Clifford Algebras and Spinors III: Bochner identity

Let $M$ be a Riemannian manifold, and let $Cl(M)$ be its Clifford bundle. Let $E \to M$ be any vector bundle with connection, and assume that $C^\infty(M, E)$ is a $Cl(M)$-module. We can define a Dirac operator $\mathcal{D}$ acting on sections of $E$ via the formula
\[ \mathcal{D} \sigma = \sum_{i=1}^n e_i \cdot \nabla_i \sigma \]
for any orthonormal frame $\{e_1, \dots, e_n\}$ on $M$, and where $\cdot$ denotes the Clifford module action. We demand that the connection on $E$ is compatible with Clifford multiplication in the following sense:
\[ \nabla_j (e_i \cdot \sigma) = (\nabla_j e_i) \cdot \sigma + e_i \cdot \nabla_j \sigma. \]

Let $R$ denote the curvature of $E$, i.e. we have
\[ [\nabla_i, \nabla_j] \sigma =  R(e_i, e_j) \sigma+ \nabla_{[e_i, e_j]} \sigma \]

We can define an endomorphism $\mathcal{R}$ on $E$ by
\[ \mathcal{R} = \frac{1}{2} \sum_{ij} R(e_i, e_j). \]

Theorem. We have the identity $\mathcal{D}^2 = -\Delta + \mathcal{R}$.

Proof. We compute
\begin{align}
\mathcal{D}^2 \sigma &= \sum_{ij} e_i \nabla_i \left( e_j \nabla_j \sigma \right) \\
&= \sum_{ij} e_i e_j \nabla_i \nabla_j \sigma + e_i ( \nabla_i e_j ) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j [\nabla_i, \nabla_j] \sigma + \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= -\Delta \sigma + \frac{1}{2}\sum_{ij}e_i e_j R(e_i, e_j) \sigma + \frac{1}{2}\sum_{ij} e_i e_j \nabla_{[e_i, e_j]} \sigma+ \sum_{ij} e_i ( \nabla_i e_j) \nabla_j \sigma \\
&= (-\Delta + \mathcal{R})\sigma + \frac{1}{2} \sum_{ij} \left( e_i e_j \nabla_{[e_i, e_j]}\sigma +  e_i (\nabla_i e_j) \nabla_j + e_j (\nabla_j e_i) \nabla_i \right)\sigma
\end{align}
We will be done provided we can show that the last term vanishes. Notice that it is fully tensorial, since it can be expressed as $\mathcal{D}^2 + \Delta - \mathcal{R}$. On the other hand, the terms $[e_i, e_j]$ and $\nabla_j e_i$ are (by definition!) proportional to Christoffel symbols. Since we can always choose a frame so that these vanish at a point, these terms must vanish identically. Hence we have $0 = \mathcal{D}^2 + \Delta - \mathcal{R}$, as desired.

Clifford Algebras and Spinors, Part II: Spin Structures and Dirac Operators

A very good reference for today's material is Dan Freed's (unpublished) notes on Dirac operators, available here.

Spin(n)

Consider the Clifford algebra \(Cl(\mathbb E^n)\) as constructed in yesterday's post. Define maps \(t, \beta: Cl(\mathbb E^n) \to Cl(\mathbb E^n)\) via
\[ (e_1 \cdots e_k)^t = e_k \cdots e_1, \ \beta(e_1 \dots e_k) = (-1)^k e_k \dots e_2 e_1 \]
 There is a natural inclusion \(\mathbb E^n \hookrightarrow Cl(\mathbb E^n)\). Given \(x \in Cl(\mathbb E^n)\) and \(v \in \mathbb E^n\), we can consider the product \(x v x^t\). In general, this might not be contained in \(\mathbb E^n \subset Cl(\mathbb E^n)\).

Definition. We define the group \(Pin(n)\) to consist of all those \(g \in Cl(\mathbb E^n)\) such that
\[ g \beta(g) = 1, \ \ g v \beta(g) \subset \mathbb E^n \ \forall\ v \in \mathbb E^n. \]
Similarly, we define the group \(Spin(n)\) to be the subgroup of \(Pin(n)\) such that \(gg^t = 1\).

Theorem. The natural action of \(Pin(n)\) on \(\mathbb E^n\) is by othogonal transformations, giving a natural map \(Pin(n) \to O(n)\). This map is a double cover. Similarly, \(Spin(n)\) is a double cover of \(SO(n)\). If \(n \geq 2\), \(Spin(n)\) is simply connected.

The importance of the spin groups is due to the following basic fact. Suppose that \(G\) is a Lie group with Lie algebra \(\mathfrak{g}\). Any representation of \(G\) induces a representation of \(\mathfrak{g}\). However,  given a representation of \(\mathfrak{g}\), it is not always possible to integrate it to a representation of \(G\). But it is always possible to integrate a representation of \(\mathfrak{g}\) to produce a representation of the universal cover of \(G\). For \(n \geq 2\), \(Spin(n)\) is the universal cover of \(SO(n)\).

Spin Structures

Let \((M^n, g)\) be a Riemannian manifold. Recall that the frame bundle \(O(M)\) is the manifold consisting of pairs \((x, \mathbb{e})\) where \(x \in M\) and \(\mathbb{e} = \{e_1, \dots, e_n\}\) is an orthonormal frame in \(T_x M\). Since the orthogonal group \(O(n)\) acts on the set of orthonormal frames, this makes \(F(M)\) into a principal \(O(n)\) bundle over \(M\). Let us assume that \(M\) is oriented, so that we may reduce its structure group to \(SO(n)\).

Suppose that \(V\) is a representation of \(SO(n)\). Then we may form the associated bundle \(SO(M) \times_{SO(n)} V\), which is a vector bundle over \(M\) with structure group \(SO(n)\). If we take the defining representation then we obtain the tangent bundle, but of course there are many others. Unfortunately, since \(SO(n)\) is not simply connected, not every representation of \(\mathfrak{so}_n\) can be integrated to a representation of \(SO(n)\). At the level of geometry, this means that in a certain sense there are certain vector bundles over \(M\) that are "missing"! Even more disturbing, is that these "missing" bundles appear to be necessary to describe many of the fundamental particles that appear in the standard model--so this has real world consequences. The solution is to equip \(M\) with a spin structure.

Definition. A spin structure on \(M\) is a principal \(Spin(n)\)-bundle \(Spin(M)\) over \(M\) together with a bundle morphism \(Spin(M) \to SO(M)\) which is a reduction of structure (i.e., satisfies the obvious axioms).

As you might expect, not every manifold admits a spin structure, and spin structures may not be unique. Loosely speaking, a spin structure is a slightly stronger notion of orientability. Spin structures may always be chosen locally, and the obstruction to consistent gluing is not too difficult to characters as a certain \(\mathbb Z_2\) cohomology class, called the second Stiefel-Whitney class.

Spin Connection and Dirac Operators

The reduction of structure \(Spin(M) \to SO(M)\) allows us to pull back the Levi-Civita connection on \(SO(M)\) to obtain a connection on \(Spin(M)\), called the spin connection. Let \(S_0\) be the spinor module described in the previous post. Then we may construct the associated bundle
\[ S = Spin(M) \times_{Spin(n)} S_0 \]
which is called the spinor bundle. Moreover, since \(S_0\) is a Clifford module, there is well-defined notion of Clifford multiplication on sections of \(S\). We may then define the Dirac operator \(\mathcal{D}\) by
\[ \mathcal{D} = \sum_{a=1}^n c(e_a) \nabla_{e_a} \]
where \(\{e_a\}\) is any orthonormal frame, \(\nabla\) is the spin connection, and \(c\) denotes Clifford multiplication.

Next time: the Weitzenböck formula, and maybe a vanishing theorem.

Clifford Algebras and Spinors

Clifford Algebras

Today I'd like to write some brief notes about Clifford algebras and spinors. A classic reference is the paper "Clifford Modules" by Atiyah-Bott-Shapiro. Clifford algebras not only useful in algebra and geometry, but are essential for the construction of theories with fermions. Let \(V\) be a vector space with a non-degenerate symmetric bilinear form \(B\). We define the Clifford algebra \(Cl(V, B)\) to be the unital associative algebra generated by \(v \in V\) subject to the relation
\[ vw + wv = -2B(v,w) \]
Equivalently, the definiting relation is \(v^2 = -B(v,v)\).

The Clifford algebra inherits a \(\mathbb Z\)-filtration as well as a \(\mathbb Z_2\)-grading from the tensor algebra. In fact, we have an analogue of the Poincare-Birkhoff-Witt theorem for Lie algebras:

Theorem The associated graded algebra of \(Cl(V,B)\) is naturally isomorphic to the exterior algebra on \(V\).

In this way, we may view the Clifford algebra as a quantization of the exterior algebra, much in the same way that \(U(\mathfrak g)\) is a quantization of the Poisson algebra of functions on \(\mathfrak g^\ast\) for a Lie algebra \(\mathfrak g\).

Example. Take (V,B) to be the Euclidean space \(\mathbb E^1\). Then we have a single generator \(e\) satisfying the relation \(e^2 =  -1\). Hence
\[ Cl(\mathbb R) \cong \mathbb R \cdot 1 \oplus \mathbb R \cdot e \cong \mathbb C \]
Where the isomorpism is given by \(e \mapsto i = \sqrt{-1}\).

Example. Take \(\mathbb E^2\). We have generators \(e_1, e_2\) both squaring to -1, and additionally we have \(e_1 e_2 = e_2 e_1\). We can define an isomorphism from \(Cl(\mathbb E^2)\) to the quaternions \(\mathbb H\) by \(e_1 \mapsto i, e_2 \mapsto j\).

Spinors

Now consider the complexified Clifford algebra, denoted \(\mathbb{C}l(V)\). Since we can now take square roots of negative numbers, the complex Clifford algebra is insensitive to the signature (as long as our bilinear form is non-degenerate). Denote by \(C_n\) the complex Clifford algebra \(Cl(\mathbb C^n)\),
where \(\mathbb C^n\) is equipped with the standard bilinear form \((x,y) = \sum_{i=1}^n x_i y_i\).

Definition. A subspace \(W \subset \mathbb C^n\) is isotropic if the restriction of the standard bilinear form to \(W\) is identically 0. A maximal isotropic subspace is an isotropic subspace that is not properly contained in any other isotropic subspace.

Theorem. Let \(W\) be a maximal isotropic subspace, and let\( \{w_1, \dots, w_k\}\) be a basis of \(W\). Let \(\omega = w_1 \cdots w_k \in C_n\), and let \(S = C_n \cdot \omega\). If n is even, then \(S\) is an irreducible Clifford module. If n is odd, then \(S=S^+ \oplus S^-\) is a direct sum irreducible Clifford modules, and \(S^+ \cong S^-\).

Irreducible Clifford modules are called spinor modules. This description of spinor modules allows one to prove straightforwardly the following complete classification of complex Clifford algebras.

Corollary. We have \(C_{2m} \cong \mathrm{End}(\mathbb C^m)\) and \(C_{2m+1} \cong \mathrm{End}(\mathbb C^m) \oplus \mathrm{End}(\mathbb C^m)\).

Note that this classification depends on n mod 2, which is closely related to Bott periodicity. There is a similar classification of real Clifford algebras.

Dirac Operators

Now we come to the real importance of Clifford algebras. Consider Euclidean space \(\mathbb{E}^n\) and let \(S\) be a spinor module for its Clifford algebra. We define the Dirac operator acting on \(S\)-valued functions as
\[ D f = \sum_{i=1}^n e_i \cdot \partial_i f \]
Now the amazing property of \(D\) is the following:
\[ D^2 = \sum_{i,j} e_i e_j \partial_i \partial_j = \sum_i e_i^2 \partial_i^2 + \sum_{i,j} e_i e_j [\partial_i, \partial_j] = -\Delta \]
hence the Dirac operator provides an algebraic (as opposed to pseudodifferential) square root of the Laplacian.

To Be Added in an Update...

Supersymmetric point particle, Dirac operators on spin manifolds, Weitzenböck formula, spinor reps of Lorentz algebra, N=1 susy.

Seiberg-Witten Theory and the Riemann-Hilbert Problem

References:

• Introduction to Seiberg-Witten theory and its stringy origin
• Seiberg-Witten and follow-up

The Classical Moduli Space of Vacua

For definiteness, we'll consider just the case of \(SU(2)\) considered by Seiberg and Witten. There is a scalar Higgs field \(\phi\). The classical vacua of the theory are given by the absolute minima of the potential energy, which in this case is proportional to
\[ \mathrm{Tr}[\phi, \phi^\dagger]^2 \]
Hence at the minimum, \([\phi,\phi^\dagger]=0\) and \(\phi\) is diagonalizable. Hence the classical moduli space of vacua \(\mathcal{M}_{cl}\) is just \(\mathbb{C}\), with complex coodinate \(a\), corresponding to the Higgs field
\[ \phi = \left( \begin{array}{rr}a & 0 \\ 0 & -a\end{array} \right) \]
Actually, due to gauge invariance, it is better to introduce another copy of the complex plane \(\mathcal{B}\) with local coordinate \(u = \frac{1}{2} Tr \phi^2 = a^2\). Then we can think of \(\mathcal{M}_{cl}\) as a branched cover of \(\mathcal{B}\), with \(a\) a (local) choice of square root of \(u\).

The goal is to understand the low energy effective theory. We introduce a cutoff \(\Lambda\) to define the quantum theory, and integrate out all degrees of freedom except for the low momentum modes of \(\phi\) (in particular, we integrate out the gauge field d.o.f.). The result is a \(\sigma\)-model with target \(\mathcal{M}_{cl}\). The kinetic term of the \(\sigma\)-model is governed by the metric on \(\mathcal{M}_{cl}\), hence the low energy effective action determines a metric on \(\mathcal{M}_{cl}\).

We'll see that 1-loop calculations introduce monodromy, so that in the quantum theory, "functions" on \(\mathcal{M}_{cl}\) are actually sections of non-trivial bundles over \(\mathcal{M}_{cl}\), and furthermore that the metric receives corrections from instantons (or BPS states). So what we really would like to understand/construct is the quantum moduli space of vacua \(\mathcal{M}\), which will be some non-trivial modification of \(\mathcal{M}_{cl}\). The key to the Seiberg-Witten solution is that susy allows us to reduce the problem to finding a specified set of holomorphic functions (in the \(u\) coordinate\) satsfying certain monodromies, and that once we know the monodromies the solution is given to us by the Riemann-Hilbert correspondence.

The Riemann-Hilbert Correspondence

Let \(X\) be \(\mathbb{P}^1\) with punctures at the points \(z_1, \ldots, z_n\). Let \(U\) be the universal cover of \(X\) and let \(G\) be the fundamental group of \(X\) (pick some basepoint away from the punctures). A set of monodromy matrices is exactly what is needed to specify a representation \(V\) of \(G\). Since \(U / G = X\), we can form the associated bundle \(E = U \times_G V\) over \(X\). The (trivial) \(G\)-connection on \(U \to X\) induces a flat connection \(\nabla\) on \(E\). This gives a map from representations of \(G\) to flat connections on \(X\).

Conversely, given a flat connection on \(X\), the monodromy about the punctures determines a representation of \(G\). Hence monodromy is a map from flat connections on \(X\) to representations of \(G\). The Riemann-Hilbert correspondence is that these two maps are bijections, modulo the natural notions of equivalence (conjugacy and gauge transformations).

Gross Overview of the Seiberg-Witten Approach

We are now ready to sketch the "big picture" idea of Seiberg and Witten, which applies not only to their \(N=2, d=4\) example but also to certain other compactifications of the \(N=1, d=6\) theory (in particular, the one considered by Gaiotto-Moore-Neitzke).

As discussed above, the theory will have a classical moduli space of vacua \(\mathcal{M}\), which turns out to be a complex manifold (or variety, and possibly with singularities). We'll let \(u\) be an abstract local complex coordinate on \(\mathcal{M}\). Supersymmetry then tells us that the main quantities we are interested in (to compute the low energy effective action) are holomorphic in \(u\) (away from the singularities/punctures of \(\mathcal{M}\)!). The general outline is as follows:

1. Identify functions \(f_i(u)\) which by susy are holomorphic in \(u\).
2. Compute the 1-loop corrections to \(f_i(u)\).
3. Compute monodromies of the corrected \(f_i(u)\).
4. Find the desired \(f_i(u)\) by solving the Riemann-Hilbert problem for these monodromies.

Now, to be more clear, it is a consequence of susy that the renormalized quantities \(f_i(u)\) are given schematically by

\[ f_{i, \mathrm{ren}}(u) = f_{i, \mathrm{cl}}(u) + f_{i,1}(\frac{u}{\Lambda}) + \sum_{k=0}^\infty c_{i,k} \left(\frac{\Lambda}{u} \right)^k \]

Here, \(f_{i,\mathrm{cl}}(u)\) is the classical function, \(f_{i,1}(u)\) is the one-loop correction, and the terms in the series are corrections coming from instantions (BPS states). Non-renormalization theorems due to susy guarantee that there are no higher loop corrections. One expects the instanton series to converge, and hence the monodromy is completely determined by the one-loop calculation. This is the key: by Riemann-Hilbert, the monodromy determines the \(f_i(u)\) uniquely--solving the Riemann-Hilbert problem is equivalent to computing the infinitely-many instantion corrections!

Now in general, solving the Riemann-Hilbert problem is difficult, so this reduction is of a theoretical but not necessarily practical nature. The second main idea of Seiberg and Witten is that we can solve this Riemann-Hilbert problem explicitly by introducing a family of curves \(\{C_u\}_{u\in\mathcal{B}}\), called Seiberg-Witten curves (or spectral curves).

Electric-Magnetic Duality

An absolutely key requirement of the Seiberg-Witten construction is electric-magnetic duality. Maxwell's equations in vacuum are
\[ dF = 0, \ \ \ d\ast F = 0. \]
Here \(F\) is a 2-form, and \(\ast F\) is its Hodge dual, a \((d-2)\)-form in \(d\)-dimensions. The first equation implies that \(F = dA\) for some 1-form \(A\), and we normally think of the second equation as the Euler-Lagrange equations for the action written in terms of \(A\). However, we could equally well take the starting point to be the second equation, taking \(\ast F = dB\), and take the first equation to be the Euler-Lagrange equations for \(B\). The problem with either of these approaches is that they allow particles of either electric or magnetic charge, but not both.

To put electric and magnetic charge on equal footing, we introduce fields \(F\) and \(F_D\) (a 2-form and a (d-2)-form\). Then the Lagrangian is (up to factors that I'm too lazy to care about)
\[ \mathcal{L} = \mathrm{Tr} F \wedge F_D \]
However, to recover Maxwell's equations, we need to impose \(\ast F_D = F\) as a constraint. So to get the right equations of motion, introduce an auxiliary field \(\lambda\) (a Lagrange multiplier), and modify the Lagrangian:
\[ \mathcal{L} = \mathrm{Tr} F \wedge F_D + \lambda(F - \ast F_D) \]
Variation with respect to \((F, F_D, \lambda)\) will reproduce Maxwell's equations exactly, but in this form the EM duality is manifest. Since EM duality exchanges electric and magnetic charges, we should consider how to modify the Lagrangian to couple the field to EM sources. Let \(J_e, J_m\) be the electric and magnetic currents, respectively. Up to conventions, Maxwell's equations read
\[ dF = J_m, \ \ \ d F_D = J_e. \]
Then we take the Lagrangian to be
\[ \mathcal{L} = \mathrm{Tr} F \wedge F_D + \lambda(F - \ast F_D) + F \wedge J_e + J_m \wedge F_D \]
to reproduce the right equations of motion.

In this form, we can consider particles with electric or magnetic charge (or both--dyons). If our gauge group has rank \(r\), then the lattice of electric charges is \(\mathbb{Z}^r\), while the lattice of magnetic charges is \((\mathbb{Z}^\ast)^r\). Hence the lattice of electromagnetic charges is
\[ \Gamma = \mathbb{Z}^r \oplus (\mathbb{Z}^\ast)^r \]
which comes with a natural symplectic pairing
\[\langle \cdot, \cdot \rangle: \Gamma \otimes \Gamma \to \mathbb{Z}.\]

(You might ask why we take the natural sympletic pairing as opposed to the natrual symmetric pairing. This is because there is actually a larger \(SL(2,\mathbb{Z})\) symmetry of the theory which preserves the symplectic pairing but not the symmetric pairing.)

Now there is an obvious source of symplectic lattices. Simply let \(C\) be a genus \(r\) compact Riemann surface. Then \(\Gamma = H_1(C, \mathbb{Z})\) is a symplectic lattice of rank \(2r\), where the symplectic pairing is now given by the intersection pairing. In fact, we can say more--if we take \(a\)- and \(b\)-cycles as generators, these form a Darboux (symplectic) basis of \(\Gamma\).

Back to the gauge theory problem. Recall that the 1-loop calculation and consideration of BPS states leads to a set of monodromy data on \(\mathcal{B}\). Suppose now that we could find a complex surface \(C \to \mathcal{B}\) whose fibers \(C_u\) are (possibly singular) genus \(r\) curves, and such that the monodromies of \(\Gamma_u := H_1(C_u, \mathbb{Z})\) agree with the given monodromies. Then we can solve the Riemann-Hilbert problem by doing geometry on this family, i.e. by finding holomorphic sections of certain associated bundles.

The SU(2) Seiberg-Witten Solution

We will now specialize to the case considered in the original paper of Seiberg and Witten. I will only construct the family--the details of the solution will follow in a subsequent post.

In this case, the group has rank \(r=1\), so we should be looking for a family of elliptic curves. In this case, the solution is almost obvious, given what I've said above. Seiberg and Witten argue that the moduli space \(\mathcal{B}\) must be \(\mathbb{C} \setminus \{\Lambda^2, -\Lambda^2\}\). The punctures at \(\pm \Lambda^2\) come from BPS states whose mass goes to zero at those values of \(u\). So the monodromy consists of three matrices, \(M_\infty, M_\pm\), the monodromies computed around \(\infty\) and \(\pm \Lambda^2\). These generate a certain modular subgroup \(G\) of \(SL(2, \mathbb{Z})\), allowing us to realize \(\mathcal{B}\) as the modular curve \(H / G\) (where \(H\) is the upper half-plane). Now, the space of elliptic curves is just \(H / SL(2, \mathbb{Z})\). So given any \(u \in \mathbb{B}\), we pick a lift \(\tilde{u}\) in \(H\) and let \(C_u\) be the corresponding elliptic curve. This is exactly the family needed to solve the Riemann-Hilbert problem!

Next time: details of this construction, including exact formulas, and some words about instanton counting.

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

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.

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.