BPS States and Wall-Crossing

This is the first in what I hope will become a series of posts on BPS state counting and wall-crossing. I'm participating in gLab, and our most immediate goal is to understand the Kontsevich-Soibelman wall-crossing formula (KSWCF) in the context of quadratic differentials on a (punctured) Riemann surface, following the lectures of Kontsevich and Neitzke at IHES.

The purpose of these posts is to keep a written record of my attempts to understand the physics behind the WCF as well as the work of Gaiotto-Moore-Neitzke.

References:

• Witten's lectures on dynamics of QFT
• Gaiotto-Moore-Neitzke
• Kontsevich-Soibelman
• Seiberg and Witten
• Distler's blogpost, and
•  This post on the n-Category cafe (note: Bridgeland's talk relates quadratic differentials to stability conditions).

Video Lectures:

• IHES Lectures by Neitzke and Kontsevich
• Lectures by Moore on the (2,0) d=6 superconformal theory
• PITP 2010 (Gaiotto, Moore, Witten, Seiberg, others!)
• Neitzke: What is a BPS state?
• Gauge fields and strings

Physics Setup:

Warning: I'm still trying to sort this all out, so a lot of this will be fuzzy and/or completely wrong. I will try to point out the points of confusion.

We will start with some kind of family of susy gauge theories (or rather, a single  "theory" with a family of vacua, depending on what asymptotic boundary conditions we specify in the path integral). We let \(\mathcal{B}\) be some kind of manifold (or variety, possibly with singularities?), and \(\{\mathcal{H}_u\}_{u \in \mathcal{B}}\) a family (bundle) of Hilbert spaces, depending on \(u \in \mathcal{B}\). Concretely, \(\mathcal{B}\) will parametrize the vacuum expectation values (VEVs) of the scalar fields of the theory. (Note, for non-scalar fields we can typically expect VEVs to vanish, for example by looking at the action of the Lorentz group.) Actually, to be more precise, \(\mathcal{B}\) parametrizes the Coulomb branch--where the VEVs break the gauge symmetry to a maximal torus (as opposed to the Higgs branch, where the VEVs just break the gauge group to a smaller subgroup).

The next ingredient is a lattice \(\Gamma\), the charge lattice, which is supposed to parametrize all possible electric and magnetic charges. Since electric and magnetic charges are dual, this lattice has a pairing \(\Gamma \otimes \Gamma \to \mathbb{Z}\) which is symplectic (or possibly just Poisson?). (Actually, maybe we should think of \(\Gamma\) as being a bundle of lattices over \(\mathcal{B}\), but this isn't completely clear to me.) The lattice gives a grading of \(\mathcal{H}\):

\[ \mathcal{H} = \bigoplus_{\gamma \in \Gamma} \mathcal{H}_\gamma \].

Now, the Hilbert spaces \(\mathcal{H}_u\) are supposed to carry representations of the \(\mathcal{N}=2\) susy algebra, with central charge \(Z\). On any state of charge \(\gamma\) above the point \(u \in \mathcal{B}\), the central charge \(Z\) acts as a scalar, which we denote by \(Z_\gamma(u)\). Manipulations with the susy algebra show the BPS bound \(M \geq |Z_\gamma(u)|\), where \(M\) is the mass of a state with charge \(\gamma\). A state is called BPS if it saturates this bound.

Finally, I'll end this post by attempting to define (or at least motivate) the walls of marginal stability. In all known examples, we have

\[ |Z_{\gamma_1 + \gamma_2}(u)|^2 = |Z_{\gamma_1}(u)|^2 + |Z_{\gamma_2}(u)|^2 +2 \mathrm{Re}(Z_{\gamma_1}(u) \bar{Z}_{\gamma_2}(u) ) \]
If the cross-term is negative, then it is possible to form stable bound states (since the mass of a BPS state of charge \(\gamma_1+\gamma_2\) is strictly less than the sum of the corresponding masses); and it is impossible to form stable bound states if the cross-term is positive. This (naive!) dichotomy tells us that there is something very special about the intermediate case. For a pair of charges \(\gamma_1, \gamma_2\) we define a wall in \(\mathcal{B}\) by
\[ W(\gamma_1, \gamma_2) = \{u \in \mathcal{B} \ | \ \mathrm{Re}(Z_{\gamma_1}(u)\bar{Z}_{\gamma_2}(u)) = 0 \} \]
and we define \(W \subset \mathcal{B}\) to be the union of all the walls.

The idea of wall-crossing is the following. We define some functions \(\Omega(\gamma; u)\) on \(\mathcal{B} \setminus W\) which are locally constant. These functions are supposed to count the number of BPS states of charge \(\gamma\) (where count really means take the trace of a particular operator over \(\mathcal{H}_{\gamma, \mathrm{BPS}}\)). The wall-crossing formula is an explicit formula that relates \(\Omega(\gamma; u_+)\) and \(\Omega(\gamma; u_-)\) for \(u_+, u_-\) on opposite sides of a wall in \(\mathcal{B}\). There are two applications of WCF:

1. We pick some particular \(u \in \mathcal{B}\) for which \(\Omega\) is particularly easy to calculate ("extreme stability"). Then by KSWCF we actually know how to compute \(\Omega\) on all of \(\mathcal{B} \setminus W\).

2. Gaiotto-Moore-Neitzke study a certain QFT whose low energy effective action is a sigma model with target space \(\mathcal{M}\), the moduli space of Higgs bundles over a Riemann surface. The invariants \(\Omega(\gamma; u)\) together with KSWCF allow them to compute the low energy effective action explicitly, giving an explicit construction of holomorphic Darboux coordinates on \(\mathcal{M}\). This is enough to recover the full hyperkahler metric on \(\mathcal{M}\), in local coordinates!


Tasklist (incomplete!):

• Define susy algebra, derive BPS bound
• Understand/construct the charge lattice and its pairing
• Sketch that 3d sigma model with \(\mathcal{N}=4\) has a hyperkahler target
• Sketch/understand why the low energy effective action has target Higgs
• Understand computation of effective action: Seiberg-Witten curves and all that
• Understand how KSWCF implies consistency of the Darboux coordinates

Seiberg-Witten Theory Video Lectures

I found some lectures by Sara Pasquetti on Seiberg-Witten theory here:

• Lecture 1
• Lecture 2
• Lecture 3

Unfortunately, it seems that the quality is so poor that it is impossible to read the blackboard!

A Toy Model for Effective Field Theory and Extra Dimensions

I wanted to see how the Fourier transform can turn field theory into many-particle mechanics. This is just silly fooling around, so you shouldn't take what follows too seriously (there are much better models of extra dimensions, to be sure!).

Take \(\phi(t, s)\) to be a field on a cylinder of radius \(R\). We consider the action

\[ S = \frac{1}{R} \int_{-\infty}^\infty \int_0^{R} |\nabla \phi|^2 ds dt \]

Expand \(\phi(t, s)\) in Fourier series:
\[ \phi(t,s ) = \sum_n \phi_n(t) e^{2 \pi i n s / R} \]
Then in Lorentzian signature, we have
\[ \int_0^{R} |\nabla \phi|^2 d\theta = R \sum_n \dot{\phi}_n^2 - \left(\frac{2\pi n}{R}\right)^2 \phi_n^2. \]

Putting this back into the action, we find
\[ S = \sum_n \int_{-\infty}^\infty \dot{\phi}_n^2- \left(\frac{2\pi n}{R}\right)^2 \phi_n^2 dt. \]

This is the action for infinitely many harmonic oscillators, with frequencies \(\omega_n = 2\pi |n| / R\). Recall that the energy levels of the harmonic oscillator are \(k\omega\) for \(k = 0, 1, \ldots\). So supposing that only a finite energy \(E\) is accessible in some particular experiment, we can only excite those modes \(\phi_n\) for which
\[ \frac{2\pi |n|}{R} < E. \]
In particular, only finitely many \(\phi_n\) may be excited at energies below \(E\), effectively reducing the field theory on the cylinder to many-particle quantum mechanics.

Generating Functions

Method of Generating Functions

Let \(X\) and \(Y\) be two smooth manifolds, and let \(M = T^\ast X, N = T^\ast Y\) with corresponding symplectic forms \(\omega_M\) and \(\omega_N\).

Question: How can we produce symplectomorphisms \(\phi: M \to N\)?

The most important construction from classical mechanics is the method of generating functions. I will outline this method, shameless stolen from Ana Cannas da Silva's lecture notes.

Suppose we have a smooth function \(f \in C^\infty(X \times Y)\). Then its graph \(\Gamma\) is a submanifold of \(M \times N\): \( \Gamma = \{ (x,y, df_{x,y}) \in M \times N \}\). Since \(M \times N\) is a product, we have projections \(\pi_M, \pi_N\), and this allows us to write the graph as
\[ \Gamma = \{ (x, y, df_x, df_y) \}\]
Now there is a not-so-obvious trick: we consider the twisted graph \(\Gamma^\sigma\) given by
\[ \Gamma^\sigma =  \{(x,y, df_x, -df_y) \} \]
Note the minus sign.

Proposition If \(\Gamma^\sigma\) is the graph of a diffeomorphism \(\phi: M \to N\), then \(\phi\) is a symplectomorphism.

Proof By construction, \(\Gamma^\sigma\) is a Lagrangian submanifold of \(M \times N\) with respect to the twisted symplectic form \(\pi_M^\ast \omega_M - \pi_N^\ast \omega_N\). It is a standard fact that a diffeomorphism is a symplectomorphism iff its graph is Lagrangian with respect to the twisted symplectic form, so we're done.

Now we have:

Modified question: Given \(f \in C^\infty(M \times N)\), when is its graph the graph of a diffeomorphism \(\phi: M \to N\)?

Pick coordinates \(x\) on \(X\) and \(y\) on \(Y\), with corresponding momenta \(\xi\) and \(\eta\). Then if \(\phi(x,\xi) = (y,\eta)\), we obtain
\[ \xi = d_x f, \ \eta = -d_y f \]
Note the simlarity to Hamilton's equations. By the implicit function theorem, we can construct a (local) diffeomorphism \(\phi\) as long as \(f\) is sufficiently non-degenerate.

Different Types of Generating Functions

We now concentrate on the special case of \(M = T^\ast \mathbb{R} = \mathbb{R} \times \mathbb{R}^\ast\). Note that this is a cotangent bundle in two ways: \(T^\ast \mathbb{R} \cong T^\ast \mathbb{R}^\ast\). Hence we can construct local diffeomorphisms \(T^\ast \mathbb{R} \to T^\ast \mathbb{R}\) in four ways, by taking functions of the forms

\[ f(x_1, x_2), \ f(x_1, p_2), \ f(p_1, x_2), \ f(p_1, p_2) \]

Origins from the Action Principle, and Hamilton-Jacobi

Suppose that we have two actions

\[ S_1 = \int p_1 \dot{q}_1 - H_1 dt, \ S_2 = \int p_2 \dot{q}_2 - H_2 dt \]

which give rise to the same dynamics. Then the Lagrangians must differ by a total derivative, i.e.

\[ p_1 \dot{q}_1 - H_1 = p_2 \dot{q}_2 - H_2  + \frac{d f}{dt} \]

Suppose that \(f = -q_2 p_2 + g(q_1, p_2, t)\). Then we have

\[ p_1 \dot{q}_1 - H_1 = -q_2 \dot{p}_2 - H_2 + \frac{\partial g}{\partial t} + \frac{\partial g}{\partial q_1}\dot{q}_1 + \frac{\partial g}{\partial p_2} \dot{p_2} \]

Comparing coefficients, we find
\[ p_1 = \frac{\partial g}{\partial q_1}, \ q_2 = \frac{\partial g}{\partial p_2}, \ H_2 = H_1 + \frac{\partial g}{\partial t} \]

Now suppose that the coordinates \((q_2, p_2)\) are chosen so that Hamilton's equations become
\[ \dot{q_2} = 0, \ \dot{p}_2 = 0 \]
Then we must have \(H_2 = 0\), i.e.
\[ H_1 + \frac{\partial g}{\partial t} = 0 \]
Now we also have \(\partial H_2 / \partial p_2 = 0\), so this tells us that \(g\) is independent of \(p_2\), i.e. \(g = g(q_1, t)\). Since \(p_1 = \partial g / \partial q_1\), we obtain
\[ \frac{\partial g}{\partial t} + H_1(q_1, \frac{\partial g}{\partial q_1}) = 0 \]
This is the Hamilton-Jacobi equation, usually written as
\[ \frac{\partial S}{\partial t} + H(x, \frac{\partial S}{\partial x}) = 0 \]
Note the similarity to the Schrodinger equation! In fact, one can derive the Hamilton-Jacobi equation from the Schrodinger equation by taking a wavefunction of the form
\[ \psi(x,t) = A(x,t) \exp({\frac{i}{\hbar} S(x,t)}) \]
and expanding in powers of \(\hbar\). This also helps to motivate the path integral formulation of quantum theory.

KAM I

In this post I want to sketch the idea of KAM, following these lecture notes.

Integrable Systems

I don't want to worry too much about details, so for now we'll define an integrable system to be a Hamiltonian system \((M, \omega, H)\) for which we can choose local Darboux coordinates \((I, \phi)\) with \(I \in \mathbb{R}^N\) and \(\phi \in T^N\), such that the Hamiltonian is a function of \(I\) only. Defining \(\omega_j := \partial H / \partial I_j\), Hamilton's equations then read
\begin{align}
\dot{I}_j &= 0, \\\
\dot{\phi}_j &= \omega_j(I).
\end{align}
Hence we obtain linear motion on the torus as our dynamics. Note in particular that the sets \(\{I = \mathrm{const}\}\) are tori, and that the dynamics are constrained to these tori. We call these tori "invariant".


Now suppose that our Hamiltonian \(H\) is of the form
\[ H(I, \phi) = h(I) + f(I, \phi) \]
with \(f\) "small". What can be said of the dynamics? Specifically, do there exist invariant tori? KAM theory lets us formulate this question in a precise way, and gives an explicit quantitative answer (as long as \(f\) is nice enough, and small enough).

I want to sketch the idea of the KAM theorem, completely ignoring analytical details.

Constructing the Symplectomorphism

Suppose we could find a symplectomorphism \(\Phi\): (I, \phi) \mapsto (\tilde{I}, \tilde{\phi})\) such that \(H(I, \phi) = H(\tilde{I}\). Then our system would still be integrable (just in new action-angle coordinates), and we'd be done. There are two relatively easy ways of constructing symplectomorphisms: integrating symplectic vector fields, and generating functions. In the lecture notes, generating functions are used, so let's take a minute to discuss them.

Proposition Let \(\Sigma(\tilde{I}, \phi)\) be a smooth function and suppose that the transformation
\[ I = \frac{\partial \Sigma}{\partial \phi}, \
\tilde{\phi} = \frac{\partial \Sigma}{\partial \tilde{I}}\]
can be inverted to produce a diffeomorphism \(\Phi: (I, \phi) \mapsto (\tilde{I}, \tilde{\phi})\). Then \(\Phi\) is a symplectomorphism.

Proof
\[ dI = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d \tilde{I} \]
\[ d\tilde{\phi} = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d\phi \]
Hence
\[ dI \wedge d\phi = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d \tilde{I} \wedge d\phi = d\tilde{I} \wedge d\tilde{\phi}. \]

We want a symplectomorphism \(\Phi\) such that
\[ H \circ \Phi(\tilde{I}, \tilde{\phi}) = \tilde{h}(\tilde{I} \]
If \(\Phi\) came from a generating function \(\Sigma\), then we have
\[ H(\frac{\partial \Sigma}{\partial \phi}, \phi) = \tilde{h}(\tilde{I}) \]
Expanding things, we have
\[ h(\frac{\partial \Sigma}{\partial \phi}) + f(\frac{\partial \Sigma}{\partial \phi}, \phi) = \tilde{h}(\tilde{I}). \]

If \(f\) is small, then we might expect \(\Phi\) to be close to the identity, and hence \(\Sigma\) ought to be close to the generating function for the identity (which is \(\langle I, \phi \rangle\)). So we take
\[ \Sigma(\tilde{I}, \phi) = \langle \tilde{I}, \phi \rangle + S(\tilde{I}, \phi) \]
where \(S\) should be "small". So we linearize the equation in \(S\):
\[ \langle \omega(\tilde{I}), \frac{\partial S}{\partial \phi} \rangle
+ f(\tilde{I}, \phi)
= \tilde{h}(\tilde{I}) - h(\tilde{I}) \]

Now we can expand \(S\) and \(f\) in Fourier series and solve coefficient-wise. This gives a formal solution \(S(\tilde{I}, \phi)\) of the equation
\[ \langle \omega, \frac{\partial S}{\partial \phi} \rangle + f(\tilde{I}, \phi) = 0. \]

Getting it to Work

Unfortunately, the Fourier series for \(S\) has no chance to converge, so instead we take a finite truncation. If we assume \(f\) is analytic, its Fourier coefficients decay exponentially fast, so this provides a very good approximate solution to the linearized equation (and we can give an explicit bound in terms of a certain norm of \(f\)). Call this function \(S_1\). We then use \(S_1\) to construct a symplectomorphism \(\Phi_1\).

Now we take
\[ H_1(I, \phi) = H \circ \Phi_1(I, \phi) = h_1(I) + f_1(I, \phi). \]
Some hard analysis then shows that \(h - h_1\) is small, and \(f_1\) is much smaller than f.

The Induction Step

The above arguments sketch a method to put the system "closer" to an integrable form. By carefully controlling \(\epsilon\)'s and \(\delta\)'s, one then shows that iterated sequence \(\Phi_1, \Phi_2 \circ \Phi_1, \ldots\) converges to some limiting symplectomorphism \(\Phi_\infty\).

Circle Diffeomorphisms I

This is the first of a series of posts based on these lecture notes on KAM theory. For now I just want to outline section 2, which is a toy model of KAM thoery.

Circle Diffeomorphisms

We consider a map \(\phi: \mathbb{R} \to \mathbb{R}\) defined by
\[ \phi(x) = x + \rho + \eta(x) \]
where \(\rho\) is its rotation number and \(\eta(x)\) is "small".

Define \(S_\sigma\) to be the strip \(\{ |\mathrm{Im} z|<\sigma\} \subset \mathbb{C}\) and let \(B_\sigma\) be the space of holomorphic functions bounded on \(S_\sigma\) with sup norm \(\|\cdot\|_\sigma\).

Goal: Show that if \(\|\eta\|_\sigma\) is sufficiently small, then there exists some diffeomorphism \(H(x)\) such that
\[ H^{-1} \circ \phi \circ H (x) = x + \rho \]
i.e. that \(\phi\) is conjugate to a pure rotation.

Linearization

The idea is that if \(\eta\) is small, then \(H\) should be close to the identity, so we suppose that
\[ H(x) = x + h(x) \]
where \(h(x)\) is small. Plugging this into the equation above and discarding higher order terms yields
\[ h(x+\rho) - h(x) = \eta(x) \]
Since \(\eta\) is periodic, we Fourier transform both sides to obtain an explicit formula for the Fourier coefficients of \(h(x)\). We have to show several things:

1. The Fourier series defining \(h(x)\) converges in some appropriate sense.

2. The function \(H(x) = x + h(x)\) is a diffeomorphism.

3. The composition \(\tilde{\phi} = H^{-1} \circ \phi \circ H\) is closer to a pure rotation than \(\phi\), in the sense that
\[ \tilde{\phi}(x) = x + \rho + \tilde{\eta}(x) \]
where \(\|\tilde{\eta}\| \ll \|\eta\|\).

Newton's Method

Carrying out the analysis, one finds that for appropriate epsilons and deltas, if \(\eta \in B_\sigma\) then \(H \in B_{\sigma - \delta}\) and that \(\|\tilde{\eta}\|_{\sigma-\delta} \leq C \|\eta\|_\sigma^2\). By carefully choosing the deltas, we can iterate this procedure (composing the \(H\)'s) to obtain a well-defined limit \(H_\infty \in B_{\sigma/2}\) such that
\[ H_\infty^{-1} \circ \phi \circ H_\infty (x) = x + \rho, \]
as desired.

So in fact the idea of the proof is extremely simple, and all of the hard work is in proving some estimates.

Gaussian Integrals: Wick's Theorem

We saw in the last update that the generating function \(Z[J]\) can be expressed as
\[ Z[J] = e^{\frac{1}{2} J \cdot A^{-1} J} \]
(at least as long as we've normalize things so that \(Z[0] = 1\). Now the wonderful thing is that this is something we can compute explicitly:
\[ Z[J] = \sum_{n = 0}^{\infty} \frac{(\frac{1}{2} A^{-1}_{ij} J^i J^j)^n}{n!}
= \sum_{n=0}^\infty \frac{(A^{-1}_{ij} J^i J^j)^n}{2^n n!} \]

For example, in the one-dimensional case (taking \(A = 1\)) we get
\[ Z[J] = \sum_{n=0}^\infty \frac{J^{2n}}{2^n n!} \]
On the other hand, by the definition of the generating function we have
\[ Z[J] = \sum_{n=0}^\infty \frac{\langle x^n \rangle}{n!} J^n \]
Comparing coefficients, we find
\[ \frac{\langle x^{2n} \rangle}{(2n)!} = \frac{1}{2^n n!} \]
so that
\[ \langle x^{2n} \rangle = \frac{(2n)!}{2^n n!}. \]
Let's give a combinatorial description. Given \(2n\) objects, in how many ways can we divide them into pairs? If we care about the order in which we pick the pairs, then we have
\[ {2n \choose 2}{2n - 2 \choose 2} \cdots {2n-(2n-2) \choose 2} = \frac{(2n)!}{2^n} \]
Of course, there are \(n!\) ways of ordering the \(n\) pairs, so after dividing by this (to account for the overcounting) we get exactly the expression for \(\langle x^{2n} \rangle\). This is the first case of Wick's theorem.

Now consider the general multidimensional case. Given \(I = (i_1, \cdots, i_{2n})\), we define a contraction to be
\[ \langle x^{j_1} x^{k_1} \rangle \cdots \langle x^{j_n} x^{k_n} \rangle \]
where \(j_1, k_1, \cdots, j_n, k_n\) is a choice of parition of \(I\) into pairs.

Theorem (Wick's theorem, Isserlis' theorem) The expectation value
\[ \langle x^{i_1} \cdots x^{i_{2n}} \rangle \]
is the sum over all full contractions. There are \((2n)!/ 2^n n!\) terms in the sum.

Proof This follows from our formula for the power series of the generating function. The reason is that the coefficient of  \(J^I\) in \((\frac{1}{2} A^{-1}_{ij} J^i J^k)^n\) is exactly given by summing products of \(A^{-1}_{ij}\) over partitions of \(I\) into pairs, and the \(n!\) in the denominator takes care of the overcounting.

Next up: perturbation theory and Feynman diagrams.