will extend $\epsilon$ to $\epsilon'$ by adding dips to the diagram from left to right and extending the augmentation and ruling as we go as follows: If a crossing $c_k$ is augmented in the original front diagram, then augment $c_k$ in the dipped diagram. In a dip, we decide whether a crossing in the $a$-lattice is augmented by following Property (R).

Property (R): At the dip between $c_k$ and $c_{k+1}$, a crossing in the $a$-lattice is augmented iff the two strands which cross there are paired in the ruling of the original front diagram between $c_k$ and $c_{k+1}$.

A crossing $b$ in the $b$-lattice between crossings $c_k$ and $c_{k+1}$ is augmented based on whether the strands which cross at $b$ are crossing or companion strands in the ruling of the original front diagram and which diagram below $c_k$ looks like .

(Figure from Sabloff’s paper.)

In particular, if $c_k$ has configuration ____ and is augmented, then $b$ is augmented if the strands crossing at $b$ are _____ strands. (Use the following to fill in this statement.)

(a), crossing
(b) or (c), crossing and companion
(d), none
(e) or (f), companion

Note that this definition relies on simultaneously building a ruling for the original front diagram. So, let us describe how to build the ruling. If we look at the left end of the ruling of a front diagram in plat position, the ruling will have strand $2k$ paired with strand $2k-1$ for all $1\leq k\leq m$. To extend the ruling over a crossing $c_k$, if $c_k$ is augmented and it looks like configuration (a), (b), or (c) just to the left of $c_k$, then switch the ruling at $c_k$. Otherwise, don’t switch the ruling.

One can check that this gives us an augmentation of the dipped diagram. For example, suppose $c_k$ is augmented and has configuration (c) just to the left. We will denote augmented crossings by a large dot. Suppose strands $i$ and $i+1$ cross at $c_k$ and that strand $i$ is paired with strand $L$ and strand $i+1$ is paired with strand $K$ between crossings $c_k$ and $c_{k+1}$.

Since $c_k$ is augmented and looks like configuration (c) to the left, our ruling is switched at $c_k$.

(I apologize for the crude drawings, but Inkscape wasn’t cooperating.)

Now let us use this diagram to compute $\epsilon'\circ\partial$. Recall that the only disks that contribute to $\epsilon'\circ\partial(c)$ are totally augmented disks, disks with a positive corner at $c$ and with all negative corners augmented by $\epsilon'$. Each such disk will contribute 1 to $\epsilon'\circ\partial(c)$, so as long as there are a even number of such disks, $\epsilon'\circ\partial(c)=0$.

There are no such disks for $c_k$, so $\epsilon'\circ\partial(c)=0$. A more interesting computation is the crossing $b_1$ in the $b$-lattice of strands $i$ and $K$. We see that there are two such disks.

So $\epsilon'\circ\partial(b_1)=0$.

We also have the following disks for the crossing $b_2$ in the $b$-lattice of strands $i+1$ and $L$. So $\epsilon'\circ\partial(b_2)=0$.

A crossing with more standard disks appearing, is the crossing $b_3$ in the $b$-lattice where strands $i+1$ and $K$ cross. It has the following disks.

There are no such disks for any crossing in the $a$-lattice, so $\epsilon'\circ\partial=0$ on the $a$-lattices. One can check that $\epsilon'\circ\partial=0$ for all remaining crossings.

While the dipped diagram makes differentials much easier, they tend to be a pain to deal with. Luckily, there’s a shortcut for finding the ruling associated to an augmentation of the original front diagram, without going through the dipped diagram:

We will modify our augmentation of the crossings of the plat position diagram while extending the ruling from left to right. Start the ruling at the left cusps like usual, and extend as follows. If a crossing $c_k$ looks like configuration (a), (b), or (c) to the left and the crossing is augmented, then switch the ruling at the crossing and update the augmentation on the crossings to the right based on the number of disks to the right which contribute $Qc_kR$ and involve the crossing strands.

Otherwise do not switch at the crossing. However, if a crossing $c_k$ looks like configuration (e), or (f) to the left and $c_k$ is augmented, then update the augmentation of the crossings to the right based on the number of disks to the right which contribute $Qc_kR$ and involve the companion strands.

This is easier to see in an example. Let’s look at the trefoil where all crossings of degree $0$ are augmented.

Start the ruling at the left.

The first crossing is augmented and the left of the crossing matches (a), so switch the ruling at $c_1$.

Now update the augmentation by looking for the disks $QaR$. We only find one.

So we get the diagram so far, with updated augmentation.

As $c_2$ is not augmented, the ruling is not switched and no updating of the augmentation is necessary.

The final crossing does not match (a), (b), or (c) to the left, so do not switch the ruling at $c_3$.

It turns out the augmentation where only $c_1$ and $c_2$ are augmented will give the same ruling. Ng and Sabloff’s paper The Correspondence Between Augmentations and Rulings for Legendrian Knots discusses how this is a many-to-one map and gives the number of $\rho$-graded augmentations going to the same $\rho$-graded ruling for $\rho\vert2r(K)$ and $\rho=0$ or $\rho$ odd. (You’ll have to read up on what a $\rho$-graded augmentation is in the paper.)

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## Setup for the Relationship Between Augmentations and Rulings

The Chekanov-Eliashberg DGA is an invariant which associates a DGA over the integers mod 2 to a Lagrangian projection of a Legendrian knot. The generators come from crossings and the differential comes from counting immersed polygons whose edges lie in the diagram of the knot and whose vertices lie at crossings. (For more on how it is defined, see Chekanov’s paper Differential Algebras of Legendrian Links.) One can then look at augmentations of the DGA which are algebra maps $\epsilon:\mathcal{A}\to\mathbb{Z}/2$ which satisfy $\epsilon\circ\partial=0$ and $\epsilon(1)=1$. If $\epsilon(c)=1$, we call $c$ augmented. If we go back to the diagram of our knot, we can define a “normal ruling” of the diagram. In the paper Chekanov-Eliashberg invariants of Legendrian knots: Existence of augmentations, Fuchs uses “splashed” diagrams to show that if a Legendrian knot has a normal ruling, then it has an augmentation. Then, in Augmentations and rulings of Legendrian knots, Sabloff uses “dipped” diagrams to show that if a knot has an augmentation, then it has a normal ruling. We will use Sabloff’s method of “dipped” diagrams to show both results. However, in this posting, I will only give the necessary notation and definitions. Sorry there are so many.

First off, a ruling of the front diagram of a knot consists of a one-to-one correspondence between the left cusps and the right cusps and, for each pair of corresponding cusps, two paths in the front diagram that join them. They must also satisfy the conditions that two paths in the ruling meet only at crossing or cusps and the interior of the two paths joining corresponding cusps are disjoint and so only meet at the cusps and so bound a topological disk. Thus, the paths in a ruling cover the front diagram. Near a crossing, we call the two ruling paths (one from each of two paired cusps) which are on strands that cross, crossing paths and the paths paired with these, companion paths. We say the ruling is switched at a crossing if one of the crossing paths lies entirely above the other near the crossing. If all of the switched crossings of a ruling are one of the following, we call the ruling normal.

(Figure from Sabloff’s paper.)

For example, the three normal rulings of the trefoil are

(Figure from Sabloff’s paper.)

Recall that the DGA for the trefoil

is $\mathbb{Z}/2\langle a_1,a_2,a_3,q_1,q_2\rangle$ with the only nontrivial differentials

$\partial q_1=1+a_1+a_3+a_3a_2a_1$
$\partial q_2=1+a_1+a_3+a_1a_2a_3$

So an example of an augmentation for the trefoil is $\epsilon(a_1)=1=\epsilon(a_2)$ and $\epsilon(a_3)=0$.

Unfortunately, we still need a few more definitions.

We will assume that the front diagram of our knot is in plat position, in other words, all the left cusps have the same x-coordinate, all the right cusps have the same x-coordinate, and none of the crossings have the same x-coordinate. The trefoil above is in plat position. We can make this assumption as a series of Legendrian versions of the Reidemeister II moves and planar isotopies will put any front diagram in plat position. Call the crossings in this diagram $c_1,\dots,c_n$.

Suppose we have $m$ right cusps and at any x-coordinate in the diagram, label the strands in the diagram from bottom to top by $1,\ldots,2m$.

A dip is constructed by a sequence of Reidemeister II moves, which looks as follows in the front projection and Lagrangian projection.

(Picture from Sabloff’s paper.)

The dipped diagram involves introducing a dip between each crossing in the plat position diagram and in between the left, respectively right, cusps and the first, respectively last, crossing. Each Reidemeister II move introduces two new crossings. In each dip, call the new crossings which appear on the left the $b$-lattice and the right the $a$-lattice. While dipped diagrams have many more crossings than the original knot diagram, the differential $\partial$ on $\mathbb{Z}$-differential graded algebra is generally much simpler. Note that in the dipped diagrams, the differential of crossings in a $b$-lattice involve at most $c_k$, crossings in the previous $a$-lattice, and crossings in the $a$-lattice, the differential of crossings in an $a$-lattice only involve crossings in the $b$-lattice, and the
differential of $c_k$ is $0$ for all $k$. This greatly reduces the totally augmented disks (disks contributing to the differential of the crossing where all negative corners are augmented) for which to look to compute whether we have an augmentation. (These are the only disks which contribute to $\epsilon\circ\partial$.)

This is most of the setup necessary to show both that augmentations imply rulings and rulings imply augmentations.