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.