WKB and Wall-Crossing a “Combinatorial” Picture of Moduli Spaces of Higgs Bundles

WKB and Wall-Crossing A “combinatorial” picture of moduli spaces of Higgs bundles

Andrew Neitzke, Harvard University (work in progress with Davide Gaiotto, Greg Moore)

Simons Center for Geometry and Physics, January 2009 Preface

Greg’s talk described the physical meaning of the Kontsevich-Soibelman WCF in terms of the interplay between four-dimensional and three-dimensional ﬁeld theories.

One begins with a four-dimensional ﬁeld theory with 8 supercharges, and reduces it on S1 of radius R.

The resulting three-dimensional theory is (on one branch) a sigma model into a hyperk¨ahler manifold M, with metric depending on R. Preface

To describe the geometry of M we introduced a family of Darboux coordinates × × Xγ : M × C → C

To prove the Kontsevich-Soibelman WCF the key property of Xγ is

I Jumps: The collection {Xγ} jumps by the symplectomorphism Ω(γ;u) Kγ along the ray `γ = {ζ : Z(γ; u)/ζ ∈ R−}.

To construct the metric on M one also needs

I Asymptotics: limζ→0 Xγ exp(−πRZ(γ; u)) exists. Preface

How should we think about the Xγ? We have a deﬁnition in terms of quantum ﬁeld theory, but how to make it mathematically intelligible?

The approach of Greg’s talk was to declare that the Xγ are deﬁned by these two desired properties.

Drawback: it’s not obvious that such functions really exist! To get them you have to solve a Riemann-Hilbert problem, which might have solution only at large R. But we think they should exist for all R. Preface

For quantum ﬁeld theories that come from branes in string theory the situation becomes more geometric and we can understand Xγ more intrinsically.

In this talk I describe one example where M is a moduli space of 1 Higgs bundles on CP with singularities. Outline

Brane constructions

The moduli space of solutions to Hitchin’s equations

The construction of Xγ

The properties of Xγ

Some examples Brane construction: four-dimensional physics

Let Q be a four-dimensional hyperk¨ahler space. (Nothing to do with M). Consider M-theory in the 11-dimensional spacetime 3 1,3 R × Q × R .

Fix one complex structure on Q. Then Q is complex symplectic. Say C ⊂ Q complex Lagrangian submanifold. Q locally looks like T ∗C.

1,3 1,3 Put N M5 branes on C × R . This gives a ﬁeld theory in R with 8 supercharges. Brane construction: four-dimensional physics

We are interested in the Coulomb branch B of our ﬁeld theory.

B is the moduli space of supersymmetric conﬁgurations where the branes are separated from one another, i.e. complex manifolds Σ ⊂ T ∗C where the projection Σ → C is generically N-fold cover. BPS states BPS states in our ﬁeld theory: M2 branes wrapping surfaces S ⊂ Q, holomorphic in some complex structure Jζ on Q, with boundary on Σ. [Mikhailov, Nekrasov, Sethi]

Requiring that S minimizes area A in its homology class, one shows ζ = eiθ and Z I I A = eiθΩ = eiθλ = |λ| S ∂S ∂S for Ω = dλ the standard complex symplectic form on T ∗C. (So in particular eiθλ = |λ| along ∂S. This gives an ODE for the curve ∂S.) BPS states

So the topological charge is

γ = [∂S] ∈ H1(Σ, Z)

with corresponding central charge I Z(γ) = λ γ The charges γ vary in a local system Γ over B. Brane construction: four-dimensional physics

We’ve introduced the data that come from four-dimensional physics.

Next, we should consider the dimensional reduction of the whole story on S1. Brane construction: three-dimensional physics

3 1,2 1 1 So now take spacetime to be R × Q × R × S with S of radius 1,2 1 R. Put N M5 branes on pt. × C × R × S .

1,2 We get a ﬁeld theory in R . We want to describe its moduli space M.

First consider the reduction of the ﬁvebrane theory on S1. This 1,2 gives a ﬁve-dimensional supersymmetric gauge theory on C × R .

1,2 Now look at conﬁgurations that are constant along R . Taking time-independent conﬁgurations gives SDYM equation F = ?F ; further reducing along two space dimensions gives Hitchin’s equations on C. [Hitchin] Hitchin’s equations

Let V be a rank N complex C ∞ vector bundle over C, with U(N)-connection D, and ϕ ∈ Ω1,0(End V ).

Hitchin’s equations:

∂¯D ϕ = 0 2 ∗ R [ϕ, ϕ ] = FD

M is the space of solutions, modulo gauge equivalence D → gDg −1, ϕ → gϕg −1.

Supersymmetry ensures M is hyperk¨ahler. Higgs bundles

How to understand M concretely as a complex manifold? Start with the special complex structure ζ = 0.

Mζ=0 is moduli space of Higgs bundles: pairs (E, ϕ) where

I E is a holomorphic rank N vector bundle over C 0 I ϕ ∈ H (End E ⊗ K) The Hitchin ﬁbration

On general grounds we expected that Mζ=0 is a complex torus ﬁbration over B. Recall B parameterizes N-fold covers (Σ ⊂ T ∗C) → C. The link is provided by the spectral curve: given a Higgs bundle (E, ϕ) deﬁne

Σ = {(z, x) : det(x − ϕ(z)) = 0} ⊂ T ∗C

This is an N-fold cover of C, given by the eigenvalues of ϕ. Σ depends only on charpoly ϕ, so the projection is

N M (E, ϕ) 7→ charpoly ϕ ∈ H0(C, K ⊗m) m=1 Flat connections

× What does Mζ look like for ζ ∈ C ?

Consider the GL(V )-valued connection:

−1 ∇z = Rζ ϕz + Dz

∇z¯ = Rζϕ¯z¯ + Dz¯

Hitchin’s equations imply ∇ is flat. This identifies Mζ with a moduli space of flat connections on V .

So holomorphic functions on Mζ are gauge-invariant quantities built out of ∇. (Our Xγ will be obtained this way.) Where are we?

We’ve introduced almost all of the structures that entered Greg’s talk:

I A complex manifold B,

I A local system Γ of lattices over (an open subset of) B, I A homomorphism Z :Γ → C varying holomorphically over B, I A collection of integers Ω(γ),

I A hyperk¨ahler torus ﬁbration M → B.

We’re almost ready to identify the holomorphic functions

× × Xγ : M × C → C

But ﬁrst we need a slight extension... Allowing simple poles

If the M5 branes intersect with some other M5 branes, the equations are modified to include singularities. So mark finitely many punctures zi on C. Consider (D, ϕ) which, near each singularity, are gauge equivalent to α dz dz¯ D = ∂ + − + ··· 2 z z¯ β dz ϕ = + ··· 2 z for some diagonal matrices α (real), β (complex). M is space of solutions to Hitchin’s equations with these boundary conditions, modulo appropriate gauge equivalence. M depends on the choices of α, β at each puncture (held fixed). A harmless restriction

One can always decouple the “center of mass” degrees of freedom to reduce from GL(V )-valued connections to SL(V )-valued ones. We’ll do that. An unfortunate restriction

From now on, we further specialize to the case where V has rank 2. We’d like to do better, but so far we can’t.

So we are considering SL(2)-valued connections, with simple poles. The monodromy eigenvalues at each pole are determined by the ﬁxed residues of ϕ and A:

µ = e±2πiν

with 1 1 ν = Rζ−1m + m − Rζm¯ 2 3 2 Cluster coordinates

Luckily, coordinates very close to the ones we need have already been studied. [Fock, Goncharov]

Choose one of the two monodromy eigenvalues at each singularity, call it µi . Let si be a ∇-ﬂat section with monodromy µi (unique up to scalar multiple).

Now suppose given any triangulation T of C, such that the set of vertices is precisely the set of singularities.

T We are going to construct a number XE for each edge E of T . Cluster coordinates

Use the ∇-parallel transport to bring all si to a common point in the interior of this quadrilateral, then

T (s1 ∧ s2)(s3 ∧ s4) XE := − (s2 ∧ s3)(s4 ∧ s1)

This is independent of the ambiguous normalization of si . Cluster coordinates

Suppose there are n singularities. Simple counting shows the number of edges is #E = n + dim M T The product of XE over edges E meeting a single singularity is the square of the monodromy µ2, held ﬁxed in the deﬁnition of M. This accounts for the extra n.

T So there are just enough XE to give (at least locally) a coordinate system in M.

We get one such coordinate system for every triangulation. Poisson structure in cluster coordinates

Write ij for the (oriented) number of faces having both Ei and Ej as edges. Deﬁne Poisson bracket

{X T , X T } = X T X T Ei Ej ij Ei Ej It is degenerate: the kernel comes from the monodromies. Dividing out by the kernel gives a holomorphic symplectic structure on Mζ . Short computation shows it agrees with the “Atiyah-Bott” symplectic structure descending from Z ω(δ1∇, δ2∇) = δ1∇ ∧ δ2∇ C

T So XE give Darboux coordinates. Mutations of cluster coordinates Any two triangulations are related by a sequence of ﬂips:

0 X T = X T (1 + X T ) E12 E12 E 0 0 T T T −1 −1 T T −1 XE = XE (1 + (XE ) ) XE = (XE ) 23 23 0 X T = X T (1 + X T ) E34 E34 E 0 X T = X T (1 + (X T )−1)−1 E41 E41 E Very close to the symplectomorphisms Kγ appearing in the Kontsevich-Soibelman WCF (stay tuned). Deﬁning our Xγ

T So far we described coordinates XE which depend on a choice of triangulation T and on a choice of monodromy eigenspace at each singularity, and are labeled by the edges E.

We wanted coordinates Xγ(ζ) which don’t depend on any choices and are labeled by γ ∈ H1(Σ, Z).

So we need to choose a canonical triangulation T (ζ), a canonical choice of monodromy eigenspaces, and also give a rule E 7→ γE . WKB ideology

Some motivation: In the limit ζ → 0, our connection is dominated by the term −1 Rζ ϕz . The WKB approximation says roughly that if we diagonalize ϕ

λ 0 ϕ = 0 −λ

then the ﬂat sections are of the form

− R R λ! ! (1) e ζ (2) 0 ψ ∼ , ψ ∼ R R λ . 0 e ζ

So as ζ → 0 computing parallel transport formally reduces to integrating λ. We want to engineer a situation where this formal expectation is realized. The WKB foliation

Start by constructing a foliation on C:

Consider the 1-form on Σ

λ = x dz

Define a WKB curve with phase ζ to be a curve on Σ whose tangents v have λ(v) ∈ Rζ These curves define an oriented foliation on Σ, which descends to an unoriented one on C. Local behavior of the WKB foliation The WKB foliation on C becomes singular at m I poles of λ (we fixed these in our definition of M: λ ∼ z )

I zeroes of ω (generically, simple zeroes) Global behavior of the WKB foliation

For generic ζ and ω, the global behavior is rather simple: [Strebel]

I A generic leaf is asymptotic to poles of ω in both directions.

I There are ﬁnitely many special leaves which are asymptotic to zeroes of ω in one direction.

The special leaves partition C up into cells of two types. The canonical triangulation

Our canonical triangulation T (ζ; u) is “dual” to this cell decomposition.

I Vertices are the poles of ω.

I Edges are the generic WKB curves (choose one from each homotopy class).

I Each face contains exactly one zero of ω. The faces are triangles (possibly degenerate). The canonical coordinates Now deﬁne coordinates on M, using the canonical triangulation. T To each edge E, associate γE ∈ H1(Σ, Z) by lifting loop from C to its double cover Σ:

We’ll deﬁne T XγE (ζ) := XE Extend to general γ by multiplicativity,

Xγ+γ0 = XγXγ0 The canonical ﬂat sections

We still have to say which of the two monodromy eigenspaces at each pole we will use.

In the limit ζ → 0, the two behave very diﬀerently: one ﬂat section is exponentially growing as we approach the pole along WKB lines, the other is exponentially decaying. We choose the exponentially decaying one.

Now we are ﬁnished deﬁning Xγ. It remains to see they have the desired properties. Jumps

Ω(γ) We claimed that the Xγ jump by the symplectomorphism Kγ along the ray `γ = {ζ : Z(γ) ∈ R−ζ}. Does it work?

Ω(γ) is counting holomorphic curves S ending on Σ, with R [∂S] = γ, and ﬁnite area given by | ∂S λ|. Moreover, ∂S is a WKB curve, with phase equal to the phase of Z(γ; u).

So when ζ lies on `γ with Ω(γ) 6= 0, there is a ﬁnite WKB curve with phase ζ.

When ζ crosses `γ, the WKB triangulation jumps. This will give the expected jump for Xγ. Jumps from holomorphic discs

Simplest example: a disc ending on Σ. The boundary ∂S projects to a saddle connection: WKB curve connecting two zeroes of λ.

This leads to a ﬂip of the WKB triangulation from T − to T + as ζ crosses the ray `γ.

T − T + Using the relation between XE and XE , ﬁnd that Xγ indeed jump by the symplectomorphism Kγ: just as expected if Ω(γ) = 1. And Ω(γ) = 1 is what we expect for an isolated holomorphic disc. Jumps from holomorphic annuli

Fancier example: an annulus ending on Σ. The boundary ∂S projects to a closed WKB curve in C. Such annuli always occur in 1-parameter families.

There is an intricate transformation of the WKB triangulation as ζ crosses `γ. After some work, ﬁnd that Xγ jump by the −2 symplectomorphism Kγ : as expected if Ω(γ) = −2. Interpret this as the contribution to Ω(γ) from the 1-parameter family of annuli. No unexpected jumps

... Asymptotics

To calculate ζ → 0 asymptotics of

(s1 ∧ s2)(s3 ∧ s4) Xγ = − (s2 ∧ s3)(s4 ∧ s1)

Idea: Each si is determined near zi by our monodromy condition. Evolve them both along the edge Eij to a common intermediate point, using the WKB approximation. This gives

R R z λ! 0 ! e ζ zi si (z) ∼ , sj (z) ∼ − R R z λ 0 e ζ zj

Combining these integrals along the four edges,

R H λ Xγ ∼ e ζ γ Asymptotics

The exact equation ∇s = 0 is not diagonal in the basis of eigenvectors for ϕ. The WKB computation amounts to pretending it is diagonal. How can this work?

Ignoring the oﬀ-diagonal pieces of ∇ gives approximate ﬂat sections: one exponentially growing as we evolve away from the singularity, one exponentially decaying. Because Eij is a WKB curve, they never exchange dominance.

We chose si to be exponentially growing. So we can neglect the oﬀ-diagonal couplings: they just introduce a mixing with the exponentially decaying section, which gives a vanishing contribution as ζ → 0. A tricky sign

... Finite examples

The combinatorially simplest examples are, strictly speaking, an extension of the formalism we developed so far.

Physically they correspond to N = 2 theories near a conformal point, where Σ develops a singularity:

Σ = {x2 = zn}

and deformations thereof.

To get this as the spectral curve, take M as moduli space of 1 solutions to Hitchin equations on CP with higher order singularity at ∞ (and no singularities anywhere else). Finite examples

So take Tr ϕ2 = P(z)dz⊗2 for polynomial P(z) of degree n + 1. B parameterizes deformations n of the bottom b 2 c coeﬃcients in P(z).

All WKB curves asymptotically approach the n + 3 rays n+3 z ∈ R+ζ as z → ∞. The A1 case

Choosing P(z) = z2 + i and varying ζ from 0 to π:

Animation

π A single BPS state appears at ζ = 2 . The A2 case

Again sweep ζ from 0 to π. P(z) = z3 + i P(z) = z3 + z

Animation Animation

3 BPS states. Total 2 BPS states. Total transformation of the Xγ is transformation of the Xγ is K0,1K1,1K1,0. K1,0K0,1. The A2 case

Animation

We come back to where we started, so

K0,1K1,1K1,0 = K1,0K0,1 The SU(2) Seiberg-Witten theory

2 dz ⊗2 1 Tr ϕ = z z + u + z u = 0 u = 2 + i

Animation Animation

(K0,1K2,1K4,1 ··· ) × −2 K2,−1K0,1 K2,0

× (···K6,−1K4,−1K2,−1) Finis

...