Andrew Neitzke

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This paper is a companion to this one in which we introduced the notion of spectral network. In that paper we showed in particular that spectral networks have associated coordinate systems on moduli spaces of flat connections on punctured surfaces C. In this paper we explain how spectral networks reproduce some special coordinate systems studied by Fock-Goncharov here. Those coordinate systems correspond to ideal triangulations of C, so part of our construction is an explanation of how to build a spectral network corresponding to each triangulation. There are many other spectral networks which do not correspond to any triangulation; in this sense spectral networks are a natural generalization of ideal triangulations.

This paper extends most of the main results of our previous work on theories of class S from the case G = A_1 to the full A series. In particular, this includes the description of the 4d and 2d-4d BPS states, the direct proof that they obey wall-crossing formulas, and the corresponding descriptions of the Darboux coordinates and hyperkahler metric on the moduli space of the S^1-compactified theory.

In the A_1 case, one of the key ingredients was a triangulation of the Riemann surface C canonically determined by a quadratic differential -- all of the main phenomena could be understood by thinking about how this triangulation varies as the quadratic differential is varied. The main obstacle to extending our results to the full A series was the lack of a suitable replacement for this triangulation. In this paper we show that the correct replacement for the triangulation is a new geometric object which we call a "spectral network".

A spectral network is a collection of paths on the Riemann surface C, obeying some simple combinatorial rules. Given a point u of the Hitchin base for G = A_N -- i.e. u is a tuple consisting of a quadratic differential, cubic differential, ..., (N+1)-differential -- we construct a canonical spectral network W(u). By studying the jumping of W(u) as u varies, we obtain 4d BPS invariants which "count" certain networks of strings on C. These BPS invariants generalize the counts of saddle connections and closed trajectories which appeared in the A_1 case.

One of the interesting uses of spectral networks is as a machine for constructing new coordinate systems on the moduli space of the dimensionally reduced theory, also known as the moduli space of flat complex G-connections over C. In the A_1 case these coordinates were studied by Fock-Goncharov. In the A_N case the coordinate systems we obtain from spectral networks include some explicitly described by Fock-Goncharov, but also include others. We conjecture that the set of coordinate systems coming from spectral networks coincides with the set of coordinate systems in the "cluster atlas" on the moduli space. If this conjecture is true it might be useful: if I understand correctly, one of the big obstacles in studying that cluster atlas is that one does not have an explicit geometric picture of what a generic coordinate system in the atlas looks like; spectral networks could be the right tool to fill that gap.

In addition to their role in the cluster story, the coordinates which come from the spectral networks W(u) have strong asymptotic properties. It appears that they are a good tool for studying the WKB approximation for higher-rank matrix differential equations -- at least for families of connections which arise from solutions of Hitchin equations. The paths making up W(u) then have an interpretation as Stokes curves for such a higher-rank system. We explore this aspect of the story only relatively briefly in this paper, but ultimately I think it will be one of the most significant points.

Physically speaking, the spectral network W(u) is most naturally understood not in terms of the pure 4d theory but in terms of a coupled 2d-4d system: indeed the curves making up W(u) are themselves walls of marginal stability, where framed 2d-4d BPS bound states form or decay.

The picture of the BPS states developed in this paper looks very natural from the point of view of M-theory -- the BPS states correspond to certain networks of strings on C, which in M-theory would naturally come from the boundaries where M2-branes end on M5-branes. I think it is worth pointing out that we derive these results starting only from relatively simple assumptions about the 4d theory and its surface defects. (These assumptions are perhaps not yet fully proven, but one can certainly imagine proving them entirely within gauge theory.) In a sense, then, what we find is that using surface defects as probes actually allows us to discover "stringy" features starting from purely gauge theory considerations!

This paper concerns an extension of the construction of hyperkahler metrics described in my earlier paper with Gaiotto and Moore, here. It appears that the hyperkahler spaces M in question come with some extra structure, namely a canonical hyperholomorphic line bundle. This bundle is nontrivial on the torus fibers of M; it should be thought of as a hyperholomorphic structure on a "theta line bundle." In this paper I explain how to construct this line bundle; however, I do not completely prove that the construction works, as I have to assume some dilogarithm identities which I really should have proven. A very similar line bundle has been discussed by Pioline et al here (indeed we originally discovered it together, although our takes on it are a bit different) and also independently by Haydys here. More recently Hitchin has studied this bundle here, with particular emphasis on its twistorial description.

(Out of an abundance of caution, I wrote above that my bundle is "very similar" to those appearing in these other papers, but presumably it is the same bundle, at least in cases where the various constructions overlap. The constructions do seem to have slightly different ranges of applicability; for example, Haydys and Hitchin use an S^1 action on the hyperkahler manifold, while my construction applies in some cases which do not have such an action; on the other hand my construction uses a lot of other structure which Haydys and Hitchin do not need, so presumably they cover some cases which I do not.)

The physical meaning of this bundle is not completely clear at the moment. In the paper I propose that it is related (in a precise way) to the IR physics of Taub-NUT centers in a four-dimensional N=2 theory compactified on an S^1 bundle.

This paper extends most aspects of the wall-crossing of BPS degeneracies in 4d N=2 theories to 4d N=2 theories coupled to 2d 1/2-BPS surface defects. Wall-crossing in N=(2,2) theories in 2d had been studied long ago by Cecotti and Vafa. Not surprisingly, what we find is that the wall-crossing in coupled 2d-4d systems is a sort of hybrid between the story of pure 4d N=2 theories and the Cecotti-Vafa 2d story.

Mathematically speaking, this means we study a specific extension of the Kontsevich-Soibelman formula. We expect that this extended "2d-4d wall-crossing formula" governs an extension of the generalized Donaldson-Thomas invariants. Very roughly speaking, this extension should be thought of as an "open" version of DT theory, in the same sense as open Gromov-Witten invariants extend ordinary Gromov-Witten theory.

Upon compactification from 4 to 3 dimensions and flowing to the IR, the coupled 2d-4d system becomes a coupled 1d-3d system, i.e. a 3d theory with a line defect. The 3d theory is a sigma model into a hyperkahler space M, and this gives a new method of constructing hyperkahler metrics (as we described in previous work here). The 1d defect turns out to be described by a hyperholomorphic vector bundle over M. Our construction thus gives a new way of constructing hyperholomorphic vector bundles.

This is a somewhat exploratory paper.

The original idea was as follows: since we know that a structure very similar to tt* geometry appears in 4d N=2 gauge theories, we might expect that many of the interesting consequences of tt* geometry will also have analogues in 4d. One of the most interesting facts derived from tt* was a link between the "BPS monodromy" (a certain matrix built from the spectrum of BPS states) and the spectrum of R-charges at a conformal point. So, is there a similar story for 4d gauge theories?

In 4d theories there is indeed a natural BPS monodromy, built from the 4d BPS states, valued in an infinite-dimensional group of symplectomorphisms. We give a physical interpretation of this BPS monodromy and investigate its properties.

Here is one surprise which emerges: at least in some examples, computing the "trace" of this monodromy (appropriately defined) leads to a modular form! (For example, the simplest case is the Argyres-Douglas CFT with Seiberg-Witten curve y^2 = x^3 + u. In this theory the modular form that arises is the Rogers-Ramanujan function.)

An interesting offshoot of this work was the identification of a new class of 4d SCFT, indexed by pairs (G,G') of ADE Lie algebras. We conjecture that the combinatorics of BPS states in these SCFTs are governed (in a precise way) by a much-studied set of cluster algebras, which also have links to a known class of integrable 2d field theories. From our construction we would expect that the BPS monodromy M obeys M^n = 1 for a specific n. This turns out to be exactly the "periodicity conjecture" which originally arose in the 2d context, and was very recently proven in full generality; so this is a kind of consistency check of our story.

This paper is a continuation of our work on wall-crossing in 4d N=2 gauge theories. In our previous work on the subject, we deduced the wall-crossing formula from a close study of analytic functions on the twistor space of the moduli space of the theory compactified on S^1. In this paper we argue that such analytic functions can be obtained as the vacuum expectation values of supersymmetric line operators wrapped around S^1.

In fact, these supersymmetric line operators turn out to be interesting objects of study in their own right. In particular the wall-crossing formula for BPS states in 4d can be proven directly using line operators, without the need for compactification on S^1.

The algebra generated by line operator vevs is a natural "big" subalgebra of the algebra of regular functions on the twistor space, conjecturally equal to the whole algebra of regular functions. The line operator vevs themselves form a particularly interesting basis for this algebra. When applied to the "theories of class S" coming from the six-dimensional (2,0) theory, this part of the story is essentially identical to considerations of Fock-Goncharov.

It turns out that not only the ordinary wall-crossing formula but also its "refined" or "motivic" generalization (keeping track of the spins of the BPS states) can be understood using line operators. The refined version arises naturally when we consider a certain noncommutative deformation of the algebra generated by the line operator vevs.

The "framed BPS states" of the title are BPS states in the Hilbert space of the 4d theory with a line operator inserted. The invariants counting these framed BPS states arise as the coefficients of the expansion of the line operator in terms of "IR line operators."

This paper is a kind of "case study" for the general story spelled out in "Four-dimensional wall-crossing via three-dimensional field theory" below.

We consider a particular class of field theories, obtained by compactification of the mysterious "(2,0) SCFT" on a punctured Riemann surface C. These theories have attracted some attention, mostly because of the starring role they play in Gaiotto's paper here and in the Alday-Gaiotto-Tachikawa conjecture. This paper actually contains the original discussion of these theories (heavily influenced by Witten's earlier paper here.) We named them "theories of class S"; they have also been called "Sicilian theories" or "Gaiotto theories" elsewhere. In this paper we pay particular attention to the simplest such theories, namely the ones that come from the (2,0) theory of type A_1.

In these theories the general wall-crossing story becomes much more concrete. The BPS states are realized as certain finite-length geodesics, in a flat (singular) metric on C. As we move around the Coulomb branch, these finite-length geodesics can appear and disappear: this is the wall-crossing phenomenon. This picture of the BPS states had appeared previously, in work of Klemm-Lerche-Mayr-Vafa-Warner.

After compactification on S^1, one gets a sigma model into a hyperkahler manifold M, which in this case is a moduli space of Higgs bundles (with tame ramification) over C. Applying our general construction in this case we obtain a new recipe for the hyperkahler metric on M. Moreover, the Coulomb branch is just the base of the Hitchin fibration (a certain space of meromorphic quadratic differentials on C.)

One of the key steps in the construction is to give a canonical local identification between a patch of M (considered as a complex manifold, in any of its generic complex structures) and a patch of a standard holomorphic symplectic torus. Here this identification can be given concretely, using coordinates written down by Fock and Goncharov. Fock and Goncharov's coordinates depend on some auxiliary data, namely an ideal triangulation of C, along with a Z_2-valued choice at each puncture; in our context this data is fixed canonically once we choose a point of the Coulomb branch and a phase. The wall-crossing formula of Kontsevich and Soibelman follows essentially from the dependence of our ideal triangulation on a point of the Coulomb branch.

The paper is rather long, because we tried to give a lot of details. In particular, at the end of the paper we discussed how our story works out for two interesting classes of field theories. One class is the SU(2) theories with 0, 1, 2, 3 or 4 flavors; the other is the Argyres-Douglas-type theories with Seiberg-Witten curve of the form y^2 = P(x), for P polynomial.

Projective superspace is one of several competing superspace formalisms for general N=2 theories in four dimensions. It is a somewhat exotic sort of superspace, in the sense that it has additional bosonic directions as well as fermionic. The extra bosonic directions roughly speaking make up a CP^1, which plays a role much like that of the twistor CP^1 of a hyperkahler manifold. (Indeed, if we consider a sigma model into a hyperkahler manifold, this CP^1 becomes literally identified with the twistor CP^1.)

In this note we describe a condition under which a Lagrangian written in projective superspace describes a conformally invariant 4d theory. The condition (which was not a big surprise, and presumably already known to many experts) is that the Lagrangian should be, in an appropriate sense, a section of the bundle O(2) over CP^1.

Argyres and Seiberg proposed a rather intriguing S-duality between two different superconformal N=2 theories in four dimensions. Theory A is the SU(3) gauge theory with N_f = 6, i.e. matter transforming in 6 copies of the fundamental representation. Theory B is the SU(2) gauge theory with N_f = 1, i.e. matter in a single copy of the fundamental representation, coupled to the Minahan-Nemeschansky SCFT with E_6 flavor symmetry by gauging an SU(2) subgroup of E_6.

Given an N=2 theory there is an associated moduli space, which generally consists of several branches. The tests of the duality given by Argyres and Seiberg involved primarily the physics of the so-called "Coulomb branch", which is a special Kahler manifold. In this paper we studied the opposite extreme, the "Higgs branch", which is hyperkahler.

The two theories have two a priori different Higgs branches. In Theory A the Higgs branch is a hyperkahler quotient F_{18} /// SU(3), where F_{18} is a flat space of quaternionic dimension 18. In Theory B the Higgs branch is (O x F_{2}) /// SU(2), where O is the minimal nilpotent coadjoint orbit of (the complex group) E_6. The two spaces both have quaternionic dimension 10. The Argyres-Seiberg duality then implies the claim that these two hyperkahler manifolds are actually the same.

In the paper we do not quite prove that the hyperkahler manifolds are the same, but we come pretty close: we prove that their twistor spaces are the same, as complex manifolds with antiholomorphic involutions and fiberwise symplectic forms. There remains a conceivable loophole -- it could happen that a single twistor space corresponds to two different hyperkahler manifolds (corresponding to two different sets of "real sections.") I am not aware of any examples where this possibility is actually realized (and more to the point, we asked some experts who also were not aware of any); but strictly speaking it has not been excluded.

Our argument is essentially by directly comparing the rings of functions on the two sides. On the B side this requires an explicit description of the ring of functions on the minimal orbit, given by results of Joseph and Kostant.

This paper argues that the wall-crossing formula for (generalized) Donaldson-Thomas invariants, written down by Kontsevich-Soibelman, governs the wall-crossing behavior of BPS degeneracies in 4d N=2 supersymmetric gauge theories.

The main idea is to study the theory compactified on a circle of radius R. On general grounds of supersymmetric field theory, the IR physics can then be described as a 3d sigma-model with target space a hyperkahler manifold M. In one of the complex structures induced by the hyperkahler structure (the "distinguished complex structure"), M is the total space of an integrable system -- in other words it is a complex symplectic manifold, fibered over the Coulomb branch of the 4d theory, with complex Lagrangian fibers, the generic fiber topologically a compact torus.

The metric on M takes a simple and explicit form in the limit R \to \infty: it approaches a "semiflat" metric, so called because it is flat when restricted to each torus fiber. The really interesting part of the story is the deviations from the semiflat metric which appear at finite R. These corrections come from a specific kind of BPS instanton, namely a BPS particle of the 4d theory going around the compactification circle. It turns out that they are weighted by the BPS index that counts these 4d particles. This index should be thought of as a physicist's version of a generalized Donaldson-Thomas invariant. The "wall-crossing phenomenon" is the fact that -- despite being protected by supersymmetry -- these BPS indices can actually jump as one varies parameters, in particular as one moves around the Coulomb branch. In our context this leads to a puzzle: the BPS indices jump, but somehow the instanton corrections to the metric on M should not jump! One of the main physical points of this paper is to explain how this puzzle is resolved; the answer is that the metric on M is actually smooth provided that the jump of the BPS indices at the wall is constrained in a very specific way, namely, it should obey the wall-crossing formula of Kontsevich-Soibelman.

The instanton-corrected M can be described exactly, but the description is a little intricate. In any complex structure except for the distinguished one, M is built by gluing together patches. Each patch looks like an open subset of a complex symplectic torus. The transition functions are certain distinguished symplectomorphisms (the same ones which appear in the Kontsevich-Soibelman wall-crossing formula). Roughly speaking we have one such symplectomorphism for each BPS particle of the theory. This part of the construction is rather parallel to what had appeared in earlier work by Kontsevich-Soibelman, in the context of the SYZ description of a K3 surface. The really new part of our story is that we build all the complex structures of M at once and keep track of how things depend on the complex structure; or, better said, we give a construction of the twistor space Z of M.

Z has the topology of M \times CP^1, and the trickiest step of our construction is to solve a certain Riemann-Hilbert problem on this CP^1. This amounts to constructing a differential equation over CP^1, valued in an infinite-dimensional bundle, with irregular singularities at the north and south poles of CP^1. (These two singularities are related to one another by a real structure which acts on the whole story.) At the irregular singularities one has Stokes phenomena, and the Stokes factors are precisely the symplectomorphisms mentioned above, corresponding to BPS particles. We solve the Riemann-Hilbert problem by relating it to finding a fixed point for a certain integral operator, which can be argued to be a contraction at least for large enough R. The behavior at smaller R is less understood although it is likely to be interesting; fortunately, if our goal is to understand the geometry of the wall-crossing formula, large R is enough.

A reader who is not interested in physics can interpret this paper as proposing a new construction of hyperkahler metrics. The data that go into the construction are, roughly speaking, a complex integrable system plus a set of integer invariants obeying the Kontsevich-Soibelman wall-crossing formula. These invariants control the quantum corrections which deform the metric away from the semiflat form. In particular, if the integer invariants are artfully chosen, these quantum corrections smooth out the behavior near the points where the semiflat metric is singular.

It may be useful to compare the output of our construction to what comes out of Gross and Wilson's work on K3 surfaces. They wrote down a metric which is not quite Ricci-flat, but a very good approximation: it is exponentially close to the Ricci-flat one in the so-called "large complex structure" limit (in our notation this is the limit of large R). Their metric is the one we would get from our approach if we ignore all of the quantum corrections except for the dominant one near each singularity of the SYZ fibration. So our method morally amounts to specifying the "improvement" that is needed to further correct their metric to the exact Ricci-flat one. I say "morally" because we have not yet applied our approach to K3 surfaces; there are some possible convergence problems which need careful consideration (although we optimistically believe they can be overcome.)

Bershadsky-Cecotti-Ooguri-Vafa famously proposed a "holomorphic anomaly equation" which governs the subtle dependence of the closed topological string partition function on a "background point" of the closed string moduli space. In previous work with Boris Pioline and Murat Gunaydin (depending heavily on prior work by Dijkgraaf, Verlinde and Vonk) we wrote down a change of coordinates which identifies this equation with the heat equation obeyed by a Siegel theta function (alas, in indefinite signature, so such "theta functions" do not literally exist.) This sharpened an analogy originally due to Witten.

Recently Walcher proposed a holomorphic anomaly equation for the open topological string partition function. In this paper we explained how this open-string modification can be translated to the heat-equation perspective. We found that it leads to a rather simple modification: indeed solutions of the original heat equation are related to the solutions of the new equation by a simple shift. The parameter of the shift is the "normal function" that specifies the open string data. This result has a whiff of "open-closed duality" about it, but unfortunately we seem to find that while the equations are related in this simple way, the particular solutions of these equations that give the topological partition functions are not.

It has to be said that this result has not found any significant application so far. I still believe quixotically that it (as well as the simpler closed string story) will eventually be part of a clearer understanding of what the holomorphic anomaly equation really means, as an exact statement about nonperturbatively defined objects. The geometry which enters this story is closely related to the integrable systems which appear in rigid N=2 theories.