Yesterday, I gave an introductory talk on Hamiltonian mechanics and symplectic geometry. The starting point is the Legendre transform. First, begin with a configuration space \(Q\). The Lagrangian \(\mathcal{L}\) is a smooth function on \(TQ\). In local coordinates \(q^i\) on \(Q\), we have coordinates \((q_i, v_i)\) on \(TQ\), where the \(v^i\) are the components of the tangent vector
\(v = v_i \partial_i \in T_q Q\). Typically, the Lagrangian will be of the form
\[ \mathcal{L}(q,v) = \frac{1}{2} g(v,v) - V(q), \]
where \(g\) is some metric on \(Q\). Now we introduce new coordinates \(p_i\) defined by
\[ p_i = \frac{\partial \mathcal{L}}{\partial v^i}. \]
If \(\mathcal{L}\) is (strictly?) convex in \(v\) then we can solve for \(v^i\) as a function of \((q^i, p_j)\). It is easy to check that the \(p_i\) transform as covectors, and so this gives a diffeomorphism \(TQ \to T^\ast Q\)(which depends on \(\mathcal{L}\)). For example, in the above Lagrangian,
\[ \frac{\partial \mathcal{L}}{\partial v} = g(v, -), \]
which is just the dual of \(v\) with respect to the metric \(g\). So for Lagrangians of this form, the map \(TQ \to T^\ast Q\) is just the one given by the metric.
Now comes the interesting part. There is a natural way to turn \(\mathcal{L}\), which is a function on \(TQ\), into a function \(H\) on \(T^\ast Q\), in such a way that if we repeat this process, we will get back the original function \(\mathcal{L}\) on \(TQ\). This is the Legendre transform:
\[ \mathcal{H} = pv - L. \]
Now suppose we have a curve \(q(t), \dot{q}(t) \in TQ\) that satisfies the Euler-Lagrange equations. Then by the identification \(TQ = T^\ast Q\), this gives a curve \((q(t), p(t)) \in T^\ast Q\). What equation does it satisfy? We have
\[ \frac{d}{dt} p = \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial v} = \frac{\partial \mathcal{L}}{\partial q} = -\frac{\partial H}{\partial q}, \]
and
\[ \frac{d}{dt}q = v = \frac{\partial H}{\partial p}. \]
These are Hamilton's equations, and they say that the curve \(\gamma = (q(t), p(t)) \in T^\ast Q\) is just an integral curve of the symplectic gradient of \(H\)! So classical mechanics is really just about flows of Hamiltonian vector fields on symplectic manifolds.
Wednesday, January 27, 2010
Thursday, January 21, 2010
Note to self
I don't have time for an update now, so this is just a reminder to make a couple of posts over the weekend about some things. On the list: math physics seminar, double vector bundles, geometric invariant theory. Possibly on the list: Legendre transforms and Kahler potentials, sigma models, and supersymmetry.
Friday, January 8, 2010
Hyperkahler reduction
Yesterday I gave a talk on hyperkahler reduction--my first invited talk ever. I thought I should summarize it here.
A hyperkahler manifold is just a Kahler manifold \( (M, g, I)\) together with two additional integrable complex structures \(J,K\) such that \(I,J,K\) satisfy the quaternion relations, and \(J\) and \(K\) are compatible with the metric. Hyperkahler manifolds are modeled on \(\mathbb{H}^n\), analogous to the way that Kahler manifolds are modeled on \(\mathbb{C}^n\)
A hyperkahler manifold is just a Kahler manifold \( (M, g, I)\) together with two additional integrable complex structures \(J,K\) such that \(I,J,K\) satisfy the quaternion relations, and \(J\) and \(K\) are compatible with the metric. Hyperkahler manifolds are modeled on \(\mathbb{H}^n\), analogous to the way that Kahler manifolds are modeled on \(\mathbb{C}^n\)
When \(G\) acts on a Kahler manifold \(M\) preserving the Kahler structure, then (under relatively weak
hypotheses) a momentum map exists and we can use this to define a quotient \(M//G\), which turns out to be Kahler. Similarly, if \(M\) is hyperkahler and \(G\) preserves the hyperkahler structure, then we have a hyperkahler quotient \(M///G\).
So far I haven't said anything interesting. There are two basic questions: (1) do any interesting manifolds actually arise in this way? and (2) what theorems about symplectic/Kahler quotients have analogues in hyperkahler quotients?
It turns out that question (1) is related to the Yang-Mills equations over \(\mathbb{R}^4\). (Of course, \(\mathbb{R}^4 = \mathbb{H}\), so maybe this is not that surprising, since moduli spaces love to inherit geometry from their parents). Depending on choices of Lagrangian, base manifold, etc. you get different moduli spaces of solutions. In general, these moduli spaces are given as quotients of infinite dimensional spaces of connections by the action of some infinite dimensional Lie group of gauge transformations. In 4-dimensions, this frequently turns out to be an infinite-dimensional hyperkahler quotient! In fact, it turns out that the Yang-Mills functional is very closely related to the momentum map for the action of the gauge transformations. Even more surprisingly, there are cases where this infinite-dimensional quotient can be expressed as a finite-dimensional quotient (see, e.g., the ADHM construction). In the case of the ADHM construction, the space you get is actually just the quotient of \(\mathbb{H}^n\) by a linear action (i.e. a group of matrices). So it turns out that even mere linear actions give you lots of interesting geometry, and lots of questions about these are still open.
Question (2) is close to my own research interests. The hyperkahler quotient is similar to but different from the symplectic quotient. In symplectic quotients, the most basic tool we have is Morse theory using the momentum map. In hyperkahler geometry, we have 3 moment maps, and it's not obvious what to do with them. Frequently, we encode them as a pair \((\mu_\mathbb{R}, \mu_\mathbb{C})\), where
\[ \mu_\mathbb{R} = \mu_1 \]
\[ \mu_\mathbb{C} = \mu_2 + i \mu_3 \]
It turns out that $\mu_\mathbb{C}$ is holomorphic, and the corresponding form $\omega_\mathbb{C} = \omega_2 + i \omega_3$ is holomorphic symplectic. It's not really clear what the best way of packaging the momentum maps is, and this is an important question because we would like to do Morse theory in some way using the momentum maps.
This leads to one possible way of bypassing these problems. Penrose introduced the idea of twistor spaces. A hyperkahler manifold has a lot of data: \(g, I, J, K, \omega_1, \omega_2, \omega_3\) with lots of relations. But there is an interesting symmetry: if \((x_1, x_2, x_3) \in S^2\), then \(I_x = x_1 I + x_2 J + x_3 K\) turns out to be another integrable complex structure. So in fact we have an \(S^2\) family of Kahler structures. But \(S^2 = \mathbb{CP}^1\), which is complex (and in fact Kahler), so we have a Kahler family of Kahler structures. Since everything in question is complex, why not try to encode our data holomorphically?
The twisor construction does just that. For \(Z = M \times \mathbb{CP}^1\). Give it the (almost) complex structure \(I = (I_x, I_0)\) where \(I_0\) is the complex structure on \(\mathbb{CP}^1\). One can check (using the Newlander-Nirenberg theorem) that \(I\) is integrable, so \(Z\) is a complex manifold (and, importantly, is not just the product \(M \times \mathbb{CP}^1\) as a complex manifold). Now we need to put some holomorphic data on \(Z\).
First, we have the projection \(p: Z \to \mathbb{CP}^1\). This is holomorphic and makes \(Z\) a fiber bundle over \(\mathbb{CP}^1\).
Second, we need the symplectic forms. Let \(T_F\) be the vertical subbundle of the tangent bundle to \(Z\). Let \(\zeta\) be a complex coordinate on \(\mathbb{CP}^1\) Now define
\[ \omega = (\omega_2 + i\omega_3) + 2\zeta \omega_1 - \zeta^2 (\omega_2 - i\omega_3).\]
This definition makes sense at least locally, and it makes sense globally thinking of \(\omega\) as a holomorphic seciton of $\Lambda^2 T_F^\ast \otimes p^\ast O(2)\). It is also symplectic along the fibers.
The third piece of data we need are the "twistor lines". Given a point \(m \in M\), the map \(\zeta \mapsto (m, \zeta)\) is a holomorphic section of \(Z\). The normal bundle to any such section turns out to be holomorphically equivalent to \(\mathbb{C}^{2n} \otimes O(1)\).
The last piece of data is a real structure \(\tau\) on \(Z\). We can define it by \(\tau(m,\zeta) = (m, \bar{\zeta}^{-1})\). This is an antiholomorphic involution. It is also compatible with 1-3 in a sense that I won't describe.
It is a theorem that these 4 pieces of data are all we need to recover M and its hyperkahler structure. (For proof, see Hitchin et al., "Hyperkahler metrics and supersymmetry".) This might seem like a convoluted construction, but the essential point is that the hyperkahler structure is encoded as holomorphic data on \(Z\)(the only piece data which is not holomorphic is the real structure \(\tau\)), and holomorphic things are very rigid and easier to work with.
In the twistor language, hyperkahler quotients correspond to a kind of twistor holomorphic symplectic quotient. Maybe this is the right way to think about things. Who knows?
So far I haven't said anything interesting. There are two basic questions: (1) do any interesting manifolds actually arise in this way? and (2) what theorems about symplectic/Kahler quotients have analogues in hyperkahler quotients?
It turns out that question (1) is related to the Yang-Mills equations over \(\mathbb{R}^4\). (Of course, \(\mathbb{R}^4 = \mathbb{H}\), so maybe this is not that surprising, since moduli spaces love to inherit geometry from their parents). Depending on choices of Lagrangian, base manifold, etc. you get different moduli spaces of solutions. In general, these moduli spaces are given as quotients of infinite dimensional spaces of connections by the action of some infinite dimensional Lie group of gauge transformations. In 4-dimensions, this frequently turns out to be an infinite-dimensional hyperkahler quotient! In fact, it turns out that the Yang-Mills functional is very closely related to the momentum map for the action of the gauge transformations. Even more surprisingly, there are cases where this infinite-dimensional quotient can be expressed as a finite-dimensional quotient (see, e.g., the ADHM construction). In the case of the ADHM construction, the space you get is actually just the quotient of \(\mathbb{H}^n\) by a linear action (i.e. a group of matrices). So it turns out that even mere linear actions give you lots of interesting geometry, and lots of questions about these are still open.
Question (2) is close to my own research interests. The hyperkahler quotient is similar to but different from the symplectic quotient. In symplectic quotients, the most basic tool we have is Morse theory using the momentum map. In hyperkahler geometry, we have 3 moment maps, and it's not obvious what to do with them. Frequently, we encode them as a pair \((\mu_\mathbb{R}, \mu_\mathbb{C})\), where
\[ \mu_\mathbb{R} = \mu_1 \]
\[ \mu_\mathbb{C} = \mu_2 + i \mu_3 \]
It turns out that $\mu_\mathbb{C}$ is holomorphic, and the corresponding form $\omega_\mathbb{C} = \omega_2 + i \omega_3$ is holomorphic symplectic. It's not really clear what the best way of packaging the momentum maps is, and this is an important question because we would like to do Morse theory in some way using the momentum maps.
This leads to one possible way of bypassing these problems. Penrose introduced the idea of twistor spaces. A hyperkahler manifold has a lot of data: \(g, I, J, K, \omega_1, \omega_2, \omega_3\) with lots of relations. But there is an interesting symmetry: if \((x_1, x_2, x_3) \in S^2\), then \(I_x = x_1 I + x_2 J + x_3 K\) turns out to be another integrable complex structure. So in fact we have an \(S^2\) family of Kahler structures. But \(S^2 = \mathbb{CP}^1\), which is complex (and in fact Kahler), so we have a Kahler family of Kahler structures. Since everything in question is complex, why not try to encode our data holomorphically?
The twisor construction does just that. For \(Z = M \times \mathbb{CP}^1\). Give it the (almost) complex structure \(I = (I_x, I_0)\) where \(I_0\) is the complex structure on \(\mathbb{CP}^1\). One can check (using the Newlander-Nirenberg theorem) that \(I\) is integrable, so \(Z\) is a complex manifold (and, importantly, is not just the product \(M \times \mathbb{CP}^1\) as a complex manifold). Now we need to put some holomorphic data on \(Z\).
First, we have the projection \(p: Z \to \mathbb{CP}^1\). This is holomorphic and makes \(Z\) a fiber bundle over \(\mathbb{CP}^1\).
Second, we need the symplectic forms. Let \(T_F\) be the vertical subbundle of the tangent bundle to \(Z\). Let \(\zeta\) be a complex coordinate on \(\mathbb{CP}^1\) Now define
\[ \omega = (\omega_2 + i\omega_3) + 2\zeta \omega_1 - \zeta^2 (\omega_2 - i\omega_3).\]
This definition makes sense at least locally, and it makes sense globally thinking of \(\omega\) as a holomorphic seciton of $\Lambda^2 T_F^\ast \otimes p^\ast O(2)\). It is also symplectic along the fibers.
The third piece of data we need are the "twistor lines". Given a point \(m \in M\), the map \(\zeta \mapsto (m, \zeta)\) is a holomorphic section of \(Z\). The normal bundle to any such section turns out to be holomorphically equivalent to \(\mathbb{C}^{2n} \otimes O(1)\).
The last piece of data is a real structure \(\tau\) on \(Z\). We can define it by \(\tau(m,\zeta) = (m, \bar{\zeta}^{-1})\). This is an antiholomorphic involution. It is also compatible with 1-3 in a sense that I won't describe.
It is a theorem that these 4 pieces of data are all we need to recover M and its hyperkahler structure. (For proof, see Hitchin et al., "Hyperkahler metrics and supersymmetry".) This might seem like a convoluted construction, but the essential point is that the hyperkahler structure is encoded as holomorphic data on \(Z\)(the only piece data which is not holomorphic is the real structure \(\tau\)), and holomorphic things are very rigid and easier to work with.
In the twistor language, hyperkahler quotients correspond to a kind of twistor holomorphic symplectic quotient. Maybe this is the right way to think about things. Who knows?
Monday, January 4, 2010
First day of 2010
It's the first (academic) day of 2010. I figured that the best use of this blog is as a log--that is, to log and plan my work for the semester/year/life. If you want to accomplish goals, the fist thing to do is to write them down! So here we go, crude outline for the next semester:
1. Work through Milnor's Morse theory book, cover to cover. This should be easy since I'm taking a class in morse theory anyway.
2. Work through Kirwan's thesis cover to cover. I've already been through quite a bit of it, and the only things that caused me any trouble last summer have since been cleared up.
3. Work though Gulliemin and Sternberg's Equivariant cohomology book. Again, quite a bit of the material I already know, so this should be doable.
4. Finish working through HKLR. Really the only remaining part is supersymmetric nonlinear sigma models.
5. The details of the ADHM construction, once and for all. I should know this already.
6. Hilbert schemes of points on a surface. Really, the hyperkahler metric for the scheme of points on \(\mathbb{C}^2\). Again, I should know this already.
We'll see how these go--this is probably ambitious, and many of these will get extended into the summer.
1. Work through Milnor's Morse theory book, cover to cover. This should be easy since I'm taking a class in morse theory anyway.
2. Work through Kirwan's thesis cover to cover. I've already been through quite a bit of it, and the only things that caused me any trouble last summer have since been cleared up.
3. Work though Gulliemin and Sternberg's Equivariant cohomology book. Again, quite a bit of the material I already know, so this should be doable.
4. Finish working through HKLR. Really the only remaining part is supersymmetric nonlinear sigma models.
5. The details of the ADHM construction, once and for all. I should know this already.
6. Hilbert schemes of points on a surface. Really, the hyperkahler metric for the scheme of points on \(\mathbb{C}^2\). Again, I should know this already.
We'll see how these go--this is probably ambitious, and many of these will get extended into the summer.
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