Riemannian Geometry

In this appendix besides introducing notation and basic deﬁnitions we recollect some of the more technical results in Riemannian geometry we exploited in these lecture notes. We freely refer to the excellent exposition of the theory provided by [4, 8], and to [27] for further details.

A.1 Notation

In what follows, Vn will always denote a C1 compact n–dimensional manifold, (n 3). Unless otherwise stated, the manifold Vn will be supposed oriented, connected, and without boundary. Let .E; Vn;/be a smooth vector bundle over Vn with projection map W E ! Vn. When no confusion arises the space of smooth 1. n; / n 1. n; / sections fs 2 C V E j ı s D idV g will simply be denoted by C V E instead of the standard .E/ (to avoid confusion (Fig. A.1) with the already too numerous s appearing in Riemannian geometry: Christoffel symbols, harmonic n i n coordinates, ...).If U V is an open set, we let fx giD1 be local coordinates for @ @=@ i n 1. ; n/ the points p 2 U. The vector ﬁelds f i WD x giD1 2 C U TV provide the n local (positively oriented) coordinate basis for TpV , p 2 U. The corresponding n 1 dual basis for the cotangent spaces Tp V , is provided by the set of –forms i n 1. ; n/ 1. n; 2 n/ fdx giD1 2 C U T V . In particular, the metric tensor g 2 C V ˝S T V , 1. n; 2 n/ where C V ˝S T V denotes the set of smooth symmetric bilinear forms over n i k V , has the local coordinates representation g D gikdx ˝dx , where gik WD g.@i;@k/, and the Einsteinp summation convention is in effect. The RiemannianR volume form is 1 ::: n . n/ d g D det gdx ^ ^dx , and we denote by Volg V WD Vn d g the volume of (the compact) .Vn; g/. The distance between points x; y 2 Vn, characterizing .Vn; g/ as a metric space, is deﬁned by setting

Fig. A.1 ...Riemanniangeometrymayhaveunexpectedsides...andmustbeusedresponsibly!

Z 1 p dg.x; y/ WD inf g..P t/; .P t// dt; (A.1) 2 C .x;y/ 0 where C .x; y/ is the space of (piecewise C1) curves W Œ0; 1 ! Vn such that .0/ D x and .1/ D y.If

2 k1:::k j1:::j jUj WD gi1h1 :::girhr U q U q g :::g (A.2) g i1:::ip hi:::hr k1j1 kqjq denotes the pointwise (squared) g–norm in the tensor bundle T.r;q/Vn WD ˝r TVn ˝q TVn, then we can endow the space of smooth .r; q/–tensor fields 1 n .r;q/ n n p p; s C .V ; T V / on V with the corresponding L .dg/ and H .dg/ Sobolev norms ÄZ 1 p : p p ;1 < ; jjUjjL .dg/ D jUjg d g Ä p 1 (A.3) Vn 1 s ÄZ X ˇ ˇ p : ˇ .i/ ˇ p H p; s. / ; jjUjj d g D r U g d g (A.4) n iD0 V where s is an integer 0, and r.i/U denotes the ith covariant derivative of the tensor field U. The completions of C1.Vn; T.r;q/Vn/ in these norms, define A Riemannian Geometry 349

Fig. A.2 The group of diffeomorphisms the Sobolev spaces of sections H p; s.Vn; T.r;q/Vn/. In particular, for p D 2,we denote by H s.Vn; T.r;q/Vn/ the Hilbert space H 2;s.Vn; T.r;q/Vn/. According to 0 > n H s. n; .r;q/ n/ Sobolev embedding theorem, if k and s 2 C k, then V T V Ck.Vn; T.r;q/Vn/. Similarly, if we denote by H s.Vn; Vn/ the Sobolev space of maps W Vn ! Vn, then H s.Vn; Vn/ Ck.Vn; Vn/ is a dense continuous inclusion for s >.n=2/ C k. Let ˚ ˇ « Diff k.Vn/ WD 2 Ck.Vn; Vn/ˇ is a bijection and 1 2 Ck.Vn; Vn/ (A.5) be the group of Ck diffeomorphisms of Vn, endowed with the compact–open Ck topology. The group of smooth diffeomorphisms Diff.Vn/ is deﬁned by the sequence of natural inclusions (Fig. A.2)

Diff 1.Vn/ Diff 2.Vn/ :::Diff k.Vn/ ::: ; (A.6) according to

. n/ 1 k. n/: Diff V D\kD1 Diff V (A.7)

Recall that under a diffeomorphism W Vn ! Vn the pull back from ..Vn/; g/ to .Vn;g/ is given for p 2 Vn by ˇ ˇ . ; / . ; / ; g V W p WD g V W j.p/ (A.8) 350 A Riemannian Geometry

n where V denotes the push forward of V deﬁned by the tangent map W TpV ! n i a i a i T.p/V . In local coordinates .U; x / and ..U/; y /, U 3 x 7! .x / WD ya..p// 2 .U/, we write @a @b g .p/ D g ..p//: (A.9) ik @xi @xk ab It is important to stress that the pull–back action of Diff.Vn/ on the metric g is a right action since under composition of any two diffeomorphisms and we have . ı /g D . g/. This latter remark plays a role when, for technical reasons, it is necessary to enlarge Diff.Vn/ to the (Hilbert manifold of the) diffeomorphisms of Sobolev class H s,fors >.n=2/ C 1, ˚ « D s.Vn/ WD 2 H s.Vn; Vn/j is a bijection and 1 2 H s.Vn; Vn/ : (A.10)

In such a framework the group of smooth diffeomorphisms can be recovered from D s.Vn/ by characterizing the smooth topology on Diff.Vn/ as the limit of the topologies of D s.Vn/, by viewing Diff.Vn/ as an Inverse Limit Hilbert group in n s n the sense of Omori [26], i.e. Diff.V / D\s>n=2D .V /. Since right multiplication is smooth, but left multiplication is not, D s.Vn/ is not “an inﬁnite–dimensional” Lie group but only a topological group [11,21].

A.2 Curvature and Scaling

Besides diffeomorphisms, the metric g is naturally acted upon also by overall rescalings according to for 2 R>0 (Fig. A.3)

g 7! g (A.11)

Fig. A.3 Besides diffeomorphisms, the metric g is naturally acted upon also by overall rescalings A Riemannian Geometry 351

g1 7! 1 g1;

1 1. n; 2 n/ where g 2 C V ˝S TV denotes the inverse metric providing the inner n n . ; i/ product in Tp V , p 2 V . In local coordinates U x , we write gik 7! gik and gik 7! 1 gik, respectively. Under the scaling (A.11) we immediately get an obvious dilation or contraction effect on distances and volume 1 2 dg.p; q/ 7! dg.p; q/ D dg.p; q/

n (A.12) n n 2 n Vol g.V / 7! Vol g.V / D Vol g.V /; which is often useful to factor out in the study of Riemannian functional. In this connection it is worthwhile to recall how the basic Riemannian deﬁnitions interact with (A.11). We start with the Levi–Civita connection rg of .Vn; g/,(“r” if there is no danger of confusion), deﬁned by

2 g .rX Y; Z/ WD X .g.Y; Z// C Y .g.Z; X// Z .g.X; Y// (A.13) C g .ŒX; Y; Z/ g .ŒY; Z; X/ C g .ŒZ; X; Y/ ; along with the torsion–free and metric compatibility conditions

rXY rY X D ŒX; Y (A.14)

X .g.Y; Z// D g .rXY; Z/ C g .Y; rXZ/ ; for all vector ﬁelds X; Y; Z 2 C1.Vn; TVn/. In a local coordinate neighborhood . ; i / @ @ k. /@ k. / U fx g we write ri j WD r@i j WD ij g k, where the Christoffel symbols ij g are given by Â Ã 1 @g ` @g` @g k.g/ D gk` j C i ij : (A.15) ij 2 @xi @xj @x`

If .Vn;g/ is the pull–back of ..Vn/; g/ under a diffeomorphism W Vn ! Vn, k. / k. / then the Christoffel symbols ij g and ij g are related by

@c @a @b @2c k.g/ D c .g/ ı C ; (A.16) ij @xk ab @xi @xj @xi@xj whereas, under the metric rescaling (A.11), we easily compute from (A.15) k. / k. / ij g D ij g , hence

r g Drg: (A.17)

1 n Let us recall that if Hessg f WD rdf denotes the Hessian of f , f 2 C .V ; R/, then n the Laplace–Beltrami operator 4g on .V ; g/ is deﬁned by 352 A Riemannian Geometry ik 4g f WD trg Hessg f D g rirk f ; (A.18) where we have used the geometer’s sign convention according to which, on P 2 .Rn;ı/ n @ Euclidean space , 4ı D iD1 @ 2 . It is useful to have the expression of xi 4g available in local coordinates Â Ã 1 @ p @ ij 4g D p det gg det g @xi @xj @2 1 @ p Á @ (A.19) D gij C p det ggij @xi@xj det g @xi @xj @2 @ D gij j ; @xi@xj @xj where we have deﬁned @ @ p Á j WD ghk j D gji gij log det g : (A.20) hk @xi @xi

From the deﬁnitions of Hessg and 4g it immediately follows that under the metric rescaling (A.11), we have

Hessg D Hessg; (A.21) 1 4 g D 4g:

The Riemann, the sectional, the Ricci and the scalar curvature operators (Fig. A.4) associated to .Vn; g/ are respectively deﬁned by Rm.g/.X; Y/Z WD rXrY rY rX rŒX;Y Z; (A.22) . . /. ; / ; / S . /. ; / g Rm g X Y Y X ; ec g X Y WD 2 (A.23) g.X; X/g.Y; Y/ .g.X; Y//

Ric.g/.Y; Z/ WD trg .X 7! Rm.g/.X; Y/Z/ ; (A.24) R.g/ WD trace Ric.g/; (A.25) for all vector ﬁelds X; Y; Z 2 C1.Vn; TVn/. They are naturally equivariant under the action of Diff.Vn/, whereas under the scaling action (A.11) we get

Rm.g/ D Rm.g/ (A.26) Sec.g/.X; Y/ D 1 Sec.g/.X; Y/ Ric.g/ D Ric.g/ A Riemannian Geometry 353

Fig. A.4 Curvature and parallel transport

R.g/ D 1 R.g/; as easily follows from their definition and (A.17). Insight in the curvature operators is often provided by the expression of their components in a judiciously chosen coordinate system .U; xi/. To this end, let us state first our conventions for the components of Rm.g/ and Ric.g/, i.e. Â Ã @ @ @ @ Rm.g/ ; WD Rh ; (A.27) @xi @xj @xk ijk @xh Xn h ; Rik WD R hik (A.28) hD1 where @h @h Rh D jk ki C h r r h; (A.29) ijk @xi @xj ir jk ki jr @h @h R D ik hk C h r r h: (A.30) ik @xh @xi ik rh hi kr Notice that when considering the Riemann tensor components in covariant form, the upper index on the Riemann tensor is lowered into the 4–th position according h . n; / to Rijk` WD R ijk gh`. We recall that, given a point p 2 V g , the exponential map n n expp W TpV ! V is defined by (Fig. A.5) 354 A Riemannian Geometry

Fig. A.5 A two–dimensional rendering of the exponential map

. / .1/; X 7! expp X WD X

n where X W Œ0; 1/ ! V is the constant speed geodesic issued from p with initial n velocity PX.t D 0/ D X.IfCut.p/ denotes the cut locus of p in .V ; g/, namely the set of points q 2 Vn such that any minimal geodesic .p; q/ connecting p to q is not properly contained in another minimal geodesic issued from p, then normal geodesic coordinates fxigWVn Cut.p/ ! Rn (based at the point p 2 Vn)are deﬁned by i . / i 1 . /; fx g q WD fX gıexpp q

i n n where fX g are the components, in an orthonormal frame fe.i/giD1 at TpV ,ofthe 1. / n vector expp q 2 TpV . In normal coordinates one gets the classical Bertrand- Diguet-Puiseaux formulas providing a geometric interpretation, (see e.g. [4]), of the sectional, Ricci and scalar curvature, viz. Â Ã S ec.X; Y/.p/ `.C.r// 2r 1 r2 O r3 ; D 6 C

p 1 g.q/ 1 .p/ xi.q/xk.q/ O r3 ; det D 6 Rik C (A.31) Â Ã R.p/ Vol .B.p; r// D !n rn 1 r2 C O.r3/ ; 6.n C 2/ A Riemannian Geometry 355

Fig. A.6 The geometrical meaning of Ricci curvature

where: (i) r WD d.g/.q; p/ is the Riemannian distance between q and p; (ii) `.C.r// is the length of the circle of radius r in the geodesic surface generated by the unit n n vectors X; Y 2 TpV ; (iii) B.p; r/ is the metric ball in .V ; g/ of radius r centered at p, and !n is the volume of the unit ball in Euclidean space. From these well– known expressions it follows that the sectional curvature measures the defect of the length of small circles C.r/ with respect to the Euclidean 2r. Similarly, the Ricci curvature measures the distortion of the solid angle, subtended by a small spherical 1. / sector in the direction X D expp q , with respect to the corresponding Euclidean value (Fig. A.6). Finally, the scalar curvature provides, as the radius varies, the distortion of the volume of small metric balls with respect to their Euclidean volume.

A.3 Some Properties of the Space of Riemannian Metrics

In the applications of Riemannian geometry discussed in these lecture notes we are not interested in a specific Riemannian metric g on Vn but rather in classes of metrics inducing on Vn the same geometric properties up to diffeomorphisms and rescalings [4]. This naturally calls into play the space of all smooth Riemannian metrics on Vn, 1. n; 2 n/ the open convex cone (in the compact–open topology on C V ˝S T V ) defined by (Fig. A.7) ˚ ˇ « M . n/ 1. n; 2 n/ˇ : et V WD g 2 C V ˝S T V g is positive definite (A.32) On M et.Vn/ there is a natural action both of the group of diffeomorphisms Diff.Vn/ and of the multiplicative group R WD .R>0; / of positive real numbers. The former acting by pull–back

Diff.Vn/ M et.Vn/ ! M et.Vn/ (A.33) .; g/ 7! g; 356 A Riemannian Geometry

Fig. A.7 The 1–dimensional cone of Riemannian metrics with its n Diff.V / orbits Og

the latter acting by homotetic rescaling

R M et.Vn/ ! M et.Vn/ (A.34) .; g/ 7! g:

The quotient space M et.Vn/=Diff.Vn/ under (A.33) is the space of smooth Riemannian structures on Vn describing Riemannian manifolds with the same geometric properties, whereas the quotient M et.Vn/=.R Diff.Vn//, associated with the composition of Diff.Vn/ with the homotheties (A.34), describes Riemannian manifolds with the same metric properties up to an overall rescaling. Note that by normalizing volume we can write Z M et.Vn/ g M et.Vn/ d 1 .Vn/; n ' 2 j g D Diff (A.35) R Diff.V / Vn n where dg denotes the Riemannian measure on .V ; g/. Most of our exploitation of the geometry of M et.Vn/ is conﬁned to its tangent n n n space T.Vn;g/M et.V / at .V ; g/. Let us recall that an element of T.Vn;g/M et.V / is an equivalence class of curves of Riemannian metrics

. ; / ! M et.Vn/ (A.36) 7! g./ ; with g.0/ D g; where any two such a curve 7! g1./ and 7! g2./ are equivalent if they have the same velocity at D 0, i.e.

Œg1 ' Œg2 (A.37)

ˇ ˇ dg1./ˇ dg2./ˇ ˇ ˇ 1. n; 2 n/: , ˇ D ˇ D h 2 C V ˝S T V d D0 d D0 A Riemannian Geometry 357

n Hence we have the identiﬁcation of the tangent space T.Vn;g/M et.V / with the space 1. n; 2 n/ of sections C V ˝S T V ,

T M . n/ 1. n; 2 n/; .Vn;g/ et V ' C V ˝S T V (A.38) which serves as the local model for the inﬁnite–dimensional (Frechét) manifold M et.Vn/. To be somewhat informal, given g 2 M et.Vn/, the elements h of n T.Vn;g/M et.V / can be interpreted as the inﬁnitesimal variations of the given metric,

. ; / ! M et.Vn/ (A.39)

7! .g.//ik D gik C hik ;0< <<1:

M . n/ 1. n; 2 n/ Note that et V , being an open cone in C V ˝S T V , is contractible and the tangent bundle TMet.Vn/ of M et.Vn/ is simply

TM . n/ M . n/ 1. n; 2 n/: et V D et V C V ˝S T V (A.40)

Applications of Riemannian geometry to quantum field theory often exploit the n Berger–Ebin decomposition [3]ofT.Vn;g/M et.V /. This decomposition allows us to discriminate between trivial (infinitesimal) variations of a metric, associated with the action of the diffeomorphisms group, and essential variations describing a change in the Riemannian structure of .Vn; g/ (in this latter case one speaks of (infinitesimal) deformations of the given g:see[4] for the proper use of the term deformation vs. variation). The construction of this decomposition is somewhat delicate and owing to its importance in the applications discussed in these lecture notes, it is worthwhile to describe the various steps involved in its proof. To begin with, let us recall that both M et.Vn/ and Diff.Vn/ are locally modeled on Frechét spaces of smooth sections and maps, where the necessary tools from analysis on Banach spaces, (basically a Picard iteration scheme), are not easily available. We need to enlarge M et.Vn/ and Diff.Vn/ to their Sobolev counterparts. To this end we consider the open subset of H s.Vn; ˝2 TVn/ defining M s. n/ > n the Riemannian metrics et V of Sobolev class s 2 , and the (topological) 0 group, D s .Vn/, defined by the set of diffeomorphisms which, as maps Vn ! Vn 0 are an open subset of the Sobolev space of maps H s .Vn; Vn/, with s0 s C 1, 0 (see (A.10)). There is a natural projection map from D s .Vn/ into M et s.Vn/

D s0 . n/ O s M s. n/ W V ! g et V (A.41) : 7! ./ D g; where n o O s M s. n/ ; D s0 . n/ g WD gQ 2 et V j gQ D g 2 V (A.42) 358 A Riemannian Geometry

0 is the D s .Vn/–orbit of a given metric g 2 M et s.Vn/, and g is the pull–back D s0 . n/ 0 1 T O s under 2 V , s s C .If .Vn;g/ g denotes the tangent space to any such T O s an orbit, then .Vn;g/ g is the image of the operator

ı H sC1. n; n/ H s. n; 2 n/ g W V TV ! V ˝ T V (A.43) : 1 w ı .w/ L g; 7! g D 2 w where Lw g denotes the Lie derivative of the metric tensor g along the vector ﬁeld w,

.L / h@ @ h @ h: w g ik Driwk Crkwi D w hgik C ghk iw C gih kw (A.44)

n Its (total) symbol in the (co)direction 2 Tp V is provided by the injective bundle homomorphism

nŒı n 2 n V g W Tp V ! ˝ STp V (A.45) 1 X dxi VnŒı.X/ . X X / dxh dxk: i 7! g D 2 h k C k h ˝

2 ı ı Note that the L adjoint g of g , with respect to the inner product (A.3), is provided by (minus) the divergence operator

s n 2 n s1 n n ıg W H .V ; ˝ T V / ! H .V ; T V / (A.46) : ij k h 7! ıg h Dg rihjk dx ; with total symbol given by

nŒı 2 n n V g W˝STp V ! Tp V (A.47) h k n ih k whk dx ˝ dx 7! V Œıg.w/ Dg iwhkdx :

ı ı ı Since g has injective symbol, Banach space theory implies that g ı g is an elliptic operator [3]. To expand in more detail, we compute the symbol of the composition ı ı g ı g according to

nŒı ı n 2 n n V g ı g W T V ! ˝ ST V ! T V (A.48) i nŒı. / nŒı nŒı. / Xi dx 7! V g X 7! V g ı V g X Á 1 VnŒı VnŒı.X/ 2ıi i X dxk; D g g D2 j jg k C k i

ı ı and the ellipticity of the operator g ı g immediately follows by observing that 2ıi i the matrix j jg k C k is non–singular with positive eigenvalues. By standard A Riemannian Geometry 359

Fig. A.8 The geometry of the Berger–Ebin decomposition of n T.Vn;g/M et.V /, the tangent space to M et.Vn/ at the metric g

2 ı T M s. n/ elliptic theory the L –orthogonal subspace to Im g in .Vn;g/ et V is spanned by the (inﬁnite–dimensional) kernel of ıg. This entails the well–known Berger–Ebin 2 n s n L .V ; dg/–orthogonal splitting [3, 8, 21] of the tangent space T.Vn;g/M et .V / (Fig. A.8),

T M s. n/ T M s. n/ ı ı H sC1. n; n/ .Vn;g/ et V Š .Vn;g/ et V \ ker g ˚ Im g V TV (A.49) T M s. n/ ı T O s; Š .Vn;g/ et V \ ker g ˚ .Vn;g/ g s n according to which, for any given tensor h 2 T.Vn;g/M et .V /, we can write T L ; hab D hab C w gab (A.50) T a T 0 where hab denotes the div–free part of h,(r hab D ), and where the vector ﬁeld w is characterized as the solution, (unique up to the Killing vectors of .Vn; g/), of the elliptic PDE

ı ı ı : g g w D g h (A.51)

We can also take into account the role of the scaling transformation (A.11) s n and reﬁne (A.49) to˚ the tangent spaceR T.Vn;g/M et« 1 .V / to the space of met- M s. n/ M s. n/ n 1 rics et1 V WD g 2 et V j V d g D of normalized total volume (cf. (A.35)). Since Z ˇ n T M s . n/ H s. n; 2 n/ˇ 0 ; .Vn;g/ et 1 V D h 2 V ˝ST V trg hd g D (A.52) V 360 A Riemannian Geometry

ik where trg h WD g hik,wehave T M s. n/ T M s . n/ ı ı H sC1. n; n/ .Vn;g/ et1 V Š .Vn;g/ et 1 V \ ker g ˚ Im g V TV (A.53) T M s . n/ ı T O s: Š .Vn;g/ et 1 V \ ker g ˚ .Vn;g/ g

A.4 Ricci Curvature

In applications of Riemannian geometry to the quantum geometry of polyhedral surface (and its interplay with Ricci ﬂow), Ricci curvature plays a central role. A deep insight in its geometric meaning is provided by the expression of its components in local harmonic coordinates .U; fxig/. These latter are deﬁned by requiring that each coordinate function xk W U ! R is harmonic. In particular, local solvability for elliptic PDE’s implies, (compare with [7], Corollary 3.30), that n n for any given point p 2 .V ; g/ there always exists a neighborhood Up V of p such that, (see (A.20)), Â Ã @2 @ 4 xi D gjk ` xi D i D 0: (A.54) g @xj@xk jk @x`

When passing from normal geodesic coordinates to harmonic coordinates we gain control on the components of the metric tensor in terms of the Ricci curvature rather than of the full Riemann tensor (cf. [14] for a particularly clear comparison of these two coordinate systems from the point of view of geometric analysis). In particular, as first stressed by C. Lanczos [20], in harmonic coordinates the components of the Ricci tensor take the suggestive form, (see e.g. [27] Lemma 2.6 for an elegant computation), 1 1 Rik D 4g .gik/ C Qik g ;@g ; (A.55) .harmonic/ 2 where for each fixed pair of indexes .i; k/, the expression 4g .gik/ denotes the Laplace–Beltrami operator applied component wise to each gik as it were a scalar ab function, (in particular 4ggik is not the tensorial (rough) Laplacian g rarb gik of the metric tensor g, a quantity that obviously vanishes identically for the Levi– Civita connection). In (A.55) Q g1;@g denotes a sum of terms quadratic in the components of g, g1 and their first derivatives, and whose explicit expression does not concern us here (compare with [27]). Hence in harmonic coordinates the Ricci curvature acts as a quasi–linear elliptic operator on the components of the metric, an observation that can be exploited to prove [9] that the metric tensor g, if not smooth, has maximal regularity in harmonic coordinates. The expression (A.55) also plays an inspiring role in Ricci flow theory, where the minus sign in front of A Riemannian Geometry 361

Fig. A.9 The Ricci curvature in harmonic coordinates as the generator of Brownian diffusion on the manifold .Vn; g/

the scalar Laplace–Beltrami 4g .gik/ suggests the right PDE candidate for the Ricci ﬂow among the many (apparently) possible ones. It is also worthwhile to observe that the factor 1=2 appearing in (A.55), motivating the conventional “2” in the Ricci ﬂow generator 2 Ric.g/, is not as incidental as is typically assumed. This is related 1 . n; / to the fact that 2 4g and not 4g is the generator of Brownian motion on V g [15] (Fig. A.9). Appendix B A Capsule of Moduli Space Theory

In this Appendix we summarize notation and a few basic deﬁnitions of Riemann Moduli theory that we have been freely using in these lecture notes, (for details we mainly refer to [12, 13, 16]).

B.1 Riemann Surfaces with Marked Points and Divisors

We shall denote by .Mg; C / a Riemann surface of genus g,(mostoftenwe simply write .M; C / if the genus is clear from the context). Recall that .M; C / is characterized by an atlas of local coordinate charts .Uk;'k/ deﬁned by maps ' C ' '1 ' . / ' . / k W Uk ! whose transition functions, h ı k W k Uk \ Uh ! h Uh \ Uk , are holomorphic maps between open subsets of C. Any such a chart .U;'/ is said to provide a local conformalp parameter, and for the generic p 2 U one sets '.p/ WD z D x C 1 y.AN0–pointed surface ..MI N0/; C /, or surface with N0 marked points, is an oriented closed, (connected), surface of genus g decorated with a distinguished set of N0 pairwise distinct points fp1;:::; pN0 g. 0 (Note that ..MI N0/; C / is distinct from the open Riemann surface .M ; C / WD

.M; C /nfp1;:::; pN0 g obtained by removing from .M; C / the points fp1;:::; pN0 g) (Fig. B.1). The tangent and cotangent spaces at p 2 .M; C / are naturally obtained by tensoring with C the tangent TpM and cotangent space TpM of the underlying real surface M, i.e., TC ;pM WD TpM˝C and TC ;pM WD TpM ˝C. The respective basis induced by the local conformal parameter z are provided by the usual expressions Â Ã Â Ã @ 1 @ p @ @ 1 @ p @ WD 1 ; WD C 1 ; (B.1) @z 2 @x @y @z 2 @x @y p p dzWD dx C 1 dy ; d z WD dx 1 dy; (B.2)

© Springer International Publishing AG 2017 363 M. Carfora, A. Marzuoli, Quantum Triangulations, Lecture Notes in Physics 942, https://doi.org/10.1007/978-3-319-67937-2 364 B A Capsule of Moduli Space Theory

Fig. B.1 A N0–pointed Riemann surface ..M; N0/; C / of genus g obtained by injecting a string of N0 pairwise distinct points in .M; C /.Hereg D 1 and N0 D 9, we typically assume that 2g 2 C N0 >0, in such a case the automorphism group, Aut .M; N0/, of the resulting pointed Riemann surface is ﬁnite. Recall that Aut .M; N0/ is the largest group of conformal automorphisms that the Riemann surface ..M; N0/; C / can admit

p 1 dx dy dz dz: ^ D 2 ^ (B.3)

0 C @=@ One can naturally split TC ;pM into the holomorphic TpM WD f zg and 00 C @=@ . ; C / antiholomorphic Tp M WD f zg tangent spaces at p 2 M according to 0 00 00 0 TC ;pM D TpM ˚ Tp M, with Tp M D TpM, ( where the overline denotes complex conjugation). In this connection it is worthwhile recalling that a smooth map f W M ! N between two surfaces M and N is holomorphic if and only if 0 the corresponding tangent map f W TC ;pM ! TC ;f .p/N is such that f TpM 0 Tf .p/N. This implies that there is a linear isomorphism between TpM and the 0 C @=@ holomorphic tangent space TpM D f zg, an isomorphism, this latter, which provides a natural dictionary between geometrical quantities on the surface M and their corresponding realizations on .M; C /. Holomorphic maps also preserve the holomorphic–antiholomorphic decomposition of the cotangentL space TC ;pM D 0 00 ˝. / ˝.p;q/. / TpM ˚ Tp M , and the corresponding splitting M D pCqÄ2 M of the space of differential forms on .M; C /. Explicitly, the spaces ˝.1;0/.M/, ˝.0;1/.M/, and ˝.1;1/.M/ of forms of type .p; q/ are locally generated by the monomials '.z/ dz, '.z/ dz, and '.z/ dz ^ dz, respectively. Corresponding to such a splitting we have the Dolbeault operators @ W ˝.p;q/.M/ ! ˝.pC1;q/.M/ and @ W ˝.p;q/.M/ ! ˝.p;qC1/.M/, with @@ D 0, @ @ D 0, @@ D@@, and d D @ C @, where d denotes the exterior derivative. Locally,

@ @ @ WD dz; @ WD d z; (B.4) @z @z

In particular, a form 2 ˝.p;0/ is holomorphic if @ D 0. A metric on a Riemann surface .M; C / is conformal if locally it can be written as

ds2 D h2.z/ dz ˝ dz; (B.5) B A Capsule of Moduli Space Theory 365 for some smooth function h.z/>0. The corresponding Kähler form is given by p 1 $ h2.z/ dz dz: WD 2 ^ (B.6) The Gaussian curvature of the (smooth) metric ds2 D h2.z/ dz ˝ dz is provided by

; K WD ds2 ln h (B.7)

where ds2 ,( if is clear from the context which conformal metric we are using), is the Laplace–Beltrami operator with respect to the metric ds2, viz. Â Ã 1 @2 @2 4 @ @ WD C D : (B.8) h2 @ x2 @ y2 h2.z/ @ z @ z

Let us recall, (see e.g. the very readable presentation in [13]), that a divisor D on a Riemann surface .M; C / is a formal linear combination X D D nk zk (B.9) where nk 2 Z and the zk are points 2 M. D is required to be locally finite, namely for any q 2 M there is a neighborhood Uq M of q such that Uq \fzkg contains only a finite number of the fzkg appearing in D. Since we typically deal with compact Riemann surfaces .M; C /, this implies that the formal sum defining D is finite. The set of divisors on .M; C /Pnaturally forms an additive (abelian) group Div.M/.The degree of a divisor D D nk zk, (on a compact surface Riemann surface .M; C /), is defined according to X deg D WD nk: (B.10)

Let f be a meromorphic function, i.e. the local quotient of two holomorphic functions on .M; C /.Letk 2 N,iff has a zero of order k at z 2 .M; C /, define the order of f at z to be ordz f WD k.> 0/, conversely, if f has a pole of order k at z 2 .M; C / we define ordz f WD k. The order of f at all other points is defined to be 0. The divisor of a meromorphic function f , f 6Á 0, is conventionally denoted by .f /, and defined as X . / . / ; f WD ordzh f zh (B.11) where the sum is over all zeros and all poles of f . A divisor D is called a principal divisor if it is the divisor .f / of a meromorphic function f 6Á 0. Since a meromorphic function f on .M; C / has the same number of zeros and poles (counted with multiplicity) it immediately follows that the degree of its divisor .f / satisfies

deg .f / D 0: (B.12) 366 B A Capsule of Moduli Space Theory

More generally, if Mn is a (not necessarily compact) complex manifold of dimension n, a divisor can be naturally associated with hypersurfaces (i.e. n 1–dimensional n n submanifolds) of M . Explicitly, if Vi are irreducible analytic hypersurfaces of M , n (i.e., for each point p 2 Vi M , Vi can be given in a neighborhood of p as the e e zeros of a single holomorphic function f , and it cannot be written as Vi D V1 [ V2, e n n where V˛, ˛ D 1; 2 are analytic hypersurfaces of M ), then a divisor on M is a locally ﬁnite formal linear combination of irreducible analytic hypersurfaces of Mn X D D nk Vk: (B.13)

n Note that given any such a divisor, we can always find an open cover fU˛g of M such that in U˛, with U˛ \ Vi 6D;, the hypersurface Vi has a local defining equation Vi Dfgi ˛.z/ D 0g where gi ˛.z/ is a holomorphic function in U˛. In each U˛ the set of fgi ˛.z/g, associated with all Vi for which U˛ \ Vi 6D;, characterizes a meromorphic function f˛ according to Y ni ; f˛ WD gi ˛ (B.14) i which are the local defining functions for the divisor D. The set of local defining n functions ff˛g associated with an open covering fU˛g of M can be also used for characterizing the line bundle ŒD associated with the given divisor D,thisisthe n bundle over M defined by the transition functions f ˛ˇ WD f˛ = fˇg for any U˛ \ Uˇ 6D;. Note that the line bundle ŒD is trivial if and only if D is the divisor of a meromorphic function. 2k . n; R/ 2 Recall that if HDR M denotes the k–th DeRham cohomology group of Mn, and V Mn is an analytic subvariety of (complex) dimension k, then the n fundamental class .V/ in the homology group H2k.M ; R/ is defined by the pairing

2k n n H .M ; R/ H2k.M ; R/ ! R (B.15) DR Z ; .V/ 7! : V

2n2k. n; R/ By Poincaré duality this determinesP the fundamental class V 2 HDR M of V. Thus, given a divisor D D n V , we can introduce its fundamental class Vk k k . / 2 . n/ D and its Poincaré dual D 2 HDR M according to X .D/ WD nk .Vk/; (B.16)

X : D WD nk Vk (B.17) B A Capsule of Moduli Space Theory 367

n Given a line bundle L over M , locally deﬁned by transition functions f ˛ˇ g relative n toacoveringfU˛g of M ,let p 1 c1.L/ ; D 2 (B.18) be the (ﬁrst) Chern class of L, where jU˛ D d !˛ denotes the curvature of L associated with any locally given connection 1–form !˛. A basic result connecting line bundles, divisors and Chern classes is the observation that if L is the line bundle associated with a divisor D, then, (see [13] for a nicely commented proof),

. / .Œ / 2 . n/: c1 L D c1 D D D 2 HDR M (B.19)

T . / B.2 The Teichmüller Space g;N0 M

In discussing the connection between polyhedral surfaces and Riemann surfaces we are naturally led to consider the relation between the space POLg;N0 .M/ and Mg;N0 , the moduli space of N0–pointed Riemann surfaces of genus g.This latter features as a basic object of study in a large variety of mathematical and physical applications of Riemann surface theory, and as such it is susceptible of many possible characterizations. From our perspective it is appropriate to adopt a

Riemannian geometry viewpoint and deﬁne Mg;N0 as a suitable quotient of the space of conformal classes of riemannian metrics the surface M can carry. For details and proofs we refer to [25]. Let us consider the set of all (smooth) Riemannian metrics on the genus–g surface M, (see Sect. 1.6), : M et.M/ D fg 2 S2.M/j g.x/.u; u/>0if u ¤ 0g ; (B.20) where S2.M/ is the space of symmetric bilinear forms on M, and let us denote by 1 C .M; RC/ the group of positive smooth functions acting on metrics g 2 M et.M/ by pointwise multiplications. This action is free, smooth (and proper), and the quotient

M . / C . / et M ; onf M WD 1 (B.21) C .M; RC/ is the Fréchet manifold of conformal structures. Note that C onf .M/ naturally extends to the pointed surface .MI N0/, obtained by decorating M with N0 marked points fp1;:::; N0g, as long as the conformal class contains a smooth representative metric [29]. Let

M et1.MI N0/,! M et .MI N0/ (B.22) 368 B A Capsule of Moduli Space Theory be the set of metrics of constant curvature 1, describing the hyperbolic structures on .MI N0/.IfDiffC.M/ is the group of all orientation preserving diffeomorphisms then n o D . / . / v N0 iffC MI N0 D 2 DiffC M W preser es setwise fpigiD1 (B.23) acts by pull-back on the metrics in M et1.MI N0/.LetDiff0.MI N0/ be the subgroup consisting of diffeomorphisms which when restricted to .MI N0/ are isotopic to the identity, then the Teichmüller space Tg;N0 .M/ associated with the genus g surface with N0 marked points M is deﬁned by

M et1.MI N0/ Tg;N0 .M/ D : (B.24) Diff0.MI N0/

From a complex function theory perspective, Tg;N0 .M/ is characterized by ﬁxing a reference complex structure C0 on .MI N0/ (a marking) and considering the set of equivalence classes of complex structures .C ; f / where f W C0 ! C is an orientation preserving quasi-conformal map, and where any two pairs of complex structures .C1; f1/ and .C2; f2/ are considered equivalent if h ı f1 is homotopic to f2 via a conformal map h W C1 ! C2. Let us assume that the reference complex structure C0 admits an antiholomorophic reﬂection j W C0 ! C0. Since any orientation reversing diffeomorphism e' can be written as ' ı j for some orientation preserving diffeomorphism ', the (extended) mapping class group : Map.MI N0/ D Diff .MI N0/=Diff0.MI N0/ (B.25)

acts naturally on Tg;N0 .M/ according to

Map.MI N0/ Tg;N0 .M/ ! Tg;N0 .M/ (B.26) ( .'; .C ; // .C ; '1/; ' D . / f 7! f ı 2 iffC MI N0 ; 1 .e';.C ; f // 7! .C ; f ı j ı ' / e' 2 Diff .MI N0/ DiffC.MI N0/ where the conjugate surface C is the Riemann surface locally described by the complex conjugate coordinate charts associated with C . It follows that the

Teichmüller space Tg;N0 .M/ can be also seen as the universal cover of the moduli space Mg;N0 of genus g Riemann surfaces with N0.T/ marked points deﬁned by

Tg;N0 .M/ Mg;N0 D (B.27) Map.MI N0/

This characterization represents the moduli associated to ..MI N0/; C / by an 2 2 equivalence class of Riemannian metrics Œds on M, where two metrics ds.1/ and 2 Œ 2 ; 1. ; R / ds.2/ deﬁne the same point ds of Mg N0 if and only if there exists f 2 C M C B A Capsule of Moduli Space Theory 369

D and an orientation preserving diffeomorphism Á 2 iffg; N0 fixing each marked 2 2 point pk individually such that ds.2/ D f ds.1/ . According to the remark above, the class Œds2 may contain singular metrics provided that they are conformal to a smooth one. As emphasized by M. Troyanov, this implies that we can represent 2 Œds 2 Mg;N0 by conical metrics, as well. For a genus g Riemann surfaces with N0.T/>3marked points the complex vector space QN0 .M/ of (holomorphic) quadratic differentials is defined by tensor fields described, in a locally uniformizing complex coordinate chart .U;/,by a holomorphic function W U ! C such that D ./d ˝ d.Awayfrom the discrete set of the zeros of , we can locally choose a canonical conformal coordinatep z. / (unique up to z. / 7!˙z. / C const) by integrating the holomorphic 1-form , i.e., Z p 0 0 0 z. / D . /d ˝ d ; (B.28)

1 so that D dz. / ˝ dz. /. If we endow QN0 .M/ with the L -(Teichmüller) norm Z : jj jj D j j; (B.29) M then the Banach space of integrable quadratic differentials on M, : QN0 .M/ Df j; jj jj < C1g; (B.30) is non-empty and consists of meromorphic quadratic differentials whose only singularities are (at worst) simple poles at the N0 distinguished points of .M; N0/.

QN0 .M/ is ﬁnite dimensional and, (according to the Riemann-Roch theorem), it has complex dimension

dimC QN0 .M/ D 3g 3 C N0.T/; (B.31)

From the viewpoint of Riemannian geometry, a quadratic differential is basically a transverse-traceless two tensor deforming a Riemannian structure to a nearby inequivalent Riemannian structure. Thus a quadratic differential D ./d ˝ d also encodes information on possible deformations of the given complex structure. Explicitly, by performing an affine transformation with constant dilatation K >1, 0 one defines a new uniformizing variable z. / associated with D ./d ˝ d by deforming the variable z. / defined by (B.28) according to

0 p z. / D KRe.z. // C 1Im.z. //: (B.32)

The new metric associated with such a deformation is provided by

ˇ ˇ ˇ ˇ2 2 . 1/2 ˇ 1 ˇ ˇ 0 ˇ K C ˇ K ˇ ˇdz. /ˇ D ˇdz. / C dz. /ˇ : (B.33) 4 K C 1 370 B A Capsule of Moduli Space Theory

2 Since dz. / is the given quadratic differential D ./d˝d, we can equivalently ˇ ˇ2 ˇ 0 ˇ write ˇdz. /ˇ as ˇ ! ˇ ˇ ˇ ˇ ˇ2 ˇ ˇ2 .K C 1/2 ˇ K 1 ˇ ˇ 0 ˇ ˇ ˇ ; dz. / D j j d C 1=2 d (B.34) 4 ˇ K C 1 jj ˇ where .= jj/ d ˝ d1 is the ( Teichmüller-) Beltrami form associated with the quadratic differential . If we consider quadratic differentials D ./d ˝ d in : .1/. / ; ./ <1 ˜ the open unit ball QN0 M Df j jj jj g in the Teichmüller norm(B.29), then there is a natural choice for the constant K provided by 1 Cjj./jj K D : (B.35) 1 jj./jj In this latter case we get ˇ ! ˇ ˇ ˇ ˇ ˇ2 ˇ ˇ2 jj./jj2 ˇ 1 ˇ ˇ 0 ˇ ˇ ˇ : dz. / D 2 j j d C 1=2 d (B.36) .jj./jj 1/ ˇ jj./jj jj ˇ According to Teichmüller’s existence theorem any complex structure on can be ./ .1/. / parametrized by the metrics (B.36)as D d ˝ d varies in QN0 M .Thisis equivalent to saying that for any given .M; g/, (with .M; N0I Œg/ deﬁning a reference complex structure C0 on .M; N0/), and any diffeomorphism f 2 Diff0.MI N0/ . ; / 1 . ; / .1/. / mapping M g into M g1 , there is a quadratic differential 2 QN0 M and a biholomorphic map F 2 Diff0.MI N0/, homotopic to f such that ŒF g1 is given by the conformal class associated with (B.36). This is the familiar point of view .1/. / which allows to identify Teichmüller space with the open unit ball QN0 M in the space of quadratic differentials QN0 .M/. It is also worthwhile noticing that (B.36) .1/. / allows us to consider the open unit ball QN0 M in the space of quadratic differential as providing a slice for the combined action of Diff0.MI N0/ and of the conformal s group C onf .MI N0/ on the space of Riemannian metrics M et.MI N0/, i.e.

.1/ M et.M N0/ . /, I C s. / . / QN0 M ! ' onf MI N0 Tg;N0 M (B.37) Diff0.MI N0/ ˇ ! ˇ Äˇ ˇ ˇ ˇ2 ˇ ˇ2 jjjj2 ˇ 1 ˇ ˇ 0 ˇ ˇ ˇ ; dz 7! 2 j j d C 1=2 d .jjjj 1/ ˇ jjjj jj ˇ where ˚ « s : C s C onf .MI N0/ D f W M ! R j f 2 H .M; R/ (B.38)

1 With .M; N0I Œg1/ a complex structure distinct from .M; N0I Œg/. B A Capsule of Moduli Space Theory 371 denotes the (Weyl) space of conformal factors defined by of all positive (real valued) functions on M, f 2 Hs.M; R/, whose derivatives up to the order s exist in the sense of distributions and are represented by square integrable functions. 1 In line with the above remarks, one introduces also the space BN0 .M/ of (L measurable) Beltrami differentials, i.e. of tensor fields $ D ./d˝d1, sections 1 ./ < of k ˝k,(k being the holomorphic cotangent bundle to M), with supM j j 1. The space of Beltrami differentials is naturally identified with the tangent space to

Tg;N0 .M/, i.e.,

1 $ D ./d ˝ d 2 TC Tg;N0 .M/; (B.39)

(with C a complex structure in Tg;N0 .M/). The two spaces QN0 .M/ and BN0 .M/ can be naturally paired according to Z h j$i D ././dd: (B.40) M

In such a sense QN0 .M/ is C-anti-linear isomorphic to TC Tg;N0 .M/, and can be . / . / canonically identiﬁed with the cotangent space TC Tg;N0 M to Tg;N0 M .Onthe . / cotangent bundle TC Tg;N0 M we can deﬁne the Weil-Petersson metric as the inner product between quadratic differentials corresponding to the L2- norm provided by Z : 2 2./ ./ 2 2 ; k kWP D h j j jd j (B.41) M

. / ./ 2 where 2 QN0 M and h jd j is the hyperbolic metric on M. Note that h is a @ 3g3CN0 Beltrami differential on M, thus if we introduce a basis f @˛ g˛D1 of the vector space of harmonic Beltrami differentials on .M; N0/, we can write Z @ @ 2 G˛ˇ D h./ jdj (B.42) M @˛ @ˇ

for the components of the Weil-Petersson metric on the tangent space to Tg;N0 .M/. We can introduce the corresponding Weil-Petersson Kähler form according to

p ˛ ˇ !WP WD 1G˛ˇdZ ^ dZ ; (B.43)

˛ . / @ where fdZ g are the basis, in QN0 M , dual to f @˛ g under the pairing (B.40). Such Kähler potential is invariant under the mapping class group Map.MI N0/ to the effect that the Weil-Petersson volume 2-form !WP on Tg;N0 .M/ descends on Mg;N0 , and it has a (differentiable) extension, in the sense of orbifold, to Mg;N0 . 372 B A Capsule of Moduli Space Theory

M ; B.3 Some Properties of the Moduli Spaces g N0

It is well-known that the moduli space Mg;N0 is a connected orbifold space of complex dimension 3g 3 C N0.T/ and that, although in general non complete, it admits a stable compactification (Deligne-Mumford) Mg;N0 . This latter is, by definition, the moduli space of stable N0-pointed surfaces of genus g, where a stable surface is a compact Riemann surface with at most ordinary double points such that its parts are hyperbolic. Topologically, a stable pointed surface (or, equivalently a stable curve, in the complex sense), is obtained by considering a finite collection of 0 embedded circles in M WD M=fp1;::: pN0 g, each in a distinct isotopy class relative to M0, and such that none of these circles bound a disk in M containing at most one pk. By contracting each such a circle, and keeping track of the marked points in such a way that any component of the resulting surface can support a hyperbolic metric, we get the stable pointed surface of genus g. Thus, the closure @Mg;N0 of

Mg;N0 in Mg;N0 consists of stable surfaces with double points, and gives rise to a stratification decomposing Mg;N0 into subvarieties. By definition, a stratum of codimension k is the component of Mg;N0 parametrizing stable surfaces (of fixed topological type) with k double points. The orbifold Mg;N0 is endowed with N0.T/ natural line bundles Li defined by the cotangent space to M at the i-th marked point. A basic observation in moduli space theory is the fact that any point p on a stable curve ..MI N0/; C / 2 Mg;N0 defines a natural mapping

..MI N0/; C / ! Mg;N0C1 (B.44)

0 that determines a stable curve ..MI N0 C 1/; C / 2 Mg;N0C1. Explicitly, as long N0 as the point p is disjoint from the set of marked points fpkgkD1 one simply deﬁnes .. 1/; C 0/ .. ; /; C / N0 MI N0 C to be MI N0 fpg . If the point p D ph for some ph 2fpkgkD1, 1 0 .. 1/; C 0/ then: (i) for any Ä i Ä N0, with i ¤ h, identify pi 2 MI N0 C with 1 the corresponding pi; (ii) take a thrice–pointed sphere CP.0;1;1/, label with a sub- index h one of its marked points .0; 1; 1/,say1h, and attach it to the given ph 2 1 Œ..MI N0/; C /; (iii) relabel the remaining two marked points .0; 1/ 2 CP.0;1;1/ as 0 0 ph and pN0C1. In this way, we get a genus g noded surface 0 : 1 s Œ..M N0/; C / ..M N0 1/; C / ..M N0/; C / CP h I D I C D I h [ .0;1;1h/ (B.45) with a rational tail and with a double point corresponding to the original marked Œ.. 1/; C 0/ point ph. Finally if p happens: to coincide with a node, then MI N0 C 0 1 results from setting pj D pj for any Ä i Ä N0 and by: (i) normalizing Œ..MI N0/; C / at the node (i.e., by separating the branches of Œ..MI N0/; C /at p); (ii) inserting a copy of CP1 with f0; 1g identiﬁed with the preimage of p and : .0;1;1/ 0 1 CP1 with pN0C1 D 2 .0;1;1/. Conversely, let

W Mg;N0C1 ! Mg;N0 (B.46) B A Capsule of Moduli Space Theory 373

forget 0 Œ..MI N0 C 1/; C / ` & ! ..MI N0/; C / collapse the projection which forgets the .N0 C 1/st marked point and collapse to a point any irreducible unstable component of the resulting curve. The fiber of over ..MI N0/; C / is parametrized by the map (B.44), and if ..MI N0/; C / has a trivial 1 automorphism group AutŒ..MI N0/; C / then ..MI N0/; C / is by definition the surface ..MI N0/; C /, otherwise it is identified with the quotient

..MI N0/; C /=AutŒ..MI N0/; C /: (B.47)

Thus, under the action of , we can consider Mg;N0C1 as a family (in the orbifold sense) of Riemann surfaces over Mg;N0 and we can identify Mg;N0C1 with the universal curve C g;N0 ,

W C g;N0 ! Mg;N0 : (B.48)

Note that, by construction, C g;N0 (but for our purposes is more proﬁtable to think in terms of Mg;N0C1) comes endowed with the N0 natural sections s1;:::;sN0

sh W Mg;N0 ! C g;N0 (B.49) : 1 Œ..M N0/; C / s Œ..M N0/; C / ..M N0/; C / CP ; I 7! h I D I h [ .0;1;1h/ defined by (B.45). N0 C ; The images of the sections si characterize a divisor fDigiD1 in g N0 which has a great geometric relevance in discussing the topology of Mg;N0 . Such a study exploits the properties of the tautological classes over C g;N0 generated by the sections N0 N0 fsigiD1 and by the corresponding divisors fDigiD1. To define such classes, recall that the cotangent bundle (in the orbifold sense) to the fibers of the universal: curve W C ; ! M ; gives rise to a holomorphic line bundle ! ; D ! g N0 g N0 g N0 C g;N0 =Mg;N0 over C g;N0 (the relative dualizing sheaf of W C g;N0 ! Mg;N0 ), this is essentially the sheaf of 1-forms with a natural polar behavior along the possible nodes of the Riemann surface describing the fiber of . It is worthwhile to discuss the behavior of the relative dualizing sheaf !g;N0 restricted to the generic divisor Dh generated by . / . / the section sh. To this: end, let z1 h and z2 1h :denote local coordinates defined in the disks p Dfjz1.h/j <1g and 1 Dfjz2.1h/j <1g respectively h h 1 centered around the marked points ph 2 ..MI N0/; C /, and 12CP.0;1;1/.Let C <1 . / th Dfth 2 Wjthj g. Consider the analytic family sh th of surfaces of genus g defined over t and obtained by removing the disks jz1.h/j < jthj and h 1 jz2.1h/j < jthj from ..MI N0/; C / and CP.0;1;1/ and gluing the resulting surfaces . . /; . // . / . / ; through the annulus f z1 h z2 1h j z1 h z2 1h D th th 2 th g by identifying 374 B A Capsule of Moduli Space Theory the points of coordinate z1.h/ with the points of coordinates z2.1h/ D th=z1.h/.The . / . / . / 0 family sh th ! th opens the node z1 h z2 1h D of the section shj..MIN0/;C /. Note that in such a way we can independently and holomorphically open the distinct N0 nodes of the various sections fskgkD1. More generally, while opening the node we can also vary the complex structure of ..MI N0/; C / by introducing local complex 3g3CN0 coordinates .˛/˛D1 for Mg;N0 around ..MI N0/; C /.If . ; / ; sh ˛ th ! Mg N0 th (B.50) denotes the family of surfaces opening of the node, then in the corresponding . ; / coordinates ˛ th the divisor Dh, image of the: P section sh, is locally defined by 0 N0 the equation th DQ. Similarly, the divisor D D hD1 Dh is characterized by the N0 0 locus of equation hD1 th D . : ! ! . / The elements of the dualizing sheaf g;N0 jsh.th/ D g;N0 Dh are differential forms u.h/ D u1dz1.h/ C u2dz2.1h/ such that u.h/ ^ dth D fdz1.h/ ^ dz2.1h/, where f is a holomorphic function of z1.h/ and z2.1h/. By differentiating z1.h/z2.1h/ D th, one gets f D u1z1.h/ u2z2.1h/ which is the defining relation for the forms in

!g;N0 .Dh/. In particular, by choosing u1 D f =2z1.h/, and u2 D f =2z2.1h/ we get O . / the local isomorphism between the sheaf of holomorphic functions sh.th/ over sh th and !g;N0 .Dh/ Â Ã 1 dz1.h/ 1 dz2.1 / f 7! u.h/ D f h : (B.51) 2 z1.h/ 2 z2.1h/

If we set f D f0 C f1.z1.h// C f2.z2.1h//, where f0 is a constant and f1.0/ D 0 D f2.0/, then on the noded surface sh,(th D 0), we get from the relation z2.1h/dz1.h/ C z1.h/dz2.1h/ D 0, . . // f0 C f1 z1 h . /; uhjz2.1h/D0 D dz1 h (B.52) z1.h/

f0 C f2.z2.1h// uhjz1.h/D0 D dz2.1h/; z2.1h/ .. /; C / CP1 . / on the two branches ph \ MI N0 and 1h \ .0;1;1/ of the node where z1 h and z2.1h/ are a local coordinate (i.e. z2.1h/ D 0 and z1.h/ D 0, respectively). dz1.h/ dz2.1h/ Thus, near the node of sh, !g;N0 .Dh/ is generated by and subjected to z1.h/ z2.1h/ dz1.h/ dz2.1h/ the relation C D 0. Stated differently, a section of the sheaf !g;N0 .Dh/ z1.h/ z2.1h/ F 1 ..M N0/; C / CP s pulled back to the smooth normalization I ph .0;1;1h/ of h can be identified with a meromorphic 1-form with at most simple poles at the marked points ph and 1h which are identified under the normalization map, and with opposite residues at such marked points. By extending such a construction to all N0 sections N0 fshghD1, we can define the line bundle ! : XN0 !g;N0 .D/ D !g;N0 Di ! C g;N0 (B.53) iD1 B A Capsule of Moduli Space Theory 375

: P ! N0 as g;N0 twisted by the divisor D D hD1 Dh, viz. the line bundle locally generated dz1.h/ dz2.1h/ by the differentials for z1.h/ ¤ 0 and for z2.1h/ ¤ 0, with z1.h/ z2.1h/ . / . / 0 1;:::; . / N0 z1 h z2 1h D , and h D N0. As above, fz1 h ghD1 are local variables N0 .. /; C / . / at the marked points fphgiD1 2 MI N0 , whereas z2 1h is the corresponding 1 variable in the thrice–pointed sphere CP.0;1;1/.

If, roughly speaking, we interpret Mg;N0C1, as a bundle of surfaces over Mg;N0 , then the stratiﬁcation of Mg;N0C1 into subvarieties fVkg provides an intuitive motivation for the existence of characteristic classes (Q-Poincaré duals of the fVkg) that describe the topological properties of Mg;N0 . Two such families of classes have proven to be particularly relevant. The ﬁrst is obtained by pulling back !g;N0 to

Mg;N0 by means of the sections sk. In this way one gets the line bundle : ! L ; sk g;N0 D k ! Mg N0 (B.54) whose fiber at the moduli point ..MI N0/; C / is defined by the cotangent space T to ..MI N0/; C / at the marked point p . By taking the first Chern class .M;pk/ k .L / L N0 c1 k of the resulting bundles f kgkD1 one gets the Witten classes .g;N0/;k 2 2 H .Mg;N0 I Q/ : . / .g;N0/;k D c1 Lk : (B.55) Conversely, if we take the Chern class c1 !g;N0 .D/ of the line bundle !g;N0 .D/ and intersect it with itself j 0 times, then one can define the Mumford k.g;N0/;j classes 2j 2 H .Mg;N0 I Q/ according to [1] Á : jC1 k.g;N0/;j D c1 !g;N0 .D/ ; (B.56)

b : where denotes ﬁber integration. If N0C1 D .g;N0C1/;N0C1 denotes the Witten 2 class in H .Mg;N0C1 I Q/ associated with the last marked point, then k.g;N0/;j can be also deﬁned as Â Ã Á C1 : b j k.g;N0/;j D N0C1 (B.57)

It is worthwhile recalling that k.g;N0/;1 is the class of the Weil-Petersson Kähler form

!WP on Mg;N0 ,[1], Á 2 1 k. ; /;1 c1 ! ; .D/ Œ! : g N0 D g N0 D 22 W P (B.58) 376 B A Capsule of Moduli Space Theory

B.4 Strebel Theorem

A basic result in phrasing the correspondence between Riemann surfaces and combinatorial structures is provided by Strebel’s theory of holomorphic (and meromorphic) quadratic differentials [28]. Recall that these objects are the holomorphic (meromorphic) sections of T0M ˝ T0M, i.e. the tensor ﬁelds on .M; C / that can be locally written as ˚ D .z/ dz ˝ dz, for some holomorphic (meromorphic) .z/. Geometrically the role of holomorphic quadratic differential stemsR p from the . / q . / 2 basic observation that for a given point p 2 U, the function q WD p z dz provides a local conformal parameter in a neighborhood U0 U of p if .p/ 6D 0. ˚ 1 In terms of the coordinate we can write D d ˝ d , and the sets fz j=p z D constg, and 1fz j

Theorem B.1 Let ..MI N0/; C / be a closed Riemann surface of genus g 0 with 1 N0 2 2 <0 . ;:::; / N0 marked points fpkgkD1, where gN0 . Let us denote by L1 LN0 an ordered N0–tuple of positive real numbers. Then there is a unique (Jenkins– Strebel) meromorphic quadratic differential ˚ on ..MI N0/; C / such that:

(i) ˚ is holomorphic on M n fp1;:::;pN0 g ; (ii) ˚ has a double pole at each pk,kD 1;:::;N0 ; (iii) The union of all non–compact horizontal leaves 1fz j=z D constg form a closed subset ..MI N0/; C / of measure zero ; (iv) Every compact horizontal leaf is a simple loop circling aroundH onep of the poles. In particular, if k is the loop around the pole pk, then Lk D ˚,(the p k branch of ˚ is chosen so that the integral is positive when the circuitation along is along the positive orientation induced by ..MI N0/; C // ; (v) Every non–compact horizontal leaf of ˚ is bounded by zeros of ˚. Conversely, every zero of degree m of ˚ bounds m C 2 horizontal leaves; (vi) If N2 denotes the number of zeros of the quadratic differential ˚, then ˚ induces a unique cell–decomposition of ..MI N0/; C / with N0 2–cells, N1 1– cells, and N2 0–cells where N1 D N0 C N2 2 C 2g. The 1–skeleton of this cell decomposition is a metric ribbon graph with vertex–valency 3 . It is straightforward to check that Strebel’s theorem characterizes a map S W ; RN0 met Mg N0 C ! RGg; N0 which, given a sequence of N0 positive real numbers fL.k/g, associates to a pointed Riemann surface ..MI N0/; C / 2 Mg;N0 ametric met @ . / ribbon graph 2 RGg; N0 with N0 labelled boundary components f k g of perimeters fL.k/g. This map is actually a bijection since, given a metric ribbon graph je. /j 2 RC with N0 labelled boundary components f@ .k/g of perimeters fL.k/g, we can, out of such combinatorial data, construct a decorated Riemannian surface B A Capsule of Moduli Space Theory 377

N0 ..MI N0/; C IfL.K/g/ 2 Mg;N0 RC . The characterization of the correspondance 1 S W 7! ..MI N0/; C IfL.K/g/ is described in a cristal clear way in [24], and one eventually establishes [19]the Theorem B.2 Strebel theory deﬁnes a natural bijection

; RN0 met ; Mg N0 C ' RGg; N0 (B.59)

N0 between the decorated moduli space Mg;N0 RC and the space of all metric ribbon met graphs RGg; N0 with N0 labelled boundary components.

B.5 The Teichmüller Space of Surfaces with Boundaries

This is also the appropriate place for a few remarks on moduli space theory @ for surfaces with boundaries. The elements of the Teichmüller space Tg; N0 of hyperbolic surfaces ˝ with N0 geodesic boundary components are marked Riemann surface modelled on a surface Sg;N0 of genus g with complete ﬁnite-area metric of constant Gaussian curvature 1, (and with N0 geodesic boundary components

@S Dt@Sj), i.e., a triple .Sg;N0 ; f ;˝/where f W Sg;N0 ! ˝ is a quasiconformal homeomorphism, (the marking map), which extends uniquely to a homeomorphism from Sg;N0 [ @S onto ˝ [ @˝. Any two such a triple .Sg;N0 ; f1;˝.1// and

.Sg;N0 ; f2;˝.2// are considered equivalent iff there is a biholomorphism h W ˝.1/ ! 1 ˝.2/ such that f2 ı h ı f1: Sg;N0 [ @S ! Sg;N0 [ @S is homotopic to the identity via continuous mappings pointwise ﬁxing @S. . ;:::; / RN0 @ . / For a given string L D L1 LN0 2 0,weletTg; N0 L denote the Teichmüller space of hyperbolic surfaces ˝ with geodesic boundary components of length : . / . / N0 j@˝1j;:::;j@˝N0 j D L1;:::;LN0 D L 2 R0: (B.60)

This characterizes the boundary length map

@ RN0 L W Tg;N0 ! 0 (B.61) . / ˝ 7! L.˝/ D j@˝1j;:::;j@˝N0 j ;

@ . / 1. / and we can write Tg;N0 L WD L L . Note that, by convention, a boundary component such that j@˝jjD0 is a cusp and moreover Tg;N0 .L D 0/ D Tg;N0 , where Tg;N0 is the Teichmüller space of hyperbolic surfaces with N0 punctures, (with / 6g 6 C 2N0 0). For each given string L D .L1;:::;LN0 there is a natural action . / @ on Tg;N0 L of the mapping class group Mapg;N0 deﬁned by the group of all the isotopy classes of orientation preserving homeomorphisms of ˝ which leave each boundary component @˝j pointwise (and isotopy-wise) ﬁxed. This action changes 378 B A Capsule of Moduli Space Theory

the marking f of Sg;N0 on ˝, and characterizes the quotient space

: ; .L/ . / Tg N0 Mg;N0 L D @ (B.62) Mapg;N0 as the moduli space of Riemann surfaces (homeomorphic to Sg;N0 ) with N0 boundary components of length j@˝kjDLk. Note again that when fLk ! 0g, Mg;N0 .L/ reduces to the usual moduli space Mg;N0 of Riemann surfaces of genus g with N0 marked points. It is worthwhile noticing that, since the boundary components of a ˝ @ ˝ @ >1 surface g;N0 2 Tg;N0 are left pointwise fixed, any surface g;N0 in Tg;N0 ,forN0 , ˝0 @ can be embedded into a surface gC1;N01 2 TgC1;N01 by glueing the two legs of a pair of pants onto two of the boundary components of ˝g;N0 . The attachment of a torus with two boundary components allows also to include ˝g;1 in ˝gC1;1. Such a chain of embeddings induces a corresponding chain of embeddings of the mapping @ @ class group Mapg;N0 into MapgC1;N01. Under direct limit, this gives rise to a notion of stable mapping class group playing a basic role in the study of the cohomology of the moduli space. We conclude this notational capsule by specializing to the boundary case the characterization of the Weil–Petersson inner product. The reader will find many more details in the remarkable and very informative papers by G. Mondello, (in particular [22]). As in Sect. B.2, we introduce the real vector space of holomorphic @ .˝/ @˝ quadratic differentials QN0 whose restrictions to is real. The corresponding @ space of Beltrami differentials, identified with the tangent space TC Tg; N0 , will be denoted by BN0 .˝/ and the they are paired according to Z . d ˝ d; d ˝ d1/ 7! d d; (B.63) ˝

@ .˝/ 1/ .˝/ where d ˝ d 2 QN0 and d ˝ d 2 BN0 . As in the boundaryless case, by noticing that the ratio between a quadratic differential and the hyperbolic metric hd ˝ d on ˝ is a Beltrami differential, we can deﬁne the Weil–Petersson @ inner product on TC Tg; N0 according to Z @ @ @ ./ 2 G˛ˇ D h jd j (B.64) M @˛ @ˇ

@ 3g3C2N0 where f @˛ g˛D1 is a basis of the vector space of harmonic Beltrami differentials on ˝. The corresponding Weil-Petersson form is provided by p ! 1 @ ˛ ˇ; WP WD G˛ˇdZ ^ dZ (B.65) B A Capsule of Moduli Space Theory 379

˛ @ . / where fdZ g are the basis, in QN0 M , dual to f ˛ g under the pairing (B.63). Note in particular that Z p ˇ ˛ ˇ WD 1 h2 dZ˛dZ jdj2; (B.66) ˝ deﬁnes the Weil–Petersson Poisson tensor associated with !WP. It is important to stress that, (because of the presence of the boundaries), the Weil–Petersson form is @ degenerate (in particular it is not Kähler) on Tg;N0 . However, the Poisson structure ˛ ˇ @ deﬁned by induces a foliation in Tg;N0 whose symplectic leaves are the spaces @ . / RN0 Tg;N0 L , of Riemann surfaces with given boundary length vector L 2 0, endowed with !WP. G. Mondello [22] has provided a nice geometrical characterization of the Poisson structure ˛ ˇ in terms of ideal hyperbolic triangulations of the surfaces ˝ 2 @ Tg;N0 . As we have seen, this characterization plays an important role in Chap. 3, and it hints to even deeper connections between the geometry of the space of polyhedral surfaces and hyperbolic geometry. Appendix C Spectral Theory on Polyhedral Surfaces

In this appendix we brieﬂy discuss the basic facts of spectral theory of Laplace type operators on polyhedral surfaces that we have exploited in these lecture notes.

C.1 Kokotov’s Spectral Theory on Polyhedral Surfaces

Spectral theory for cone manifolds has a long standing tradition, (a ﬁne sample of classical works is provided by [2, 5, 6, 10, 18]). Here we shall mainly refer to the elegant results recently obtained by A. Kokotov [17]. They provide a rather complete analysis of the determinant of the Laplacian on polyhedral surfaces. It must be stressed that whereas the study of the determinant of the Laplacian in the smooth setting is a well–developed subject, results in the polyhedral case have been sparse and often subjected to quite restrictive hypotheses [10, 21]. As a consequence of the presence of the conical points fp1;:::; N0g,the Laplacian on the Riemann surface ..M; N0/; Csg/ is not an essentially self-adjoint operator. There are (inﬁnitely) many possible self–adjoint extensions of , with domains typically determined by the behavior of functions (formally) harmonic at the conical points. To take care of this extension problem in a natural way, let us 1 0 recall that the minimal domain Dmin of the Laplacian on C0 .M ; R/ consists of 1 0 the graph closure on the set C0 .M ; R/, where a function u 2 Dmin if there is a 1 0 2 sequence fukg2C0 .M ; R/ and a function w 2 L .M/ such that uk ! u and 2 2 uk ! w in L .M/, where L .M/ denotes the space of square summable functions on ..M; N0/; Csg/. Similarly, the maximal domain Dmax for , corresponding to the 2 domain of the adjoint operator to on Dmin, is the subspace of functions v 2 L .M/ 2 such that for all u 2 Dmin there is a function f 2 L .M/ with hv; ui D hf ; ui, 2 where h; i denotes the L .M/ pairing on ..M; N0/; Csg/. All possible distinct self– adjoint extensions of on ..M; N0/; Csg/ are parametrized by a domain D , with Dmin Â D Â Dmax [23]. Explicitly, we can consider without loss of generality the

© Springer International Publishing AG 2017 381 M. Carfora, A. Marzuoli, Quantum Triangulations, Lecture Notes in Physics 942, https://doi.org/10.1007/978-3-319-67937-2 382 C Spectral Theory on Polyhedral Surfaces case in which we have just one conical point fpg with conical angle , and denote by z the local conformal parameter in a neighborhood of p. One introduces[17]the functions (formally harmonic on ..M; p/; Csg/), deﬁned by

˙ 2k p 2 k Vk .z/ z 1 z ; k >0; ˙ WD j j exp arg (C.1)

0 0 VC WD 1; V WD ln jzj: (C.2)

2. / < These functions are in L M as long as k 2 , and we can consider the linear E 2. / ' k . / 0 < subspaces of L M generated by the functions V˙ z , with Ä k 2 , where ' is a C1–molliﬁer of the characteristic function of the cone with vertex at p, e.g., '.z/ WD c exp .jzj2 1/1, jzj <1, and '.z/ D 0,forjzj1, with '.z D 0/ D 1, (assuming that the region isometric to the cone corresponds to jzj <1). The self– adjoint extension of are then parametrized by the subspaces of functions E such that[17] I Â Ã @v @u h u ;vi hu ; vi D lim u v D 0; (C.3) "&0C @r @r for all u;v 2 E , and where r WD jzj, (note that in the above characterization one exploits the delicate fact that any u 2 Dmin is such that u.z/ D O.r/ as r & 0–see section 3.2.1 of [17] for details). The Friedrics extension, which is the relevant one for the results to follow, is associated with the subspace E generated by the functions ' k . / 0 < D D E VC z , Ä k 2 . The associated domain WD min C comprises functions which are bounded near the apex of the cone. Denote by H .z; z0I / the heat kernel for the Friedrichs extension of the Laplacian .ThisisadistributiononM M Œ0; 1/ suchR that, for all u 2 D WD 0 0 D E H . ; / . / 0 min C away from the conical points, the convolution M z z I u z dAz is smooth for all >0, and Â Ã @ .z/ H .z; z0 / 0; @ C I D (C.4)

lim H .z; z0I / D ı.z; z0/; &0C where .z; z0I / 2 .M MnDiag .M M// Œ0; 1/, and .z/ denotes the Laplacian with respect to the variable z. The Dirac initial condition is under- 1 0 stoodR in the distributional sense, i.e., for any smooth u 2 C0 .M ; R/ \ D , 0 0 C H . ; / . / 0 . / 0 M z z I u z dAz ! u z ,as & , where the limit is meant in the 1 0 uniform norm on C0 .M ; R/. We are now ready to state Kokotov’s main results

Theorem C.1 (Kokotov[17], Th.1) Let ..M; N0/; C / be the Riemann surface, P Ásg v. / N0 .k/ 1 with conical singularities Di T WD kD1 2 , associated with the polyhe- C Spectral Theory on Polyhedral Surfaces 383 dral manifold .PT ; M/, and let be (the Friedrichs extension) of the corresponding Laplace operator. Then

(i) has a discrete spectral resolution, the eigenvalues 0 D 0 <1 Ä 2 Ä ::: !1have ﬁnite multiplicities, and the associated spectral counting function N./ WD Card Œk 1 W k Ä D O./,as !1. (ii) If Tr e denotes the heat trace of the heat kernel H .z; z0I / associated with . ; D /, then, for some ">0, the asymptotics Z Area ..M; N0/; C / Tr e WD H .z; zI / dA D sg (C.5) M 4

1 XN0 2 .k/ C C O.e "= /; 12 .k/ 2 kD1

holds in the uniform norm on C1.M0; R/. Let X 1 .s/ WD (C.6) s k k >0 denote the –function associated with the positive part of the spectrum of the operator . ; D /, then we have

Theorem C.2 (Kokotov[17]) The function .s/ is holomorphic in the half–plane f< s >1g, and there is an entire function e.s/ such that

1 Area ..M; N0/; Csg/ .s/ D (C.7) .s/ 4 .s 1/

" # ) 1 XN0 2 .k/ 1 C 1 C e.s/ ; 12 .k/ 2 s kD1 where .s/ denotes the Gamma function. .s/ is a regular at s D 0, and one can deﬁne the Ray–Singer –regularized determinate of . ; D / according to

0 ˚ « 0 det WD exp .s D 0/ : (C.8)

As a direct consequence of this representation, one has[17]the B Corollary C.1 Let ..M; N0/; C / and ..M; N0/; C / two homothetic Riemann sg sg P Á v. / N0 .k/ 1 0. / surfaces, with (the same) conical singularities Di T WD kD1 2 k , and let ds2 and dse2 D ds2, with k >0a positive constant, be the respective 384 C Spectral Theory on Polyhedral Surfaces conical metrics, (see Theorem 2.1). If we denote by det0 and det0 e the associated –regularized determinants, then one has the rescaling

Á P n o 0 .M/ 1 1 N0 2 C .k/ 2 0 det e D 6 12 kD1 .k/ 2 det : (C.9)

For 2 Œ0; 1, let us consider two distinct families of polyhedral surfaces

7! .T.1/; M/ 2 POLg; N0 .M/ (C.10)

7! .T.2/; M/ 2 POL .M/; (C.11) g;bN0

b (note that generally N0 6D N0, and that may be allowed to vary on a smooth param- 0 . ; / N0 eter manifold [17]). We assume that the corresponding vertex sets f .1/ k I gkD1 b 0 . ; /N0 Œ0; 1 and f .2/ h ghD1 are disjoint for all 2 , and that they support distinct –independent conical singularities f.1/.k/g and f.2/.h/g. We also assume that .T.1/; M/ , and .T.2/; M/ , 2 Œ0; 1, deﬁne the same ( –independent) conformal .1/ .2/ b .. ; /; C / .. ; b /; C / . / N0 . / N0 structure M N0 sg ' M N0 sg .Letfpk gkD1 2 M and fqh ghD1 2 M be the disjoint sets of points associated with the divisors Â Ã XN0 .1/.k/ Div.T.1/; / 1 p . /; WD 2 k (C.12) kD1 and Â Ã XbN0 .2/.h/ Div.T.2/; / 1 q . /: WD 2 h (C.13) hD1

2 .. ; /; C .1// According to (2.70) the conical metric dsT.1/ of M N0 sg around the generic conical point pk. / is given, in term of a local conformal parameter t.k; /,by Â Ã 2 .k/ Œ . / 2 .1/ 2 L k . ; / 2 . ; /2 ; ds ;. / WD 2 jt k j jdt k j (C.14) T.1/ k 42 jt.k; /j

2 .. ; b /; C .2// whereas the conical metric dsT.2/ of M N0 sg around the generic conical point qh. / is given, in term of a local conformal parameter z.h; /,by Â Ã 2 . / Œ 0. / 2 .2/ h 2 L h . ; / 2 . ; /2 : ds ;. / WD 2 jz h j jdz h j (C.15) T.2/ k 42 jz.h; /j C Spectral Theory on Polyhedral Surfaces 385

b 2 . / N0 Since in the metric dsT.1/ the points fqh ghD1 2 M, supporting the conical 2 singularities of dsT.2/ , are regular points, we can assume that there are smooth . / . ; / . / 2 functions g.2; h/ of z h such that in a neighborhood of qh the metric dsT.1/ takes the form ˇ ˇ ˇ ˇ2 2 ˇ ˇ . . ; //ˇ . ; /2 : dsT.1/; WD g.2; h/ z h jdz h j (C.16) qh. /

Similarly, we can assume that there are smooth functions f.1; k/. / such that in a . / 2 neighborhood of pk the metric dsT.2/ takes the form ˇ ˇ ˇ ˇ2 2 ˇ ˇ . . ; //ˇ . ; /2 : dsT.2/; WD f.1; k/ t k jdt k j (C.17) pk. /

With these notational remarks along the way, we have the following Theorem C.3 (Kokotov [17]) If det .1/ and det .2/ respectively denote the – regularized determinants of the (Friedrichs extension of the) Laplacian associated 2 2 with the conical metrics dsT.1/ and dsT.2/ , then there is a constant C independent of 2 Œ0; 1 such that Â Ã .2/ Q ˇ ˇ 1 .h/ 1 0 .1/ .1/ bN0 ˇ ˇ 6 2 det Area ..M; N0/; Csg / D1 g.2; h/ h Â Ã ; 0 .2/ D C .2/ .1/ (C.18) b 1 .k/ det .. ; 0/; C / Q ˇ ˇ 1 Area M N sg N0 ˇ ˇ 6 2 kD1 f.1; k/

.j/ where Area ..M; N0/; Csg /,j D 1; 2, denotes the area of the Riemann surface .1/ 2 ..M; N0/; C / in the corresponding conical metric ds , and where we have set sg T.j/ f.1; k/ WD f.1; k/.t.k; /D 0/ and g.2; h/ WD g.2; h/.z.h; /D 0/. As emphasized by Kokotov, this results extends to polyhedral surfaces Polyakov’s formula describing the scaling, under a conformal transformation, of the determinant of the Laplacian on a smooth Riemann surface.

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Symbols R–matrix, 327 3j symbol, 265, 278 Kaul representation, 328 double, 284 representation in an algebra, 319 quantum, 283 trace function, 320 6j symbol, 265 algebraic identities, 267, 271 as a quantum gate, 311, 337 C in terms of 4F3, 266 Center of mass quantum, 273, 288 interpretation of NLM action, 157 symmetries, 266 Chern and Euler classes ˇ–function, 154 of the space of polyhedral surfaces, 80 Chern class, 367 Chern–Simons–Witten functional, 274 A Chern-Simons invariant, 258 Askey hierarchy Collapse-expansion deformation, 22 q-deformed, 268 combinatorial moves, 271 q-deformed, 288 bistellar type, 272 Askey hierarcy, 267 elementary shellings, 279 automorphism group relation between bistellar moves and of the dual polytope, 63 shellings, 279 Automorphisms group Combinatorial properties, 4 of a polyhedral surface, 20 Complex structure, 371 complexity class, 311 BQP, 324 B NP, 318 Background ﬁeld P, 313, 318 in the QFT analysis of NLM, 159 #P, 322 Beltrami differentials, 371 Jones invariant, 322 Belyi’s theorem, 107 problems in group theory, 317 Berger–Ebin decomposition problems in knot theory, 313 for TT tensors, 214 Cone, 5 Berger-Ebin decomposition Conformal parameter properties, 359 local, 363 Bertrand-Diguet-Puiseaux formulas, 354 conformal structure braid group, 315 around conical singularities, 55

Conformal structures Discretized Liouville fields, 239 manifold of, 367 Distance function Conical angles on a Riemannian manifold, 347 and divisors, 76 Divergence operator representation on the unit sphere, 29 on a Riemannian manifold, 358 Conical defect, 13 the principal symbol, 358 Conical defects Divisor propagation of, 252 on a Riemann surface, 365 Conical singularities dual polytope existence of preassigned, 14 themetricgeometryofthe,61 invisibility from the conformal viewpoint , Dynamical triangulations, 9, 23, 230 71 as an approximating net, 230 Core geodesics, 252 Correlation functions, 149 Coupling E dilatonic, 212 edge tachyonic, 212 barycenter, 57 Coupling parameters, 156 Edge lengths, 10 controlling an Action, 145 edge refinement Couplings of a ribbon graph, 62 as coordinates for an action, 147 of an abstract graph, 64 Curvature and scaling Edge–replicating, 44 on a Riemannian manifold, 350 Effective action, 162, 170 Cut-off scale, 151 background field, 162 Cylindrical metric, 72 Effective action functional, 170 Effective area, 243 Effective theory, 152 D Euclidean length structure, 11 Deficit angle, 14 Euler class and curvature, 14 of circle bundles over space of polyhedral Deformation surfaces, 41 of a fiducial action, 147 Euler number, 77 degenerations Euler- Poincaré equation, 6 of polyhedral surfaces, 49 Exponential map Degree of a divisor, 365 on a Riemannian manifold, 353 Dehn- Sommerville equations, 6 Determinant of Laplacian, 381 for conical metrics, 238 Developing map, 254 F Diffeomorphisms f-vector, 6 group of, 17 Face, 2 of Sobolev class on a Riemannian manifold, Face pairing, 252 350 Fattening triangles, 26 On a manifold, 349 Fiducial action Diffeomorphisms group deformation of, 149 fixing the vertex set, 19 Filtration, 150 dihedral angles First Chern class, 249 ideal tetrahedra and, 87 Fixed connectivity, 8 Dilation holonomy, 13 Flip move, 21 Dilaton and equilateral triangulations, 22 and Riemannian metric measure spaces, isometric, 21 140 Fluctuations Dilaton field, 141 averaging, 152 linear, 222 spectrum of, 150 Index 389

Functional integration horospheres over the space of Riemannian structures, and null vectors, 84 213 Hyperbolic cone-manifolds Functional Jacobian, 218 three dimensional, 252 Functional measure hyperbolic distance, 86 in non-critical string theory, 218 hyperbolic space in quantum Liouville theory, 220 3-dimensional, 83 upper half-space model, 86 Hyperbolic structure, 252 G hyperbolic structure Gaussian measure complete, 95 reference, 160 incomplete, 99, 102 geodesic boundaries Hyperbolic surface decorated , 101 Incomplete, 255 hyperbolic surface with, 99 hyperbolic surface Glueings, 5 obtained by glueing triangles, 97 Group of confeomorphisms, 213 with punctures, 102 Hyperbolic volume, 255

H h-length, 91 I half–edges, 57 ideal simplex handle–pinching degeneration, 49 ideal triangle, 87 Handlebody, 255 ideal symplex Harmonic map ideal tetrahedron, 87 action functional deformation, 130 ideal triangle as a center of mass functional, 125 and horocyclic segments, 91 conformal invariance, 124 h-lengths associated with a, 93 energy density, 124 ideal triangles energy functional, 122 glueing, 93 energy functional and ﬂuctuating ﬁelds, rigidity, 94 139 sliding, 94 Harmonic map action incidence relations, 57 deformation of, 147 Intersection numbers Heegard splitting, 300 Mirzakhani approach, 248 Turaev–Viro state sum, 301 intersection theory hierarchies of state sums, 285 Witten-Kontsevich, 83 Holography, 258 Isometry group, 17 Holonomy complex valued, 14 rotational, 14 Holonomy representation, 12, 253 J horocycle Jones polynomial, 257 Thurston invariant, 99 Jones polynomials, 276, 290, 291, 298 horocycles additive approximation, 323, 338 signed hyperbolic distance between two, as Markov traces, 321 98 in the CSW framework, 321 horocyclic segment, 91 universality, 322 horocyclical decoration, 93 horosphere, 84 as a local screen, 89 K Euclidean triangle cut by a, 87 Knot–link, 255 Euclidean triangles in a, 87 KPZ exponent visual, 88 for random polyhedral surfaces, 246 390 Index

KPZ relations, 225 Non-Linear Sigma Model, 118 KPZ scaling action, 128 In the conformal gauge, 225

O L Orbifold lambda length, 85 in the sense of Thurston, 19 computation of, 90 Laplace operator on conical manifolds, 383 P Lie derivative, 358 Pachner theorem, 21 conformal, 214 Partition function the principal symbol, 358 fixed area, 225 Link Path of a simplex, 5 admissible, 9 link invariants, 313 Perelman’s entropy linking number, 340 generating functional, 183 Liouville action, 240 Piecewise-Linear(PL) discretized, 240 manifold, 4 regularized, 228 properties, 4 Liouville free vertex area, 243 Pillow tail Liouville mode, 217, 232 polyhedral, 45 Liouville theory role of, 46 quantum, 217 stability of, 47 Liouvuille action, 216 pillow tail component, 52 Lobachevsky function, 256 pillow tail degeneration, 49 Logarithmic dilation, 13 Pillow-tail pinching, 44 complexified, 16 Pinching node, 45 pinching node, 52 behavior of the combinatorial WP form at M a, 108 Manin and Zograf’s asymptotics, 234 Pointed Riemann-surface Mapping class group stable, 372 for surfaces with boundary, 99 Pointed surface, 363 Markov moves, 316 Poisson bivector, 103 Matrix model Polyakov action, 212 and Eynard and Orantin’s approach, 231 Polyakov’s formula, 385 medians Polygonal bundle, 28 of a triangle, 58 Polyhedral arc-length map, 34 Metric Polyhedral cone, 25 piecewise flat, 11 and spherical polygons, 26 Metric geometry directrix of a, 25 of polyhedral surfaces, 10 Polyhedral cotangent cone, 26, 33 Milnor’s formula, 256 Polyhedral cotangent cones Minimal subtraction, 165 and circle bundles over the space of Minimum incidence, 7 polyhedral surfaces, 35 Moduli, 368 Polyhedral structures, 18 and singular metrics, 369 as singular Euclidean structures, 18 Moduli space parametrization of the space of, 25 Deligne-Mumford compactification, 372 set of all, 19 Polyhedral surface, 8 and Regge surface, 8 N local Euclidean structure of, 11 Non-linear model, 212 open, 9 Index 391 polyhedral surface Ribbon graphs null vector on a, 88 and differentiable orbifolds, 68 Polyhedral surfaces metric, 67 curvature structure of, 11 topology of the space of, 67 deformations of, 25 Ricci curvature distance between two points, 11 in harmonic coordinates, 360 infinitesimal deformations of, 25 Ricci Flow similarity class of the triangles of, 12 as one loop perturbative RG flow, 166 slit-open transformation on, 43 Ricci flow stable, 47 as a dynamical system on M et.Vn/, 171 stable degenerations of, 43 as a weakly-parabolic flow, 171 tangent vector to the space of, 40 commutation rules, 199 the various set of , 20 conjugation and factorization, 188 polyhedral surfaces Hamilton–DeTurck version, 176 the space of stable, 50 Hodge-DeRham-Lichnerowicz operator, Polyhedron, 4 194 underlying, 3 non-dissipative directions, 204 polytope on Riemannian metric measure spaces, barycentrically dual, 56 177 the conical, 56 volume normalized, 174 Ponzano–Regge model, 269 Ricci solitons, 169 for a closed 3–manifold, 270 and quasi-Einstein metrics, 169 for a simplicial pair, 277 Ricci tensor, 352 holographic projection, 286 Riemann moduli space, 215 Pseudo-manifolds, 6 Riemann tensor, 352 Pure gravity critical exponent, 235 Riemannian metrics Space of, 355 space of, 17 Q Riemannian structures QFT analysys orbifold of, 18 of the NLM, 156 space of, 18 Quadratic differentials, 72, 369 and Strebel theorem, 376 Jenkins–Strebel, 72 S quantum invariants of 3–manifolds, 339 Scalar curvature, 352 Quantum invariants of colored fat graphs, Schläfli formula, 41 290 Sectional curvature, 352 Quasi-Einstein metrics, 167 similarity structure, 97 obtained by glueing, 97 Simplex, 1 R Simplices R–matrix, 295 equilateral, 9 Random polyhedral surfaces Simplicial complex, 3 Partition function for, 244 Simplicial division, 4 Regge surfaces, 8 Simplicial isomorphism, 4 Regular point Simplicial manifolds, 4 of a polyhedron, 6 Simplicial map, 4 Reidemeister moves, 313 Singular Euclidean metric, 70 regular isotopy, 314 Singular Euclidean strutures Renormalization Group Flow, 151, 153 conically complete, 13 Renormalization group flow Singular point for the NLM, 165 of a polyhedron, 6 ribbon graph, 64 Skeleton, 2 metrized, 66 sky mapping, 89 392 Index

Slice theorem Triangle for the group of confeomorphisms, complex modulus of a, 14 213 similarity class, 14 Sobolev norm Triangle inequalities, 10 on a Riemannian manifold, 348 Triangular pillow Sobolev space of maps, 120 visualization of, 46 Space of Actions, 147 Triangulations, 3 Space of localizable maps, 121 combinatorially equivalent, 7 Space of polyhedral cones distinct triangulations of the same over a vertex, 31 PL-manifolds, 7 Space of polyhedral surfaces triangulations tangent space to, 25 complex geometry and, 55 Space of Ribbon graphs Turaev–Viro quantum invariants, 273 orbicell, 68 ﬁxed boundary triangulation, 277, 289 Space of spherical polygons, 31 initial data, 287 Spacetime ﬁelds, 223 Tutte, 7 SPherical polygons the space of, 29 Spherical polygons, 28 V spin network, 309 vertex q-deformed automata, 325, 334 complex coordinates around a, 55 quantum simulator, 309, 343 Vertex angle structure, 26 spiral staircase, 96 Vertex angle structures and similarity structure, 97 the space of, 28 Star Vertex angles, 10 of a simplex, 3 Vertex angles vector, 27 open, 3 visual diractions, 89 Stirling’s formula, 234 visual directions, 250 Strebel theory, 376 Volume conjecture Subdivision, 4 hyperbolic, 257 Susceptibility exponent, 225, 227 Symplectic form over the space of polyhedral surfaces, W 38 Weil–Petersson volume symplectic volume, 109 of moduli space of Riemann surfaces with boundaries, 251 Weil-Petersson two-form, 102 T for polyhedral surfaces, 102 Tautological classes, 373 Poisson structure associated with the, 103 Teichmüller space, 102 Weil-Petersson form, 83 decorated, 102 Weil-Petersson measure, 215 Teichmüller space Weil-Petersson metric, 371 decorated, 102 Weil-Petersson volume, 371 The dilatonic NLM action Weyl rescalings for polyhedral surfaces, 142 group of, 213 The space of polyhedral structures Lie algebra of, 213 as a differentiable orbifold, 20 Whitehead expansion and collapse, 67 Thinning triangles, 26 Wiener measure, 148 Thrice-punctured sphere, 46 Wilson loop, 276, 328 Thurston, 5 topological knot theory, 312 Alexander theorem, 314 Y closures of knot diagrams, 316 Yang–Baxter relation, 288, 296, 315, 320