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1 ASYMPTOTIC FAITHFULNESS OF THE QUANTUM SUènè REPRESENTATIONS OF THE MAPPING CLASS GROUPS JçRGEN ELLEGAARD ANDERSEN Abstract. We prove that the sequence of projective quantum SUènè representations of the mapping class group obtained from the projective æat SUènè-Verlinde bundles over Teichmíuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus of the underlying surface is at least 3. For the genus 2 case, we prove that this intersection is exactly the order two subgroup, generated by thehyper-elliptic involution, in the case of n = 2 and for né2 the intersection is trivial. 1. Introduction In this paper we shall study the ænite dimensional projective quantum SUènè representations of the mapping class group of a genus g surface. These form the only rigorously constructed part of the gaugetheoretic approach to Topological Quantum Field Theories in dimension 3, which Witten proposed in his seminal paper ëw1ë. We discovered the asymptotic faithfulness property for these representations by studying this approach, which we will now brieæy describe, leaving further details to section 2, 3 and the references given there. By applying geometric quantization to the moduli space M of æat SUènè-connections on an oriented closed surface æ, one gets a certain ænite rank bundle over Teichmíuller space T, which we will call the Verlinde bundle Vk at level k, where k is any positive integer. The æber of this bundle over a point ç 2 T is Vk;ç = H 0 èm ç ; L k è, where M ç is M equipped with a complex structure induced from ç and L is an ample generator of the Picard group of M ç. The main result pertaining to this bundle Vk is that its projectivization PèVkè supports a natural æat connection. This is a result proved independently by Axelrod, Della Pietra and Witten ëadwë and by Date :15'th of April This research was conducted for the Clay Mathematics Institute at University of California, Berkeley and for MaPhySto í Centre for Mathematical Physics and Stochastics, funded by The Danish National Research Foundation. 1

2 2 JçRGEN ELLEGAARD ANDERSEN Hitchin ëhë. Now, since there is an action of the mapping class group, of æ on Vk covering its action on T, which preserves the æat connection in PèVkè, we get for each k a ænite dimensional projective representation, say ç k, of,, namely on the covariant constant sections of PèVkè over T. This sequence of representations ç k is the quantum SUènè representations of the mapping class groups. In this paper we prove asymptotic faithfulness of the quantum SUènè representations ç k : Theorem 1. We have that 1ë k=1 kerèç k è= è f1g gé2 or èg =2, né2è f1;hg g =2, n =2: where g is the genus of the surface æ and H is the hyperelliptic involution. This theorem states that for any element ç of the mapping class group,, which is not the identity element, there is an integer k such that ç k èçè isnot amultiple of the identity. The key ingredient in our proof is the use of the endomorphism bundle EndèVkè and the construction of sections of this bundle via Toeplitz operators associated to smooth functions on the moduli space M. By showing that these sections are asymptotically æat sections of EndèV k è èsee theorem 7 for the precise statementè, we prove that any element in the above intersection of kernels acts trivial on the smooth functions on M, hence act by theidentity onm èsee the proof of Theorem 8è. Theorem 1 now follows directly from knowing which elements of the mapping class group acts trivially on the moduli space M. The abelian case, i.e. the case where SUènè is replaced by Uè1è, was considered in ëa2ë, before we consider the case discussed in this paper. In this case, with the use of Theta-functions, explicit expressions for the Toeplitz operators associated to holonomy functions can be obtained. From these expressions it follows that the Toeplitz operators are not covariant constant either, even in this much simpler case èalthough the relevant connection is the one induced from the L 2 -projection as shown by Ramadas in ër1ëè. However, they are asymptotically covariant, in fact we ænd explicit perturbations to all orders in k, which in this case, we argue, can be summed and use to create actual covariant constant sections of the endomorphism bundle. The result as far as the mapping class group goes, is that the intersection of the kernels over all k in this case is the Torelli group.

3 ASYMPTOTIC FAITHFULNESS 3 We remark that it is an interesting problem to understand how the Toeplitz operator constructions used in this paper are related to the deformation quantization of the moduli spaces described in ëamr1ë and ëamr2ë. In the abelian case, the resulting deformation quantization was explicitly described in ëa2ë. From the above theorem 1 follows directly that, Corollary 1. If the quantum SUènè representations ç k then we have for all ç 2, that are unitary, j Trèç k èçèèjçdim ç k with equality for all k, if and only if ç 2 è f1g gé2 or èg =2, né2è f1;hg g =2, n =2: By the work of Laszlo ëla1ë, it is known that PèVkè with its æat connection is isomorphic to the bundle of conformal blocks for slèn; Cè with its æat connection over T as constructed by Tsuchiya, Ueno and Yamada ëtuyë. This means that the quantum SUènè representations ç k is the same sequence of representations as the one arising from the space of conformal blocks for slèn; Cè. By the work of Reshetikhin-Turaev, Topological Quantum Field Theories have been constructed in dimension 3 from the quantum group U q slèn; Cè èsee ërt1ë, ërt2ë and ëtëè or alternatively from the Kauæman bracket and the Homæy-polynomial by Blanchet, Habegger, Masbaum and Vogel èsee ëbhmv1ë, ëbhmv2ë and ëb1ëè. In ongoing work of Ueno joint with this author èsee ëau1ë and ëau2ëè, we are in the process of establishing, that the TUY construction of the bundle of conformal blocks over Teichmíuller space for slèn; Cè gives a modular functor, which in turn gives a TQFT, which is isomorphic to the U q slèn; Cè-Reshetikhin-Turaev TQFT. A corollary of this is that the quantum SUènè representations are isomorphic to the ones that are part of the U q slèn; Cè-Reshetikhin-Turaev TQFT. Since it is well known that the Reshtikhin-Turaev TQFT is unitary one will get unitarity of the quantum SUènè representations from this. We note that unitarity is not clear from the geometric construction of the quantum SUènè representions. Furthermore, one will get that the norm of the Reshtikhin-Turaev quantum invariant for all k and n = 3 can separate the mapping torus of the identity from the rest of the mapping tori as a purely TQFT consequence of Corollary 1. This paper is organized as follows. In section 2 we give the basic setup of applying geometric quantization to construct the Verlinde

4 4 JçRGEN ELLEGAARD ANDERSEN bundle over Teichmíuller space. In section 3 we review the construction of the æat connection in the projective Verlinde bundle. In section 4 we review the general results about Toeplitz operators, we need. In section 5 we prove that the Toeplitz operators for smooth functions on the moduli space give asymptotically æat sections of the endomorphism bundle of the Verlinde bundle. Finally, in section 6 we prove the asymptotic faithfulness Theorem 1 above. After the completion of this work, we have learned from private communication with Freedman and Walker, that they have a proof of the Asymptotic faithfulness property for the SUè2è-BHMV-representations which uses BHMV-technology. For the SUè2è-BHMV-representations it is already known by the work of Roberts ëroë, that they are irreducible for k large enough prime and that they have inænite image for k large enough by the work of Masbaum ëmë. We would like to thank Nigel Hitchin for valuable discussion and the referee for his suggestions. 2. The gauge theory construction Let æ be a closed oriented surface of genus g ç 2 and p 2 æ. Let P be a principal G = SUènè-bundle over æ. Clearly, all such P are trivializable. Let ç be an element of the center of G. Let M be the moduli space of æat G -connections in Pj æ,p with holonomy ç around p. We can identify M = Hom ç èç 1 èæ, pè; G è= G ; where Hom ç means the space of homomorphism of ç 1 èæ,pèto G which sends a little loop around p to ç. Wechoose an invariant bilinear form fæ; æg on g =Lieè G è, normalized fè ^ ëè ^ èëg is a generator of the image of the integer such that, 1 6 cohomology in the real cohomology in degree 3 of G, where è is g-valued Mauer-Cartan 1-form on G. This bilinear form induces the structure of a symplectic stratiæed space on M. Let M 0 ç M consist of the irreducible representations. It is well-known that M 0 is smooth and dense in M and represents the top stratum in this stratiæcation. In fact T ëaë M 0 ç = H 1 èæ;d A è; where A is any æat connection in P representing a point in M 0 and d A is the induced covariant derivative in the associated adjoint bundle. Using this identiæcation, the symplectic structure on M 0 is very easily

5 described: ASYMPTOTIC FAITHFULNESS 5!è' 1 ;' 2 è= Z æ f' 1 ^ ' 2 g; where ' i are d A -closed 1-forms on æ with values in ad P. We have a natural action of,, the mapping class group of æ, on M induced by pull back. Following Ramadas, Singer and Weitsman ërswë and Freed ëfrë, we will now construct a Hermitian line bundle with a connection L over M with a natural lift of the,-action on M. We only review the case ç = 1 here and refer the reader to ëfrë for the general case. Choose a trivialization of P. Let A be the space of SUènè-connections in P and AF the subset of æat connections. The trivialization of P gives a base point A 0 in A. Let ~L = A æ C with the natural Hermitian structure inherited from C. Let ç be the 1-form on A given by ç A èaè = Z æ fèa, A 0 è ^ ag; where a 2 T A A = æ 1 èæ; ad P è. Then ç determines a Hermitian connection in ~ L. Let æ:a æ G! Uè1è be the Wess-Zumino-Witten cocycle, see ëfrë. We remark in passing that the WZW-cocycle can be expressed in terms of the Chern-Simons functional. Now, deæne where G acts on ~L by L =è~lj AF è= G gèa; zè =èg æ A; æèa; gèzè: The Hermitian connection is invariant under this lift of the action of G on A, hence we get a topological line bundle over M, which is smooth over M 0 and a Hermitian connection r in L j M 0. By an almost identical construction, we can lift the action of, on M to act on L such that the Hermitian connection is preserved èsee e.g. ëa1ëè. Let now ç 2 T be a complex structure on æ. Let us review how ç induces a complex structure on M which is compatible with the symplectic structure on this moduli space. The complex structure on æ induces a æ-operator on 1-forms and via Hodge theory we get that H 1 èæ;d A è ç = kerèda + æd A æè:

6 6 JçRGEN ELLEGAARD ANDERSEN On this kernel, consisting of the harmonic 1-forms with values in ad P, the æ-operator acts and its square is,1, hence we get an almost complex structure on M 0 by letting I =,æ. It is a classical result by Seshadri, that this actually makes M 0 a smooth Kíahler manifold, which as such, we denote M 0 ç. By using the è0; 1è part of r in L, we get an induced holomorphic structure in the bundle L j M 0. From a more algebraic geometric point of view, we let M ç be the moduli space of S-equivalence class of semi-stable bundles of rank n and determinant isomorphic to dëpë, where d is an integer representing ç under the natural identiæcation of the center of SUènè with Z =n Z. By using Mumford's Geometric Invariant Theory, Narasimhan and Seshadri èsee ënsëè showed that M ç is a complex algebraic projective variety homeomorphic to M, such that the quasi-projective sub-variety M s ç consisting of stable bundles is isomorphic as a Kíahler manifold to M 0 ç. Referring to ëdnë we recall that Theorem 2 èdrezet & Narasimhanè. The Picard groups of M ç Mç 0 agree, i.e. PicèM ç è=picèm 0 çè=hli; where L is the holomorphic line bundle over M 0 ç constructed above. and It follows from this theorem, that the line bundle L extends to all of M ç and as such we also denote it L. By Hartog's theorem it follows that H 0 èm 0 ç; L k è ç = H 0 èm ç ; L k è; for all integers k. In particular, we see that this is a ænite dimensional vector space. Deænition 1. The Verlinde bundle Vk over Teichmíuller space, is by deænition, the bundle, whose æber over ç 2 T is H 0 èm ç ; L k è, where k is a positive integer. 3. The projectively flat connection In this section we will review Axelrod, Della Pietra and Witten's and Hitchin's construction of the projective æat connection over Teichmíuller space in the Verlinde bundle. Let Hk be the trivial C 1 èm;l k è-bundle over T which contains Vk, the Verlinde sub-bundle. If we have a smooth one-parameter family of complex structures ç t on æ, then that induces a smooth one-parameter family of complex structures on M, say I t. In particular we get ç 0 t 2 T çt èt è, which gives an I 0 t 2 H 1 èm;tè èhere T refers to the holomorphic tangent bundle of Mè.

7 ASYMPTOTIC FAITHFULNESS 7 Suppose s t is a corresponding smooth one-parameter family in C 1 èm;l k è such that s t 2 H 0 èm çt ; L k è: By diæerentiating the equation we see that Hence, if we have an operator è1 + ii t èrs t =0; ii 0 trs +è1+ii t èrs 0 t =0: uèvè :C 1 èm;l k è! C 1 èm;l k è for all v 2 T èt è, varying smoothly with respect to v, and satisfying ii 0 tr 1;0 s t + r 0;1 uèç 0 tèès t è=0; then we get a connection induced in Vk by letting ^r v = ^r t v,uèvè; è1è for all v 2 T èt è, where ^r t is the trivial connection in Hk. In order to specify the particular u we are interested in, we use the symplectic structure on! 2 æ 1;1 èmè to deæne the tensor G = Gèvè 2 æ 0 èm;s 2 èt èè by I 0 = G!: Following Hitchin, we give an explicit formula for G in terms of v 2 T èt è: The holomorphic tangent space to Teichmíuller space at ç 2 T is given by T 1;0 ç èt è ç = H 1 èæ ç ;K,1 è: Furthermore, the holomorphic co-tangent space to the moduli space of semi-stable bundles at the equivalence class of a stable bundle E is given by T æ ëeëm ç ç = H 0 èæ ç ; End 0 èeè æ Kè: Thinking of G 2 æ 0 èm;s 2 èt èè as a quadratic function on T æ = T æ M, we have that Gèvèèæ; æè = where v 0 is the è1; 0è-part of v. Z æ Trèæ 2 èv 0 From this formula it is clear that G 2 H 0 èm;s 2 èt èè. Furthermore, we observe that ^r agrees with ^r t along the anti-holomorphic directions T 1;0 èt è. The particular uèvè =u G èvè =u G we are interested in is given by 1 u G èsè = 2èk + nè èæ G, 2G@Fr + ikf G ès:

8 8 JçRGEN ELLEGAARD ANDERSEN The leading order term æ G is the 2'nd order operator given by æ G : C 1 èm;l k è r,,,! 1;0 C 1 èm;t æ æ L k è r 1;0 æ1+1ær,,,,,,,,,! 1;0 C 1 èm;t æ æ T æ L k è G,,,! C 1 èm;t æ L k è Tr,,,! C 1 èm;l k è: The function F is the Ricci potential uniquely determined as the real function with zero average over M, which satisæes the following equation Ric = 2n! +2i@ We observe that F 2 C 1 èt æmè, and we deæne the function f G = f Gèvè 2 C 1 èmè for any v 2 T èt è by f G =,ivëf ë: Theorem 3 èaxelrod, Della Pietra & Witten; Hitchinè. The expression è1è above deænes a unique connection in the bundle PèVkè, which is æat. Faltings has established this theorem in the case of general semisimple Lie groups G,èseeëFalëè. We remark about genus 2, that ëadwë covers this case, but ëhë excludes this case, however, the work of Van Geemen and De Jong ëvgdjë extends Hitchin's approach to the genus 2 case. As discussed in the introduction, we see by Laszlo's theorem that this particular connection is the relevant one to study. 4. Toeplitz operators on compact Kíahler manifolds In this section èm 2d ;!è will denote a compact Kíahler manifold on which we have a holomorphic line bundle L admitting a compatible Hermitian connection, whose curvature is the symplectic form!. For each f 2 C 1 èmè consider the associated Toeplitz operator f given as the composition of the multiplication operator M f : H 0 èm;l k è! C 1 èm;l k è with the orthogonal projection ç : C 1 èm;l k è! H 0 èm;l k è. Sometimes we will suppress the superscript èkè and just write T f = f. We remark that we can also consider T f as an operator from L 2 èm;l k èto H 0 èm;l k è. Let us here give an explicit formula for ç: On L 2 èm;l k è we have the deæning inner product: hs 1 ;s 2 i = 1 d! Z M ès 1 ;s 2 è! d where s i 2 L 2 èm;l k è and èæ; æè is the æberwise Hermitian structure in L k. Let now h ij = hs i ;s j i, where s i is a basis of H 0 èm;l k è. Let h,1 ij

9 be the inverse matrix of h ij. Then ASYMPTOTIC FAITHFULNESS 9 çèsè = X i;j hs; s i ih,1 ij s j: This formula will be useful, when we have to compute the derivative of ç along a one-parameter curve of complex structures. Suppose we have a smooth section X 2 C 1 ètmè of the holomorphic tangent bundle of M. We then claim that the operator çr X is a zero-order Toeplitz operator. Suppose s 1 2 C 1 èm;l k è and s 2 2 H 0 èm;l k è, then we have that Xès 1 ;s 2 è=èr X s 1 ;s 2 è: Now, calculating the Lie derivative alongx of ès 1 ;s 2 è!d d! and using the above, one obtains after integration that hr X s 1 ;s 2 i =,hædèi X!ès 1 ;s 2 i; where æ denotes contraction with!. Thus è2è çr X = f X ; as operators from C 1 èm;l k è to H 0 èm;l k è, where f X =,ædèi X!è. Iterating this, we ænd for all X 1 ;X 2 2 C 1 ètmè that è3è çr X1 r X2 = èx 2 èf X1 è+f X2 f X1 è again as operators from C 1 èm;l k è to H 0 èm;l k è. We need the following theorems on Toeplitz operators. The ærst is due to Bordemann, Meinrenken and Schlichenmaier èsee ëbmsëè. Theorem 4 èbordemann, Meinrenken and Schlichenmaierè. For any f 2 C 1 èmè we have that èkè lim ktf k = sup jf èxèj: k!1 x2m Since the association of the sequence of Toeplitz operators T k f, k 2 Z+ is linear in f, we see from this theorem, that this association is faithful. Theorem 5 èschlichenmaierè. For any pair of smooth functions f 1 ;f 2 2 C 1 èmè, we have an asymptotic expansion f 1 f 2 ç 1X l=0 c l èf 1 ;f 2 è k,l ;

10 10 JçRGEN ELLEGAARD ANDERSEN where c l èf 1 ;f 2 è 2 C 1 èmè are uniquely determined since ç means the following: For all L 2 Z+ we have that k f 1 f 2, Moreover, c 0 èf 1 ;f 2 è=f 1 f 2. LX l=0 c l èf 1 ;f 2 è k,l k = Oèk,èL+1è è: This theorem is proved in ëschë, where it is also proved that the formal generating series for the c l èf 1 ;f 2 è's gives a formal deformation quantization of the Poisson structure on M induced from!. By examining the proof in ëschë of this theorem, one observes that for families of functions, the estimates in theorem 5 are uniform over compact parameter spaces. Theorem 6 èkarabegov &Schlichenmaierè. For any pair of smooth functions f 1 ;f 2 2 C 1 èmè the bilinear maps c l èf 1 ;f 2 è 2 C 1 èmè, l 2 Z+ are given by bi-diæerential operator of order at most èl; lè, in particular c l èf 1 ;f 2 èèpè only depends on the l'th jet of f 1 and f 2 at p for each p 2 M. This theorem is proved in ëksë, where furthermore the resulting æ- product mentioned above is identiæed in terms of Karabegov's classiæcation of æ-products with separation of variables on Kíahler manifolds. 5. Toeplitz operators on moduli space and the projective flat connection Let f 2 Cc 1 èm 0 è be a smooth function on the moduli space with compact support on M 0. We consider f as a section of the endomorphism bundle EndèVkè. The æat connection ^r in the projective bundle PèVkè, induces a æat connection ^r e in the endomorphism bundle EndèVkè. We shall now establish that the sections f are in a certain sense asymptotically æat by proving the following theorem. Theorem 7. Let ç 0 and ç 1 be two points in Teichmíuller space and P ç0 ;ç 1 be the parallel transport in the æat bundle EndèVkè from ç 0 to ç 1. Then kp ç0 ;ç 1 f;ç 0, f;ç 1 k = Oèk,1 è; where kækis the operator norm on H 0 èm;l k è. In order to prove this theorem, one needs of course estimates on k ^r e ç t 0 f;ç t k along a smooth one-parameter family of complex structures ç t èt 2 J = ë0; 1ëè from ç 0 to ç 1. We will see èproposition 2 belowè

11 ASYMPTOTIC FAITHFULNESS 11 that ^r e is asymptotically well related to the tensor product norm on èh 0 èm;l k èè æ æ H 0 èm;l k è induced from èa slightly twisted version ofè the L 2 -norm on H 0 èm;l k è. However, the operator norm and the tensor product norm on H 0 èm;l k è æ èh 0 èm;l k èè æ are related by a factor equal to the square root of dim H 0 èm;l k è, which is a polynomial in k of degree d = èn 2, 1èèg, 1è èsee è5è belowè. Hence we really need estimates on the covariant derivative, which decays faster than k,d=2 to establish Theorem 7. In order to obtain that, one needs to perturb the function f by adding on suæciently many terms of the form èh l è t k,l èl = 1;:::;mè, where èh l è t 2 C 1 c èm 0 è, t 2 J, to obtain a smooth one-parameter family of functions èf m è t = f + P m l=1 èh lè t k,l for an méèn 2, 1èèg, 1è=2, 1which satisæes that k ^r e ç 0 t èf mè t;ç t k is Oèk,m,1 è. This is the content of Proposition 1 below. However, in order to choose these functions h l correctly, we need to do a few elementary calculations. Along any smooth one-parameter family of complex structures ç t èindex by Jè, consider a basis of covariant constant sections s i =ès i è t, i =1;::: ;N of Vk over ç t : s 0 i = u G ès i è; i =1;:::N: The projection ç, as we saw above, is then given by çèsè = X i;j hs; s i ih,1 ij s j: Then ç 0 èsè = X i;j hs; s 0 iih,1 ij s j + X i;j + X i;j hs; s i ièh,1 ij è 0 s j hs; s i ih,1 ij s0 j: An easy computation gives that èh,1 ij è0 =, X h,1 il èhs 0 l;s m i + hs l ;s 0 mièh,1 mj ; l;m

12 12 JçRGEN ELLEGAARD ANDERSEN so çç 0 èsè = X i;j, X hu æ Gs; s i ih,1 ij s j i;l;m;j hs; s i ih,1 il hs l ;s 0 mih,1 = çu æ G, X m;j hçs; s 0 mih,1 mj s j: mj s j Hence we conclude that Lemma 1. çç 0 = çu æ G, çu æ Gç: Let m be a non-negative integer and let èf m è t = mx i=0 èh i è t k,i ; where the h i 's are arbitrary smooth one-parameter family of functions Cc 1 èm 0 è indexed by J, except, we require that h 0 = f. We will from now mostly suppress the subscript t on all quantities. We have that f m is a section of EndèVkè over the curve ç. Let A m = ^r e ç 0èT f m è=èt fm è 0, ëu G ;T fm ë=çf 0 m + ç 0 f m, ëu G ;çf m ë: Since ^r e ç 0èT f m è is a section of EndèVkè, we have of course that ça m ç = A m ç : H 0 èm çt ; L k è! H 0 èm çt ; L k è: Proposition 1. Given f and a non-negative integer, there exists unique smooth one-parameter families of functions èh i è t 2 C 1 c èmè, i =1;::: ;m and t 2 J such that and èh i è 0 =0, i =1;::: ;m. sup ka m çk = Oèk,m,1 è; t2j Proof. For any choice of h i 's, i =1;::: ;m, we compute that ça m ç = mx i=1 +, çh 0 içk,i mx mx i=0 i=0 èçu æ Gh i ç, çu æ Gçh i çèk,i èçu G çh i ç, çh i u G çèk,i :

13 ASYMPTOTIC FAITHFULNESS 13 Pulling L k back to the desingularization èwhich we also denote M ç è of M ç, if M ç is singular and using è2è and è3è we deæne the function H G,asfollows çèæ G, 2G@Frè =çh G : We also deæne H i 2 C 1 c èm 0 è which depends on h i and G by Then we have that ça m ç = çh i èæ G, 2G@Frè =çh i : mx i=1,, + + çh 0 içk,i mx mx i=0 mx i=0 mx i=0 i=0 1 2èk + nè èçh Gçh i ç, çh G h i çèk,i 1 2èk + nè èçh Gçh i ç, çh i çèk,i k 2èk + nè ièçf Gçh i ç, çf G h i çèk,i k 2èk + nè ièçf Gçh i ç, çh i f G çèk,i : We will now argue that we can for arbitrary integer m uniquely determine the functions h i with the properties stated in Proposition 1. In order to do that, replace each product of Toeplitz operators on the left hand side of the above expression by their asymptotic expansion as given by Theorem 6. In the last two terms we notice after this substitution, that the terms in the bracket is now proportional to k,1,since the leading order term in the expansion of the product of two Toeplitz operators, is just the Toeplitz operator associated to the product of the two functions. Suppose now that we have uniquely determined èh i è t 2 C 1 c èmè, i =1;::: ;m, 1 and t 2 J such that sup t2j ka m,1 çk = Oèk,m è; and èh i è 0 = 0, i = 1;::: ;m, 1. By considering an arbitrary h m 2 C 1 c èm 0 è and collecting up terms, we ænd that the total coeæcient of k,m of the right hand side is the Toeplitz operator associated to h 0 m minus an expression in h i ;H i, i =1;::: ;m, 1andH G. By theorem 4 we now see that A m satisæes the required norm-estimate if and only if h 0 m equals this expression in h i ;H i, i =1;::: ;m, 1andH G, which is a smooth one-parameter family of functions in C 1 c èm 0 èby the locality

14 14 JçRGEN ELLEGAARD ANDERSEN of the c l 's as provided by Theorem 6. E.g. considering the coeæcient of k,1, we ænd that h 1 has to satisfy that h 0 1 =, 1 2 èh Gh 0, H 0, c 1 è ç f G ;h 0 è, c 1 èf G ;h 0 èè; where the right hand side is seen to be a smooth one-parameter family of functions in C 1 c èm 0 è, since h 0 = f 2 C 1 c èm 0 è. By simply integrating the resulting expression for h 0 m from 0 to t, we get the unique solution èh m è t 2 C 1 c èm 0 è, which satisæes that èh m è 0 = 0. By its very construction, this h m and the h i, i =1;::: ;m, 1 gives an f m, whose associated A m satisæes the required norm estimate. è4è Consider now the following Hermitian structure on Hk: hs 1 ;s 2 i F = 1 d! Z M ès 1 ;s 2 èe 1 2 F! d We claim that hæ; æi F and hæ; æi are equivalent uniformly in k on Vk over any compact subset K of T. We clearly have that jsj F = j èkè sjçkt k jsj; e 1 4 F e 1 4 F so by Theorem 4 we see that there exists a constant C èdepending on Kè such that jsj F ç Cjsj; for all k. Conversely, we have that jsj 2 = hçe, 1 4 F e 1 4 F çs; si çjhèçe, 1 4 F e 1 4 F ç, çe, 1 4 F çe 1 4 F çès; sij + jhçe, 1 4 F çe 1 4 F çs; sij çkçe, 1 4 F e 1 4 F ç, çe, 1 4 F çe 1 4 F çkjsj 2 + kçe, 1 4 F çkjçe 1 4 F çsj jsj: By Theorem 4 and 5 we see there exists constants C 0 and C 00 èagain depending on Kè such that jsjç C0 k jsj + C00 jsj F : But then we have for all suæciently large k that jsjç2c 00 jsj F : Hence we have established the claimed equivalence. æ

15 ASYMPTOTIC FAITHFULNESS 15 Along any smooth one-parameter family of complex structures ç t, we have that d dt hs 1;s 2 i F = h^r t ç s 1;s t 0 2 i F + hs 1 ; ^r t ç s 2i t 0 F + h s 1;s 2 i F : So, if we let Eèsè = d dt jsj2 F,h^r ç 0 t s; si F,hs; ^r ç 0 t si F we have for all sections s of Vk that Eèsè = 1 2èk + nè èhçe1 2 F èæ G, 2G@Fr,inf G ès; si + hs; çe 1 2 F èæ G, 2G@Fr,inf G èsiè Hence by combining Theorem 4, è2è and è3è we have proved that Proposition 2. The Hermitian structure è4è is asymptotically æat with respect to the connections ^r, i.e. for any compact subset K of T, there exists a constant C such that for all sections s of Vk over K, we have that jeèsèjç C k + n jsj2 F over K. We note that this proposition implies the same proposition for sections of the EndèVkè with respect to the induced inner product on EndèVkè =V æ k æ Vk, which we also denote hæ; æi F. We denote the analogous quantity of E for the endomorphism bundle by E e. Proof of Theorem 7. Let ç t, t 2 J be a smooth one-parameter family of complex structures such thatç t is a curve int between the two points in question. Choose an méd=2andf m as provided by Proposition 1. Deæne n k : J! R + by Then dn k dt n k ètè =jp ç0 ;ç t, j 2 F = h^r e ç 0 t èp ç 0 ;ç t, è;p ç0 ;ç t, i F + hp ç0 ;ç t, ; ^r e ç 0 t èp ç 0 ;ç t, èi F + E e èp ç0 ;ç t, è =,h^r e ç t 0 ;P ç0 ;ç t, i F,hP ç0 ;ç t, ; ^r e ç t 0 i F + E e èp ç0 ;ç t, è:

16 16 JçRGEN ELLEGAARD ANDERSEN Recall that we have the following inequalities between the tensor product norm and the operator norm on Vk: è5è kækçjæjç q P g;n èkèkæk; where P g;n èkè is the rank of Vk given by the Verlinde formula. By the Lefschetz-Rieman-Roch Theorem this is a polynomial in k of degree d=2. Since jæj and jæj F are equivalent uniformly in k, we get that there exists a constant C such that j dn k dt jç2j ^r e ç t 0 j F jp ç0 ;ç t, j F + je e èp ç0 ;ç t, T q èkè èj ç 2C P g;n kn 1=2 k + je e èp ç0 ;ç t, èj: Consequently we can apply Proposition 1 and 2 to obtain there exists a constant C 0 such that j dn k This estimate implies that dt jçc0 k èn1=2 k + n k è: n k ètè ç èexpè C0 t 2k è, 1è2 : Which therefore implies that kp ç0 ;ç 1 èf mè0 ;ç 0, èf mè1 ;ç 1 kçc 00 n k è1è 1=2 ; for some constant C 00. The theorem then follows from this estimate, since èf m è 0 = f and k èf mè1 ;ç 1, f;ç 1 k = Oèk,1 è. 6. Asymptotic faithfulness Recall that the æat connection on the bundle PèVkè gives the projective representation of the mapping class group ç k :,! AutèPèV k èè; where PèV k è = covariant constant sections of PèVkè over Teichmíuller space. Theorem 8. For any ç 2,, we have that ç 2 1ë k=1 ker ç k if and only if ç induces the identity on M. æ

17 ASYMPTOTIC FAITHFULNESS 17 Proof. Suppose we have a ç 2,. Choose a ç in T. If f 2 Cc 1 èm 0 è then f æ ç 2 Cc 1 èm 0 è and we get the following commutative diagram H 0 èm ç ; L k è f;ç? y ç,,,! æ H 0 èm çèçè ; L k è f æç;çèçè? y P çèçè;ç,,,,! H 0 èm ç ; L k è?? yp çèçè;ç f æç;çèçè H 0 èm ç ; L k è ç,,,! æ H 0 èm çèçè ; L k è P çèçè;ç,,,,! H 0 èm ç ; L k è; where P çèçè;ç : H 0 èm çèçè ; L k è! H 0 èm ç ; L k è on the horizontal arrows refer to parallel transport in the Verlinde bundle itself, whereas it refers to the parallel transport in the endomorphism bundle EndèVkè on the last vertical arrow. Suppose now ç 2 T 1 ker ç k=1 k, then P çèçè;ç æ ç æ = ç k èçè 2 C Id and we get that f;ç = P çèçè;ç. By theorem 7 we fæç;çèçè get that èkè èkè lim kt k!1 f,fæç;çk = lim ktf;ç, k!1 fæç;çk = lim kp çèçè;ç, fæç;çèçè k!1 fæç;çk =0: By Bordemann, Meinrenken and Schlichenmaier's theorem 4, we must have that f = f æ ç. Since this holds for any f 2 Cc 1 èm 0 è, we must have that ç acts by the identity on M. Proof of Theorem 1. Our main theorem 1 now follows directly from Theorem 8, since it is known that the only element of,,which acts by the identity on the SUè2è moduli space M is the identity, if g é 2, and if g = 2, it is contained in the sub-group generated by thehyper-elliptic involution. A way to see this using the moduli space of æat SLè2; C è- connections goes as follows: For n =2wehave that M is a real slice in the SLè2; C è moduli space M. Hence if ç acts by the identity on M, it will also act by the identity on an open neighbourhood of M in M, since it acts holomorphicly on M. But since M is connected, ç must act by the identity on the entire SLè2; C è-moduli space M. Now T Teichmíuller space is also included in M, hence we get that ç acts by the identity ont. But then the statement about ç follows by classical theory of how, acts on T. For n é 2, we just have to check that the hyper-elliptic involution act non trivial for all n é 2, since the SUè2è-moduli space embeds into the SUènè-moduli space. However, it is a simple exercise in the use of the Lefschetz æxed point theorem, to check that the dimension of the æxed point set of the hyper-elliptic involution in genus 2 in the æ

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