Discriminant circle bundles over local models of Strebel graphs and Boutroux curves

0.8 MB·

by  Marco Bertola

Discriminant circle bundles over local models of Strebel graphs and
Boutroux curves
M.Bertola and D.Korotkin
February 3, 2017
7
1 Abstract
0
We study special ”discriminant” circle bundles over two elementary moduli spaces of meromorphic
2
quadraticdifferentialswithrealperiodsdenoted byQR(−7)andQR([−3]2). ThespaceQR(−7)isthe mod-
0 0 0
b uli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real
e
periods;itappearsnaturallyinthestudyofaneighbourhoodoftheWitten’scycleW inthecombinatorial
F 1
model based on Jenkins-Strebel quadratic differentials of M . The space QR([−3]2) is the moduli space
g,n 0
2 of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most 3 with real
periods;itappearsindescriptionofaneighbourhoodofKontsevich’sboundaryW ofthecombinatorial
−1,−1
] model. TheapplicationoftheformalismoftheBergmantau-functiontothecombinatorialmodel(withthe
G
goalofcomputinganalyticallyPoincaredualcyclestocertaincombinationsoftautologicalclasses)requires
A the study of special sections of circle bundles over QR(−7) and QR([−3]2); in the case of the space QR(−7)
0 0 0
. a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces
h
QR(−7) and QR([−3]2), also called the spaces of Boutroux curves in detail, together with corresponding
t 0 0
a circle bundles.
m
[
Contents

2
v
1 Motivation and summary of results 2
4
1
2 Combinatorial model of via Jenkins-Strebel differentials 4
7 g,n
M
7
0 3 A local model near Witten’s cycle W1: the space QR0(−7) 7
. 3.1 Resolution of a 5-valent vertex by flat surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1
0 3.2 Plumbing construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7
1 4 A local model near Kontsevich’s boundary W and space QR([ 3]2) 11
: 4.1 Resolution of two one-valent vertices by flat sur−ge1r,−y1 . . . . . . . .0. .−. . . . . . . . . . . . . . . 12
v
i 4.2 Plumbing construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
X
r 5 Modular discriminant ∆ on the space R( 7) 14
a 1 Q0 −
5.1 Space ( 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0
Q −R
5.2 Real slice ( 7): Boutroux curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Q0 − R
5.3 Variation of arg∆ on ( 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1 Q0 −
6 A6.1rguSmpaecnetsQo(f[∆3±−]21),−1.o.n.t.he. .sp.a.ce. .Q.R0(.[−.3.]2.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255
0
6.2 Real slice Q−R([ 3]2): Boutroux curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.3 Variation of0arg−∆±−1,−1 on QR0([−3]2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A Proof of Proposition 2 29
1
1 Motivation and summary of results
The combinatorial model comb of the moduli spaces (for n 1) based on Jenkins-Strebel differentials
Mg,n Mg,n ≥
[11, 25, 7, 8] was proven to be very fruitful in studies of geometry of the moduli space. In particular, it was
used in Kontsevich’s proof [12, 13] of Witten’s conjecture [26] about intersection numbers of ψ-classes.
InthisapproachthecombinatorialmodelisaCWcomplexinwhicheachcellishomeomorphictoasimplex
labelledbyfatgraphsembeddedonaRiemannsurface. Itwasshownin[9,10,22,23],followingearlierwork[24]
(where the combinatorial model of based on hyperbolic geometry was used) and [1] that natural cycles in
g,n
M
thecombinatorialmodelcanbeinterpretedasPoincar´edualstocertaincombinationsofMiller-Morita-Mumford
classes,orkappa-classes. Thesenaturalcyclesconsistofcellscorrespondingtofatgraphswithfixedoddvalencies
of vertices: cells of highest dimension correspond to three-valent fatgraphs. If the fatgraph contains only one
vertexofvalencydifferentfrom3thensuchcyclesarecalled”Witten’scycles”;ifthenumberofsuchnon-generic
vertices is bigger than one, the corresponding union of cells is called ”Kontsevich cycle”. A particular role is
played by the following two cycles; the first cycle is the Witten’s cycle W which contains fatgraphs with one
1
vertexofvalency5andotherverticesofvalency3. ThesecondcycleistheKontsevich’sboundaryW ofthe
1, 1
− −
combinatorialmodel, containingfatgraphswhichhavetwoverticesofvalency1andallotherverticesofvalency
3. The Kontsevich’s boundary W is a subsetofDeligne-Mumford boundaryof (some components of
1, 1 g,n
− − M
the DM boundary are mapped to a point in W ).
1, 1
− −
In general, the Kontsevich–Witten cycle W consists of cells corresponding to Strebel graphs where
(k1,...,kr)
r vertices have valency 2k +3 respectively (k 1) and all the others have valency 3 (see [9]).
j j
≥−
The starting point of computations relating combinatorial cycles to κ-classes and their combinations is the
explicitexpressionfortheconnectionformsincirclebundles(orU(1)bundles)whichcorrespondtotautological
line bundles associated to the k–th puncture, see [12, 28] and [9, 22, 23] for details.
k
L
Themainmotivationofthisworkistoprovidetheanalyticalbackgroundforanapplicationofthetechniques
of the Bergman tau-function to the combinatorial model comb of . The Bergman tau-function first
Mg,n Mg,n
appeared in the theory of isomonodromic deformations [17] and Frobenius manifolds [5] in the context on
Hurwitz spaces. It was then extended to the moduli spaces of Abelian [14] and quadratic differentials [15] over
Riemann surfaces. Geometrically, the Bergman tau-function is a (holomorphic or meromorphic) section of the
productofHodgelinebundleandothernaturallinebundlesoverthecorrespondingmodulispace. Therefore,in
aholomorphicframework,ananalysisofitssingularitystructureallowstoderivevariousrelationsinthePicard
groupsofthesemodulispaces[20,18,19]. Thedirectapplicationofthistechniqueinthecontextofcombinatorial
model of based on JS differentials is problematic due to the absence of a natural holomorphic structure
g,n
M
of the cells. The Bergman tau-function appropriately defined on the Jenkins-Strebel combinatorial model of
isonlyreal-analyticineachcell;however,itsargumentcanstillbeusedtogetasmoothsectionofacircle
g,n
M
bundle. Inturn,studyofmonodromyofthesesectionsallowsustofindcyclesofthecombinatorialmodelwhich
are Poincar´e dual to certain combinations of standard tautological classes on [3]. The implementation of
g,n
M
thisidearequiresthestudyofthelocalbehaviouroftheBergmantau-functionandcorrespondingcirclebundle
in a neighbourhood of the cycles W and W .
1 1, 1
HereafterwedescribeneighbourhoodsofW− −andW in combbyapplyinganappropriate”flatwelding”
1 −1,−1 Mg,n
which equivalently can be interpreted via the ”plumbing” construction using the flat structure introduced on a
Riemann surface by the JS differential.
Combinatorial model near W and W . The study of a neighbourhood of the Witten’s cycle W in
1 1, 1 1
comb leads to the appearance of the rea−l s−lice R( 7) (space of ”Boutroux curves”) of the moduli space,
Mg,n Q0 −
( 7), of meromorphic quadratic differentials on Riemann sphere with one pole of order 7.
0
Q T−he space R( 7) appears in the study of a neighbourhood of W comb as follows; cells forming the
Q0 − 1 ⊂ Mg,n
cycle W are obtained from cells of highest dimension of comb by contraction of two edges having exactly
1 Mg,n
one common vertex; this contraction gives rise to the creation of a “plumbing zone” as shown in Fig. 6 which
separates a component CP1 containing the three zeros of Q and a pole of Q of degree 7 at the nodal point.
R
Thus the quadratic differential Q arising on the separated Riemann sphere belongs to the space Q ( 7).
Thespace R([ 3]2)appearsinaneighbourhoodofKontsevich’sboundaryW : itisthemod0u−lispaceof
Q0 − −1,−1
quadraticdifferentialsontheRiemannspherewithtwopolesoforder3. ThecellsformingW areobtained
1, 1
− −
2
from the cells of the highest dimension by simultaneous contraction of two edges having two common vertices.
Such a contraction creates two “plumbing zones” as shown in Fig. 10; the Riemann sphere arising between the
plumbing zones contains two simple zeros of Q and two poles of degree 3 at the arising nodal points. Thus the
quadratic differential Q arising on the separating Riemann sphere is an element of the space QR([ 3]2,[1]2).
Both spaces R( 7) and R([ 3]2) have real dimension 2. Surprisingly enough, we did not0fin−d a complete
Q0 − Q0 −
self-contained description of these two elementary moduli spaces in the literature; one of the goals of the paper
is to fill this gap.
The stratification of these spaces as well as their complexified versions ( 7) and ([ 3]2), respectively,
0 0
Q − Q −
is studied in detail in Sections 5.1, 6.1.
Space R( 7). The generic element of the complex space ( 7) can be represented by the quadratic
Q0 − Q0 −
differential
Q=(x x )(x x )(x x )(dx)2 , x +x +x =0. (1.1)
1 2 3 1 2 3
− − −
The local period (or homological) coordinates can be defined as integrals of v = √Q over two independent
homologycyclesontheellipticcurve(the”canonicalcovering”) definedbytheequationv2 =Q(seeSect. 5.1
R C
for details). The real slice ( 7) of ( 7) is defined by the requirement that all periods of v are real and
R Q0 − Q0 −
thenthespace ( 7)stratifiesintocellsof(real)dimensions2(cid:98)(thetopcellofgenericelements), ofdimension
Q0 −
1 i.e. the cell where two zeros of Q coincide) and zero dimensional (three coincident roots x =x =x =0).
1 2 3
The reality of all periods of v implies that the period σ of the curve belongs to a one-dimensional subset
C
of the modular curve shown in Fig.11, left pane.
1
R The space R( 7,[1]3) is fibered over the set with fiber R . T(cid:98)he points at of correspond to
R Q0 − R1 + ∞ R1
the space ( 7,1,2). The point of intersection of with the real axis in the plane of the J–invariant
Q0 − R1
(Fig.11, right pane) is J 940.34 and it corresponds to the Boutroux–Krichever curve [27, 16, 2], which is the
uniqueBoutrouxcurvein(cid:39) R( 7,[1]3)possessingarealinvolution(cid:63) whichleavestheJenkins-Strebeldifferential
Q0 −
invariant: Q(x(cid:63))=Q(x).
The zeros x , x and x are connected by two horizontal (i.e. parallel to the real line in the plane of
1 2 3
x
coordinate z(x) = v) geodesics in the metric Q. We will always denote by x the ”central” zero which is
2
| |
connected by these geodesics to two others. Then the remaining zeros x and x can also be labeled such that
1 3
(cid:82)
in the positive direction around the origin x goes after x and x after x .
3 2 1 3
The lengths of the two geodesics are given by the absolute values A = x2√Q , B = x3√Q . The
x1 x2
lengths (A,B) define the map of the space R( 7,[1]3) to R2. (cid:12) (cid:12) (cid:12) (cid:12)
Q0 − + (cid:12)(cid:82) (cid:12) (cid:12)(cid:82) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Discriminant circle bundle on R( 7). Consider the section ϕ of a U(1) bundle over R( 7) given by
Q0 − 1 Q0 −
the argument of the modular discriminant
ϕ =Arg∆ (1.2)
1 1
where
∆ =[(x x )(x x )(x x )]2. (1.3)
1 1 2 3 1 3 2
− − −
Although∆ itselfiswell-definedonlyonthestratum R([1]3, 7),itsargumentϕ canbecontinuedsmoothly
1 Q0 − 1
through the boundaries A=0, B =0.
R
We will show (Theorem 2) that then increment of ϕ on ( 7) between the boundaries A=0 and B =0
1 Q0 −
equals π/5; this computation is technically non-trivial. Therefore, π/5 is the monodromy of ϕ around the zero
1
dimensional cell (represented by Q=x3dx2) on the space R(3, 7).
Q0 −
Description of R([ 3]2). The complex moduli space ([ 3]2) consists of quadratic differentials Q on the
Q0 − Q0 −
Riemann sphere with two poles of degree 3. The generic element of this space is represented by
(x x )(x x )
Q= − 1 − 2 (dx)2 , x =x , x =0=x . (1.4)
x3 1 (cid:54) 2 1 (cid:54) (cid:54) 2
The integrals of √Q along paths two arbitrary independent homology classes on the canonical covering y2 =
x(x x )(x x ) are local period coordinates on ([ 3]2).
1 2 0
− − Q −
3
The space R([ 3]2) is defined by the condition that all periods of v =√Q on are real; it is fibered over
the set Qo0f r−eal dimension 1 within the modular curve (Fig. 15, left pane) wiCth the fiber R .
1, 1 +
TheRp−oin−ts at of correspond to the “wall” i.e. to the space R(2,[ 3](cid:98)2).
∞ R−1,−1 Q0 −
The set intersects the real axis in the plane of the J–invariant at J 1690,586,7791 (Fig.15,
1, 1
R− − (cid:39) {− }
right pane). The points J 1690 and J 586 correspond to curves with a real involution, however the
(cid:39) − (cid:39)
differentialQisnotinvariantwithrespecttothisinvolution. ThepointJ 7791(x 1.8037, x = 0.3797)
1 2
(cid:39) (cid:39) −
corresponds to a curve possessing a real involution (cid:63) (acting as a standard complex conjugation) such that
Q(x(cid:63))=Q(x). This is the natural analog of the Boutroux–Krichever curve in the space R([ 3]2).
Q0 −
One of the zeros, which we denote by x , is connected by an infinite horizontal geodesics in the metric Q
1
| |
tothepolex=0, andanotherzeros(x )tothepolex= . Inthiswaywegetthelabelingofthezerosx and
2 1
∞
x in this case. The zeros x and x are connected by two horizontal geodesics which we denote by e and e .
2 1 2 1 2
Thegeodesicse ande willbelabelledasfollows: whenonearrivestox alonghorizontalgeodesicsemanating
1 2 1
from x = 0, turning right one follows the geodesics e , and turning left one follows the geodesics e (see Fig.
1 2
7). The length of e will be denoted by A and the length of e by B.
1 2
The lengths (A,B) define the map of the space R([ 3]2,[1]2) to R2.
Q0 − +
DRis(c[ri3m]2i)naarnetgicviernclbeybthuenfdollelowoinngQeR0xp([r−es3s]i2o)n.s:The sections ϕ±−1,−1 of the two natural U(1) bundles over
Q0 −
ϕ±1, 1 =Arg∆±1, 1 (1.5)
− − − −
∆+−1,−1 :=x61x62(x1−x2)2, ∆−−1,−1 :=x61x62(x1−x2)26 . (1.6)
The analysis contained in Section 6 shows that the increments of ϕ±1, 1 between the boundary A=0 and the
boundary B =0 on QR([ 3]2) equals 13π and 25π, respectively. − −
25πT, rheesrpeefocrteiv,emlyon(soeder0oTmh−mies. o4f).ϕ±−1,−1 along a simple closed, non-contractible loop in QR0([−3]2) equal 13π and
In summary, the main results of this paper are the following. First, we show how the moduli spaces
QR( 7) and QR([ 3]2) appear under a degeneration of a generic Strebel graph via shrinking of two adjacent
ed0ge−s. Second,0we−study analytically the spaces QR( 7) and QR([ 3]2) and natural circle bundles over them.
0 − 0 −
These results provide the analytical tools necessary to apply the formalism of the Bergman tau-function in the
description of various tautological classes to the combinatorial model of [3].
g,n
M
Thepaperisorganizedasfollows. InSection2werecallthecombinatorialmodelof basedonJenkins-
g,n
M
Strebeldifferentials. Insection3wedescribeaneighbourhoodofWitten’scycleW inthecombinatorialmodel
1
R
and show how the moduli space Q ( 7) appears in this context. In Section 4 we describe a neighbourhood of
Kontsevich’s boundary W of 0 −comb and demonstrate the appearance of the space QR([ 3]2). In Section
−1,−1 Mg,n R 0 −
5 we study the geometry of the space Q ( 7) in detail and compute the increment of the argument of the
modulardiscriminant∆ onthisspace. Fi0na−lly, inSection6westudythegeometryofthespaceQR([ 3]2)and
1 0 −
compute the increment of the arguments of ∆±1, 1 on this space.
− −
2 Combinatorial model of via Jenkins-Strebel differentials
g,n
M
HerewebrieflyrecallthemainingredientsofthecombinatorialmodelofthemodulispacesofRiemannsurfaces
based on Jenkins-Strebel differentials. The moduli space of Riemann surfaces of genus g with n marked points
is denoted by and its Deligne-Mumford compactification by . Let be a Riemann surface of genus
g,n g,n
M M C
g and Q be a meromorphic quadratic differential with second order poles at the points y ,...,y . Zeros of Q
1 n
are denoted by x ,...,x and their multiplicities by d=(d ,...,d ); we have m d =4g 4+2n. Denote
1 m 1 m i=1 i −
by d the moduli space of such quadratic differentials; its dimension equals 2g 2+n+m. For the stratum
Qg,n (cid:80)−
of highest dimension, when all zeros are simple and m=4g 4+2n, this dimension equals 6g 6+3n.
− −
4
Canonical cover. The equation
v2 =Q
in T defines a two-sheeted covering of , called canonical covering in Teichmu¨ller theory ( is known
∗
C C C C
under the name of ”spectral covering” in the theory of Hitchin’s systems or under the name of ”Seiberg-Witten
curve” in the theory of supersymmetric(cid:98)Yang-Mills theory). The branch points of the covering coi(cid:98)ncide with
zeros of odd multiplicity of Q, whose number we denote by m : then the genus of the canonical covering is
odd
gˆ = 2g+ modd 1. For a generic Q, which has 4g 4+2n simple zeros, gˆ = 4g 3+n. It is convenient to
2 − − −
introduce the notation
m
odd
g =gˆ g =g+ 1
− − 2 −
Denote by µ the holomorphic involution interchanging the sheets of . The Abelian differential of the third
kind v on has 2n simple poles at the points y ,yµ (slightly abusiCng the notations we use y to denote
C { i i} i
positions of poles both on and ). Consider the decomposition of H(cid:98)( y ,yµ n ,R) (whose dimension
C C 1 C \{ i i}i=1
equals 2gˆ+(cid:98)2n 1) into the direct sum of even and odd subspaces:
−
(cid:98) (cid:98)
H H , dimH =2g+n 1, dimH =2g +n. (2.1)
+ +
⊕ − − − −
Notice that dimH = dim d . Consequently, one can introduce a system of local coordinates on the space
d called “homo−logical coQorgd,ninates” by choosing a set of independent cycles s , i=1,...,dimH in H and
Qg,n i − −
integrating v over these cycles:
= v . (2.2)
i
P
(cid:90)si
The following definition of Jenkins-Strebel differential is equivalent to the standard one [25]:
Definition 1 The quadratic differential Q is called Jenkins-Strebel differential if all homological coordinates
i
P
are real.
Therealityofall impliesthatthebiresiduesofQatitspolesy arerealandnegativei.e. thereexistsuch
i j
p R that in anyPlocal coordinate ζ near y one has:
j + j
∈
(p /2π)2
Q(ζ)= − j (1+ (ζ))(dζ)2 (2.3)
ζ2 O
Results of Jenking and Strebel provide the existence and uniqueness of a differential Q on a given Riemann
surface with n marked points with given constants p R and all real periods v for s H (combinations
i ∈ + si i ∈ −
of cycles surrounding z also form a part of H ; the reality of periods of v around these cycles is guaranteed by
i
− (cid:82)
the form of biresidues in (2.3)). This statement provides a basis for the combinatorial description of .
g,n
For each given vector p = (p ,...,p ) Rn one can construct a combinatorial model of the modMuli space
1 n ∈ +
which will be denoted by [p]. Namely, for each Riemann surface of genus g with n marked points
g,n g,n
M M C
z ,...,z consider the unique Jenkins-Strebel differential Q whose singular part at z is as in (2.3).
1 n i
The stratum d [p] of [p] consists of punctured Riemann surfaces such that the Jenkins-Strebel
Mg,n Mg,n
differential Q has zeros of multiplicities d = (d ,...,d ). The real dimension of the stratum d [p] is given
1 m Mg,n
by
dimRMdg,n[p]=2g−2+m (2.4)
The largeststratum correspondsto Jenkins-Strebeldifferentials with simplezeros andhas real dimensionequal
to 6g 6+2n (in this case m=4g 4+2n) which coincides with the real dimension of Rn.
− − Mg,n× +
Theorientedhorizontaltrajectoriesofthe1-formv on (whichprojectdowntothenon-orientedhorizontal
C
trajectories of Q on ) connect zeros x .
i
C
These horizontal trajectories on form the edges of a(cid:98)n embedded graph (also called fatgraph, or ribbon
C
graph) Γ on with vertices at x ,...,x and valencies d +2,...,d +2, respectively. Therefore, the stratum
1 m 1 m
C
of highest dimension 6g 6+2n corresponds to fatgraphs on with trivalent vertices.
− C
5
‘3 ‘2
p p
‘3 ‘1
p
‘1
‘2 p
‘1 ‘p2 ‘p3
Figure 1: Fatgraph on a genus 1 Riemann surface representing a point in [p]. Two simple zeros x
1,1 1
M
and x of Q are connected by 3 edges of lengths (cid:96) , (cid:96) and (cid:96) . The fatgraph has only one face of perimeter
2 1 2 3
p=2((cid:96) +(cid:96) +(cid:96) ). The lengths (cid:96) and (cid:96) can be used as coordinates on [p]
1 2 3 1 2 1,1
M
Thelength(cid:96) ofanedgee E(Γ)connectingtwoverticesv ,v isequaltotheabsolutevalueoftheintegral
e 1 2
∈
of v along the horizontal trajectory connecting the two vertices. In turn, up to a factor of 1/2, this length
coincides with the integral of v over the integer cycles in H consisting of the trajectory e on one sheet of the
projection andthesametrajectoryintheoppositedir−ectionontheothersheet. Fixingthevectorp Rn
C →C ∈ +
one imposes n linear constraints on the lengths of the edges: for the j–th face f F(Γ), j =1,...,n we have
j
∈
(cid:96) =(cid:98)p . AsimpleexampleoffatgraphonaRiemannsurfaceofgenus1withonemarkedpointisshown
e∈∂fj e j
in Fig. 1. In the flat metric Q on the neighbourhoods of a pole y is an infinite cylinder of perimeter p .
(cid:80) | | C i i
The union of all strata d [p] for fixed p Rn forms the combinatorial model [p] of . This
Mg,n ∈ + Mg,n Mg,n
combinatorial model is set-theoretically isomorphic to .
g,n
M
Thecompactification [p]ofthecombinatorialmodelisconstructedbyadditionoftheso–calledKontse-
g,n
M
vich boundary to [p]: the stratum W of real co-dimension 2 of the Kontsevich boundary corresponds
g,n 1, 1
M − −
to fatgraphs with exactly two one–valent vertices (i.e. two simple poles of the JS differential Q), while all other
vertices remain trivalent.
The Kontsevich boundary is “smaller” than the Deligne-Mumford boundary of since the Jenkins-
g,n
M
Strebel combinatorial model is not applicable to non-punctured Riemann surfaces.
Each face F of the fatgraph Γ (the face F contains the pole y ) can be mapped to the unit disk via the
j j j
map
2πi x
w (x)=exp v ; (2.5)
j
p
(cid:32) j (cid:90)xj (cid:33)
where x is an arbitrarily chosen vertex of the face F . Under the map (2.5) y is being mapped to the origin
j j j
and x to 1. One can always choose the branch of the differential v =√Q which has residue pj at y . Within
j 2πi j
the face F the flat metric Q = v 2 on is expressed as:
j
| | | | C
p2 dw 2
j j
Q = . (2.6)
| | 4π2 w
(cid:12) j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thefatgraphcorrespondingtothestratumofhighest(cid:12)dimen(cid:12)sioninthecombinatorialmodel comb[p]shown
M1,1
in Fig. 1 has only one face; this face can be mapped to the unit disk. The constraint between lengths in this
case reads 2((cid:96) +(cid:96) +(cid:96) )=p.
1 2 3
Inverting this logic it is possible to construct a polyhedral Riemann surface (i.e. Riemann surface with flat
metric with conical singularities) from a fatgraph equipped with lengths of all edges by the conformal welding
[25].
All strata of [p] can be obtained from the stratum of highest dimension W (which corresponds to
g,n
M
fatgraphs with all tri-valent vertices) by contraction of one or more edges. Various components of W are glued
along (real) codimension one boundaries which correspond to fatgraphs with one vertex of valency 4 and all
other vertices of valency 3 (i.e. the Jenkins-Strebel differential corresponding to the boundary has one double
zero and other zeros are simple). The procedure of crossing the boundary from one cell of W to another
6
(sometimes in this way one can get into the same cell; then the cell is ”wrapped on itself”) is described by the
so-called Whitehead move shown in Fig.3.
Our goal will be to study in detail the contraction of two edges having one vertex in common. Then
depending on geometry one gets either cells forming the Witten’s cycle W (corresponding fatgraphs have one
1
5-valent vertex and other vertices of valency 3), or cells forming the Kontsevich’s boundary W of [p]
1, 1 g,n
− − M
(whichcorrespondstofatgraphshavingtwoverticesofvalency1andallotherverticesofvalency3). Thesetwo
types of contraction are shown in Fig.2
1 1
1
2 4
4 2
4 2
3
3 3
Figure 3: The Whitehead move.
Figure 2: A typical embedded Strebel graph. The two highlighted
regions are where the contraction of edges leads either to the cell
W (left region) or to the cell W (right region).
1 1, 1
− −
R
3 A local model near Witten’s cycle W : the space Q ( 7)
1 0
−
Cells of the cycle W in the combinatorial model [p] (the vector p will be kept fixed in all constructions
1 g,n
M
below)correspondtofatgraphswhichhaveallverticesofvalency3exceptonevertexwhichhasvalency5. Here
we shall understand an n-punctured Riemann surface as an element of the combinatorial model [p] i.e.
g,n
C M
we assume that uniquely defines the JS differential Q.
C
A point of W can be obtained from a trivalent fatgraph via contraction of two edges having one vertex in
1
common;conversely,anyRiemannsurfacefromaneighbourhoodof inW isatwo(real)-parameterdeformation
C
of . We are going to show how this deformation can be described via ”flat surgery” of a Riemann surface
C C
and a Riemann sphere equipped with an appropriate real-normalized quadratic differential. This flat surgery
also explicitly gives the Jenkins-Strebel differential and the fatgraph on the deformed Riemann surface.
Furthermore, the flat surgery can also be understood in terms of a ”plumbing construction” connecting
C
with a Riemann sphere equipped with an appropriate flat metric.
Denote the zero of order 3 of the JS differential on by x ; denote also the fatgraph corresponding to by
1
C C
2/5
Γ. Let ζ be a “distinguished” local parameter on near x : ζ(x) = 5 x v (ζ(x) is defined up to a fifth
C 1 2 x1
rootofunity); thusQ(x)canbewrittenintermsofζ asQ(ζ)=ζ3(dζ(cid:104))2. The(cid:105)localcoordinateonthecanonical
(cid:82)
cover near x is given by ζ1/2.
1
C
Introduce also the “flat” coordinate (both on and ):
C C
(cid:98)
x
z(x)= (cid:98) v ;
(cid:90)x1
in a neighbourhood of x the flat coordinate is expressed via the local coordinate ζ as z = 2ζ5/2. We have
1 5
Q(x) = (dz(x))2 on and v = dz on . The metric Q(x) has a conical point at x with cone angle 5π while
1
C C | |
at all other vertices the cone angle equals 3π.
(cid:98)
3.1 Resolution of a 5-valent vertex by flat surgery
There are 5 edges of the fatgraph Γ emanating from the vertex x ; their directions are given by the angles 2πk
1 5
intheζ-plane. Denotetheirlengths(startingfromtheedgegoingalongpositiverealline)by(cid:96) ,...,(cid:96) . Weare
1 5
7
going to construct an explicit deformation α,β of with two real (”small”) parameters, α and β, by changing
C C
these lengths as follows:
((cid:96) ,...,(cid:96) ) ((cid:96) (α+β), (cid:96) +β, (cid:96) β, (cid:96) α, (cid:96) +α) (3.1)
1 5 1 2 3 4 5
→ − − −
The triple zero, x, of Q on splits into three simple zeros (x ,x ,x ) of the JS differential Qα,β on α,β. We
1 2 3
C C
assume that the new edges of length (cid:96) (α+β) and (cid:96) +β meet at x ; x is connected to x by the edge of
1 2 1 1 2
−
length α, x is the endpoint of the edge (cid:96) +β and it is also connected to x by the edge of length β; the edges
2 (cid:101) 2 3
(cid:96) α and (cid:96) +α meet at x . The resulting configuration is shown in Fig.4.
4 5 3
−
x
‘2 ‘1 b
‘
3 ‘ β
5 +
x R
‘4 R−β x1 α R −(α + β)
e x2
β
R −α x3 R+α
Figure 4: The Riemann surface α,β is obtained by replacing the five-valent vertex with three regular vertices
C
and two new edges of length α and β between them with appropriate adjustment of the lengths of the original
fatgraph. The limit α,β 0 produces the original Riemann surface. The five arcs on the boundaries of the
→
plug-in region are five semicircles or radius R in the flat coordinates of either metrics.
We then construct a two–(real) parameter family α,β of deformations of as explained below.
C C
Fix some radius R which satisfies the conditions R < (cid:96) for i = 1,...,5. Consider the disk given by
i R
D
z(x) R on (see Fig. 4); in terms of the coordinate ζ the disk is given by ζ 5R 2/5. Five edges of
| |≤ C DR | |≤ 2
thefatgraphΓwithinDR aregivenbythesegments 0, 52R 25 e2i5πk,k =0,...,4intheζ(cid:0)-pla(cid:1)ne. Theperimeter
of R in the flat metric Q equals 5πR. (cid:104) (cid:0) (cid:1) (cid:105)
D | |
Let us now excise the region from the Riemann surface and obtain a Riemann surface with boundary,
R
D C
which we denote by . Denote by the canonical cover with the disk deleted on both copies of . Let
R R R
C C C D C
us assume that the branch cut connecting the triple zero x with some other zero on (say, x) goes along the
C
positive real line in the ζ - coordinat(cid:98)e. (cid:98)
We are going to attach to another disk α,β which we now describe. Consider an element of the moduli
space QR[ 7], that is, a quadrCaRtic differential QDRon CP1 g(cid:101)iven by (cid:98)
0 − 0
(cid:98)
Q (x)=(x x )(x x )(x x )dx2 (3.2)
0 1 2 3
− − −
such that x +x +x =0. The Abelian differential v =√Q is defined on the canonical cover of CP1 that
1 2 3 0 0 0
C
is the elliptic curve with branch points at x ,x ,x and .
1 2 3
Assumethatallperiodsofv on arereal. Thenthe∞reexisttwohorizontaltrajectoriesofv (cid:98)onCP1 which
0 0 0
C
connect x , x and x . Assume that x is the “central” zero i.e. it is connected by the horizontal trajectories
1 2 3 2
(cid:98)
8
to x and x . Choose the branch cuts on to connect x with x , x and along the horizontal trajectories.
1 3 0 2 1 3
C ∞
Assume also that x and x are enumerated such tat the branch cuts [x , ), [x ,x ] and [x ,x ] meet at x
1 3 2 2 1 2 3 2
∞
in the counterclockwise order as shown in(cid:98)Fig.5, left (this picture is drawn in the coordinate x).
Introduce on a canonical basis of cycles (a,b) such that the a-cycle encircles the branch points x and x
0 1 2
C
and b-cycle encircles the branch points x and x . The real periods of v over cycles a and b are expressed as
2 3 0
follows via the len(cid:98)gths α and β of the branch cuts [x ,x ] and [x ,x ]:
2 1 2 3
v =2α , v =2β . (3.3)
0 0
(cid:12)(cid:90)a (cid:12) (cid:12)(cid:90)b (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
The conditions (3.3) determine the bra(cid:12)nch po(cid:12)ints x up to(cid:12)mult(cid:12)iplication of all x by a fifth root of unity (see
i i
Th. 1).
The quadratic differential Q which hasthree simple zerosand onepole ofdegree 7on CP1 is anelement of
0
the space R([1]3, 7); it is an analog of the Jenkins-Strebel differential in the case of a pole having an order
Q0 −
higher than 2.
Π1 R
β α
Π2 R
Figure 5: Left: the ribbon graph on CP1
of the quadratic differential Q and the
β 0
R−βΠ5 R+ gneaotedexsi.cIdnidskicaotferdadairuestRheddriaswtanncinescionortdhie-
x1 aΠ4 Π3 R metric Q along the trajectories. Right:
α | 0|
the five half-disks in the upper half plane,
Π1 xR2−(α+β) β uniformized by the flat coordinate.
β
α b
R − x3 Π3 Π4 R
R
Π2 +
α
α
Π5 R
We choose the branch cuts to go along the edges [x ,x ], [x ,x ] and [x , ] (the thick lines in Fig. 5, left
1 2 3 2 2
∞
pane). The edges of the fatgraph Γ split the x-plane into 5 regions as shown in Fig.5 (left). Five horizontal
0
trajectories of v which connect x , x and x to approach along the rays argx = 2πk/5. In the flat
0 1 2 3
metric Q on CP1 each of these five regions is unif∞ormized by t∞he flat coordinate z to a half-plane Imz >0.
0
| |
On each of the five rays connecting the point at infinity with the points x we cut segments whose lengths
i
(starting from the ray [x , ) and counting couterclockwise) are chosen to be R α β, R+β, R β, R α
2
∞ − − − −
and R+α. Then we consider a half-circle (in the flat metric Q ) in each region which connects the endpoints
0
| |
ofthechosensegments(Fig. 5,right). Itiseasytoseethatthediametersofallofthesefivehalf-circlescoincide
andareallequalto2R. Inthiswayweobtainfiveregionswhicharehalf-disksofradiusRintheflatcoordinate
z; they will be denoted by Π ,...,Π , and their union by α,β.
1 5 DR
TheflatcoordinatesineachoftheregionsΠ aredefineduptosignandtranslations;theglobalflatcoordinate
i
on the whole α,β is defined as follows. Choose the initial point of integration to be x and choose the system
DR 2
9
of three branch cuts within α,β as shown in Fig.5 (left): then the flat coordinate on α,β with three deleted
DR DR
branch cuts is defined by
x
w(x)=α+β+ v (3.4)
0
(cid:90)x2
where the determination of v has to be chosen so that the image of Π is in the upper half plane of the
0 4
z–variable. One can easily verify that, according to the orientation of the a- and b- cycles on shown in Fig.5,
0
C
left pane, the values of the coordinate w(x) on different sides of the branch cut [x , ) differ by a sign.
2
∞
The canonical two-sheeted cover of the domain α,β with branch cuts going along horizontal trajectories
DR
[x ,x ], [x ,x ] and [x , ) will be denoted by α,β.
2 1 2 3 2 ∞ DR
Theshapeandperimeteroftheboundaryof α,β intheglobalflatcoordinateisacirclewindingbyanangle
DR
of 5π. The positions of the points where the out(cid:98)going edges (in z-coordinate) and the branch cut intersect the
boundary of α,β, coincide with those of . In particular, the perimeter of α,β also equals 5πR. Therefore,
DR DR DR
we can identify the boundary of with the boundary of α,β (this procedure is called ”conformal welding”
CR DR
in [25]) to get the new Riemann surface α,β together with its canonical cover α,β. The five lengths of the
C C
edges originally connected to the zero of order 3 on are then changed according to (3.1). Notice that this
C
deformation preserves perimeters of all faces of the fatgraph; therefore the Ri(cid:98)emann surface α,β and the
C
corresponding fatgraph belong to the same combinatorial model [p] as the original surface .
g,n
M C
Moreover, the result of such ”conformal welding” does not depend on the choice of radius R of the excised
disk as long as it remains sufficiently small in comparison with (cid:96) ,...,(cid:96) and sufficiently large in comparison
1 5
with α and β.
3.2 Plumbing construction
To study the limit α,β 0 in the previous ”conformal welding” scheme one needs to degenerate the quadratic
differential(3.2)onCP1→. Theequivalent”plumbing”constructionpresentedhereallowstokeep and fixed.
0
C C
To deform into α,β (and conversely, to study the limit α,β 0 of α,β) we introduce the ”plumbing
C C → C
parameter” t=α+β and the parameters
α α β α
A= = B = = (3.5)
t α+β t α+β
such that A+B =1.
x1
x2
Plumbingzone
x3
| {z }
x1
x2
x3 +3
7
−
Figure6: SeparatingoftheRiemannspherewiththreesimplezerosandapoleoforder7ofQinaneighbourhood
of W
1
10