CONSTRUCTING BUILDINGS AND HARMONIC MAPS

Transcription

1 CONSTRUCTING BUILDINGS AND HARMONIC MAPS LUDMIL KATZARKOV, ALEXANDER NOLL, PRANAV PANDIT, AND CARLOS SIMPSON Happy Birthday Maxim! Abstract. In a cntinuatin f ur previus wrk [17], we utline a thery which shuld lead t the cnstructin f a universal pre-building and versal building with a φ-harmnic map frm a Riemann surface, in the case f tw-dimensinal buildings fr the grup SL 3. This will prvide a generalizatin f the space f leaves f the fliatin defined by a quadratic differential in the classical thery fr SL 2. Our cnjectural cnstructin wuld determine the expnents fr SL 3 WKB prblems, and it can be put int practice n examples. 1. Intrductin Let X be a Riemann surface, cmpact fr nw. The mduli spaces f representatins, vectr bundles with cnnectin, and semistable Higgs bundles dented M B, M DR and M H respectively, are ismrphic as spaces, and we dente the cmmn underlying space just by M. This space has three different algebraic structures, and the Betti and de Rham nes share the same cmplex manifld. These algebraic varieties are nncmpact, indeed M B is affine, but they have cmpactificatins. First, M DR and M H are natural rbifld cmpactificatins, and when M itself is smth, the rbifld cmpactificatins are smth. The Betti mduli space, therwise usually knwn as the character variety, admits many natural cmpactificatins. Indeed the mapping class grup acts n M B but it desn t stabilize any ne f them. Grss, Hacking, Keel and Kntsevich [10] have recently studied mre clsely the prblem f cmpactifying M B and shw that there are indeed sme ptimal chices with gd prperties. Mrgan-Shalen [21] fr SL 2, and recently Parreau [22] fr grups f higher rank, cnstruct a cmpactificatin f M where the pints at 1

2 2 KATZARKOV, NOLL, PANDIT, AND SIMPSON are actins f the fundamental grup n R-buildings. This cmpactificatin is related t the inverse limit f the algebraic cmpactificatins f M B, and we shall call it M P B. Let M T SC be the Tychnff-Stne-Čech cmpactificatin. It is universal, s it maps t the ther nes. The pints at infinity may be identified with nn-principal ultrafilters 1 ω n M. We may therefre cnsider the maps M H M T SC M P B. M DR Given a nn-principal ultrafilter ω, cnsider its image ω DR in M DR (resp. ω H in M H ) and its image ωb P in M P B. Our basic questin is t understand the relatinship between ω DR (resp. ω H ) and ωb P. In what fllws we cncentrate n ω DR but there are als cnjectures fr ω H as mentined in [17], with recent results by Cllier and Li [5]. The divisrs at infinity in M DR and M H are bth the same. They are D = MH/C where MH := M H Nilp is the cmplement f the nilptent cne. Recall that Nilp = f 1 (0) where f : M H A N is the Hitchin fibratin. Hence ω DR (and similarly ω H ) may be identified as an equivalence class f pints in MH mdul the actin f C. We may therefre write ω DR = (E, ϕ) as a semistable Higgs bundle, such that the Higgs field ϕ is nt nilptent, and this identificatin hlds up t nnzer cmplex scaling f ϕ. It turns ut that the essential features f the crrespndence with M P B depend nt n ϕ up t cmplex scaling, but up t real scaling. We therefre intrduce the real blwing up M DR f M DR alng the divisr D. It is still a cmpactificatin f M DR. The bundary at infinity is D which is an S 1 -bundle ver D, with D = M H/R +,. Let ω DR dente the image f ω in D M DR. We may again write ω DR = (E, ϕ) 1 Here, by an ultrafilter n a nrmal (T 4 ) tplgical space, we mean a maximal filter cnsisting f clsed subsets, as were cnsidered by Wallman [28].

3 CONSTRUCTING BUILDINGS 3 but this time ϕ is defined up t psitive real scaling. We dente by φ the spectrum f ϕ. It cnsists f a multivalued tuple f hlmrphic differential frms n X. This is equivalent t saying that it is a pint in the Hitchin base A N, r again equivalently, a spectral curve Σ T X. Fr us, it will be mst useful t think f writing φ = (φ 1,..., φ r ) lcally ver X, where φ i are hlmrphic differential frms. There will generally be sme singularities at pints ver which Σ X is ramified, and as we mve arund in the cmplement f these singularities the rder f the φ i may change. Dente by X the cmplement f the set f singularities. Gaitt-Mre-Neitzke [8] define the spectral netwrk assciated t φ. This is ne f the main players in ur stry. We refer t [8, 9] fr many pictures, and t ur paper [17] fr sme specific pictures related t the BNR example. These structures cnstitute ur understanding f ω DR. On the ther side, the pint ωb P may be identified with an actin f π 1 (X) n an R-building Cne ω. The thery f Grmv-Schen and Krevaar-Schen allws us t chse an equivariant harmnic mapping h : X Cne ω. This is mst ften uniquely determined by the actin. The metric n the building, and hence the differential f the harmnic mapping, are well-defined up t psitive real scaling. Here, we use the terminlgy harmnic map t an R-building t mean a map such that the dmain, minus a singular set f real cdimensin 2, admits an pen cvering where each pen set maps int a single apartment, and these lcal maps are harmnic mappings t Euclidean space. The differential f a harmnic map is the real part f a mutlivalued differential frm. In [17] we used the grupid versin f Parreau s thery [22] t cnstruct the harmnic mapping h, and the classical lcal WKB apprximatin t shw that its differential is φ. Therem 1.1 ([17]). The differential f the harmnic mapping h is the real part f the multivalued differential φ cnsidered abve: dh = Rφ. We nte that Cllier and Li have prven the crrespnding result fr the Hitchin WKB prblem in sme cases [5]. This therem suggests the fllwing questin. Questin 1.2. T what extent des φ determine h and hence ω P B? In what fllws we shall assume that we have passed t the universal cver f the Riemann surface s X = X. Furthermre, since ur

4 4 KATZARKOV, NOLL, PANDIT, AND SIMPSON cnsideratins nw, abut hw t integrate the differential φ int a harmnic map h, cncern mainly bunded regins f X, we may envisin ther examples. Such might riginally have cme frm nncmpact Riemann surfaces and differential equatins with irregular singularities. The BNR example in [17] was f that frm. A φ-harmnic map will mean a harmnic map h frm X t a building B, such that its differential is dh = Rφ. The gal f this paper is t sketch a thery which shuld lead t the answer t Questin 1.2, at least fr the grup SL 3 and under certain genericity hyptheses abut the absence f BPS states Cntents. We start by reviewing briefly the thery f harmnic maps t trees, viewed as buildings fr SL 2, with the universal map given by the leaf space f a fliatin defined by a quadratic differential. The remainder f the paper is devted t generalizing this classical thery t higher rank buildings and in particular, as discussed in Sectin 3, t tw-dimensinal buildings fr SL 3. In rder t g twards an algebraic viewpint, we indicate in Sectin 4 hw ne can view a building as a presheaf n a certain Grthendieck site f enclsures. This allws ne t frmulate a small-bject argument cmpleting a pre-building t a building. In Sectin 5 we describe the initial cnstructin assciated t a spectral curve φ. This is the building-like bject, admitting a φ-harmnic map frm X, btained by glueing tgether small lcal pieces. The initial cnstructin will have pints f psitive curvature, and the main wrk is in Sectins 6 and 8 where we describe a sequence f mdificatins t the initial cnstructin designed t remve the psitive curvature pints. One f the main difficulties is that a pint where nly fur 60 sectrs meet, admits in principle tw different fldings. Infrmatin cming frm the riginal harmnic map serves t determine which f the tw fldings shuld be used. In rder t keep track f this infrmatin, we intrduce the ntin f scafflding. The result f Sectins 6 and 8 is a sequence f mves leading t a sequence f tw-manifld cnstructins. Our main cnjecture is that when the spectral netwrk [8] f φ desn t have any BPS states, this sequence is well defined and cnverges lcally t a tw-manifld cnstructin with nly nnpsitive curvature. The universal pre-building is btained by putting back sme pieces that were trimmed ff during the prcedure. In Sectin 7, cming as an interlude between the tw main sectins, we present an extended example. We shw hw t treat the BNR example which we had already cnsidered rather extensively in [17],

5 CONSTRUCTING BUILDINGS 5 but frm the pint f view f the general prcess being described here. It was thrugh cnsideratin f this example that we arrived at ur prcess, and we hpe that it will guide the reader t understanding hw things wrk. In Sectin 9 we present just a few pictures shwing what can happen in a typical slightly mre cmplicated example. This example and thers will be cnsidered mre extensively elsewhere. In Sectin 10 we cnsider sme cnsequences f ur still cnjectural prcess, and indicate varius directins fr further study Cnventin. This paper is intended t sketch a picture rather than prvide cmplete prfs. All stated therems, prpsitins and lemmas are actually quasi-therems : plausible and tractable statements fr which we have in mind a ptential methd f prf. Statements which shuld be cnsidered as pen prblems needing cnsiderably mre wrk t prve, are labeled as cnjectures Acknwledgements. The ideas presented here are part f ur nging attempt t understand the gemetric picture relating pints in the Hitchin base and stability cnditins. S this wrk is initiated and mtivated by the wrk f Maxim Kntsevich and Yan Sibelman, as exemplified by their many lectures and the discussins we have had with them. It is a great pleasure t dedicate this paper t Maxim n the ccasin f his 50 th birthday. The reader will nte that many ther mathematicians are als cntributing t this fast-develping thery and we wuld als like t thank them fr all f their input. We hpe t present here a small cntributin f ur wn. On a smewhat mre specific level, the idea f lking at harmnic maps t buildings, generalizing the leaf space tree f a fliatin, came up during sme fairly wide discussins abut this develping thery in which Fabian Haiden als participated and made imprtant cmments, s we wuld like t thank Fabian. We wuld like t thank many ther peple including Brian Cllier, Gergis Daskalpuls, Mikhail Kapranv, Françis Laburie, Ian Le, Chikak Mese, Richard Schen, Richard Wentwrth, and the members f the Gemetric Langlands seminar at the University f Chicag, fr interesting discussins cntributing t this paper. We thank Drn Puder fr an interesting talk at the IAS n the thery f Stallings graphs. The authrs wuld like t thank the University f Miami fr hspitality and supprt during the cmpletin f this wrk. The furth

6 6 KATZARKOV, NOLL, PANDIT, AND SIMPSON named authr wuld in additin like t thank the Fund Fr Mathematics at the Institute fr Advanced Study in Princetn fr supprt. The first named authr was supprted by the Simns Fundatin as a Simns Fellw. The first, secnd, and third named authrs were funded by an ERC grant, as well as the fllwing grants: NSF DMS FRG, NSF DMS , NSF DMS FRG, NSF DMS , NSF DMS , FWF P N25, FWF P20778, FWF P N25. The furth named authr was supprted in part by the ANR grant 933R03/13ANR002SRAR (Tfigru). The first named authr was als funded by the fllwing grants: DMS Wall Crssings in Gemetry and Physics, DMS Spectra, Gaps, Degeneratins and Cycles, and OISE PASI On Wall Crssings, Stability Hdge Structures & TQFT. The secnd named authr was in additin funded by the Advanced Grant Arithmetic and physics f Higgs mduli spaces N f the Eurpean Research Cuncil. We wuld all like t thank the IHES fr hsting Maxim s birthday cnference, where we did a significant part f the wrk presented here. 2. Trees and the leaf space Let us first cnsider the case f representatins int SL 2. Exact WKB analysis and its relatinship with the gemetry f quadratic differentials has been cnsidered extensively by Iwaki and Nakanishi [13]. In the Mrgan-Shalen-Parreau thery, the limiting building Cne ω is then an R-tree. In saying R-tree we include the data f a distance functin which is the standard ne n any apartment. The chice f metric is well-defined up t a glbal scalar n Cne ω. The spectral curve Σ X is a 2-sheeted cvering defined by a quadratic differential q H 0 (X, K 2 X ). The multivalued differential frm φ is just the set f tw square rts: φ i = ± q. The singularities are the zers f q, and we shall usually assume that these are simple zers. It crrespnds t saying that Σ is smth which, fr a 2-sheeted cver, implies having simple branch pints. The differential frm Rφ is well defined up t a change f sign. Therefre, it defines a single real directin at every nnsingular pint f X, which we call the fliatin directin. These directins are the tangent directins t the leaves f a real fliatin, which we call the fliatin defined by φ r equivalently the fliatin defined by the quadratic differential q. Suppse T is an R-tree and h : X T is a φ-harmnic map. Then, the clsed leaves f the fliatin defined by φ map t single pints

7 CONSTRUCTING BUILDINGS 7 A D B C Figure 1. A fliatin with BPS state in T. This is clear frm the differential cnditin at smth pints. By cntinuity it als extends acrss the singularities: the clsed leaves are defined t be the smallest clsed subsets which are invariant by the fliatin. In pictures, a leaf entering a three-fld singular pint therefre generates tw branches f the leaf ging ut in the ther tw directins, and all three f these branches have t map t the same pint in T. We assume that the space f leaves f the fliatin is well-defined as an R-tree. Dente it by T φ. The pints f T φ are by definitin just the clsed leaves f the fliatin. In rder t cnsider the universal prperty, we say that a flding map u : R T is a map such that there is a lcally finite decmpsitin f R int segments, such that n each segment u is an ismetric embedding. A map u : T T between R-trees is a flding map if its restrictin t each apartment is a flding map. Nte that an ismetric embedding is a particularly nice kind f flding map which desn t actually fld anything. Crllary 2.1. The prjectin map h φ : X T φ is a φ-harmnic map, universal amng φ-harmnic maps t R-trees. That says that if h : X T is any ther φ-harmnic map, there exists a unique flding map T φ T making the diagram cmmute. X T φ T

8 8 KATZARKOV, NOLL, PANDIT, AND SIMPSON A D B C Figure 2. The qutient tree In general, we have t admit the pssibility that T φ T be a flding map rather than an ismetric embedding. Let us lk at an example which illustrates this. Suppse that ur quadratic differential r equivalently φ, has tw singular pints which are n the same leaf f the fliatin. See Figure 1. The leaf segment between the tw singularities is emphasized, it is called a BPS-state [8]. These play a key rle in the wallcrssing stry. Here, it is the existence f the BPS state which leads t nnrigidity f the harmnic map. The space f leaves f the fliatin is a tree T φ with fur segments. In Figure 2 these segments are labeled in the same way as the crrespnding regins in the previus picture f the fliatin. Suppse we have a φ-harmnic map h t anther tree T. The universal prperty gives a factrizing map u : T φ T. The map h is suppsed t send small pen sets in X t single apartments in T. The small pen sets in X prvide sme small segments in the tree T φ, and the segments in T φ which are images f these neighbrhds, have t map by ismetric embeddings (i.e. nt be flded) under u. Little neighbrhds alng the segments jining the regins fr example A t B, B t C, C t D and D t A, as well as a neighbrhd alng the BPS state, are shwn in Figure 3. The fur uter neighbrhds prject t fur crner segments as shwn in Figure 4. Therefre u cannt fld these segments. The little neighbrhd alng the central leaf prjects t the segment shwn in Figure 5 jining the upper edge A f the tree and the lwer

9 CONSTRUCTING BUILDINGS 9 A D B C Figure 3. Sme little neighbrhds A D B C Figure 4. Nn-flded segments edge C. It fllws that u is nt allwed t fld the tw segments A and C tgether. Hwever, u culd very well fld the segments B and D tgether since they are nt cnstrained by any neighbrhds in X. Therefre, we can have a harmnic 2 map t the tree T shwn in Figure 6. The central dt ges t the central dt, the segments A and C g t A and C, and the segments B and D g t the paths frm the central dt ut t B and D, which are flded tgether alng sme shrt segment. 2 R. Wentwrth pinted ut t us that ur cmpsed map t the tree in Figure 6, while lcally harmnic, will nt hwever be glbally energy-minimizing.

10 10 KATZARKOV, NOLL, PANDIT, AND SIMPSON A D B C Figure 5. Nn-flded segment frm central neighbrhd A D B C Figure 6. A flded tree We see in this example that the prjectin t the leaf space f the fliatin is nt rigid, and this nn-rigidity lks clsely related t the presence f the BPS state. On the cntrary, if there are n BPS states, then h φ is rigid: Therem 2.2. Suppse that the spectral curve is smth, i.e. the quadratic differential has simple zers. If there are n BPS states, then any φ-harmnic map h : X T factrs as h = u h φ thrugh a unique map u : T φ T which is an ismetric embedding. Thus, T is just T φ plus sme ther edges nt tuched by the image f h.

11 CONSTRUCTING BUILDINGS 11 In the situatin f the therem, any tw nnsingular pints f X are jined by a strictly nncritical path, that is t say a path transverse t the fliatin. The distance between the tw pints in T φ is the length f the path using the transverse measure defined by Rφ. This nncritical path als has t g t a nncritical path in T, s the distance in T is the same as in T φ. If there is a BPS state, then sme distances might nt be welldetermined. Fr example the distance between pints in regins B an D may change with the family f maps pictured in Figure 6. Nn-uniqueness f the distance is smething that has t be expected frm the pint f view f the Vrs resurgent expressin fr the transprt functin [27]. The resurgent expressin may be viewed, rughly speaking, as a cmbinatin f cntributins f different expnential rders e λit. See [24] fr a mre precise discussin f hw this wrks. When there is a BPS state, then tw expnents have the same real part, Rλ i = Rλ j. If these are the leading terms, then we btain a functin which becmes scillatry alng the psitive real directin in t. A simple example wuld be 2 cs(t) = e it + e it. If the transprt functin T P Q (t) is scillatry, then the asympttic expnent ν ω P Q = lim ω 1 t lg T P Q(t) culd very well depend n the chice f ultrafilter ω used t define the limit. Thus, when there is a BPS state we can expect that the distance functin defined by the harmnic map culd depend n the chice f ultrafilter. In particular, it wuldn t be uniquely determined by the spectral differential φ. These cnsideratins, fitting perfectly well with the illustratin f Figure 6, mtivate the hypthesis abut BPS states in Therem Tw-dimensinal buildings We feel that there shuld be a similar picture fr higher-dimensinal buildings. The basic philsphy and mtivatins were described in [17]. Our idea at the current stage f this prject is t cncentrate n mappings t 2-dimensinal buildings. These buildings are asympttic cnes fr symmetric spaces SL 3 /SU(3), and the mappings are limiting pints in the sense f Parreau s thery [22] [17]. The spectral curves fr such mappings are triple cverings Σ X. Fr this SL 3 case, ur gal is t sketch the utlines f a thery which shuld lead t a generalizatin f Therem 2.2. It will say that fr generic φ in a chamber where there are n BPS states, there shuld

12 12 KATZARKOV, NOLL, PANDIT, AND SIMPSON be a versal map t a building, such that the resulting distance functin is uniquely determined by φ and preserved under φ-harmnic maps. The SL 3 situatin is in many ways an intermediate case. In the SL 2 case, the mapping h φ t the tree T φ was surjective. Fr SL r with r 4, the crrespnding buildings have dimensin 3, s the map frm X has n hpe f being surjective. We will present sme speculatins abut that higher rank situatin at the end f the paper. In the SL 3 case, the dimensin f the building is tw, which is the same as the dimensin f X. Therefre, we can expect that X will surject nt a subset which at least has nnempty interir. S, it presents sme similarity with the case f trees, and this simplifies the gemetrical aspects. We are able t develp a fairly precise althugh still cnjectural picture. The image f X will be a qutient f X, glueing tgether pints ver certain prtins. This was seen in ur BNR example f [17] which shall be recalled in detail in Sectin 7 belw. Our gal is t cnstruct a map h φ : X B φ which shuld play the same rle as the prjectin t the tree f leaves in the SL 2 case. The cnstructin has several steps. The main part will be the cnstructin f a map t a pre-building h pre φ : X B pre φ such that the building B φ can then be btained frm B pre φ by adding n sectrs nt tuched by the image f X. Already in the BNR example f [17] there were infinitely many additinal sectrs t be added here. They seem t be smewhat less related t the gemetry f the situatin. The pre-building will itself be a qutient f an initial cnstructin. The initial cnstructin is btained by glueing tgether small pieces. Recall that ne f the main characteristics f a building is its nnpsitive curvature prperty. The initial cnstructin will, hwever, have sme psitively curved pints: thse are pints where the ttal surrunding angle is 240 rather than 360. As we shall see belw, it leads t a prcess f successive pasting tgether f parts f the cnstructin. We cnjecture that after a lcally finite number f steps this prcess shuld stp and give a pre-building. 4. Cnstructins as presheaves n enclsures In rder t get started, we need a precise way t manipulate the building-like bjects invlved in the cnstructin. The idea fr passing frm B pre φ t B φ will be t apply the small bject argument. Als, the

13 CONSTRUCTING BUILDINGS 13 p Figure 7. Sme enclsures cnstructin f B pre φ itself will invlve successively impsing a bigger and bigger relatin n the initial cnstructin. S, it appears that we are wrking with algebraic rather than tplgical r metric bjects. This makes it desirable t have an algebraic framewrk. We prpse t cnsider a Grthendieck site E f the basic building blcks, called enclsures. Then cnstructins will be sheaves f sets n the site E, satisfying basic lcal presentability and separability prperties. Intuitively, a cnstructin is a space btained by glueing tgether the basic pieces such as shwn in Figure 7. Prfs are nt yet given, hwever we hpe that they will be reasnably straightfrward. The general thery is described fr buildings f any dimensin. Let A be the affine space n which ur buildings will be mdeled. A rt half-space is a half-space bunded by a rt hyperplane. An enclsure is a bunded clsed subset defined by the intersectin f finitely many rt half-spaces. Fr buildings crrespnding t SL 3, the affine space is 3 A = R 2 and sme examples f enclsures are shwn in Figure 7. An affine map f enclsures E E is a map which is the restrictin f an autmrphism f A given by an affine Weyl grup element. Let E be the categry f enclsures and affine maps between them. There is an bject pint dented p cnsisting f a single pint in A. 3 Mre precisely A is the space f triples (x 1, x 2, x 3 ) with x i = 0 and the rt half-spaces are defined by x i x j c.

14 14 KATZARKOV, NOLL, PANDIT, AND SIMPSON We define a Grthendieck tplgy n E as fllws: a cvering f E is a finite cllectin f affine maps f enclsures E i E such that E is the unin f their images. The categry E admits fiber prducts, but nt prducts and in particular there is n terminal bject. Prpsitin 4.1. The cverings define a Grthendieck tplgy n E. If E is an enclsure, we dente by Ẽ the sheaf assciated t the presheaf represented by E. It is different frm the presheaf: if E is anther enclsure, then the sectins f Ẽ ver E, which is t say the maps E Ẽ r equivalently the maps Ẽ Ẽ, are the flding maps frm E t E. These are the cntinuus maps which are piecewise affine fr a decmpsitin f E int finitely many pieces which are themselves enclsures. We can give a nrmalized frm fr cverings. Suppse H 1,..., H k is a sequence f parallel Weyl hyperplanes in rder. Then we btain a sequence f strips S 0,..., S 2k cvering A. The strip S 2i is the clsed subset cnsisting f everything between and including H i and H i+1, with S 0 and S 2k being the tw uter half-planes (we assume k 2 s there is n questin abut the rdering f these). The strip S 2i+1 is just H i 1 itself. Suppse we are given a cllectin f such sequences f strips S 1,..., S a fr varius directins f the Weyl hyperplanes. Then fr J = (j 1,..., j a ) with 0 j i 2k i we may cnsider the enclsure These cver E. U J := E S 1 j 1 S a j a. Lemma 4.2. Suppse {V k } is a cvering f E. Then it may be refined t a standard cvering, that is t say a cvering f the frm {U J } cnstructed abve. We remark that in a standard cvering {U J }, the intersectins f elements are again elements, since intersectins f strips are included as strips t (that was why we included the H i themselves). We may nw give a mre explicit descriptin f the flding maps. Crllary 4.3. Suppse E and F are enclsures. Any flding map, that is t say a map t the assciated sheaf E F, is given by taking a standard cvering {U J } and assigning fr each J {H + i, H i } an affine map a J : U J F, subject t the cnditin that if U K U J then a J UK = a K. Tw flding maps are the same if and nly if they are the same pintwise, which is equivalent t saying that they are the same n a cmmn refinement f the tw cvers; a cmmn refinement may be btained by taking the unins f the sequences f hyperplanes.

15 CONSTRUCTING BUILDINGS 15 Crllary 4.4. A flding map between enclsures is finite-t-ne. We nw cnsider a sheaf F n E. We say that it is finitely generated if there is a finite cllectin f maps E i F frm enclsures, such that the map f presheaves Ei F is surjective in the sheaf-theretical sense, i.e. it induces a surjectin f assciated sheaves. We say that F is finitely related if, fr any tw maps frm enclsures E, E F, the fiber prduct E F E is finitely generated. We say that F is finitely presented if it is finitely related and finitely generated. Lemma 4.5. If E is an enclsure, then any subsheaf F Ẽ is finitely related. Mre generally if {E i } i=1,...,n is a nnempty cllectin f enclsures, then any subsheaf F i Ẽ i is finitely related. Prf. We need t cnsider the fiber prduct G F G fr tw maps frm enclsures G, G F. These maps crrespnd t sequences (ζ 1,..., ζ n ) with ζ i : G Ẽi, resp. (ζ 1,..., ζ n) with ζ i : G Ẽi. Suppse G = G J (resp. G = G J ) are (finite) cverings by enclsures. Suppse we can prve that G J F G J are finitely generated. One can then cnclude that G F G is finitely generated. Apply this t a cmmn refinement f the cverings needed t define the flding maps ζ i, in the abve standard frm. We cnclude that it suffices t cnsider the case where ζ i are affine maps. Nw in this case (and n lnger using the ntatin fr the cverings), the fiber prduct is expressed as G F G = {(x, x ) s.t. ζ i (x) = ζ i(x ) fr i = 1,..., n}. This expressin is smewhat heuristic as we are really talking abut sheaves but it serves t indicate the prf. Since ζ 1 is an ismrphism we may assume that it is the identity and same fr ζ 1. Therefre, the first cnditin says that x = x and with this nrmalizatin, we may write G F G = {x A s.t. x G G, ζ i (x) = ζ i(x) fr i = 2,..., n}. The cnditins ζ i (x) = ζ i(x) define Weyl hyperplanes r are always true, s this represents G F G as an enclsure. This cmpletes the prf.

16 16 KATZARKOV, NOLL, PANDIT, AND SIMPSON Nte that E E will never be finitely generated as sn as dim(a) 1. Therefre, the empty direct prduct which is t say the terminal bject in sheaves n enclsures, is nt finitely related. The abve prf used n 1 in an essential way. One shuld nt cnfuse the terminal bject with the enclsure p when p A is a pint. There are n maps E p fr E different frm a pint r the empty set. A cnstructin is a finitely related sheaf n the site f enclsures. Therem 4.6. The categry f cnstructins is clsed under finite climits, and fiber prducts. We can define a tplgical space underlying a cnstructin. If F is a cnstructin, let F (p) dente the set f pints, that is t say the set f maps p F where p A is a pint (recall frm abve that this is different frm the terminal sheaf ). Give F (p) a tplgy as fllws: fr any enclsure E, E(p) (which is equal t Ẽ(p)) has a tplgy as a subset f the affine space A. Then we say that a subset U F (p) is pen if its pullback t E(p) is pen, fr any enclsure E and any map E F. Cnjecture 4.7. If F is a cnstructin then the tplgical space F (p) Hausdrff; furthermre it is a CW-cmplex. It might be necessary t add additinal hyptheses n F in rder t insure that F (p) is a CW cmplex Spherical thery. We wuld like t cnsider the lcal structure f a cnstructin at a pint. Fr this we need a spherical versin f the abve thery. It seems like we prbably dn t need t cnsider enclsures in the spherical building but nly sectrs. A sectr is a minimal clsed chamber f a given dimensin in the spherical cmplex assciated t A. The set f sectrs is partially rdered by inclusin and the spherical Weyl grup acts n it. It is a finite tplgical space, in particular it has a structure f site (the nly cverings f a sectr must include that sectr itself). Let S be the categry f sectrs. A spherical cnstructin is a presheaf r equivalently sheaf n S. We have a map frm S t the filters n A lcated at any given pint. Suppse F is a sheaf n E and x F (p). We wuld like t assciate the spherical cnstructin F x, defined as fllws: if σ S then σ crrespnds t a filter f enclsures, that is t say a filtered categry f enclsures E with p n the bundary f E and whse lcal crner at p lks like σ. Call this categry σ. Fr E σ, cnsider the set

17 CONSTRUCTING BUILDINGS 17 F (E) x cnsisting f maps E F such that p maps t x. Then we set F x (σ) := lim F (E) x.,e σ An element f F x (σ) therefre cnsists f a germ f map E F sending p t x, such that the crner f E at p is σ. These germs are up t equivalence that if tw maps agree n a smaller enclsure als cntaining p and having σ as crner, then the tw germs are said t be equivalent R-trees. When the grup is SL 2, the standard apartment is just R and the enclsures are clsed bunded segments. The categry f cnstructins gives a gd pint f view fr the thery f R-trees. Fr example, if q is a quadratic differential defining a spectral multivalued differential φ i = ± q n a cmpact Riemann surface X, then the tree T ϕ f leaves f the fliatin Rφ n X may be seen as a sheaf n E SL2 as fllws: fr a segment E let T pre φ (E) be the space f differentiable maps E X which are transverse t the fliatin such that the pullback f Rφ is the standard differential dx n E R. These maps are taken mdul the relatin that tw maps are the same if they map pints f E t the same leaves f the fliatin. Then T φ is the assciated sheaf. Thus T φ (E) is the space f maps frm E t the leaf space, which are represented n finitely many segments cvering E by differentiable maps int X The SL 3 case. We nw specialize t the case f buildings fr the grup SL 3. The affine space is A = R 2. The spherical Weyl grup is the symmetric grup acting thrugh its irreducible 2-dimensinal representatin. Sme enclsures were pictured in Figure 7. There are three directins f reflectin hyperplanes. These divide the vectr space at the rigin int six 2-dimensinal sectrs, acted upn transitively by the Weyl grup. On the ther hand, there are tw rbits fr the 1-dimensinal sectrs, the even vertices f the hexagn and the dd vertices. Therefre ur categry f sectrs S is equivalent t the fllwing categry S : there are an bject η, crrespnding t the 2- dimensinal sectrs, and tw bjects ν + and ν crrespnding t the 1-dimensinal sectrs. We chse ne f these dented ν + which we say has psitive rientatin. The mrphisms are ν η ν +. A spherical cnstructin H is a sheaf n S. This cnsists f three sets H(η), H(ν + ) and H(ν ) with mrphisms H(ν ) H(η) H(ν + ).

18 18 KATZARKOV, NOLL, PANDIT, AND SIMPSON Such a structure may be viewed as a graph whse edges are H(η) with vertices gruped int the psitive nes H(ν + ) and negative nes H(ν ). Each edge jins a psitive vertex t a negative vertex. A spherical cnstructin is equivalent t such a graph. If we have a cnstructin F fr this Weyl grup and if x F (p) is a pint then we btain a spherical cnstructin F x which is a graph as abve. Fllwing the simple characterizatin which was given fr example by Abramenk and Brwn [1], we say that a spherical cnstructin is a spherical building if any tw vertices are at distance 3, every pair f vertices is cntained in a hexagn, and if there are n lps f length 4 (the length f a lp has t be even because f the parity prperty f edges). A spherical cnstructin is a spherical pre-building if it is cnnected, if every nde is cntained in at least ne edge, and if it has n lps f length 4. The cnstructin f [17] gives a way f ging frm a spherical pre-building t a spherical building. In a spherical pre-building, say that tw ndes are ppsite if they have ppsite parity and are at distance 3. If it is spherical building this means that they are ppsite ndes f any hexagn cntaining them. A segment is a 1-dimensinal enclsure S A. We nte that a segment has a natural rientatin. At any pint x S(p) in the interir, the spherical building S x has tw elements, S x,+ and S x,, nt jined by any edge; they are f ppsite parity and the psitive directin in S(p) is defined t be the directin ging twards S x,+. If x is an endpint then S x has ne element, riented psitive r negative respectively at the tw endpints f S. Let us dente by S t,+ the segment f length t based at the rigin, such that the parity f the single element f S t,+ 0 is psitive. We assume that these segments are all in the same line s that S t,+ S t,+ when t < t. Let S t, dente the segment with the ppsite rientatin at the rigin. Remark 4.8. Suppse F is a cnstructin. If ϕ : S t,+ F is a mrphism then fr x S t,+, the image f the psitive (resp. negative) element f Sx t,+ under ϕ is dented ϕ x,+ (resp. ϕ x, ) and it is a psitive (resp. negative) nde in the spherical building F ϕ(x). Same fr S t,+. Definitin 4.9. Suppse E is an enclsure, and F a cnstructin satisfying SPB-lc. We say that a map f : E F is immersive at a pint x E(p), if the map f spherical cnstructins E x F f(x) preserves distances. The map is immersive if it is immersive at all pints f E(p).

19 CONSTRUCTING BUILDINGS 19 Here, the distances in E x are calculated by cnsidering E x A x. In particular, if E is a segment then we cnsider the distance between the tw elements f E x t be 3. Thus a map frm a segment ϕ : S F is immersive at an interir pint x S if ϕ x,+ and ϕ x, are ppsite in the spherical pre-building F ϕ(x). We als use the terminlgy straight fr an immersive segment, and say that a map frm a segment is angular therwise. Intuitively, a map f : E F is immersive if and nly if f(p) : E(p) F (p) is lcally injective. Lemma Suppse that the spherical cnstructins F x are at least spherical pre-buildings. A map ϕ is immersive at all but at mst finitely many angular pints. We say that ϕ is immersive if there are n angular pints. We nw list sme extensin cnditins fr a cnstructin F (let us reiterate that we are wrking here in the SL 3 situatin). SPB-lc : that fr any x F (p) the spherical cnstructin F x is a spherical pre-building. SB-lc : fr any x F (p) the spherical cnstructin F x is a spherical building. Ex-Seg : let t dente the ther endpint f S t,+. Assuming SPBlc, fr any immersive map ϕ : S t,+ F and any element ν F ϕ(t) ppsite t ϕ t, and fr any t > t we ask that there exist an extensin f ϕ t an immersive map ϕ : [0, t ] F such that (ϕ ) + t = ν. Similarly fr segments S t, in the ppsite rientatin. We next get t ur main extensin statement, fr btuse angles. Sme ntatin will be needed first. Let P a,b,+ be parallelgrams centered at the rigin with side lengths a and b, and psitive rientatin f the tw edges at the rigin (resp. P a,b, with negative rientatin). We may assume that the first edge is the segment S a,+, and dente the secnd edge by ωs b,+ (it is btained by rtating by 120 degrees). Similarly let T a dente the triangle with edge length a. We cnsider bth segments S a,+ and > S a, t be edges f T a starting frm the rigin. Ex-Obt : Assuming SPB-lc, suppse we are given maps ϕ : S a,+ F and ψ : ωs b,+ F such that ϕ(0) = ψ(0) = x. Suppse that the distance between the elements ϕ 0,+ and ψ 0,+ in the spherical building F x is 2 (i.e. they are distinct). Suppse that the maps ϕ and ψ are immersive. Then there exists an immersive map ζ : P a,b,+ F cinciding with the given maps ϕ and ψ n the edges. We als ask the same cnditin with the ther rientatin, fr P a,b,.

20 20 KATZARKOV, NOLL, PANDIT, AND SIMPSON Ex-Side : given a segment ϕ : S F f length a with x = f(0) ne f the endpints, and an edge in the spherical building ν F x, ϕ extends t an immersed triangle T a F such that ν is nt in the image f ϕ 0. Lemma Suppse F is a cnstructin satisfying SPB-lc, Ex-Obt and Ex-Side. Then F satisfies SB-lc. Prf. Suppse x F (p). By hypthesis F x is a spherical pre-building. We wuld like t shw it is a spherical building. Frm Ex-Obt, we get that any tw ndes f the same parity in F x have distance 2. By hypthesis SPB-lc, the F x are cnnected graphs and all ndes are cntained in edges. It fllws that any tw ndes have distance 3. The cnditin Ex-Side implies that any nde in any f the spherical pre-buildings F x, is cntained in at least tw distinct edges. It nw fllws that any tw ndes are cntained in a hexagn. Thus F x is a spherical building. Lemma Suppse F is a cnstructin satisfying SB-lc, Ex-Obt and Ex-Seg. Then it satisfies Ex-Side. Furthermre, the sectr f the triangle can be specified at either f the endpints f the segment. Therem Suppse F is a cnstructin satisfying SB-lc, Ex-Seg, and Ex-Obt, such that F (p) is cnnected. Suppse x, y F (p). Then there exists an immersive map frm an enclsure f : E F such that x, y are in the image f f(p). Prf. (Sketch) Chse a path frm x t y, that is t say a cntinuus map frm [0, 1] t the tplgical space F (p). We assume a tplgical result saying that the path may be cvered by finitely many intervals which map int single enclsures. Frm this, we may assume that the path has the fllwing frm: it ges thrugh a series f triangles which are immersive maps ϕ i : T a F, such that ϕ i and ϕ i+1 share a cmmn edge. Ntice here that we may assume that all the triangles have the same edge length. We have x in the image f ϕ 1 and y in the image f ϕ k. We btain in this way a strip f triangles. We next nte that there are mves using Ex-Obt which allw us t assume that there are nt cnsecutive turns in the same directin alng an edge. Frm this cnditin, we may then extend using Ex-Obt and Ex-Side (which fllws frm Ex-Seg) t turn this strip f triangles, int a single immersive map frm an enclsure. We let A dente the sheaf represented by the affine space f the same ntatin. Mre precisely, fr an enclsure E a map E A cnsists f a factrizatin E E A where E A is an enclsure in its

21 CONSTRUCTING BUILDINGS 21 standard psitin, and E E is a flding map (that is, a map t Ẽ ). Thus A is a direct limit f its enclsures. In particular we knw what it means t have an immersive map frm A t smewhere, the map shuld be immersive n each f the enclsures in A. If F is a cnstructin, an apartment is an immersive map A F. Prpsitin Suppse F is a cnstructin satisfying SB-lc, Ex- Obt and Ex-Seg. Then any immersive map frm an enclsure E F extends t an apartment A F. Crllary If F is a cnstructin satisfying SB-lc, Ex-Obt and Ex-Seg, such that F (p) is cnnected, then any tw pints x, y F (p) are cntained in a cmmn apartment. Prf. Cmbine the previus prpsitin with Therem Prpsitin Suppse F is a cnstructin satisfying SB-lc. Then an immersive map E F frm an enclsure, is injective. The same is true fr an apartment A F. Definitin Say that F is a building if it is a cnstructin with F (p) cnnected and simply cnnected, satisfying SB-lc, Ex-Obt and Ex-Seg. Cnjecture If F is a building, then the tplgical space F (p) has a natural structure f R-building mdeled n the affine space A Pre-buildings. Definitin A pre-building is a cnstructin F which satisfies SPB-lc such that F (p) is simply cnnected. Suppse F is a pre-building. Recall that a segment s : S F is straight if it is immersive, that is fr any y S(p) which is nt an endpint, the tw directins f S y map t tw vertices f F s(y) which are separated by three edges. Definitin A map V a,b F t a pre-building is standard if the tw edges are straight and their tw directins at the rigin are nt the same. We wuld nw like t investigate the prcess f adding a parallelgram P a,b t a pre-building F alng a standard inclusin V a,b F t get a new pre-building. The prblem is that a similar parallelgram, r a part f it, might already be there. Fr example if we add P a,b t itself by taking a pushut alng V a,b then we generate a furfld pint at the rigin.

22 22 KATZARKOV, NOLL, PANDIT, AND SIMPSON Thus the need fr a prcess which we call flding in, f jining up the new P a,b with any part f it that might have already been there. This will be used in the small-bject argument belw as well as in ur general mdificatin prcedure later. Suppse F is a pre-building and d : V a,b F is a standard inclusin. Define R d P a,b t be the unin f all f the P c,d P a,b based at the rigin, such that there exists a map n the bttm making the fllwing diagram cmmute: V c,d V a,b P c,d F. One can shw that the map, if it exists, is unique. We btain a subsheaf R P a,b and there is a map R F cmbining all f the previusly mentined maps. The flding-in f P a,b alng d is the pushut G := F R P a,b. The relatin R is designed s that G again satisfies SPB-lc. Since R was defined as a very general unin f things, there remains pen the smewhat subtle technical issue, which we haven t treated, f shwing that G is a cnstructin. The flding-in G accepts a map F V a,b : P a,b G and we have the fllwing universal prperty: Lemma The flding-in G is a pre-building. Suppse B is a prebuilding and f : F B is a map such that the image f d is again a standard inclusin int B (as will be the case fr example if f is nn-flding). Suppse that V a,b B cmpletes t a map P a,b B. Then these factr thrugh a unique map G B. We als cnjecture a versality prperty fr maps t buildings: if F B is any map t a building then there exists an extensin t G B, hwever this extensin might nt be unique. Because f the nn-uniqueness n the interir pieces making up R, this cnjecture requires a study f cnvergence issues and the statement might need t be mdified The small-bject argument. The cnditins Ex-Seg, Ex-Obt and Ex-Side are extensin cnditins f the fllwing frm: we have a certain arrangement R cnsisting f sme pints r segments, and a map R E t an enclsure putting R n the bundary f E. Then the cnditins state that fr any immersive map R F there exists an extensin t an immersive map E F.

23 CONSTRUCTING BUILDINGS 23 These cnditins may be ensured using the small bject argument. Given F we may define Ex(F ) t be the cnstructin btained by a successive infinite family f pushuts alng the inclusins R E, using the flding-in prcess described in the previus subsectin fr Ex Obt. Therem If F is a pre-building, and if we frm F := Ex(F ) by iterating up t the first infinite rdinal ω, then F is a building. Prf. (Sketch) We first nte that F is a cnstructin, and F (p) is simply cnnected. Als F again satisfies SBP-lc. By the small bject argument, F satisfies the extensin prperties Ex-Seg, Ex-Obt and Ex-Side. Fr Ex-Obt, the flding-in prcess nly adds parallelgrams alng standard inclusins, but this turns ut t be gd enugh t get it in general. Hence, F satisfies SB-lc, s it is a building. In particular any tw pints f F (p) are cntained in a cmmn apartment. Cnjecture 4.18 then says that F (p) has a natural structure f R- building. Cnjecture Suppse F is a pre-building, and let F be the building btained by the small-bject argument. Then it is versal: fr any ther map t a building F B there exists a factrizatin thrugh F B. The difficulty is that we needed t use the flding-in cnstructin t cnstruct F, and the versality prperty fr the flding-in cnstructin is nt easy t see because f the pssibly infinite nature f the relatins R cupled with nn-uniqueness f extensins t P a,b fr nnstandard maps V a,b B. The small-bject cnstructin may need t be mdified in rder t get t this versality cnjecture. 5. An initial cnstructin The next step is t t relate cnstructins t φ-harmnic maps. Cnsider a Riemann surface X tgether with a spectral curve, defining a multivalued differential φ. Recall that lcally n X, φ may be written as (φ 1,..., φ n ) with φ i hlmrphic, and φ i = 0. In the present paper we cnsider spectral curves fr SL 3 s φ = (φ 1, φ 2, φ 3 ). The fliatins are defined by Rφ ij = 0. In ur SL 3 case there are three fliatin lines ging thrugh each pint f X. Fr any pint x X there exists a neighbrhd U x, with the prperty [17] that U x has t map t a single apartment under any φ- harmnic map. Thus lcally, a φ-harmnic map t a building factrs thrugh the map h x : U x A

24 24 KATZARKOV, NOLL, PANDIT, AND SIMPSON t the standard apartment given by integrating the frms Rφ i with basepint x. The fliatin lines defined by Rφ ij = 0 are just the preimages in U x f the reflectin hyperplanes f A. The preimage f a small enclsure near the rigin in A will therefre be a dmain in U x whse bundary is cmpsed f fliatin lines fr the three fliatins. We may shrink U x s that its clsure is itself such a dmain The Ω pq argument. Recall briefly the prf f [17] shwing existence f such a neighbrhd U x. A path γ : [0, 1] X is nncritical if the differentials γ (Rφ i ) remain in the same rder all alng the path. Fr any φ-harmnic map t a building h : X B, the image f a nncritical path has t be cntained in a single apartment. One shws this using the prperty f a building that says tw ppsite sectrs based at any pint are cntained in a single apartment. Suppse p, q X are jined by sme nncritical path. Let Ω pq dente the subset swept ut by all f these paths, in ther wrds it is the set f all pints y X such that there exists a nncritical path γ with γ(0) = p, γ(1) = q and γ(s) = y fr sme s [0, 1]. We claim that Ω pq maps int a single apartment. Suppse y, y Ω pq and let γ and γ dente the crrespnding paths. We have apartments A, A B such that h γ ges int A and h γ ges int A. Nw A A is a Finslercnvex subset in either f the tw apartments, and it cntains p and q. In particular it cntains the parallelgram with ppsite endpints p and q. It fllws that it cntains the images f the tw paths, s h(y) and h(y ) are bth in A. Letting y vary we get that h(ω pq ) A. The φ i admit a unifrm determinatin ver Ω pq, and the map Ω pq A is determined just by integrating the real ne-frms Rφ i. Nw if x X, we may find nearby pints p, q X such that x is in the interir f Ω pq. This is easy t see away frm the caustic lines. The caustic lines are transverse t all f the differentials s if x lies n a caustic ne can chse p and q n the caustic itself, as will shw up in ur pictures later. This still wrks at an intersectin f caustics t. Our neighbrhd U x is nw chsen t be any neighbrhd f x cntained in sme Ω pq. This cnstructin is unifrm, independent f the harmnic map h Caustics. A pint x X is n a caustic if the three fliatin lines are tangent. This is equivalent t saying that the three pints φ i (x) T X x are aligned, i.e. they are n a single real segment. The caustics are the curves f pints in X satisfying this cnditin. We include als the branch pints in the caustics. There is a single caustic cming ut f each rdinary branch pint.

25 CONSTRUCTING BUILDINGS 25 The caustics play a fundamental rle in the gemetry f harmnic maps t buildings, specially in the SL 3 case. Let C X dente the unin f the caustic curves. Then, fr any x X C the map h x : U x A is etale at x. Hence, if h : X B is a φ-harmnic map t a building, h is etale nt the lcal apartments utside f C. In ther wrds, the lcal integrals f any tw f the differentials prvide lcal crdinate systems n X C. On the ther hand, h flds X alng C. We may als view this as determining a flat Riemannian metric n X C, pulled back frm the standard Weyl-invariant metric by the lcal maps h x t the standard apartment btained by integrating Rφ i. The metric has a distributinal curvature cncentrated alng C. Frm the pictures it seems that the curvature is everywhere negative, and frm ur prcess we shall see that the ttal amunt f curvature alng a single caustic jining tw branch pints gives an excess angle f Nn-caustic pints. If x X C is a nn-caustic pint, then we may assume that the neighbrhd U x maps ismrphically t the interir f an enclsure in the standard apartment. The enclsure E x culd be chsen as a standard hexagn r perhaps a standard parallelgram fr example. We have chsen U x such that fr any harmnic φ-map t a building h : X B, the map U x B factrs thrugh an apartment A B via the map h x : U x A given by integrating the Rφ i. The map h x factrs thrugh an affine (nn-flding) map E x A. Altgether we get a factrizatin (1) U x E x B and the map E x B desn t fld alng any edges passing thrugh x. When x is a branch pint r n a caustic, we can still get a lcal factrizatin f the frm (1) thrugh a cnstructin E x, as will be discussed next Smth pints f caustics. Suppse x C is a pint in the smth lcus f a caustic curve. Then the lcal integratin map flds every neighbrhd f x in tw alng C. The image f U x by h x is flded alng C. By intersecting with a smaller enclsure cntaining the image f x, we may assume that U x has the prperty that there is an enclsure E x with h x : U x E x being a prper 2 t 1 cvering flded alng C. We may als assume that E x is the cnvex hull f the clsed h x (U x ).

26 26 KATZARKOV, NOLL, PANDIT, AND SIMPSON Fr the generic situatin, there are tw ther types f pints that need t be cnsidered: the intersectins f caustics, and the branch pints Branch pints. Fr the branch pints, lcally tw f the differentials say φ 1 and φ 2 cme tgether, and the third ne culd be cnsidered as independent. Therefre, a φ-harmnic map lks lcally like the prjectin t the tree f leaves f a quadratic differential, crssed with a real segment. It means that we are frced t cnsider a singular cnstructin rather than an enclsure. This singular cnstructin still dented E x may be taken fr example as the unin f three half-hexagns jined alng their diameters. Figure 8. Three half-hexagns If x = b is a branch pint then we may chse the neighbrhd U x tgether with a map h x : U x E x such that E x is the cnvex hull f the image. The image f U x, shaded in abve, lks lcally like the ne that we saw in the BNR example [17] Crssing f caustics. The case where x is a crssing pint f tw caustics is new, nt appearing in the BNR example. We have therefre lked fairly clsely at this situatin in ne f the next basic examples. A spectral netwrk with the tw crssing caustics will be shwn later as Figure 22 f Sectin 9. The Ω pq argument wrks als here, s we have a neighbrhd U x f x which has t map t a single apartment in any φ-harmnic map t a building. The tw caustics divide U x int fur sectrs. The map h x : U x A is 1 t 1 in the interir f tw f the ppsing sectrs; it is 3 t 1 in the interir f the ther tw ppsing sectrs; it is 2 t 1 alng the caustics and 1 t 1 at x. This is emphasized by including the images f tw circles in the picture shwn in Figure 9.

27 CONSTRUCTING BUILDINGS 27 Figure 9. A neighbrhd f the crssing We chse U x as a hexagnal shaped regin shwn in Figure 9 n the left. The image E x in a standard apartment is a hexagn-shaped enclsure, shwn n the right in Figure 9. As said abve, the map is 3 : 1 ver the thin middle regins between the tw caustics. The map h x : U x E x is prper and U x is the inverse image f E x The initial cnstructin. We nw put tgether the abve neighbrhds t get a gd cvering f X. Therem 5.1. There exists a finite cvering f X by pen sets U i, and cnstructins E i as cnsidered abve (either an enclsure r the unin f three half-hexagns), tgether with maps h i : U i E i such that fr any harmnic φ-map t a building h : X B, there is an ismetric embedding E i B such that h factrs thrugh h i and indeed U i is a cnnected cmpnent f h 1 (E i ). Fr the intersectins U ij := U i U j, we btain cnstructins E ij with inclusins t bth E i and E j, such that E ij is the cnvex hull f h i (U ij ) in U i (resp. the cnvex hull f h j (U ij ) in U j ). Define Z := i E i where the relatin is btained by identifying the E ij E i with E ij E j. Therem 5.2. This defines a cnstructin Z. We have a φ-harmnic map h Z : X Z and it has the fllwing universal prperty: fr any φ-harmnic map t a building h : X B, there is a factrizatin h = h h Z fr a unique map f cnstructins h : Z B. The cnstructin Z is called ur initial cnstructin. It is nt a prebuilding because it will nt, in general, have the required nnpsitive

28 28 KATZARKOV, NOLL, PANDIT, AND SIMPSON curvature prperty. Fr example the initial cnstructin described in Sectin 7 will have furfld pints a 1 and a 3. Lemma 5.3. We can nnetheless insure that the lcal spherical cnstructins f Z dn t have cycles f length 2. In the next sectin we lk at hw t mdify the initial cnstructin in rder t remve the psitively curved pints, that is t say the pints where the spherical building has a cycle f length Mdifying cnstructins In this sectin we cnsider hw t g frm the initial cnstructin t a pre-building by a sequence f mdificatin steps. The reader is referred t Sectin 7 fr an illustratin f the varius peratins t be described here. These peratins are very similar in spirit t fldings and trimmings in the thery f Stallings graphs [25, 15, 23], and the tw-manifld cnstructin that cmes ut at the end shuld be cnsidered as a cre Scafflding. In rder t keep track f what kind f flding happens, we first lk at sme extra infrmatin that can be attached t a cnstructin. In this subsectin we remain as usual in the SL 3 case. Let F be a cnstructin. An edge germ f F is defined t be a quadruple (x, v, a, b) where x F (p) and v is a vertex in the spherical cnstructin F x, and a and b are edges in F x sharing v as a cmmn endpint. There is a change f dimensin when passing frm the spherical building t the cnstructin itself, s the vertex v crrespnds t a germ f 1-dimensinal segment based at x, and the edges a and b crrespnd t germs f 2-dimensinal sectrs based at x that are separated by the segment. Suppse f : F G is a map f cnstructins. If (x, v, a, b) is an edge germ f F, then the images f(a) and f(b) are edges in G f(x) sharing the vertex f(v). We say that f flds alng (x, v, a, b) if f(a) and f(b) cincide. We say that f pens alng (x, v, a, b) if it desn t fld. Let EG(F ) dente the set f edge germs f F. A scafflding f a cnstructin F is a pair σ = (σ, σ f ) such that σ and σ f are disjint subsets f EG(F ). The first set σ is said t be the set f edge germs which are marked pen, and the secnd set σ f is said t be the set f edge germs which are marked fld. If f : F G is a map f cnstructins, and σ is a scafflding f F, we say that f is cmpatible with σ if f flds alng the edge germs in σ f and pens alng the edge germs in σ.

29 CONSTRUCTING BUILDINGS 29 Implicit in this terminlgy is that G was prvided with a fully pen scafflding (such as will usually be the case fr a pre-building). Mre generally a map between cnstructins bth prvided with scaffldings is cmpatible if it maps the pen edge germs in F t pen nes in G, and fr edge germs marked fld in F it either flds them r else maps them int edge germs marked fld fr G. Definitin 6.1. A scafflding is full if σ σ f = EG(F ). A scafflding is cherent if there exists a building B and a map h : F B cmpatible with σ. It shuld be pssible t replace the definitin f cherence with an explicit list f required prperties, but we dn t d that here. Recall that we will be wrking under the assumptin f existence f sme φ-harmnic map s the abve definitin is adequate fr ur purpses. One f the main prperties fllwing frm cherence is a prpagatin prperty. A neighbrhd f a hexagnal pint cannt be flded in an arbitrary way. One may list the pssibilities, the main ne being just flding in tw; nte hwever that there is anther interesting case f three fld lines alternating with pen lines. Certain cases are ruled ut and we may cnclude the fllwing prperty: Lemma 6.2. If σ is cherent, and x is a hexagnal pint, then if tw adjacent edge germs at x are in σ it fllws that the tw ppsite edge germs are als in σ. This will mainly be used t prpagate the pen r fld edge germs alng edges which are straight in the fllwing sense. This definitin cincides with Definitin 4.9 fr a pre-building prvided with the fully pen scafflding. Definitin 6.3. Suppse S is a segment and ϕ F (S). Suppse σ is a scafflding fr F. We say that ϕ is straight if, at any pint x in the interir f ϕ (that is t say x F (p) is the image f a pint a S(p) which is nt an endpint), the frward and backward directins alng ϕ in the spherical cnstructin F x are separated by three edges a, b, c such that the edge germs ab and bc are in σ. Here the edge germ dented ab crrespnds t the vertex separating the edges a and b in the spherical cnstructin. Crllary 6.4. Suppse ϕ is a straight edge and σ a cherent scafflding. Then the marking f bth frward and backward edge germs f ϕ is the same all alng ϕ. Straight edges are mapped t straight edges:

30 30 KATZARKOV, NOLL, PANDIT, AND SIMPSON Lemma 6.5. Suppse F is a cnstructin with scafflding σ and ϕ F (S) is an edge. If h : F B is any map t a building cmpatible with σ, and if ϕ is straight with respect t σ, then the image f ϕ is cntained in a single apartment f B and is a straight line segment in that r any ther apartment. Scafflding is used t keep track f the additinal infrmatin which cmes frm a φ-harmnic map, namely that small pen neighbrhds in X map withut flding t apartments in a building. Fr trees, this was illustrated in Figures 3, 4 and 5. Prpsitin 6.6. The initial cnstructin Z f Therem 5.2 is prvided with a cherent full scafflding σ Z, such that if x X C is a nn-caustic pint, then all edge germs in the hexagn in Z hz (x) image f the lcal spherical cnstructin f E x at the rigin, are in σ Z. If B is a building, there is a ne-t-ne crrespndence between harmnic φ-maps X B and cnstructin maps Z B cmpatible with σ Z. Prf. (Sketch) All edge germs based at any pint x X C are in σ by the remark at the end f Sectin 5.3. In the regins near caustics, intersectins f caustics r branch pints, the lcal cnstructins E x map int any building withut flding, s all f their edge germs are in σ. The remaining edges are always straight segments attached t furfld pints, and at these furfld pints tw f the edges are already marked pen s the ther tw must be marked fld. This is illustrated in the BNR example in Sectin 7. The scafflding is cherent because we are assuming that there exists at least ne φ-harmnic map such as can be btained frm Parreau s thery [22] [17] Cutting ut. We nw describe an imprtant peratin. Suppse (F, σ) is a cnstructin with cherent scafflding. Suppse P a,b is a parallelgram (with either rientatin), and dente by V a,b the unin f tw segments based at the rigin n the bundary f P a,b. Suppse that F may be written as a pushut F = G V a,b P a,b alng a map g : V a,b G. Let σ G be the induced scafflding f G. Suppse that the tw segments making up V a,b, f lengths a and b respectively, are straight in G with respect t σ G. Therem 6.7. In the abve situatin, if B is a building, then any map h G : G B cmpatible with σ G extends uniquely t a map h : F B cmpatible with σ. Furthermre, under this map the image f the parallelgram P a,b is ismetrically embedded in B and desn t fld any edge germs f EG(P a,b ).

31 CONSTRUCTING BUILDINGS 31 In the abve situatin, we refer t G as being btained frm F by cutting ut the image f the parallelgram P a,b. Ntice that all f the edge germs in EG(P a,b ) EG(F ) are either marked pen, r nt marked. We may extend the scafflding σ t ne σ 1 with σ f 1 = σ f and σ 1 = σ EG(P a,b ), and the therem implies that any map frm F t a building cmpatible with σ must als be cmpatible with σ 1. If σ is full then f curse cherence implies that σ 1 = σ. Nte als that if σ is full then σ G is full. We say that (F, σ) is trimmed if it is nt pssible t cut ut any parallelgrams in the abve way. Therem 6.8. The initial cnstructin Z f Therem 5.2, prvided with the scafflding σ Z f Prpsitin 6.6, may be trimmed. The result is a tw-manifld cnstructin Z 0 again prvided with a cherent full scafflding σ 0. If B is a building, there is a ne-t-ne crrespndence between: Harmnic φ-maps X B; Maps Z B cmpatible with σ Z ; and Maps Z 0 B cmpatible with σ Pasting tgether. The cnstructin Z 0 nw btained will have, in general, many furfld pints f psitive curvature. These cannt appear in the image f a map Z 0 B, s sme flding must ccur. The directin f the flds is determined by the scafflding, and indeed that was the reasn fr intrducing the ntin f scafflding: withut it, there is nt a priri any preferred way f determining the directin t fld a furfld pint. Hwever, nce the directin is specified, we may prceed t glue tgether sme further pieces f the cnstructin accrding t the required flding. This is the pasting tgether prcess. Put W a,b := P a,b V a,b P a,b. We have a prjectin π : W a,b P a,b given by the identity n each f the tw pieces. Here we may use either f the tw pssible rientatins, that desn t need t be specified but f curse it shuld be the same fr bth parallelgrams. The lcal spherical cnstructin f W a,b at the rigin 0 W a,b is a graph with a single cycle f length 4, in ther wrds the rigin is a furfld pint. In particular, if W a,b B is any map t a building, tw f the fur edge germs at 0 must be flded. There is a chice here: at least tw ppsite edge germs must be flded, but the ther tw culd

32 32 KATZARKOV, NOLL, PANDIT, AND SIMPSON either be pened r als flded. We are interested in the case when the riginally given parallelgrams are nt flded. Let v 1, v 2 be the edge germs at the rigin in W a,b which are in the middle f the spherical cnstructins f the tw pieces P a,b (the spherical cnstructin f P a,b at the rigin is a graph with tw adjacent edges and three vertices, the middle edge germ crrespnds t the middle vertex). Prpsitin 6.9. Suppse we are given a cherent scafflding σ W f W a,b. We assume that v 1 and v 2 are in σw, and that the tw edges cmprising V a,b are straight. Then any map h W : W a,b B t a building, cmpatible with σ W, factrs thrugh the prjectin W a,b π P a,b h P B via a map h P which is an ismetric embedding f the parallelgram int a single apartment f B. Let σ W,max be the full scafflding defined by taking all edge germs f each P a,b t be pen, and letting the edge germs alng the edges f V a,b be flded. A crllary f the prpsitin is that any cherent scafflding f W a,b satisfying the hyptheses has t be cntained in σ W,max. Nw suppse F is a cnstructin with cherent scafflding σ. Suppse we have an inclusin W a,b F and suppse that the induced scafflding σ W f W a,b satisfies the hyptheses f the prpsitin, namely the angle between v 1 and v 2 is pen and the edges f V a,b are straight. Define the qutient F := F W a,b P a,b which amunts t pasting tgether the tw parallelgrams which make up W a,b. Crllary Under the abve hyptheses, any map t a building h : F B cmpatible with σ factrs thrugh a map F B. This peratin may be cmbined with the cutting-ut peratin. Let V a,b dente the ppsite cpy f V a,b inside P a,b. We may write W a,b = V a,b p p V a,b, it is a rectangular ne-dimensinal cnstructin frmed frm tw edges f length a and tw edges f length b (again as usual, we shuld chse

33 CONSTRUCTING BUILDINGS 33 ne f the tw pssible rientatins fr everything). We may define a prjectin W a,b V a,b as the identity n the tw pieces. The cmbinatin f pasting tgether and cutting ut may be summarized in the fllwing therem. Therem Suppse (F, σ) is a cnstructin with cherent scafflding. Suppse that F can be written F = G W a,b W a,b. Suppse the scafflding σ W induced by σ n W a,b satisfies the hyptheses f Prpsitin 6.9. Define Ĝ := G W a,b V a,b. We may write F = Ĝ V a,b P a,b. Furthermre, σ induces scaffldings σ G n G and σĝ n Ĝ. Let σ F be the scafflding f F btained frm σĝ by declaring all edge germs f P a,b t be pen. Assume that V a,b Ĝ is standard with respect t the scafflding σ Ĝ. Then there is a ne-t-ne crrespndence between the sets { } maps h : Ĝ B t a building cmpatible with σ Ĝ and {maps h : F B t a building cmpatible with σ, } bth being the same as { maps h : F } B t a building cmpatible with σ F via the natural maps Ĝ F F. The scafflding σĝ is never full, because it desn t say what t d alng the new edges which have been intrduced by glueing tgether the apprpriate edges in W a,b. This is the main prblem fr iterating the cnstructin, and it eventually leads t the need t lk fr BPS states.

34 34 KATZARKOV, NOLL, PANDIT, AND SIMPSON 6.4. Tw-manifld cnstructins. A tw-manifld cnstructin is a cnstructin F such that the tplgical space F (p) is a 2-dimensinal manifld. Lemma A cnstructin F is a tw-manifld if and nly if its lcal spherical cnstructins F x are plygns. The plygns have an even number f edges because f the rientatins f enclsures. Suppse F is a tw-manifld cnstructin. A pint x F (p) is a flat pint if F x is a hexagn; it is psitively curved pint if F x has tw r fur sides, and it is a negatively curved pint if F x has eight r mre sides. Usually we will deal with just three kinds, the rectangular pints, the flat r hexagnal pints, and the ctagnal pints. We have the fllwing imprtant bservatin which shws that the tw-manifld prperty is preserved by the cmbinatin f pasting tgether and cutting ut. Lemma In the situatin f Therem 6.11, suppse F is a twmanifld cnstructin. Then Ĝ is als a tw-manifld cnstructin. Suppse F is a tw-manifld cnstructin with n twfld pints. Suppse x F (p) is a rectangular pint (i.e. F x has fur edges), and suppse σ is a full cherent scafflding such that tw f the edge germs at x are in σ. Then there exists a cpy f W a,b F sending the rigin t x, and by fullness and cherence f σ we may assume that we are in the situatin f Therem 6.11, meaning that the tw edges f V a,b are straight and the angle between them is pen. We may assume that it is a maximal such cpy. The prcess f Therem 6.11 yields a new tw-manifld cnstructin Ĝ The reductin prcess. We may nw schematize the reductin prcess btained by iterating the abve peratin. We are ging t define a sequence f tw-manifld cnstructins F i with full cherent scaffldings σ i. The first ne F 0 will be btained by trimming sme initial cnstructin. We assume that there are never any twfld pints. We will be happy and the sequence will stp if we reach a cnstructin F i which has n furfld pints, s it is a pre-building. Suppse we have cnstructed the (F i, σ i ) fr i k. We suppse that F k has n twfld pints, and that at every rectangular pint, tw f the edges are marked pen. Chse a rectangular pint x F k and let W a,b F k be a maximal cpy with the rigin crrespnding t x, such that the edges f V a,b are straight. Applying the pasting tgether

35 CONSTRUCTING BUILDINGS 35 and cutting ut prcess f Therem 6.11, we btain a new tw-manifld cnstructin F k+1 := Ĝ. Hypthesis We assume Ĝ has n twfld pints, and that there is a unique way t extend the scafflding σĝ t a full cherent scafflding σ k+1 n F k+1 such that there are tw pen edges at any new rectangular pints. Crllary If this hypthesis is satisfied at each stage, then the reductin prcess may be defined. One f the main ideas which we wuld like t suggest is that this hypthesis will be true if there are n BPS states. Hw the reductin prcess wrks, and the relatinship with BPS states, will be discussed further in Sectin 8 belw Recvering the pre-building. Suppse we start with a cnstructin F and scafflding σ, then trim it t btain a tw-manifld cnstructin F 0 withut twfld pints. Suppse then that Hypthesis 6.14 hlds at each stage s we can define the reductin prcess, and suppse that the prcess stps after a finite time with sme F k which has n psitively curved pints. Then we wuld like t say that this allws us t recver a pre-building with a map frm F. We explain hw this wrks in the present subsectin. The first stage is t und the initial trimming. Put G 0 := F. Then apply Therem 6.7 successively t btain a sequence f cnstructins G i, with G i 1 playing the rle dented F in Therem 6.7. Hence, G i 1 is a pushut f G i with a standard parallelgram. If this prcess stps at a trimmed cnstructin F 0 = G k then we may successively d the pushuts t get G k 1, G k 2,..., G 0 = F. Thus F is btained frm F 0 by a sequence f pushuts with standard parallelgrams. Next, starting with F 0 we d a sequence f pasting and cutting peratins t yield tw-manifld cnstructins F i. Here, F i is btained frm F i 1 by applying Therem In particular, we have a cnstructin F i 1 which is n the ne hand the result f a pasting peratin applied t F i 1, but n the ther hand is a pushut f F i alng a standard parallelgram. Define inductively a sequence f cnstructins H i, starting with H 0 := F, as fllws. At each stage, we will have F i H i, and H i is btained frm F i by a sequence f pushuts alng standard parallelgrams (t be precise this means pushuts alng the inclusin V a,b P a,b ). This hlds fr H 0.

36 36 KATZARKOV, NOLL, PANDIT, AND SIMPSON Suppse we knw H i 1. Then F i 1 is a pasting f F i 1 as in Crllary D the same pasting t H i 1 instead f F i 1, in ther wrds H i := H i 1 F i 1 Fi 1. By the inductive hypthesis, H i 1 is a pushut f F i 1 by a series f standard parallelgrams, thus it fllws that H i is a pushut f F i 1 by the same series. On the ther hand, F i 1 was a pushut f F i by a standard parallelgram. Therefre we btain the inductive hypthesis that H i is a pushut f F i alng a series f standard parallelgrams. We have a sequence f maps H i 1 H i. These are qutient maps. Cmpsing them we btain a series f maps F H i. If F was the initial cnstructin fr (X, φ) then we wuld get in this way a φ-harmnic map X H i. Under the hypthesis that the prcess leading t a sequence f F i stps at sme finite k, we btain a cnstructin H k. If F k has n furfld pints, we wuld like t say that H k is the pre-building; hwever, in putting back in the pushuts with standard parallelgrams, we might be intrducing new furfld pints. Therefre, as was dne previusly fr the small bject argument, we shuld apply the fldingin prcess f Lemma Precisely, having F k H k btained by a sequence f pushuts by standard parallelgrams, apply the flding-in prcess at each stage t transfrm the pushut int a gd pushut as described in Lemma 4.21 preserving the pre-building prperty. The resulting cnstructin is a pre-building H accepting a map F H. It is universal fr maps t pre-buildings. Therem Suppse that there are n BPS states and Cnjectures 8.4 and 8.5 hld. Then we btain a pre-building B pre φ := H and a φ- harmnic map h φ : X B pre φ which is universal fr φ-harmnic maps t pre-buildings cntaining extensins t the enclsures E x used fr the initial cnstructin. The small bject argument f Therem 4.22 yields a versal φ-harmnic map t a building. The map t the building is nly versal. Indeed, at a stage f the cnstructin f B pre φ where we add a pushut with a standard parallelgram, if we are given a map V a,b B which is nt itself standard (say fr example the tw edges cincide) then the extensin t a map P a,b B is nt unique.

37 CONSTRUCTING BUILDINGS The BNR example, revisited In rder t illustrate the prcedure described abve, we shw hw it wrks in the BNR example frm [17]. It was by cnsidering this example in a new way that we came upn the abve prcedure. Recall the picture frm [17, Figure 3] f the spectral netwrk cntaining tw singular pints b 1, b 2. At each singularity, there are tw spectral netwrk lines which meet lines frm the ppsite singularity, and the resulting fur segments delimit a diamnd-shaped zne in the middle f the picture. There is a single caustic C jining b 1 t b 2 acrss the middle f this regin. Recall that we had fund that a φ-harmnic map t a building wuld fld tgether this middle regin alng the caustic, identifying the tw cllisin pints. The resulting pre-building is cnical with a single singularity at the rigin. Here, eight sectrs frm a tw-manifld, with the cllisin being a negatively curved singular pint f ttal angle 480. There are tw additinal sectrs, frming a link acrss the middle f the ctagn. The image f the middle regin ges int these sectrs but is nt surjective due t the flding alng the caustic, see [17, Figure 4]. We nw illustrate hw this end picture cmes abut using the prcess described abve. The first step is t create the initial cnstructin. Fr this, let us cver the caustic by sme regins as illustrated in Figure 10. The regins which cver the caustic are the nes which are bunded by the tw fliatin lines ging frm a 0 t a 0, resp. a 2 t a 2, resp. a 4 t a 4. The first and last nes include the singularities b 1 and b 2. Each f these regins is flded under any φ-harmnic map. They are in fact Ω pq regins (except at the tw endpints). The image f the regins frm Figure 10 under the integratin map t an apartment, are shwn in Figure 11. The enclsures crrespnding t these flded pieces are sketched int the picture cmpleting the shaded regins t full parallelgrams. The initial cnstructin Z cnsists f glueing tgether these enclsures cvering the caustics, and then adding enclsures arund the remaining pints which are nt fr the mment glued tgether. The illustratin f Figure 11 may be viewed as a picture f Z, where the nn-shaded regins f the curve crrespnd t tw different sheets, an upper and a lwer ne crrespnding t the upper and lwer regins in Figure 10. The cmpletins f parallelgrams just belw the caustic are part f the initial cnstructin Z (these pieces shuld be cnsidered as shaded having nly ne sheet) but they are nt images f pints in X.

38 38 KATZARKOV, NOLL, PANDIT, AND SIMPSON Figure 10. Regins cvering the caustic The zne belw the dashed line is empty, that is t say it desn t crrespnd t any pints in Z, in Figure 11 and similarly fr the subsequent nes. Nw Z may be trimmed by taking ut the regins cntaining flded pieces, leaving Z 0 as shwn in Figure 12. Again there are tw sheets, which are jined tgether alng the edge frm b 1 t b 2 (and als alng the rays pinting utward frm b 1 and b 2 ). In this case the edge where the tw sheets are jined crrespnds t the dtted line, and as said abve, the zne belw the dtted line is empty. In Z 0 we have five pints a 0, a 1, a 2, a 3, a 4 which are nt hexagnal. The pints a 0, a 2, a 4 are eightfld whereas a 1, a 3 are furfld pints. The segments b 1 a 0 and a 4 b 2 are marked pen, due t the trivalent edges in the cnstructins which are placed at the singularities. Indeed, n the uter side f b 1 ne gets an pen edge, because we are in the image f a neighbrhd in x as was illustrated in Figures 3 and 4. This pen edge then prpagates int the segment b 1 a 0 by Lemma 6.4, because in Z 0 all the pints alng this edge are hexagnal, including b 1, and up t (but nt including) a 0. The segment is straight because n either side we are in the image f neighbrhds frm X. We may nw shw that the fur segments a 0 a 1, a 1 a 2, a 2 a 3 and a 3 a 4 are marked fld. Cnsider fr example the furfld pint a 1. Let u 1

39 CONSTRUCTING BUILDINGS 39 Figure 11. Images f the regins Figure 12. After trimming

40 40 KATZARKOV, NOLL, PANDIT, AND SIMPSON and u 1 be the pints which cmplete the tw parallelgrams spanned by a 0, a 1, a 2. The fur sectrs arund a 1 are a 1 a 0 u 1, a 1 a 0 u 1, a 1 a 2 u 1 and a 1 a 2 u 1. Suppse we are given a map Z 0 B t a building, cming frm a φ-harmnic map h : X B. The tw sectrs a 1 a 0 u 1 and a 1 a 2 u 1 cme frm tw adjacent sectrs at a pint in X (ne f the lifts f the pint a 1 ). Since the map h desn t fld anything in X C, we cnclude that there is n flding alng the edge a 1 u 1. Similarly fr a 1 u 1. But since there are n cycles f length 4 in the lcal spherical buildings f B, the fur sectrs have t be cllapsed smehw. Therefre, ur map must fld alng the segments a 1 a 0 and a 1 a 2. The same argument at a 3 shws that the map must fld alng a 3 a 2 and a 3 a 4. Therefre the fur segments a 0 a 1, a 1 a 2, a 2 a 3 and a 3 a 4 are marked fld. All ther edges f Z 0 crrespnd t edges in X, s we cnclude that all edges except fr these fur are marked pen. We may nw lk at hw the pasting-tgether prcess wrks. The first step is t paste tgether the tw parallelgrams a 0 a 1 a 2 u 1 and a 0 a 1 a 2 u 1. Similarly we paste tgether a 2 a 3 a 4 u 3 and a 2 a 3 a 4 u 3. The result is shwn in Figure 13. Figure 13 Figure 14 Then we can cut ut the parallelgrams btained frm the abve pasting. This gives the picture shwn in Figure 14 which is again a tw-manifld cnstructin Z 1. As befre there are tw sheets jined alng the edge. Nw there is ne remaining furfld pint at a 2 (ntice that what was previusly an eightfld pint has nw becme a furfld pint). Dente by u 1 and u 3 the images in Z 1 f the pairs f pints (u 1, u 1) and (u 3, u 3) respectively. By the same reasning as befre, the segments a 2 u 1 and a 2 u 3 are marked fld whereas all the ther edges are marked pen. The next and last step f the pasting-tgether prcess is t paste tgether the tw parallelgrams u 1 a 2 u 3 c and u 1 a 2 u 3 c where c and c are the tw cllisin pints frm X. This gives Figure 15.

41 CONSTRUCTING BUILDINGS 41 Figure 15 Figure 16 Cutting ut the parallelgram btained frm the pasting-tgether, gives the tw-manifld cnstructin Z 2 shwn in Figure 16. It has nly a single eightfld pint at the image f the tw cllisin pints which are nw identified. There are n furfld pints. Therefre we may back up and add back in all f the parallelgrams which were remved by the cutting-ut prcesses. This gives the picture shwn in Figure 17 which is the universal pre-building accepting a map frm X. Here as befre, the unshaded regins abve and t the left crrespnd t tw sheets; the shaded regins as well as the remaining pieces f the parallelgrams just next t the dtted line, are single sheets, and everything belw the dtted line is empty. Figure 17. The pre-building When we turn this int a building using the small bject argument, we get back the universal building cnstructed in [17].

42 42 KATZARKOV, NOLL, PANDIT, AND SIMPSON This was a first illustratin f hw the reductin prcess intrduced in Sectin 6.5 leads t a universal pre-building. In the next sectin we discuss sme further aspects f the prcess in the general case. 8. The prcess in general In general we have the fllwing setup: starting with (X, φ) we first make the initial cnstructin Z, then trim it t get a tw-manifld cnstructin Z 0. Then we g thrugh a sequence f steps f pastingtgether then cutting-ut described in Sectin 6.5. This yields a sequence f tw-manifld cnstructins Z i (these were dented F i in Sectin 6 but we change t the ntatin Z i here in rder t think f them as mdificatins f the riginal surface Z). The cnstructin Z 0 has a cherent full scafflding by Lemma 6.6. In rder t prceed with the cnstructin at each step, we wuld like t knw that after each peratin the new Z i may still be given a uniquely defined full cherent scafflding, which is Hypthesis If this hypthesis hlds at each stage, then we can cntinue the peratin and hpe that it cnverges lcally at least. In this sectin we explain sme ideas fr hw t understand mre precisely the sequence f cnstructins, and hw t see Hypthesis 6.14 saying that full scaffldings will be determined at each step, if there are n BPS states Prperties f scaffldings. Let us first aximatize sme prperties f ur tw-manifld cnstructins and their scaffldings. Our tw-manifld cnstructins Z i are prvided with scaffldings in which almst all f the edges are marked pen r. Thse vertices in the lcal spherical buildings which are marked fld r f, are parts f straight edges (that is, edges such that at each pint, at least ne side f the edge has bth directins transverse t that edge, marked ). The marking is the same alng the straight edge. Furfld pints We require that, at any furfld pint there shuld be tw ppsite edges marked, and tw ppsite edges marked f: f f

43 CONSTRUCTING BUILDINGS 43 The cnfiguratin with all fur edges flded is admissible frm the pint f view f cherence, but thrughut ur prcedure we cnjecture that it shuld nt ccur. Hexagnal pints At a hexagnal pint, the pssibilities are as fllws. First, either all edges are pen r tw ppsite edges are flded. f f It is als pssible t have three and three f alternating; this cnfiguratin shuld actually be cnsidered as a superpsitin f a 4v pint plus an 8v pint starting a new pstcaustic curve. Furthermre, ne additinal edge can als be flded. f f f f f f f Finally, a hexagnal pint can have all edges flded. Again, we cnjecture that this desn t appear in ur prcedure. Eightfld pints At an eightfld pint, there are the fllwing main pssibilities. First, a BNR pint with all edges pen. Then, a pint with a single fld line, as happens at the end f a pstcaustic curve, initially cming frm a singularity. f

44 44 KATZARKOV, NOLL, PANDIT, AND SIMPSON There are tw ways f having a pst-caustic pass thrugh the eightfld pint, either tw ppsite fld lines, as is mst standard; r tw adjacent fld lines,. f f f f Ntice that when the tw adjacent fld lines get flded up, the image is a hexagn, with ne sectr cvered three times by passing ver and back. This picture with tw adjacent fld lines cmes up, in particular, in the initial cnstructin when we have a crssing f tw caustic lines. It can als arise smetime later in the prcedure. Other eightfld pints several ther cnfiguratins are als admissible frm the pint f view f cherence, althugh we feel that they prbably will nt ccur in the prcedure. It is left t the reader t enumerate these. This finishes the descriptin f the expected prperties f pen and fld markings f ur scafflding. The set f pssible lcal pictures listed abve cvers the situatins which we meet alng the varius steps f the prcess. A scafflding f a tw-manifld cnstructin will be called admissible if its lcal pictures are in the abve list (cmpleted as per the preceding paragraph). Prpsitin 8.1. If Z i is a tw-manifld cnstructin with a full cherent scafflding, then the scafflding is admissible. Crllary 8.2. In the tw-manifld cnstructin btained by trimming the initial cnstructin, and then in the tw-manifld cnstructins which are met alng the way f ur prcess (assuming that there are n BPS states and Hypthesis 6.14 hlds), the full cherent scafflding is always admissible Pst-caustics. It is a cnsequence f the abve list f lcal pictures allwed fr an admissible scafflding, that the cllectin f pints which have an edge marked fld in the scafflding is rganized int a cllectin f piecewise linear curves marked as flded edges. Furthermre, it fllws that the endpints f the curves are 8v pints (r 6v triple pints), and alng a curve the successive segments have

45 CONSTRUCTING BUILDINGS 45 endpints which alternate as 8v, 4v, 8v,..., 4v, 8v, 4v, 8v. All pints utside f these are hexagnal i.e. flat. Call these curves pst-caustic curves, because they arise in the initial cnstructin as apprximatins t the caustics. We may think f Z i as a lcally flat surface but with singularities: psitive curvature at the 4v pints (ttal angle 240 ) and negative curvature (ttal angle 480 ) at the 8v pints. The ttal curvature f a pst-caustic curve is equivalent t ne 8v vertex Determining new markings. The next step is t start trying t remve the furfld pints by using the pasting-tgether cnstructin f Sectin 6.3. Suppse we have gtten t a tw-manifld cnstructin Z i. At a furfld pint we chse a standard pair f parallelgrams t be glued tgether. It may be seen frm the Ω pq argument that the tw parallelgrams must g t the same place in any harmnic map t a building. Here, by a standard parallelgram we mean ne in which all f the interir edges, including interir edges alng the sides, are labeled as pen. Thus in a map t a building the paths which lk nncritical, must actually map t nncritical paths. This applies in particular t the tw uter edges which are cmmn t bth parallelgrams. They frm a nncritical path jining the tw endpints f the parallelgrams p and q, and the Ω pq argument wrks as usual. Bth parallelgrams are swept ut by nncritical paths s the tw images in any map t a building must be identified. Let Y i+1 dente the result f pasting tgether the tw parallelgrams. The next tw-manifld cnstructin Z i+1 is then btained by cutting ut the resulting parallelgram frm Y i+1. Dente by the cmmn edges f the parallelgrams the tw edges which are cmmn and which start frm the furfld pint; the glued edges are the ther tw edges f each parallelgram. We assume that at least ne f the tw edge lengths f the parallelgram is maximal. It means that a nn-hexagnal pint, r pssibly a hexagnal pint with nntrivial marking, was encuntered alng sme glued edge f at least ne f the parallelgrams. In the new tw-manifld cnstructin Z i+1, the nly edges which dn t benefit frm the previus scafflding are the tw glued edges. Therefre, the main prblem is hw t determine the new marking alng these edges. We illustrate this in the next sectin A sample step. Let us cnsider an example f pasting with a singularity n the edge. We draw this in Figure 18. The tw parallelgrams t be pasted tgether are P AQB and P AQB. They share

46 46 KATZARKOV, NOLL, PANDIT, AND SIMPSON P A P R R B B Q B Q Figure 18 Figure 19 tw cmmn edges, P A and QA. These edges cme tgether at a furfld pint A, and the ther tw edges at A are marked pen, s the edges P A and QA are marked fld ; hence they have been drawn as thick lines. The interir edges f bth parallelgrams are all marked pen, in particular the fur sides f each ne are straight. Thus, the tw parallelgrams must be mapped t the same place under any harmnic map t a building, and we will paste them tgether and cut them ut. Pasting tgether identifies the pints B and B, we dente the image just by B t. After cutting ut the parallelgram we get a new tw-manifld shwn in Figure 19 The eightfld pint R lies n the edge P B f the first parallelgram, this is cnstraining the width f the parallelgram t be maximal. Let R dente the crrespnding pint n the ther parallelgram; we are assuming that R is a regular hexagnal pint. Nw R and R are als identified in the new tw-manifld and we let their image be called R t. In the picture we have drawn, we are assuming that P and Q are hexagnal pints and the flded edges P A and QA extend beynd the pints P and Q respectively. In the new tw-manifld, P and Q are furfld pints, frm which it fllws that the edges QB and P R are flded. Main questin: what is the marking f the edge RB? Let us lk in the spherical cnstructin at the pint R. At R we have a hexagn with all directins marked pen, whereas at the riginal pint R we are assuming there is an eightfld pint. The tw directins which are interir t the parallelgram are marked pen by the hypthesis that the parallelgram is standard. We may picture the riginal pint R as fllws, with the edges ging twards P and B marked p and b respectively.

47 CONSTRUCTING BUILDINGS 47 s 1 p() s 2 s 3 s 4 Ntice that the edge p shuld be marked pen therwise we wuld have a nn-admissible cnfiguratin at P. There are several different pssibilities fr the labels s i as well as that f b, as will be discussed shrtly. At the new pint R, the three interir sectrs have been replaced by the three sectrs which were exterir t the parallelgram at the pint R. The tw directins inside here are again marked pen because f ur hypthesis that R was a flat hexagnal pint. We are left with a diagram f the fllwing frm: s 1 p(f) b s 2 s 3 s 4 b(?) The directin p is nw marked fld as discussed previusly. Lemma 8.3. In the abve situatin, the markings f s 2, s 3 and s 4 determine the marking f b. Prf. Since p is marked fld, it has t be flded leading t a hexagnal pint. The edge germs t the left and right f p are jined tgether int a single ne marked s belw, and we dn t knw hw that ne will be further flded. The hexagn may be pictured as: s 2 s s 3 s 4 b

48 48 KATZARKOV, NOLL, PANDIT, AND SIMPSON If s 2, s 3 and s 4 are pen, then it fllws that the full hexagn must be pen. If either s 2 r s 4 is flded, then frm the given pen edge we cannt have a triple fld, s the ppsite edge must als be flded. If s 2 is flded then it fllws that b must be flded. If s 4 is flded then s must be flded. The hexagn then flds t a germ f half-apartment; and ne can see b wuld be flded if and nly if s 3 is flded. On the ther hand, if s 3 is flded, then since the edge ppsite t it is pen, it fllws that s and b must be flded. This cmpletes the determinatin f the marking f b. Of curse this lemma treats just ne f the varius pssible situatins which can arise. We expect that similar cnsideratins shuld hld in the ther cases. In ur example, since the edge RB is straight, the marking f b prpagates alng the full edge and we cnclude that the marking f RB is determined; fr ur example, that cmpletes the determinatin f the full scafflding n the new tw-manifld. By cherence ne sees that there were nly a few pssibilities fr the flded edges at R, and the reader may find it interesting t study what happens t the pst-caustic curves in these cases. Suppse fr example that the marking f b determined by the lemma is pen. The pssibilities fr R are: a BNR pint with n flding; a single flded edge s 1 ; tw ppsite flded edges s 1 and b; r tw adjacent flded edges s 4 and b. In the latter tw cases, the edge BR was flded in the riginal cnstructin, and this fld must cntinue past B. In the new cnstructin, the edge BR is n lnger flded but the part f the pst-caustic after B will still be present. Therefre, at the pint B in the new cnstructin we btain tw adjacent flded edges, this cntinuatin tgether with the new fld alng BQ. Als in these tw cases, at the new pint R we have tw flded edges, p and either s 1 (adjacent) r s 4 (ppsite). The new pst-caustic cming frm the pint P thus attaches nt the piece f pst-caustic that used t cme frm ne side f B, whereas the ld pst-caustic n the ther side f B is nw attached t the piece ging thrugh Q The indeterminate case. In a less generic situatin, the new marking might nt be well determined. Cnsider a picture similar t the previus ne, but where the pst-caustic bunding the parallelgram ends at an eightfld pint P as shwn in Figure 20. Nw when we paste and cut ut, the picture is in Figure 21, nting specially that the edge P R is nw pen. The argument used t cnclude that RB shuld be pen, is n lnger valid and we dn t knw hw t

49 CONSTRUCTING BUILDINGS 49 P A P R B B Q R? B Q Figure 20 Figure 21 mark RB (hwever that might be determined fr sme cnfiguratins f the markings f b and the s i ). It crrespnds t a BPS state: the eightfld pint P lies n a spectral netwrk curve ging in the upward directin, whereas in the cases where the markings f s i dn t tell us what t d fr b, the pint R is als n a spectral netwrk curve. These tw spectral netwrk curves meet head-n alng the segment P R and we get a BPS state. One can als see that the resulting pre-building can admit flding maps t ther buildings. Here tw eightfld pints are cnnected by an edge (which is nt marked pen ). This is the first and mst basic example where can be a nntrivial chice f maps t buildings, sme f which fld and hence d nt preserve distances. Such a tw-manifld cnstructin, which we dente nw by C, is analgus t the tree with a BPS state f Figure 2. c 1 c 2 In this picture there are tw sheets belw the three edges, and n sheets abve. The tw eightfld pints c 1 and c 2 are jined by an edge which we have emphasized; it crrespnds t a BPS state. This cnstructin C is itself a pre-building, s it can be cmpleted t a building C B cntaining it ismetrically as a subset. Hwever, it als admits a map t anther cnstructin, flding alng the edge c 1 c 2. T see this, chse tw trapezids sharing the edge c 1 c 2, and

50 50 KATZARKOV, NOLL, PANDIT, AND SIMPSON paste them tgether. c 1 c 2 View this as a single sheet ver the trapezid and tw sheets belw. The resulting cnstructin C is again a pre-building s it can be cmpleted t a building C B. Cmpsing with the prjectin C C therefre gives a map t a building h : C B. Nw h flds alng the segment c 1 c 2, and it is nt an ismetric embedding. This family f maps h is the analgue fr SL 3 f the family f flding maps f trees pictured in Figure Keeping track f BPS states. Our main cnjecture is that if there are n BPS states, then an indeterminate case desn t ccur. In rder t set up the pssibility t cnsider BPS states, we will need t pull alng the spectral netwrk (SN) lines in the prcess f ur sequence f Z i. Let us cnsider the SN lines n the Z i as being additinal markings, which shuld cnjecturally be subject t the fllwing cnditins. Sme pints will be marked as singularities. Each singularity will have a single SN line cming ut f it. Each eightfld pint which is at the end f a pstcaustic, that is t say, having a single fld line, will have a single SN cming ut f it. It is either ppsite t the fld line, r adjacent t it (this is similar t the picture fr the cases f tw fld lines at an eightfld pint). Each eightfld pint with zer fld lines, that is t say a BNR pint, will have tw SN lines cming ut f it. The eightfld pints with tw fld lines dn t have SN lines. The SN lines shuld nt intersect. The SN lines, in the middle, g thrugh regular hexagnal pints and they are straight. The steps f ur prcess cnsist f pasting tgether tw parallelgrams and cutting them ut. The tw edges which are cmmn t the tw parallelgrams are fld lines, meeting in a furfld pint.

51 CONSTRUCTING BUILDINGS 51 We chse parallelgrams which have singular pints smewhere n the bundary. Each mve will either decrease the ttal length f the pstcaustics i.e. the fld lines, r diminish the number f furfld pints. In the case f integer edgelengths this wuld prve the finiteness cnjecture. In general we need sme ther argument. One f the basic mves, when applied t the first furfld pint alng a pstcaustic, will glue tgether sme new pen edges. In the case that the SN curve was ppsite the fld line, the new glued edges are added t the SN curve, this way we build ut the SN curve. In the adjacent case, n the cntrary, the SN curve gets eaten up. One thing that can happen here is that as the SN curve gets eaten up, we can end up passing the singularity. Then it changes t the ther ppsite case and we start spinning ut the SN curve in the ppsite directin. This is why all the SN directins at a singular pint can actually have the pprtunity t cntribute. The main prblem is t analyze the transfrmatins f this picture which can ccur when the varius types f singularities ccur n the bundary f ur parallelgram. The markings r f f the new edges glued in after we cut ut the tw parallelgrams, are mstly determined; hwever, because f the singularity there is ne segment where it isn t determined. This was illustrated in the previus subsectin. At least in a first stage, we will prbably als want t assume sme genericity f the psitins f the singularities s that we dn t reach several singularities n the bundary at nce. The basic idea is t say that if we reach a psitin where the new markings r f f the indeterminate edge are nt well determined by the cnfiguratin, then it must be that we had tw SN curves which jin up, and these frm a BPS state. This phenmenn was illustrated in the example f Figures 20 and 21. It is pssible t envisin finishing the prcess with n mre furfld pints, but still having a segment marked fld jining tw eightfld pints. In this case, the pre-building is nt rigid and there might be several harmnic mappings inducing different distance functins. We are cnjecturing that this can nly ccur if there is a BPS state, indeed the fld line between eightfld pints shuld crrespnd t a piece f the BPS state. Understanding all f the pssible cases currently lks cmplicated, but we hpe that it can be dne. We frmulate the resulting statement as a cnjecture.

52 52 KATZARKOV, NOLL, PANDIT, AND SIMPSON Cnjecture 8.4. Suppse that (X, φ) has n BPS states. Then the sequence f tw-manifld cnstructins Z i is well-defined until we get t ne which has n mre furfld pints, r sixfld pints with triple fld lines. Furthermre, this prcess stps in finite time, at least lcally n X. Finally, at the end f the prcess there are n mre fld markings in the scafflding. Assciated with this is a finiteness cnjecture. Cnjecture 8.5. On bunded regins the prcess stabilizes in finitely many steps. These cnjectures say that the series f tw-manifld cnstructins will stabilize t a nnpsitively curved ne which can be called the cre. It plays a rle analgus t Stallings cre graphs [25, 15, 23]. Starting frm the cre we can then put back everything that had been trimmed ff, fllwing the discussin in Sectin 6.6, t get a pre-building. Crllary 8.6. If there are n BPS states, then n a bunded regin we btain a well-defined pre-building B pre φ. It has a φ-harmnic map h φ : X B pre φ such that if h : X B is any φ-harmnic map t a building, then there exists a unique factrizatin thrugh a nn-flding map f : B pre φ B Distances. One f the main statements which we wuld like t btain thrugh this thery is that the factrizatin frm the pre-building is an ismetric embedding. Recall that the Finsler distance is defined n an apartment A with crdinates x 1, x 2, x 3 subject t x i = 0, by d((x 1, x 2, x 3 ), (x 1, x 2, x 3)) := max{x i x i }. It is nt symmetric, and indeed the tw numbers d(x, x ) and d(x, x) serve t define the vectr distance which is the series f numbers x i x i arranged in nnincreasing rder. Cnjecture 8.7. In the situatin f Crllary 8.6, given a φ-harmnic map h : X B t a building, the factrizatin f : B pre φ B is an ismetric embedding fr the Finsler distance. T apprach this statement, we shuld define a ntin f nncritical path in a cnstructin, with respect t a scafflding. Suppse F is a cnstructin prvided with scafflding, and suppse S is a straight segment jining its tw endpints x and y. Let s x and s y dente the vertices crrespnding t the directins f S in F x and F y respectively. We say that F is Finsler cncave alng S if there exist vertices u and v in F x and F y respectively, such that u is at distance 4

53 CONSTRUCTING BUILDINGS 53 frm s x and v is at distance 4 frm s y. Include als the pssibility f a segment f length zer at a pint x with tw vertices u, v f F x at distance 5. Say that F is Finsler nncncave if there is n segment r pint at which it is Finsler cncave. The idea t shw Cnjecture 8.7 wuld be t d the fllwing steps. The Finsler (r vectr) distance between pints in a (pre)building is calculated by adding up the distances alng a nncritical path. If (F, σ) is a cnstructin with scafflding and F B is a cmpatible map t a building, then the image f a nncritical path in F is a nncritical path in B with the same Finsler length. In a cnnected cnstructin with fully pen scafflding, which is Finsler nncncave, any tw pints are jined by a nncritical path. When there are n BPS states and assuming the previus cnjectures, the pre-building B pre φ is Finsler nncncave. It shuld be pssible t check these prperties fr any given example Isperimetric cnsideratins. We clse this sectin by mentining an imprtant aspect. Under the pasting-tgether prcess, we glue tgether tw parallelgrams which are riginally jined alng tw edges. The tw parallelgrams are suppsed t be disjint except fr sharing these tw edges. We need t chse a maximal such situatin. In particular, when increasing the size f the parallelgram up t its maximal value (determined by when we meet sme kinds f singularities alng the bundary) we wuld like t make sure that we dn t suddenly hit tw pints which are already previusly identified. This is insured using an isperimetric argument. Namely, suppse we have pasted tgether the parallelgrams just up t a pint right befre the extra jined pints. Cut ut this pasted parallelgram, leaving as usual a tw-manifld cnstructin. The eightfld pint frm the vertex f the pasted parallelgram, is very near t a pair f pints n the tw sheets which are already identified. This gives a very shrt lp in ur twmanifld cnstructin (by very shrt here, it means arbitrarily shrt depending n hw clse we gt t the already-jined pints). By hypthesis, ur Riemann surface X is simply cnnected (if we started with an riginal prblem n a nn-simply cnnected surface the first step wuld have been t pass t the universal cver). It fllws that the successive tw-manifld cnstructins Z i are als simply cnnected. S, we are nw at a simply cnnected tw-manifld cnstructin which has a very shrt lp. This lp cuts the surface int tw pieces, at

54 54 KATZARKOV, NOLL, PANDIT, AND SIMPSON least ne f which is a disk. We claim that this is impssible. T see that, ntice that since we are ging arbitrarily clse t the jinedtgether pint, there are nly tw pssible ways fr a pst-caustic curve f fld lines t pass frm utside the disk t inside the disk: either thrugh the eightfld vertex f the pasted parallelgram, r thrugh the jined-tgether pint. Frm this maximum f tw pst-caustics passing thrugh the bundary f the disk, and using the fact that alng any pst-caustic the singular pints alternate eightfld and furfld pints starting with eightfld pints at the endpints, we cunt that the difference (number f furfld pints) (number f eightfld pints) is at mst 1. It fllws that the ttal angle deficit due t the curvature inside the disk is at mst 120 f psitive curvature (whereas an arbitrary amunt f negative curvature). Therefre the isperimetric inequality fr this disk cannt be wrse than that f a single furfld pint, where a small bundary implies that the disk is small itself. We get a cntradictin t the arbitrarily small nature f ur lp. This cmpletes the prf that any tw parallelgrams t be pasted tgether thrugh ur prcess, are jined slely n the tw edges cming ut f the furfld pint. 9. The A2 example We include a few pictures cncerning the next class f examples after BNR. It illustrates the phenmenn f crssing f caustics, and there are angular rtatins leading t a BPS state. Because f space cnsideratins we dn t have rm fr a full analysis here, that will be dne elsewhere. The spectral netwrk fr this example is shwn in Figure 22. Nte that there are tw caustics intersecting in the middle. The assciated tw-manifld cnstructin is, in this case, the same as the pre-building. It will lk as shwn in Figure 23. There are tw eightfld pints c and c. The zig-zag bundary lines are shwn t emphasize that the picture represents three sheets ver the triangular regins and a single sheet elsewhere. The image f the map frm X t the pre-building is shwn in Figure 24. At the tp is again the spectral netwrk, reprising Figure 22, with sme hrizntal dashed lines intrduced fr visualizatin. Belw it is the image, in the pre-building that was pictured befre. One can fllw the hrizntal dashed lines that lp arund in rder t understand what is happening.

55 CONSTRUCTING BUILDINGS 55 Figure 22. A spectral netwrk Figure 23. The pre-building The cllisin pints c 1 and c 6 g t the pint labeled c in the prebuilding; the cllisin pints c 3 and c 4 g t the pint labeled c. The cllisin pints c 2 and c 5 dn t g t singular pints in the pre-building.

56 56 KATZARKOV, NOLL, PANDIT, AND SIMPSON Figure 24. The φ-harmnic map Nte hw the regins cntaining pints c 2 and c 5 are flded ver t ppsite sides as cmpared t the dmain picture. As the spectral differential φ changes, fr example just by being multiplied by an angular cnstant e iθ r sme mre cmplicated defrmatin, the relative psitins f the tw singular pints c and c will change. The BPS states ccur when the line segment cc ges in