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Teaching Differential Geometry I

Posted on February 16, 2014 by guangbox

This quarter I am teaching undergraduate differential geometry at UC Irvine. We are spending two quarters on this course, which is slightly longer than a semester course, such as the one I TAed at Princeton. So we have more time to study various topics which I didn’t know when I was in college.

We are using the book “Elements of Differential Geometry” by Richard S. Millman and George D. Parker, different from the more popular do Carmo’s “Differential Geometry of Curves and surfaces“. Indeed do Carmo’s was once regarded as an “easy” book in some sense. Indeed the core materials won’t differ too much but details vary.

So the purpose of writing (maybe several) posts is to clarify some understanding of about materials of the book/course, and to fill some gaps existing in many places in the book. Hopefully this can attract my students occasionally.

In the first post of this series, we consider the rotation index theorem in plane curve theory.

Basic notations

Definition. A regular plane curve is a $C^3$ -map $\alpha: (a, b) \to {\mathbb R}^2$ such that $\alpha'(s) \neq 0$ for all $s\in (a, b)$ . It is a unit speed curve if

$|\alpha'(s)| = 1$ for all $s\in (a, b)$ .

Now we consider only unit speed curves. In fact all regualr curves can be reparametrized to be a unit speed curve.

Definition: The tangent vector field is the vector-valued $C^2$ -map $t(s) = \alpha'(s)$ . The normal vector field $n(s)$ is basically “i” times $t(s)$ , where i is the imaginary unit. The curvature is then the $C^1$ -function

$k(s) = \langle t'(s), n(s) \rangle$ .

Now we consider closed curves, which are equivalent to a periodic map $\alpha: (-\infty, +\infty) \to {\mathbb R}^2$ . If $\alpha$ is of unit speed, then the least positive period is equal to the perimeter, usually denoted by $L$ , of the curve. A closed curve is simple, if

$\alpha(s_1) = \alpha(s_2)$ implies that $s_1 - s_2 = n L$ .

Rotation index

Now it is a bit cumbersome to define the notion of rotation index. Basically, it counts within one period, how many 360 degrees you have turned.

Indeed, if you know some topology, the rotation index can be defined as follows. The tangent vector field $t(s)$ defines a periodic map to the unit circle of ${\mathbb R}^2$ . So if we glue 0 and L to get a circle of perimeter L, then $t(s)$ induces a continuous map between two circles. The rotation index is the topological degree of this map.

If you know something about differential forms, then on the unit circle (which is the target space of the map $t(s)$ ), there is a differential 1-form, called $d\theta$ . We can integrate this 1-form on the 1-dimensional circle, and we have

$\int_{S^1} d\theta = 2\pi$

which agrees with our convention that the perimeter of a unit circle is $2\pi$ . Now the pull-back of $d\theta$ by the map $t(s)$ is a 1-form on $[0, L]$ , which is actually equal to $k(s) ds$ , where $k(s)$ is the curvature. The rotation index is then defined to be

$i_{\alpha}:= {1\over 2\pi} \int_0^L k(s) ds$ .

However, in the textbook, if we don’t want to introduce the language of topological degree or differential forms, then we have to define a function $\theta(s)$ along the curve, which is difficult to give a concise expression.

Rotation index theorem

Anyway, the main theorem, called rotation index theorem is

“If $\alpha$ is a simple closed plane curve, then its rotation index is either 1 or -1.”

The proof given in the textbook was due to Heinz Hopf. I think a key point here is to make use of the condition that the curve is simple.

Indeed, we can define a vector valued function $a(u, v)$ for $0 \leq u \leq v \leq L$ . If $0<u<v<L$ , then $a(u, v)$ is equal to the unit vector pointing to the direction of $\alpha(v) - \alpha(u)$ . That is

$a(u, v) = {\alpha(v) - \alpha(u) \over |\alpha(v) - \alpha(u)|}$ .

We define $a(u, u) = t(u)$ , the tangent vector at $\alpha(u)$ . And we notices that $a(0, L) = - a(0, 0) = - a(L, L)$ . If we write down the expression of $a(u, v)$ using coordinates, then we can see that $a(u, v)$ is a $C^2$ -map. And only for simple curves this map is well-defined.

Now, the differential form $d\theta$ on $S^1$ is pull-backed to a 1-form on the domain of $a$ , which is a triangle with three sides $\overline{AB}, \overline{BC}, \overline{CA}$ . Here

$A = (0, 0), B= (L, L), C = (0, L) .$

We can orient the boundar so that it goes in the counterclockwise direction.

By our “fancy” definition of rotation index, we see that the rotation index is equal to the integral of this 1-form on the diagonal $\overline{AB}$ . However, by the great “Stokes-Green” formula, we can reduce this integral to the integral of the 1-form on the other two sides of the triangle, plus the contribution of the interior of the triangle. However, the form $d \theta$ is closed, which means the pull-backed 1-form is also closed. This means the contribution from the interior is zero.

On the other hand, if we choose good starting point of the simple closed curve, it is easy to see that the integral of the 1-form on the other two sides-one horizontal and one vertical-are either both $\pi$ or both $- \pi$ . Therefore the rotation index should be either 1 or -1.

On the other hand, thinking a little bit more, if you know that the topological degree is homotopy invariant, then the triangle also gives a concrete homotopy which relates the rotation index to the topological degree of another continuous map.

(New update: I am supposed to give a undergraduate MathClub talk in a few days. Originally I would like to discuss Morse theory. But now I feel that differential forms is a good topic and I can include this as an example.)

Posted in Uncategorized | Tagged differential forms, differential geometry, teaching | Leave a comment

Two equivalent formulations of Morse-Smale transversality

Posted on July 16, 2012 by guangbox

I. some differential topology

Let $M, N_1, N_2$ be smooth manifolds and let $f_i: M \to N_i,\ i=1, 2$ be smooth maps. Let $L_i \subset N_i$ be smooth submanifolds such that $f_i$ is transverse to $L_i$ .

Now being transverse means the following: for any $x_i \in M$ such that $y_i= f_i(x_i) \in L_i$ , we have

${\rm Image} df_i(x_i)+ T_{y_i} L_i = T_{y_i} M$

Its consequence is $Y_i:= f_i^{-1}(L_i) \subset M$ are smooth submanifolds; for any $x_i \in Y_i$ , the tangent space $T_{x_i} Y_i$ consists of those tangent vectors $V\in T_{x_i} M$ such that $df_i (V) \in T_{f_i(x_i)} L_i$ .

Now we consider the maps

$f_1: Y_2 \to N_1$ and $f_2: Y_1 \to N_2$

Question: Is (I) $f_1|_{Y_2}$ is transverse to $L_2$ equivalent to (II) $f_2|_{Y_1}$ is transverse to $L_1$ ?

I never took a graduate course on differential topology, except for spending some weeks reading Milnor’s “Topology from a differentiable viewpoint”. So I try to give an answer.

An observation(or guess) is that: the two statements are both equivalent to (III) that the map $f: M \to N_1\times N_2$ is transverse to $L_1 \times L_2$ .

Indeed, for $(y_1, y_2) = (f_1(x), f_2(x)) \in L_1 \times L_2$ , take normal vectors $V_i \in (N_i)_{y_i} L_i$ . Because $f_2$ is transverse to $L_2$ , there exists $W_1 \in T_x M$ such that $df_2(W_1) - V_2 \in T_{y_2} L_2$ . And If $f_1|_{Y_2}$ is transverse to $L_1$ , there exists $W_2 \in T_x Y_2$ such that $df_1(W_2) - (V_1 + df_1(W_1)) \in T_{y_1} L_1$ . Then we consider the vector $W= W_1 + W_2 \in T_x M$ , we see

$df(W) = ( df_1(W_1+ W_2), df_2(W_1 + W_2) ) \in (V_1, V_2) + T_{y_1} L_1 \oplus T_{y_2} L_2$ .

This proves that ${\bf (I)} \Longrightarrow {\bf (III)}$ . The same argument for ${\bf (II)} \Longrightarrow {\bf (III)}$ . And it is easier to see that ${\bf (III)} \Longrightarrow {\bf (I)}, {\bf (II)}$ .

II. An infinite dimensional case

In Morse theory, there is a transversality condition which topologists favor: let $M$ be a smooth manifold and $f: M \to {\mathbb R}$ be a Morse function and $g$ be a Riemannian metric on $M$ . Then we can define the negative gradient flow of the pair $(f, g)$ , to be

$\phi_t: M \to M, {d\phi_t \over dt}(x) = - \nabla f ( \phi_t(x)),\forall t\in {\mathbb R}, x\in M.$

For any critical point $p \in M$ of $f$ , the unstable(resp. stable) manifold of $p$ is

$W^u(p) = \{ x\in M | \lim_{t\to -\infty} \phi_t(x) = p\}$ (resp. $W^s(p) = \{ x\in M | \lim_{t\to +\infty} \phi_t(x) = p\}$ ).

We say that $(f, g)$ is a Morse-Smale pair, if the following condition holds:

(MSI) For any two critical points $p, q$ , $W^u(p)$ and $W^s(q)$ intersect transversely in $M$ .

There is another transversality formulation which analysts favor. Let’s fix the two critical points $p, q$ . Consider the Banach manifold ${\mathcal B}$ , whose elements are maps $x: {\mathbb R} \to M$ satisfying

1. $x$ is of class $W^{1, 2}_{loc}$ , i.e., if we take a smooth local coordinates near $x(t)$ , then locally $x(t)$ is a vector-valued $W^{1, 2}$ function.

2. There exists $T >0$ such that $x|_{(-\infty, -T]} = \exp_p V(t)$ , $x|_{[T, +\infty)} = \exp_q W(t)$ with $V \in W^{1, 2}( (-\infty, -T], T_p M )$ , $W \in W^{1, 2}( [T, +\infty), T_qM )$ .

For each $x\in {\mathcal B}$ , the tangent space $T_x {\mathcal B}$ is the Banach space $W^{1, 2} ( {\mathbb R}, x^*TM )$ .

Also, consider a Banach space bundle ${\mathcal E}\to {\mathcal B}$ , whose fibre over $x\in {\mathcal B}$ is the Banach space $L^2({\mathbb R}, x^*TM )$ . Consider a section ${\mathcal S}: {\mathcal B} \to {\mathcal E}$ defined by

${\mathcal S}(x)(t) = x'(t) + \nabla f (x(t))$ .

This is a Fredholm section, meaning that its linearization is a Fredholm operator. The transversality that analysts favor is

(MSII): For each $x \in {\mathcal S}^{-1}(0)$ , the linearization $D_x {\mathcal S}: T_x {\mathcal B} \to {\mathcal E}_x$ is surjective.

If this condition holds, then ${\mathcal S}^{-1}(0)$ will be a smooth, finite dimensional submanifold of ${\mathcal B}$ .

Observe that: ${\mathcal S}^{-1}(0)$ can be identified with $W^u(p) \cap W^s(q)$ , by the evaluation $x\mapsto x(0)$ .

~~QUESTION~~: Is (MSI) equivalent to (MSII)?

This can be answered in the spirit of the first part of this post.

We consider a larger Banach manifold. Let ${\mathcal B}_+$ be the space of $W^{1, 2}_{loc}$ -maps from $[0, +\infty)$ to $M$ having the same assymptotic condition we required for an element in ${\mathcal B}$ when $t \to +\infty$ ; and ${\mathcal B}_-$ be the space of $W^{1,2}_{loc}$ -maps from $(-\infty, 0]$ to $M$ , having the same assymptotic condition we required for an element in ${\mathcal B}$ when $t \to -\infty$ .

Then because of the Sobolev embedding theorem: $W^{1, 2}\hookrightarrow C^0$ , the evaluation map

$e: {\mathcal B}_- \times {\mathcal B}_+ \to M \times M$

is smooth, and transverse to the diagonal $\Delta \subset M \times M$ . Now we see

${\mathcal B}= e^{-1} (\Delta) \subset {\mathcal B}_- \times {\mathcal B}_+$

is a smooth submanifold by the transversality.

On the other hand, there are analogous Banach space bundles ${\mathcal E}_\pm \to {\mathcal B}_\pm$ , and the analogous smooth sections ${\mathcal S}_\pm: {\mathcal B}_\pm \to {\mathcal E}_\pm$ . Using some knowledge of ODE, we know that ${\mathcal S}_\pm$ are transverse to the zero sections.

We see that we have a natural identification

$W^u(p) \times W^s(q) = {\mathcal S}_-^{-1}(0) \times {\mathcal S}_+^{-1}(0) \subset {\mathcal B}_-\times {\mathcal B}_+$ .

Now the condition (MSI) is just: $e$ restricted to ${\mathcal S}_-^{-1}(0) \times {\mathcal S}_+^{-1}(0)$ is transverse to the diagonal $\Delta$ . The condition (MSII) is just: $({\mathcal S}_- , {\mathcal S}_+ )$ restricted to $e^{-1}(\Delta)$ is transverse to $(0, 0)$ . So by the result of the first part, we derive the equivalence between (MSI) and (MSII), modulo some technicality because we have jumped from finite dimension to infinite dimension.

Posted in Uncategorized | Tagged differential topology, Morse theory | Leave a comment

A simple result of abelian vortices

Posted on May 28, 2012 by guangbox

UPDATE: Indeed, the example I gave below was treated by Jaffe and Taubes in their book “Vortices and monopoles”. They argued directly over ${\mathbb C}$ , by looking at the PDE

$\Delta u + ( e^{-u} -1 ) = 4 \pi \sum_j \delta(x- x_j)$

where $\delta(x-x_j)$ is the $\delta$ distribution centered at $x_j$ .

1. The concept of stable pairs and abelian vortices

Consider a Riemann surface $\Sigma$ . Let $L \to \Sigma$ be “the” smooth line bundle of degree $d>0$ . An abelian(or $U(1)$ ) holomorphic pair of degree $d$ is a pair $( \overline\partial_L, \phi )$ , where $\overline\partial_L$ is a holomorphic structure on $L$ and $\phi$ is a holomorphic section. We can identify holomorphic pairs up to isomorphism, that is, $( \overline\partial_L, \phi ) \sim ( \overline\partial_L', \phi')$ if there exists $g: \Sigma \to \mathbb{C}^*$ such that

$\overline\partial_L' = g^{-1}\circ \overline\partial_L \circ g = \overline\partial_L + \overline\partial \log g, \phi'(z) = g^{-1}(z) \phi(z).$

Actually the moduli space of such objects(up to isomorphism) is quite simple: for $\phi \neq 0$ , the holomorphic structure(up to isomorphism) of $L$ is completely determined by the zero locus of $\phi$ , which is a divisor(of degree $d$ ) in $\Sigma$ . So to study this moduli space, we consider those “stable” objects, i.e., those pairs with $\phi \neq 0$ , and they are parametrized exactly by the $d$ -th symmetric product of $\Sigma$ , which is denoted by $S^d \Sigma$ .

Here the “stability” condition is not naive. It has generalizations to non-abelian situations.

Now, we look at Hermitian metrics on $L$ . A metric $H$ and the Cauchy-Riemann operator $\overline\partial_L$ determines a unique CHERN connection(great Chern!!), denoted by $A_{\overline\partial_L, H}$ on $L$ . And we know(choosing a smooth Kahler form $\Omega$ on $\Sigma$ with area 1) that for each Cauchy-Riemann operator $\overline\partial_L$ , there exists a unique Hermitian metric such that

$-{1\over 2\pi \sqrt{-1}} F_{A_{\overline\partial_L, H}} = d\cdot \Omega.$

Note that the coefficient $d$ is the only constant that guarantees the existence of a solution.

When coupled with the section $\phi$ , the “correct” equation should be

$-{1\over 2\pi \sqrt{-1}} F_{A_{\overline\partial_L, H}} + ( |\phi|_H^2 - c ) \Omega = 0.$

Now the constant $c$ can be taken to be anything that no less than $d$ (if it is less than $d$ then it is easy to show the contradiction.) This equation is called the vortex equation.

2. The Hitchin-Kobayashi correspondence

THEOREM For any stable pair $(\overline\partial_L, \phi)$ , and any Kahler form $\Omega$ , there exists a unique smooth Hermitian metric $H$ on $L$ which solves the above equation.

This is an example of the principle called “Hitchin-Kobayashi correspondence”, by which one can identify the moduli of holomorphic objects(the stable pairs), and the moduli of symplectic/gauge-theoretic objects(the solutions to the vortex equation).

3. The adiabatic limit

Now for some purpose, we would like to resale the Kahler form $\Omega$ by $\lambda^2 \Omega$ with $\lambda >>0$ . Let’s fix the holomorphic objects $( \overline\partial_L, \phi)$ . The theorem says that there exists a unique metric solving the vortex equation for the Kahler form $\lambda^2 \Omega$ , which is denoted by $H_\lambda$ . We want to know the assymptotic behavior of $H_\lambda$ as $\lambda \to \infty$ . Indeed, there exists a smooth function $h_\lambda$ such that

$|v|_{H_\lambda} = e^{-2 h_\lambda}|v|_{H_1}$

and the question is asking the behavior of the function $h_\lambda$ . Rewriting the vortex equation in terms of $h_\lambda$ , denoting $F_{\overline\partial_L, H_1} = - g_1 2\pi \sqrt{-1}$ , $|\phi|_{H_1}^2 = g_2$ , it reads

${2\over \pi} \Delta h_\lambda + \lambda^2 ( g_2 e^{-2h_\lambda} - 1) + g_1 = 0.$

Set $f_\lambda = h_\lambda - {1\over 2} \log g_2$ (at least away from the zero of $\phi$ ), we have

${2\over \pi} \Delta f_\lambda + \lambda^2 ( e^{-2 f_\lambda} -1 ) + g_1 - {1\over \pi} \Delta \log g_2 = 0 .$

But it is a trivial fact that although $\log g_2$ is singular at zeros of $\phi$ , $\Delta \log g_2$ is smooth everywhere. Then it is easy to show(by maximal principle) that as $\lambda \to + \infty$ , $h_\lambda \sim {1\over 2} \log g_2$ , and the metric blows up like $g_2^{-1} = |\phi|_{H_1}^{-2}$ near zeroes of $\phi$ . And away from the zeroes, the curvature converges to zero hence the connection is flat away from the zeroes.

This is a qualitative version of a result I learned from Tim Perutz in his lecture series in the Northwestern Gauge theory master class last winter. Later I will post something which slightly generalizes this result.

4. The planary vortex

The energy of a vortex is defined to be the sum of the $L^2$ energy of the three parts: $d_A \phi, F_A, {1\over 2}|\phi|^2 - c$ with respect to the metric determined by $\lambda^2 \Omega$ . By some calculation we will see that the energy doesn’t depend on $\lambda$ and is equal to $cd$ . Hence the energy should be preserved in the limit. But away from the zeroes, everything converges to trivial objects, so one can guess that the energy is stored in the singularities.

Indeed, if we zoom in at $p\in \phi^{-1}(0)$ by a factor $\lambda$ , the radius $\lambda^{-1} \log \lambda$ disk is dilated to a disk of radius $\log \lambda$ , which will exhaust the complex plane. This zoom-in process is quite like the bubbling of holomorphic spheres, but this time one cannot fill in the point at infinity by removal of singularity: the limit object is something only defined on the plane. Indeed, one can show that after the zooming-in, the sequence will converge in $C^\infty_{loc}$ topology to an object $( B, u )$ , where $B$ is a smooth connection on the trivial bundle ${\mathbb C} \times {\mathbb C}\to {\mathbb C}$ , and $u$ is a smooth section of the trivial bundle, such that

$\overline\partial_B u = 0,$ $-{1\over 2\pi \sqrt{-1}} F_B + ( |u|^2 - c ) dsdt = 0.$

Here $ds dt$ is the standard area form on the plane. Such an object is called a planary vortex. This object has been studied in a quite general setting by Fabian Ziltener.

Posted in Uncategorized | Tagged bubbling, Gauge Theory, Moduli space, PDE, stable pairs, Symplectic Geometry, vortices | Leave a comment

Verlinde algebra

Posted on January 8, 2012 by guangbox

The motivation of writing this notes is that I am studying Witten’s paper “The Verlinde algebra and the quantum cohomology of Grassmannians“, and trying to understand this subject from the most naive perspective.

First, let me briefly explain the two objects. The Verlinde algebra, named after the Dutch physicist Erik Verlinde, is certain quotient of the representation ring of a compact Lie group. Just think of $U(k)$ or $SU(k)$ . The quantum cohomology, is a ring structure on the cohomology group of a symplectic manifold $(M, \omega)$ , which deforms the usual cup product by the “quantum” corrections given by the genus-zero Gromov-Witten invariants of $(M, \omega)$ . They are both introduced by physicists but actually I know little about their physical background. Luckily, there have been smart mathematicians who were able to translate those material into more accessible language.

There is a quite elementary introduction to both of the two objects in Witten’s paper, and compared with a generic paper of Witten, these parts are quite “mathematical”. In the following I just try to write down something I was not quite clear, most of which is on Verlinde algebra.

1. The Verlinde numbers

The Verlinde algebra is closed related to the representation of loop groups and moduli space of vector bundles over Riemann surfaces. For simplicity, take $G= U(k) or SU(k)$ . Let $\Sigma$ be a Riemann surface and let ${\mathcal M}(\Sigma, G)$ be the moduli space of stable $G_{\mathbb C}$ -bundles over $\Sigma$ . Here $G_{\mathbb C}$ is the complexification, namely, $GL(k) or SL(k)$ . There is a canonical line bundle ${\mathcal L} \to {\mathcal M}(\Sigma, G)$ . Note that we are in holomorphic category, holomorphic sections

$s\in H^0({\mathcal M}(\Sigma, G), {\mathcal L}^{\otimes l})$

are called “non-abelian $\theta$ -functions of level $l$ “, which is a reasonable, but nontrivial generalization of the classical $\theta$ -function(for holomorphic line bundles).

The story can be described both in algebraic geometric language and gauge-theoretical language, and the Hitchin-Kobayashi correspondence translates between these two languages.

In 1988, Verlinde discovered a remarkable formula computing the dimension of $H^0({\mathcal M}(\Sigma, G), {\mathcal L}^{\otimes l})$ . This was later proved by mathematicians(Faltings, Beauville).

If we replace $\Sigma$ by a Riemann surface with boundary, say, a pair-of-pants, then, we can pick three irreducible representations $\alpha_i, i=1, 2, 3$ of $G$ , two of which are “input” and the third is the “output”. This still gives us a corresponding moduli space(moduli of stable bundles with parabolic structure) and a bundle ${\mathcal L}$ . Then the dimension of the corresponding theta-functions(or called Verlinde numbers), $N_{\alpha_1, \alpha_2, \alpha_3}$ can be used to construct a product, i.e.

$\alpha_1 * \alpha_2 = \sum_{\alpha_3} N_{\alpha_1, \alpha_2, \alpha_3} \alpha_3 .$

Indeed, the generators $\alpha_i$ are understood as representations of $G$ , but those of the loop group

$LG= Map(S^1, G).$

And not all representations, but those called “positive energy representation of level $l$ “. However, irreducible ones can be classified by representations of $G$ .

On the other hand, quantum cohomology is given by similar structure constants $GW_{\beta_1, \beta_2, \beta_3}$ , which are the “genus-zero three point Gromov-Witten invariants”. Instead of bundles over a pair-of-pants, they are given by counting holomorphic maps from a pair-of-pants into the Grassmannian, and $\beta_i$ are cohomology classes of $Gr(k, N)$ .

Actually, one can consider n-point function $N_{\alpha_1, \ldots, \alpha_n}$ or $GW_{\beta_1, \ldots, \beta_n}$ and now these numbers are independent of the conformal structure(how we place the $n$ marked points/holes). Then by stretching a sphere with 4 punctures, the invariance gives us the associativity of the two products.

Now it is quite mysterious to me that these two stories are equivalent, which was first observed by Witten in the case of $U(k)$ (despite of the case of projective spaces). More precisely, if we take the complex Grassmannian $Gr(k, N)$ , then the level of the Verlinde algebra should be $N-k$ . Take $P\to Gr(k, N)$ be the tautological $U(k)$ -bundle. A concrete map is: for each irreducible representation $\alpha$ of dimension $d$ , the top Chern class $P\times_{U(k)} \alpha$ is a $2d$ dimensional cohomology class and this correspondence should be a ring isomorphism, between Verlinde algebra and quantum cohomology. And even more concretely, such representations are labeled by integers

$0 \leq \lambda_1 \leq \cdots \leq \lambda_k \leq N-k$

as the anti-dominant weight of this representation. The bound $N-k$ is the level constrain. But remember that such a sequence also determines a Schubert cycle of the Grassmannian! It can be shown that the top Chern class of the representation is exactly the Poincare dual to this Schubert cycle(this is rather classical result.)

The essential part of Witten’s argument is physical, using a lot of path-integrals and other physical terms which I don’t understand at all. And I noticed that so far there is still no good mathematical explanation of this fact(after nearly 20 years!)

Posted in Uncategorized | Tagged genus-zero, Gromov-Witten, Representation theory, Verlinde algebra, Witten | Leave a comment

Hitchin system and HyperKahler metric

Posted on November 6, 2011 by guangbox

I attended the conference “Mirror symmetry in Midwest” held in Kansas State University. Among all the talks, Andrew Neitzke gave a minicourse on a new, explicit construction of hyperkahler metrics on certain SYZ torus fibrations. I used to learn something about hyperkahler geometry some time ago, as well as some very superficial understanding of Hitchin system(see the previous post), so I hope to understand his talk which can makes me think I didn’t waste my time on some useless stuff.

The basic scheme is: on a torus fibration ${\mathcal M} \to {\mathcal B}$ , we can naively write down a metric on the locus ${\mathcal M}'$ of regular fibres, called the semi-flat metric $g^{sf}$ . Now to extend the metric to the whole fibration, we need to add certain “quantum correction term”, which arises from certain integer-valued enumerative invariant. A local example is called the Ooguri-Vafa metric, constructed near an $I_1$ elliptic singularity. Then they managed to apply this idea to the Hitchin system. Actually, the metric is known to exist, but it is good to have such an explicit expression.

In the following I only expand the above scheme a little bit.

Hitchin system for $SU(2)$

Let’s fix a Riemann surface $C$ with punctures $p_1, \ldots, p_n$ , and the trivial $SU(2)$ -bundle over $C\setminus \{ p_1, \ldots, p_n\}$ . Fix the assymptotic data $m_i\in {\mathbb C}$ , $m_i^{\zeta} \in {\mathbb R}$ .

Consider the Hitchin equation on pairs $(A, \phi)$ :

$D_A^{0, 1}\phi=0;\ F_A+ R^2 [\phi, \overline{\phi}] =0$

with assymptotic conditions near $p_i$ , described by the numbers $m_i, m_i^{\zeta}$ (holonomies of $A$ and residues of $\phi$ ).

Here $A$ is an $SU(2)$ connection, and $\phi \in \Omega^{1, 0}(\mathfrak{sl}(2))$ .

Then the moduli space of solutions to the above equation, modulo gauge transformations, is isomorphic to the space of flat $SU(2)$ -connections on the punctured Riemann surface, and carries a hyperKahler metric!(due to N. Hitchin).

There is a map $(A, \phi) \mapsto \phi_2:= {\rm Tr} (\phi^2) \in {\mathcal B}$ , where ${\mathcal B}$ is the space of quadratic differentials on $C$ . This gives our torus fibration, and a generic fiber will be the Jacobian of the spectral curve(see below). Then $\phi_2$ will have poles at $p_i$ and zeroes somewhere else. Also, $\phi_2$ gives a flat metric on the Riemann surface away from the poles and zeroes.

Inside $T^*C$ , each $\phi_2$ gives a spectral curve $\Sigma = \{ \lambda \in T^*C\ | \ \lambda^2 = \phi_2 \}$ , which is a branched double cover of $C$ , whose branching locus is exactly the zero locus of $\phi_2$ . Now take $\Gamma_{\phi_2} : = H_1( \Sigma, {\mathbb Z})$ a local system of lattice over ${\mathcal B}$ . Define the counting invariant $\Omega(\gamma), \ \gamma\in \Gamma$ .

$\Omega(\gamma)=1$ , if there is a saddle connection between two different zeroes of $\phi_2$ whose two lifts to $\Sigma$ is of class $\gamma$ ;
$\Omega(\gamma)=-2$ , if there is a closed geodesic loop in $C$ whose lift is of class $\gamma$ ;
$\Omega(\gamma)=0$ otherwise.

Note that the counting $\Omega$ is not continuous: they obey certain wall-crossing formula due to Kontsevich-Soibelman, and the wall-crossing formula is exactly what they need to legally extend the corrected metric to the singular fibers. Now this set of numbers can be used to correct the naive metric away from the singular locus of the fibration. Actually the so-called Fock-Goncharov coordinates on the moduli space of flat connections can be used to write down the corrected metric.

Remark: if we consider Lie groups of higher rank, then we don’t have the simple correspondence of quadratic differentials and saddle connections, etc. But Andrew told me that they have found a way to generalize.

Unfortunately, I could get through all the details of their big project, through merely a 3-hour lecture. But it is still amazing to see such a connection between the “geometric” hyperkahler metric and a “topological” enumeration.

Posted in Uncategorized | Tagged Gauge Theory, Hitchin, Hyperkahler, Symplectic Geometry | Leave a comment

Zinger’s sharp compactness theorem of genus-one pseudoholomorphic curves

Posted on October 10, 2011 by guangbox

I am reading Aleksey Zinger’s paper A sharp compactness theorem for genus-one pseudoholomorphic maps. Hopefully I will post a series of notes in the next few weeks, explaining this paper. Let’s start with the motivation.

The basic motivation is that: in the usual(or Naive) theory of pseudoholomorphic curves, the moduli space is not compact and we have Gromov’s compactness theorem. So let ${\mathcal M}_{g, n}(X, A; J)$ be the moduli space of $J$ -holomorphic maps of genus $g$ Riemann surfaces with $n$ marked points representing the homology class $A\in H_2(M)$ . The most well-known compactification $\overline{\mathcal M}_{g, n}(X, A; J)$ is the moduli of stable maps, introduced by Kontsevich. This moduli can be used to defined the so-called Gromov-Witten invariants of symplectic manifolds.

Let’s first clarify in what sense $\overline{\mathcal M}_{g, n}(A)$ is a compactification of ${\mathcal M}_{g, n}(A) .$

“Definition” 1. A sequence of $J$ -holomorphic curves converges in the sense of Gromov, if the underlying Riemann surface converges correspondingly in the Deligne-Mumford moduli space of stable curves(this gives us an identification of the smooth loci of the domains), and the maps correspondingly converges in $C^\infty$ -topology on each compact subset of the smooth locus of the limit domain.

We shall ask if every stable map really arises as the limit of a sequence of pseudoholomorphic maps with smooth domain. In genus-zero, it is correct. We can prove the corresponding gluing theorem with no constrains. However, even in genus-one, this is not the case. That is to say, the moduli of stable maps is not a “sharp” compactification of those maps with smooth domains, roughly in a $C^\infty$ -topology. Zinger’s paper explicitly gives the closure of ${\mathcal M}_{1, n}(X, A; J)$ in the moduli of stable maps, or, a sharp compactification $\overline{\mathcal M}_{1, n}^0(X, A; J)$ such that

${\mathcal M}_{1, n}(X, A; J) \subset \overline{\mathcal M}_{1, n}^0 (X,A;J) \subset \overline{\mathcal M}_{1, n}(X, A; J)$ .

The domain of a genus-one stable maps consists of two parts: the principal component $\Sigma_0$ , which is a torus or a circle of rational curves; and the bubble components, which are rational curves. The closure $\overline{\mathcal M}_{1, n}^0(X, A; J)$ consists of two types of maps:

1) Those whose restriction to $\Sigma_0$ are not constant;

2) If $u : \Sigma \to X$ represents an element in $\overline{\mathcal M}_{1, n}^0(X, A; J)$ and $u|_{\Sigma_0}$ is constant $p\in X$ , then denote $\Sigma_i, i=1, \ldots, k$ be the first bubbles in each branch which are not constant, and $D_i u$ the complex line in $T_p X$ given by the tangent map of $u$ at the node between $\Sigma_i$ and inner components. The constrain is that $D_i u, i=1, \ldots, k$ should be complex linearly dependent.

It seems that the only way to prove the sharpness of this compactification is to directly construct a gluing map, which is the most sophisticated part of the theory of pseudholomorphic curves. Moreover, this paper is quite notationally involved. Hopefully I can get the point.

The outline:

1. The set-up and balanced stable holomorphic spheres

2. A power series expansion appeared in gluing genus-zero maps

3. Closedness of $\overline{\mathcal M}_{1, n}^0(X, A; J)$

4. TBD

Maybe some pictures are to be added.

Posted in Uncategorized | Tagged Aleksey Zinger, compactness, Gang Tian, Gromov-Witten, Symplectic Geometry | Leave a comment

Morse inequality and Lagrangian intersection

Posted on October 1, 2011 by guangbox

This Thursday I gave the “kick-off” talk of this semester’s informal geometry seminar(Tian’s seminar). The word “kick-off” is what I learned yesterday because recently there have been so many “kick-off”.

This talk is a report of Frauenfelder’s thesis on “moment Floer homology”. Since some of the attendants were new-comers or outsiders of symplectic geometry, I added a recollection of Morse theory and Floer’s Lagrangian intersection homology, with the hope that I wouldn’t get people lost at the beginning. But even I spend one third time on this recollection, people still started to get confused at some point. And also I was wasting the lives of others who are so familiar with these stuff. It’s still hard for me to design a 2-hour talk and carry it out successfully. Any way, the notes of this talk has been put on my homepage

Click to access momentfloer.pdf

So now I would like to repeat the recollection part here, which shows a good motivation of Arnold conjecture which was not known to me before I started to prepare this talk.

Suppose $M$ is a compact oriented manifold, $f: M \to {\mathbb R}$ is called a Morse function if its differential $df$ , viewed as a section of the cotangent bundle $T^*M$ , is transverse to the zero section. Then using Morse theory one can study the topology of the manifold, by looking at the change of homotopy type when passing a critical point of $f$ . On the other hand, which is not quite straightforward, assuming one knows the topology of $M$ , we can estimate the number of critical points.

Note that, the set of critical points of $f$ is exactly the zero locus of the 1-form $df$ . We know from Poincare-Hopf theorem, the sum of the local indices of the zeroes of a 1-form gives the Euler characteristic of the bundle $T^*M$ . Hence the number of critical points is at least the absolute value of the Euler characteristic. This is a topological estimate.

However, even from the topological aspects of Morse theory, we know that we have a much stronger estimate, i.e., the Morse inequality: if we denote by

$M_t(f):= \sum_i t^i m_i(f)$

the Morse polynomial of $f$ , where $m_i(f)$ is the number of critical points of $f$ with Morse index $i$ , and denote by

$P_t(M):= \sum_i t^i \beta_i (M)$

the Poincare polynomial of $M$ , where $\beta_i(M)$ is the $i$ -th Betti number of $M$ (using ${\mathbb Z}_2$ -coefficients), then there exists a polynomial $R(t)$ with nonnegative integer coefficients such that

$M_t(f) - P_t(M) = (1+t) R(t).$

Then let $t=1$ , we have

$\# {\rm Crit} (f) \geq \sum_i \beta_i (M).$

Why we have this stronger estimate for $df$ ? We see that $T^*M$ is the standard model of a symplectic manifold, the phase space of classical mechanics, with the standard symplectic structure

$\omega= \sum_i dq_i \wedge dp_i.$

Here $p_1, \ldots, p_n$ is any local coordinates on $M$ and $q_1, \ldots, q_n$ is the bundle coordinates of $T^*M$ when using the local frame $dp_1, \ldots, dp_n.$ Then a 1-form $\alpha$ , viewed as a submanifold of $T^*M$ is a Lagrangian submanifold if and only if $\alpha$ is closed. Moreover, if $\alpha = df$ , then $\alpha$ is obtained from a deformation of the zero section. More precisely, the Hamiltonian vector field of $f$ is the vector field

$X_f := \sum_i {\partial f \over \partial p_i} {\partial \over \partial q_i } \in \Gamma (T T^*M)$

The time-1 map of the flow of the above vector field translates the zero section of $T^*M$ exactly to the submanifold defined by $df$ , which we denote by $\phi_f( L_0) = L_f$ . Now if the two submanifolds intersect transversely, then Morse inequality gives a lower bound of the number of points in $L_0 \cap L_f$ .

I think the Lagrangian property of closed 1-forms is why we have Novikov’s Morse theory for closed 1-forms, and the above should also be one of the motivations of the Arnold conjecture, although I never read either of the original literature.

Now this post is already rather long, and I leave the Floer homology part for the future. The series of attacks of Arnold conjecture by Floer started with the paper Morse theory for Lagrangian intersections in 1988. A story from Hofer said that when this paper appeared(or Floer’s work was announced), Gromov thought it would’t be correct since this is a variational problem for which the indices(like Morse index) are infinite from both sides.

Posted in Uncategorized | Tagged Arnold conjecture, Floer homology, Gang Tian, Helmut Hofer, Lagrangian intersection, Morse theory, Symplectic Geometry | Leave a comment

Carleman similarity principle

Posted on September 23, 2011 by guangbox

We know that analytic functions on the complex plane locally have power series expansion. This implies that its zero locus must be isolated and it cannot vanish at some point in infinite order, unless that this function vanishes identically.

This phenomenon is a special case of the so-called “unique continuation property” of certain elliptic differential operators. Analytic functions are solutions to the differential equation

$\overline\partial f := {1\over 2} ({\partial f\over \partial s} + i {\partial f\over \partial t} ) =0.$

We know from complex analysis, if two analytic function agrees at infinitely many points which have an accumulation point, then the two functions are equal. If they are not equal, then they must differ by some function in the order $O(|z|^k)$ for some finite $k$ .

This fact can be generalized to the case where we have different complex structures. Let’s still look at a smooth function $f: {\mathbb C}\to {\mathbb C}$ . The $\overline\partial$ is just the conjugate-linear part of the differential of $f$ , which is

$\overline\partial f := 1/2 ( df + J df j )$

where $J$ is the almost complex structure on the target ${\mathbb C}$ and $j$ is that on the domain.

We may replace the standard almost complex structures by nonconstant ones which still satisfies $J^2 =-id$ . Then the operator will be nonlinear. More generally, we may consider maps into ${\mathbb C}^n$ but the domain is still 1-dimensional. Then such a $\overline\partial$ -equation, or Cauchy-Riemann equation can be written as

${\partial f \over \partial s } + J {\partial f \over \partial t} + C(f) f =0$

where $C: {\mathbb C}^n \to {\mathbb C}^{n\times n}$ is a family of matrices, which we assume to be smooth, and $(s, t)$ is a local complex coordinate on the domain such that $j \partial_s = \partial_t$ .

Now the Carleman similarity principle says that solutions to such an equation are “similar” to holomorphic functions, in the sense that

If $f(0, 0)=0$ , then there exists a map $\Phi: D_\epsilon \to {\rm Hom}_{\mathbb R} ({\mathbb C}^n, {\mathbb R}^{2n} )$ where $D_\epsilon \subset {\mathbb C}$ is a small neighborhood of the origin in the complex plane, such that

$J_0 \Phi(z)= \Phi(z) J(f(z))$ and $\Phi(z) f(z)$ is holomorphic in the usual sense.

Since $\Phi(z)$ is linear for each $z\in D_\epsilon$ it immediately implies that $f$ cannot vanishes to infinite order at 0, and 0 must be an isolated zero of $f$ , unless it vanishes identically.

Applications

Since this principle is local, it applies to maps from subset of ${\mathbb C}$ into an almost complex manifold which satisfies the Cauchy-Riemann equation, i.e., $J$ -holomorphic curves. The implications includes a unique continuation theorem, and the existence of injective points. It also implies that if a $J$ -holomorphic curve intersects a subvariety infinitely many times locally, then it is mapped entirely into that subvariety. The last statement can be viewed in parallel with the fact that if the first $k$ coordinates of a holomorphic map into ${\mathbb C}^n$ vanishes in infinite order near the origin, then this component vanishes identically.

Proof of the similarity principle

Although the statement is local, we can prove it using a very “global” theorem, i.e., the Riemann-Roch theorem on the Riemann sphere for the trivial holomorphic vector bundle.

Basically, we may first find a $\Phi(z)$ which transforms $J$ to the standard one. Then after doing that the required $\Phi(z)$ will be a solution to a inhomogeneous Cauchy-Riemann equation on the rank-( $n\times n$ ) trivial bundle. The inhomogeneous term is determined by the given functions locally, and it can be extended arbitrarily onto the Riemann sphere. Hence if we require that $\Phi(0)= id$ then by Riemann-Roch we know that there exists a unique solution, which satisfies the required condition locally near the origin.

Posted in Uncategorized | Tagged Cauchy-Riemann operators, pseudoholomorphic curves, Riemann-Roch, Symplectic Geometry | Leave a comment

Hofer-Zehnder capacity and the existence of periodic orbits

Posted on March 14, 2011 by guangbox

The existence of solutions of certain variational problem always attracts attentions. Here in the case Hamiltonian dynamics, the existence of periodic orbits has some miraculous relation with some far-reaching invariants: symplectic capacities.

An abstract capacity should be a function(functor) $c$ from the category of all symplectic manifolds to $\mathbb R$ , which is denoted by $c(M, \omega)$ , such that if $(M, \omega)$ can be symplectically embedded into $(N, \tau)$ , then we should have $c(M, \omega)\leq c(N, \tau).$

The first capacity came from Gromov’s nonsqueezing theorem.

So let $(M, \omega)$ be a symplectic manifold, not necessarily compact. A Hamiltonian $H$ is called admissible, if it is compactly supported, has maximum 0, achieves its minimum on some open subset, and most importantly, has all nonconstant periodic orbit of $H$ has periods bigger than 1. The set of all admissible Hamiltonians is denoted by ${\mathcal H} (M, \omega)$ .

Definition The Hofer-Zehnder capacity of $(M, \omega)$ is defined to be the supreme of $-\min H$ for all admissible Hamiltonian $H\in {\mathcal H}(M, \omega)$ .

which could be $+\infty$ .

Theorem. Let $S\subset M$ be a closed hypersurface enclosing a open submanifold $\Omega$ , and the Hofer-Zehnder capacity $c(M, \omega)<\infty$ . Suppose we have a Hamiltonian $H$ defined near $S$ such that $S$ is an energy hypersurface and $H|_S$ is a regular value, so that a tubular neighborhood of $S$ looks like $S\times (-1 , 1)$ . Then the set of all $\epsilon\in (-1, 1)$ such that $S\times \{\epsilon\}$ has periodic orbits has measure 2.

Idea of Proof. Use the monotonicity of capacities. Let $\Omega_\epsilon$ be the open subset of $M$ enclosed by the hypersurface $S\times \{\epsilon\}$ . Then $c(\Omega_\epsilon, \omega)$ is nondecreasing in $\epsilon$ , hence differentiable almost everywhere. Then one can make estimates to show that at those differentiable $\epsilon$ ‘s, there exist periodic orbits on $\partial \Omega_\epsilon$ .

Posted in Uncategorized | Tagged Helmut Hofer, Periodic orbits, symplectic capacities, Symplectic Geometry | Leave a comment