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We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{RP}^2}$. We give a conjectural method for determining $Y(P)$ by solving a fixed-point problem for a certain integral operator. The technology of spectral networks and BPS state counts is a key input to the formulation of this fixed-point problem. We work out two families of examples in detail.

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We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), whenever k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). The action is induced by certain rational maps on Gr(k,n). Each of these maps is a quasi-automorphism of the cluster structure, a notion we defined in previous work. We also describe a quasi-isomorphism between Gr(k,n) and a certain Fock-Goncharov space of SL_k-local systems in a disk. The quasi-isomorphism identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy have proposed a description of the cluster combinatorics for Gr(3,n) in terms of Kuperberg's basis of non-elliptic webs. As our main application, we prove these conjectures for Gr(3,9). The proof relies on the fact that Gr(3,9) is of finite mutation type.

A new construction of BPS monodromies for 4d ${\mathcal N}=2$ theories of class S is introduced. A novel feature of this construction is its manifest invariance under Kontsevich-Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve $C$ of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations, and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For $A_1$-type theories, the graphs encoding the monodromy are "dessins d'enfants" on $C$, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg-Witten curve.

We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.

We associate to each stable Higgs pair (A0, Φ0) on a compact Riemann surface X a singular limiting configuration (A∞, Φ∞), assuming that det Φ has only simple zeroes. We then prove a desingulariza-tion theorem by constructing a family of solutions (At, tΦt) to Hitchin's equations which converge to this limiting configuration as t → ∞. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

We associate to each stable Higgs pair $(A_0,\Phi_0)$ on a compact Riemann
surface $X$ a singular limiting configuration $(A_\infty,\Phi_\infty)$,
assuming that $\det \Phi$ has only simple zeroes. We then prove a
desingularization theorem by constructing a family of solutions $(A_t,t\Phi_t)$
to Hitchin's equations which converge to this limiting configuration as $t \to
\infty$. This provides a new proof, via gluing methods, for elements in the
ends of the Higgs bundle moduli space and identifies a dense open subset of the
boundary of the compactification of this moduli space.

We study the BPS spectra of N=2 complete quantum field theories in four
dimensions. For examples that can be described by a pair of M5 branes on a
punctured Riemann surface we explain how triangulations of the surface fix a
BPS quiver and superpotential for the theory. The BPS spectrum can then be
determined by solving the quantum mechanics problem encoded by the quiver. By
analyzing the structure of this quantum mechanics we show that all
asymptotically free examples, Argyres-Douglas models, and theories defined by
punctured spheres and tori have a chamber with finitely many BPS states. In all
such cases we determine the spectrum.

We find a relation between the spectrum of solitons of massive N=2 quantum field theories in d=2 and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric N=2 conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A-D-E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.

Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing the ground state metric on a given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that the ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.

We explore the relationship between four-dimensional N=2 quantum field
theories and their associated BPS quivers. For a wide class of theories
including super-Yang-Mills theories, Argyres-Douglas models, and theories
defined by M5-branes on punctured Riemann surfaces, there exists a quiver which
implicitly characterizes the field theory. We study various aspects of this
correspondence including the quiver interpretation of flavor symmetries,
gauging, decoupling limits, and field theory dualities. In general a given
quiver describes only a patch of the moduli space of the field theory, and a
key role is played by quantum mechanical dualities, encoded by quiver
mutations, which relate distinct quivers valid in different patches. Analyzing
the consistency conditions imposed on the spectrum by these dualities results
in a powerful and novel mutation method for determining the BPS states. We
apply our method to determine the BPS spectrum in a wide class of examples,
including the strong coupling spectrum of super-Yang-Mills with an ADE gauge
group and fundamental matter, and trinion theories defined by M5-branes on
spheres with three punctures.

We show how aspects of the R-charge of N=2 CFTs in four dimensions are encoded in the q-deformed Kontsevich-Soibelman monodromy operator, built from their dyon spectra. In particular, the monodromy operator should have finite order if the R-charges are rational. We verify this for a number of examples including those arising from pairs of ADE singularities on a Calabi-Yau threefold (some of which are dual to 6d (2,0) ADE theories suitably fibered over the plane). In these cases we find that our monodromy maps to that of the Y-systems, studied by Zamolodchikov in the context of TBA. Moreover we find that the trace of the (fractional) q-deformed KS monodromy is given by the characters of 2d conformal field theories associated to the corresponding TBA (i.e. integrable deformations of the generalized parafermionic systems). The Verlinde algebra gets realized through evaluation of line operators at the loci of the associated hyperKahler manifold fixed under R-symmetry action. Moreover, we propose how the TBA system arises as part of the N=2 theory in 4 dimensions. Finally, we initiate a classification of N=2 superconformal theories in 4 dimensions based on their quiver data and find that this classification problem is mapped to the classification of N=2 theories in 2 dimensions, and use this to classify all the 4d, N=2 theories with up to 3 generators for BPS states. Comment: 161 pages, 4 figures; v2: references added, small corrections

Let X be a Calabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integers DT^a(t) which count stable sheaves with Chern character a on X, with respect to a Gieseker stability condition t. They are defined only for Chern characters a for which there are no strictly semistable sheaves on X. They have the good property that they are unchanged under deformations of X. Their behaviour under change of stability condition t was not understood until now. This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) are rational numbers, defined for all Chern characters a, and are equal to DT^a(t) if there are no strictly semistable sheaves in class a. They are deformation-invariant, and have a known transformation law under change of stability condition. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M. We compute our invariants in examples, and make a conjecture about their integrality properties. We extend the theory to abelian categories of representations of a quiver with relations coming from a superpotential, and connect our ideas with Szendroi's "noncommutative Donaldson-Thomas invariants" and work by Reineke and others. This book is surveyed in the paper arXiv:0910.0105. Comment: (v6) 212 pages. Minor changes. Final version, to appear in Memoirs of the A.M.S

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic;
the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When
S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they
carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant
under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have
positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their
positive real points provide the two higher Teichmller spaces related to G and S, while the points with values in the tropical
semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces.
It is related to the motivic dilogarithm.

In a continuation of our previous work [21], we outline a theory which should lead to the construction of a universal pre-building and versal building with a φ-harmonic map from a Riemann surface, in the case of twodimensional buildings for the group SL3. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for SL2. Our conjectural construction would determine the exponents for SL3 WKB problems, and it can be put into practice on examples.

In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperk\"{a}hler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration $\mathcal{M}$ over a base space $\mathcal{B}$, except for a divisor $D$ in $\mathcal{B}$, in which the torus fiber degenerates into a nodal torus. The hyperk\"{a}hler metric $g$ is obtained via solutions $\mathcal{X}_\gamma$ of a Riemann-Hilbert problem. We interpret the Kontsevich-Soibelman Wall Crossing Formula as an isomonodromic deformation of a family of RH problems, therefore guaranteeing continuity of $\mathcal{X}_\gamma$ at the walls of marginal stability. The technical details about solving the different classes of Riemann-Hilbert problems that arise here are left to a second article. To extend this construction to singular fibers, we use the Ooguri-Vafa case as our model and choose a suitable gauge transformation that allow us to define an integral equation defined at the degenerate fiber, whose solutions are the desired Darboux coordinates $\mathcal{X}_\gamma$. We show that these functions yield a holomorphic symplectic form $\varpi(\zeta)$, which, by Hitchin's twistor construction, constructs the desired hyperk\"{a}hler metric.

Given a Riemann surface $X = (\Sigma, J)$ we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,t\omega_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tilde{\Sigma},\tilde{J})$ to the symmetric space SL$_n(\mathbb{C}) / \mbox{SU}(n)$ and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of $\omega_n$.

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S). We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution * of the space X(m,S) provided by dualising a local system on S. We calculate the DT-transformation of the moduli space X(m,S), with few exceptions. Namely, let C(m,S) be the transformation of the space X(m,S) given by the product of three commuting maps: the involution *, the product, over all punctures of S, of the longest element of the Weyl group action corresponding to the puncture, and the "shift of the special points on the boundary by one" map. Using a characterisation of a class of DT-transformations due to Keller, we prove that C(m,S) = DT. We prove that, burring few exceptions, the Weyl group and the involution * act by cluster transformations of the dual moduli space A(m, S). So the formula C(m,S) = DT is valid for the space A(m,S). Our main result, combined with the work of Gross, Hacking, Keel and Kontsevich, deliver a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S), with few exceptions.

The "abelianization" process introduced by Gaiotto, Hollands, Moore, and
Neitzke turns $\operatorname{SL}_K \mathbb{C}$ local systems on a punctured
surface into $\mathbb{C}^\times$ local systems, giving coordinates on the
decorated $\operatorname{SL}_K \mathbb{C}$ character variety that are known to
match Fock and Goncharov's cluster coordinates in the $\operatorname{SL}_2
\mathbb{C}$ case. This paper extends abelianization to $\operatorname{SL}_2
\mathbb{R}$ local systems on compact surfaces, using tools from dynamics to
overcome the technical challenges that arise in the compact case. The approach
taken here seems to complement the one recently used by Bonahon and Dreyer to
arrive at a similar construction in a different geometric setting.

Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree $0$
on a compact connected Riemann surface. Once we fix the flat metric
$h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$
$(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,t\theta)$ such
that $\det(h_t)=h_{\det(E)}$. We study the behaviour of $h_t$ when $t$ goes to
$\infty$. First, we show that the Hitchin equation is asymptotically decoupled
under the assumption that the Higgs field is generically regular semisimple. We
apply it to the study of the so called Hitchin WKB-problem. Second, we study
the convergence of the sequence $(E,\overline{\partial}_E,\theta,h_t)$ in the
case where the rank of $E$ is two. We introduce a rule to determine the
parabolic weights of a "limiting configuration", and we show the convergence of
the sequence to the limiting configuration in an appropriate sense.

In a continuation of our previous work, we outline a theory which should lead
to the construction of a universal pre-building and versal building with a
$\phi$-harmonic map from a Riemann surface, in the case of two-dimensional
buildings for the group $SL_3$. This will provide a generalization of the space
of leaves of the foliation defined by a quadratic differential in the classical
theory for $SL_2$. Our conjectural construction would determine the exponents
for $SL_3$ WKB problems, and it can be put into practice on examples.

We give a characterization for an orientation preserving harmonic diffeomorphism from ℂ into a complete, simply connected, negatively pinched surface to have a polynomial growth Hopf differential. In particular, we prove that an orientation preserving harmonic diffeomorphism from ℂ into the Poincaré disk ℍ has a polynomial growth Hopf differential of degree m if and only if its image is an ideal polygon with m+2 vertices on ∂ℍ, with the assumption that the conformal metric on ℂ with the ∂-energy density as the conformal factor is complete. We will describe the geometric behavior of this harmonic diffeomorphisms in terms of the trajectories of their Hopf differentials. We will also construct all harmonic diffeomorphisms in this class, and prove that there is an m-1 parameter family of nontrivially distinct harmonic diffeomorphisms from the complex plane to a fixed ideal polygon with m+2 vertices in the hyperbolic plane. In particular, such harmonic maps are not unique, answering a question of Schoen.

Recently it has become apparent that [Formula: see text] supersymmetric quantum field theory has something to do with cluster algebras. I review one aspect of the connection: supersymmetric quantum field theories have associated hyperkähler moduli spaces, and these moduli spaces carry a structure that looks like an extension of the notion of cluster variety. In particular, one encounters the usual variables and mutations of the cluster story, along with more exotic extra variables and generalized mutations. I focus on a class of examples where the underlying cluster varieties are moduli spaces of flat connections on surfaces, as considered by Fock and Goncharov [Fock V, Goncharov A (2006) Publ Math Inst Hautes Études Sci 103:1-211]. The work reviewed here is largely joint with Davide Gaiotto and Greg Moore.

Using Hitchin's parameterization of the Hitchin-Teichm\"uller component of
the $SL(n,\mathbb{R})$ representation variety, we study the asymptotics of
certain families of representations. In fact, for certain Higgs bundles in the
$SL(n,\mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian
metric solving the Higgs bundle equations. This analysis is used to estimate
the asymptotics of the corresponding family of flat connections as we scale the
differentials by a real parameter. We consider Higgs fields that have only one
holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We
also study the asymptotics of the associated family of equivariant harmonic
maps to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ and relate it
to recent work of Katzarkov, Noll, Pandit and Simpson.

The notion of a universal building associated with a point in the Hitchin
base is introduced. This is a building equipped with a harmonic map from a
Riemann surface that is initial among harmonic maps which induce the given
cameral cover of the Riemann surface. In the rank one case, the universal
building is the leaf space of the quadratic differential defining the point in
the Hitchin base.
The main conjectures of this paper are: (1) the universal building always
exists; (2) the harmonic map to the universal building controls the asymptotics
of the Riemann-Hilbert correspondence and the non-abelian Hodge correspondence;
(3) the singularities of the universal building give rise to Spectral Networks;
and (4) the universal building encodes the data of a 3d Calabi-Yau category
whose space of stability conditions has a connected component that contains the
Hitchin base.
The main theorem establishes the existence of the universal building,
conjecture (3), as well as the Riemann-Hilbert part of conjecture (2), in the
case of the rank two example introduced in the seminal work of
Berk-Nevins-Roberts on higher order Stokes phenomena. It is also shown that the
asymptotics of the Riemann-Hilbert correspondence is always controlled by a
harmonic map to a certain building, which is constructed as the asymptotic cone
of a symmetric space.

The limiting behavior of high-energy harmonic maps between closed hyperbolic surfaces is analyzed. In general a measured foliation on the domain is shown to be mapped very nearly (exponentially in the energy) to its geodesic representative in the range. This foliation is in fact the horizontal foliation φh of the Hopf differential φ of the harmonic map.φh is also characterized as nearly maximizing (up to an additive constant) the ratio of squared hyperbolic length in the range to extremal length in the domain, among all simple closed curves in the domain. The same ratio gives the energy of the map up to an additive constant. This can be viewed as an analogy to other canonical maps between surfaces, for which different optimization problems are characterized by corresponding length-ratio maximizations. In addition, the asymptotics of a family of harmonic maps obtained when the domain surface is varied along a classical Teichmuller ray are studied. As expected, the limiting Hopf foliation and the foliation determining the ray are equivalent as topological (not measured) foliations. © 1992, International Press of Boston, Inc. All Rights Reserved.

Si M est une surface de Riemann compacte, on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d'auto-dualite associees a une representation irreductible: p:Π 1 (M)→PSL(2,C)

The WKB theory for differential equations of arbitrary order or integral equations in one dimension is investigated. The rules previously stated for the construction of Stokes’ lines for Nth‐order differential equations, N⩾3, or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path. This paradox is resolved by the demonstration that new Stokes’ lines can arise when previously defined Stokes’ lines cross. A new formulation of the WKB problem is given to justify the new Stokes’ lines. With the new Stokes’ lines, the asymptotic forms can be shown to be independent of path. In addition, the WKB eigenvalue problem is formulated, and the global dispersion relation is shown to be a functional of loop integrals of the action.

We study a certain type of wild harmonic bundles in relation with a Toda
equation. We explain how to obtain a classification of the real valued
solutions of the Toda equation in terms of their parabolic weights, from the
viewpoint of the Kobayashi-Hitchin correspondence. Then, we study the
associated integrable variation of twistor structure. In particular, we give a
criterion for the existence of an integral structure. It follows from two
results. One is the explicit computation of the Stokes factors of a certain
meromorphic flat bundle. The other is an explicit description of the associated
meromorphic flat bundle. We use the opposite filtration of the limit mixed
twistor structure with an induced torus action.

The rings of SL(V) invariants of configurations of vectors and linear forms
in a finite-dimensional complex vector space V were explicitly described by
Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these
rings carries a natural cluster algebra structure (typically, many of them)
whose cluster variables include Weyl's generators. We describe and explore
these cluster structures using the combinatorial machinery of tensor diagrams.
A key role is played by the web bases introduced by G.Kuperberg.

We introduce new geometric objects called spectral networks. Spectral
networks are networks of trajectories on Riemann surfaces obeying certain local
rules. Spectral networks arise naturally in four-dimensional N=2 theories
coupled to surface defects, particularly the theories of class S. In these
theories spectral networks provide a useful tool for the computation of BPS
degeneracies: the network directly determines the degeneracies of solitons
living on the surface defect, which in turn determine the degeneracies for
particles living in the 4d bulk. Spectral networks also lead to a new map
between flat GL(K,C) connections on a two-dimensional surface C and flat
abelian connections on an appropriate branched cover Sigma of C. This
construction produces natural coordinate systems on moduli spaces of flat
GL(K,C) connections on C, which we conjecture are cluster coordinate systems.

We consider a class of line operators in d=4, N=2 supersymmetric field
theories which leave four supersymmetries unbroken. Such line operators support
a new class of BPS states which we call "framed BPS states." These include halo
bound states similar to those of d=4, N=2 supergravity, where (ordinary) BPS
particles are loosely bound to the line operator. Using this construction, we
give a new proof of the Kontsevich-Soibelman wall-crossing formula for the
ordinary BPS particles, by reducing it to the semiprimitive wall-crossing
formula. After reducing on S1, the expansion of the vevs of the line operators
in the IR provides a new physical interpretation of the "Darboux coordinates"
on the moduli space M of the theory. Moreover, we introduce a "protected spin
character" which keeps track of the spin degrees of freedom of the framed BPS
states. We show that the generating functions of protected spin characters
admit a multiplication which defines a deformation of the algebra of functions
on M. As an illustration of these ideas, we consider the six-dimensional (2,0)
field theory of A1 type compactified on a Riemann surface C. Here we show
(extending previous results) that line operators are classified by certain
laminations on a suitably decorated version of C, and we compute the spectrum
of framed BPS states in several explicit examples. Finally we indicate some
interesting connections to the theory of cluster algebras.

Motivated by the computation of scattering amplitudes at strong coupling, we
consider minimal area surfaces in AdS_5 which end on a null polygonal contour
at the boundary. We map the classical problem of finding the surface into an
SU(4) Hitchin system. The polygon with six edges is the first non-trivial
example. For this case, we write an integral equation which determines the area
as a function of the shape of the polygon. The equations are identical to those
of the Thermodynamics Bethe Ansatz. Moreover, the area is given by the free
energy of this TBA system. The high temperature limit of the TBA system can be
exactly solved. It leads to an explicit expression for a special class of
hexagonal contours.

We consider BPS states in a large class of d=4, N=2 field theories, obtained
by reducing six-dimensional (2,0) superconformal field theories on Riemann
surfaces, with defect operators inserted at points of the Riemann surface.
Further dimensional reduction on S^1 yields sigma models, whose target spaces
are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In
the case where the Higgs bundles have rank 2, we construct canonical Darboux
coordinate systems on their moduli spaces. These coordinate systems are related
to one another by Poisson transformations associated to BPS states, and have
well-controlled asymptotic behavior, obtained from the WKB approximation. The
existence of these coordinates implies the Kontsevich-Soibelman wall-crossing
formula for the BPS spectrum. This construction provides a concrete realization
of a general physical explanation of the wall-crossing formula which was
proposed in 0807.4723. It also yields a new method for computing the spectrum
using the combinatorics of triangulations of the Riemann surface.

Thesis (Ph.D., Dept. of Mathematics)--Harvard University, 2004. Includes bibliographical references (leaves 75-78).

We consider minimal surfaces in three dimensional anti-de-Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. We describe in detail the mathematical problem that needs to be solved. This problem is mathematically the same as the one studied by Gaiotto, Moore and Neitzke in the context of the moduli space of certain supersymmetric theories. Using their results we can find the explicit answer for the area of a surface that ends on an eight-sided polygon. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling. In addition, we give fairly explicit solutions for regular polygons.

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

. We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L 2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L 2 complex. As a consequence, we obtain a Hard Lefschetz-type theorem. Rsum (Mtriques harmoniques et connexions singularits irrgulires) Nous identions le complexe de de Rham de l'extension minimale d'un br mromorphe connexion sur une surface de Riemann compacte X au complexe L 2 associ ce br sur la surface de Riemann prive des ples, lorsqu'on munit celui-ci d'une mtrique convenable et la surface pointe d'une mtrique complte. En appliquant des rsultats de C. Simpson, nous montrons l'existence d'une mtrique harmonique sur ce br, donnant lieu au mme complexe L 2 . Nous en dduisons un thorme de typ...

F. Labourie and the author independently have shown that a convex real projective structure on an oriented closed surface S of genus at least two is equivalent to a pair of a conformal structure plus a holomorphic cubic differential. Along certain paths, we find the limiting holonomy of convex real projective structures on a surface S corresponding corresponding to a given fixed conformal structure S and a holomorphic cubic differential λ U
0 as \(\lambda\to\infty\) . We explicitly give part of the data needed to identify the boundary point in Inkang Kim’s compactification of the deformation space of convex real projective structures. The proof follows similar analysis to that studied by Mike Wolf is his application of harmonic map theory to reproduce Thurston’s boundary of Teichmüller space.

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a minimal surface in a symmetric space.

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate
ring of the Grassmannian G(k, n) is a cluster algebra of geometric type. Those Grassmannians that are of finite cluster type are identified and their cluster variables are interpreted geometrically in terms of configurations of points in C P2. 2000 Mathematics Subject Classification 22E46, 05Exx.

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of Goldman. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of Cheng-Yau and C.P. Wang and a result of Dunkel on the asymptotics of systems of ODEs.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that they carry hyperKahler metrics, which are complete when the residue of the connection if semisimple.

First an `irregular Riemann-Hilbert correspondence' is established for meromorphic connections on principal G-bundles over a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic deformations of such connections are considered and it is proved that the classical actions of quantum Weyl groups found by De Concini, Kac and Procesi do arise from isomonodromy (and so have a purely geometrical origin). Finally a certain flat connection appearing in work of De Concini and Toledano Laredo is derived from isomonodromy, indicating that the above result is the classical analogue of their conjectural Kohno-Drinfeld theorem for quantum Weyl groups.

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation. Comment: 7 pages, 4 figures