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Quasidisks and string theory
Volume 252, number 4
PHYSICS LETTERS B
27 December 1990
Quasidisks and string theory Osmo Pekonen Department of Mathematics, Universityof Jyvdskyld, Seminaarinkatu 15, SF-40100 Jyv6skyld, Finland Received 10 September 1990
A heuristic model of non-perturbative bosonic string theory on the Bers universal Teichmiiller space of normalized quasidisks is discussed. It is suggested that the infinite-dimensional analogue of the Polyakov energy might be the quasidisk area.
1. Introduction
A formulation of bosonic string theory over a certain notion of"universal Riemann moduli space" or rather over its compactification, the universal moduli space o f stable Riemann surfaces, has been proposed by Friedan and Shenker [ 1 ]. However, the classical notion of Bers' universal Teichmiiller space [ 2 ], usually denoted by T ( 1 ), has more mathematical tenacity than universal Riemann moduli space, which in the classical set-up reduces to a point. In this heuristic note, we wish to put on record one remark about Bers' universal Teichmiiller space that might be relevant for a model of non-perturbative string theory over T ( 1 ). Namely, we describe a simple realvalued functional on T ( 1 ) which might turn out to play the role o f "universal Polyakov energy" although, for the time being, we cannot produce any evidence for our conjecture. Briefly, if T( 1 ) is described as a space o f quasidisks, then the functional in question is the area of the area of the quasidisk.
2. Universal Teichmiiller space as a space of quasidisks
Denote the interior and the exterior of the unit circle S ~by A and A*, respectively. We suppose that the reader is familiar with the notion o f a quasiconformal homeomorphism in the complex plane [ 3 ]. It is well known that these are obtained as the solutions o f the Beltrami equation
Wz=UWz,
(1)
where the Beltrami differential # is in the unit ball of essentially bounded complex functions in the domain of interest, in our set-up A. We are interested in a special solution of the Beltrami equation whose existence and uniqueness is guaranteed by a classical theorem of Ahlfors and Bers [2]. First of all, we extend the Beltrami differential /t as zero to A*. This simple idea is so useful that it carries the name "the trick o f Bers". N o w we know by the theorem of Ahlfors and Bers that ( 1 ) admits a unique global solution, classically denoted by w u, that fixes any three given boundary points, let us say + 1 and i. Notice that the Bers trick reduces the Beltrami equation to the C a u c h y - R i e m a n n equation in A* so that the solution is a univalent function [4] (a holomorphic injection) outside the unit disk. The image of A under any global quasiconformal map is called a quasidisk. A normalized quasidisk is an equivalence class of quasidisks under the action of the M6bius group of the circle (which allows us to move any three points to three prescribed positions). N o w one possible definition, often called the geometric one [5], of the universal Teichmiiller space T ( 1 ) is as the space of normalized quasidisks. This turns out to be an infinite-dimensional complex Banach manifold via the famous Bers embedding theorem [ 2 ]. The universal Teichmtiller space, as the name suggests, is a huge space that contains all the finite-dimensional Teichmiiller spaces T ( G ) where G is an arbitrary fuchsian group acting on A. (Recall
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
555
V o l u m e 252, n u m b e r 4
PHYSICS LETTERS B
that T (G) parametrizes the complex structures of the Riemann surface A/G. ) More precisely, let us conjugate G to a quasi-fuchsian group GU=wUG(wU)-I so that G ~ acts discontinuously on the quasidisk wU(A). The Teichmtiller space T ( G ) then is the space of those normalized quasidisks on which some quasi-fuchsian conjugate of G acts discontinuously. It can be shown that T (G) is a complex submanifold o f T ( 1 ) but the issues concerning the complex structure of Teichmfiller spaces are deep. For a thorough exposition, see ref. [ 2 ]. Quasidisks constitute a rich family of plane configurations that admits surprisingly many independent characterizations [6]. Conceivably, a generic quasidisk has a fractal boundary. More precisely, Bowen [7] showed that every quasidisk in any T ( G ) , A itself excepted, has a boundary of Hausdorff dimension greater than one. On the other hand, several authors [ 5,8-12 ] have studied the space M of normalized quasidisks with smooth boundaries. This again is a complex submanifold of T ( 1 ). More precisely, M is one leaf of a holomorphic foliation o f T ( 1 ) [9]. Moreover, M is an infinite-dimensional K~ihler manifold which actually carries a 2-parameter family of K~ihler structures. The simplest of them can be described in geometric terms by saying that its K~ihler potential is the logarithm of the analytic capacity of the quasidisk [5 ]. More precisely, given a smooth normalized quasidisk, we may use the Bers trick to find a unique univalent function f=w~'lA, on A*. Then the analytic capacity is the expression I f ' ( ~ ) I. In finite dimensions, the Ricci curvature usually does not make sense. Strikingly, the Ricci curvature of the space M can be computed and equals - 2 6 times its second cohomology generator [8,12]. The occurrence of the number 26 is reminiscent of the critical dimension of bosonic string theory but this coincidence has never been satisfactorily explained in intrinsic geometric terms. This is an indication that further research into the meaning of universal Teichmfiller space in string theory is needed.
3. The area of the quasidisk as an infinitedimensional Polyakov energy?
Besides the analytic capacity which relates to the 556
27 D e c e m b e r 1990
K~ihler structure, there is another simple numerical invariant of a quasidisk: the area. Using the Bets trick, we may associate to every normalized quasidisk a uniquely defined univalent function outside the unit disk. There will be a simple pole at infinity so that the function is of the general form
f(z)=z+bo+ ~ b,z-"
(2)
n=l
for [z l > 1. In the general theory of univalent functions, such functions are known as the class Z. It is not difficult to compute that the area of the corresponding quasidisk is A=zc(l-
n=, ~
nlbn]2) "
(3)
Of course, this is non-negative so that we deduce the classical area theorem [4] about the coefficients b, in the class Z:
~ nlb,]2<~l. n=l
We may think of the coefficients bn as coordinates on T ( 1 ). A refinement of the area theorem shows that b , = O ( n - l/2-~/300) but the coefficients in the class Z still retain many mysteries. It is known that [b~ I ~ 1, Jb2] ~<2, and [b3] ~< ½+ e -6. These bounds are sharp but there is no "Bieberbach conjecture" about the general sharp upper bound for lb, I. Now we turn to Polyakov's approach to the quantization of the bosonic string [ 13 ]. The central issue is to give a meaning for the partition function Z. Stating things as concisely as possible, this means the formal sum of formal integrals
g = 0 /,'g
Here S is the Polyakov energy, i.e., the Dirichlet energy of an arbitrary embedding of the propagating string into a background spacetime which, as a first approximation, can be taken to be the euclidean space of dimension d. The integration is with respect to the so-called Polyakov measure over the moduli space .~f/g of each genus g. The Polyakov measure exhibits conformal anomaly cancellation in the critical spacetime dimension d = 2 6 . Integration is followed by sum-
Volume 252, number 4
PHYSICS LETTERS B
m a t i o n over the genus but it is well k n o w n that the perturbative series (4) is infrared divergent. Perturbative string theory also suffers from the philosophical drawback that the topology and geometry of a background space need to be given. Merely the d i m e n s i o n arises as a constraint. Perhaps we are just scratching the surface of some underlying intrinsic geometric principle that would imply more stringent conditions on the global properties of spacetime. Our heuristic proposal is to replace the perturbarive series (4) by integration over the universal Teichmtiller space T ( 1 ) and to take the area of the quasidisk as the "universal Polyakov energy". Explicitly, we want to give a m e a n i n g to the formal integral Z=
f
e -A ,
T(I)
where A is given by (3). This proposal is hardly more than a shot in the dark as we are unable to provide any mathematical evidence for its justification. Notice, however, the formal resemblance of the quadratic expression in (3) to a "Dirichlet energy". Specifically, we do not know how to construct a "universal Polyakov measure" on T ( 1 ). In ref. [ 11 ] a formal measure on the l e a f M was discussed. As the curvature structure of M involves the critical n u m b e r 26, we suspect that the conformal a n o m a l y cancellation p h e n o m e n o n has its counterpart in the present universal set-up. Yet, so far we are unable to bring the dangling ends together.
27 December 1990
Acknowledgement These thoughts came to my m i n d during the special year 1989-1990 on quasiconformal mappings at Institut Mittag-Leffier, Sweden. I thank Professor Mark Bowick, Professor Clifford Earle, Professor Jouko Mickelsson, and Professor Robert Penner for discussions and the Academy of F i n l a n d for a scholarship.
References [ 1] D. Friedan and S. Shenker, Phys. Left. B 175 (1986) 287. [2] S. Nag, The complex analytic theory of Teichmiillerspaces (Wiley, New York, 1988). [3 ] O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane (Springer, Berlin, 1973). [4] P.L. Duren, Univalent functions (Springer, Berlin, 1983). [ 5 ] A.A. Kirillovand D.V. Yurev, Funct. Anal. Appl. 20 ( 1986) 322. [6 ] F.W. Gehring, Characteristicproperties ofquasidisks, S6m. Math. Sup. 84 (Presses Universit6 de Montr6al, Montreal, 1982). [ 7 ] R. Bowen, Publ. Math. IHES 50 (1979) 259. [ 8] M.J. Bowickand S.G. Lahiri,J. Math. Phys. 29 (1988) 1979. [9 ] S. Nag and A. Verjovsky,Commun. Math. Phys. 130 (1990) 123. [ 10] T. Ratiu and A. Todorov, An infinite-dimensionalpoint of view on Weil-Peterssonmetric, Max-PlanckInstitut preprint (1990). [ 11 ] H.-W. Wiesbrock, The mathematics of the string algebra, preprint DESY 90-003. [12] B. Zumino, The geometry of the Virasoro group for physicists, in: Proc. Carg6se Summer School on Particle physics (1987), ed. R. Gastmans (Plenum, New York). [ 13] E. D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917.
557
PHYSICS LETTERS B
27 December 1990
Quasidisks and string theory Osmo Pekonen Department of Mathematics, Universityof Jyvdskyld, Seminaarinkatu 15, SF-40100 Jyv6skyld, Finland Received 10 September 1990
A heuristic model of non-perturbative bosonic string theory on the Bers universal Teichmiiller space of normalized quasidisks is discussed. It is suggested that the infinite-dimensional analogue of the Polyakov energy might be the quasidisk area.
1. Introduction
A formulation of bosonic string theory over a certain notion of"universal Riemann moduli space" or rather over its compactification, the universal moduli space o f stable Riemann surfaces, has been proposed by Friedan and Shenker [ 1 ]. However, the classical notion of Bers' universal Teichmiiller space [ 2 ], usually denoted by T ( 1 ), has more mathematical tenacity than universal Riemann moduli space, which in the classical set-up reduces to a point. In this heuristic note, we wish to put on record one remark about Bers' universal Teichmiiller space that might be relevant for a model of non-perturbative string theory over T ( 1 ). Namely, we describe a simple realvalued functional on T ( 1 ) which might turn out to play the role o f "universal Polyakov energy" although, for the time being, we cannot produce any evidence for our conjecture. Briefly, if T( 1 ) is described as a space o f quasidisks, then the functional in question is the area of the area of the quasidisk.
2. Universal Teichmiiller space as a space of quasidisks
Denote the interior and the exterior of the unit circle S ~by A and A*, respectively. We suppose that the reader is familiar with the notion o f a quasiconformal homeomorphism in the complex plane [ 3 ]. It is well known that these are obtained as the solutions o f the Beltrami equation
Wz=UWz,
(1)
where the Beltrami differential # is in the unit ball of essentially bounded complex functions in the domain of interest, in our set-up A. We are interested in a special solution of the Beltrami equation whose existence and uniqueness is guaranteed by a classical theorem of Ahlfors and Bers [2]. First of all, we extend the Beltrami differential /t as zero to A*. This simple idea is so useful that it carries the name "the trick o f Bers". N o w we know by the theorem of Ahlfors and Bers that ( 1 ) admits a unique global solution, classically denoted by w u, that fixes any three given boundary points, let us say + 1 and i. Notice that the Bers trick reduces the Beltrami equation to the C a u c h y - R i e m a n n equation in A* so that the solution is a univalent function [4] (a holomorphic injection) outside the unit disk. The image of A under any global quasiconformal map is called a quasidisk. A normalized quasidisk is an equivalence class of quasidisks under the action of the M6bius group of the circle (which allows us to move any three points to three prescribed positions). N o w one possible definition, often called the geometric one [5], of the universal Teichmiiller space T ( 1 ) is as the space of normalized quasidisks. This turns out to be an infinite-dimensional complex Banach manifold via the famous Bers embedding theorem [ 2 ]. The universal Teichmtiller space, as the name suggests, is a huge space that contains all the finite-dimensional Teichmiiller spaces T ( G ) where G is an arbitrary fuchsian group acting on A. (Recall
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
555
V o l u m e 252, n u m b e r 4
PHYSICS LETTERS B
that T (G) parametrizes the complex structures of the Riemann surface A/G. ) More precisely, let us conjugate G to a quasi-fuchsian group GU=wUG(wU)-I so that G ~ acts discontinuously on the quasidisk wU(A). The Teichmtiller space T ( G ) then is the space of those normalized quasidisks on which some quasi-fuchsian conjugate of G acts discontinuously. It can be shown that T (G) is a complex submanifold o f T ( 1 ) but the issues concerning the complex structure of Teichmfiller spaces are deep. For a thorough exposition, see ref. [ 2 ]. Quasidisks constitute a rich family of plane configurations that admits surprisingly many independent characterizations [6]. Conceivably, a generic quasidisk has a fractal boundary. More precisely, Bowen [7] showed that every quasidisk in any T ( G ) , A itself excepted, has a boundary of Hausdorff dimension greater than one. On the other hand, several authors [ 5,8-12 ] have studied the space M of normalized quasidisks with smooth boundaries. This again is a complex submanifold of T ( 1 ). More precisely, M is one leaf of a holomorphic foliation o f T ( 1 ) [9]. Moreover, M is an infinite-dimensional K~ihler manifold which actually carries a 2-parameter family of K~ihler structures. The simplest of them can be described in geometric terms by saying that its K~ihler potential is the logarithm of the analytic capacity of the quasidisk [5 ]. More precisely, given a smooth normalized quasidisk, we may use the Bers trick to find a unique univalent function f=w~'lA, on A*. Then the analytic capacity is the expression I f ' ( ~ ) I. In finite dimensions, the Ricci curvature usually does not make sense. Strikingly, the Ricci curvature of the space M can be computed and equals - 2 6 times its second cohomology generator [8,12]. The occurrence of the number 26 is reminiscent of the critical dimension of bosonic string theory but this coincidence has never been satisfactorily explained in intrinsic geometric terms. This is an indication that further research into the meaning of universal Teichmfiller space in string theory is needed.
3. The area of the quasidisk as an infinitedimensional Polyakov energy?
Besides the analytic capacity which relates to the 556
27 D e c e m b e r 1990
K~ihler structure, there is another simple numerical invariant of a quasidisk: the area. Using the Bets trick, we may associate to every normalized quasidisk a uniquely defined univalent function outside the unit disk. There will be a simple pole at infinity so that the function is of the general form
f(z)=z+bo+ ~ b,z-"
(2)
n=l
for [z l > 1. In the general theory of univalent functions, such functions are known as the class Z. It is not difficult to compute that the area of the corresponding quasidisk is A=zc(l-
n=, ~
nlbn]2) "
(3)
Of course, this is non-negative so that we deduce the classical area theorem [4] about the coefficients b, in the class Z:
~ nlb,]2<~l. n=l
We may think of the coefficients bn as coordinates on T ( 1 ). A refinement of the area theorem shows that b , = O ( n - l/2-~/300) but the coefficients in the class Z still retain many mysteries. It is known that [b~ I ~ 1, Jb2] ~<2, and [b3] ~< ½+ e -6. These bounds are sharp but there is no "Bieberbach conjecture" about the general sharp upper bound for lb, I. Now we turn to Polyakov's approach to the quantization of the bosonic string [ 13 ]. The central issue is to give a meaning for the partition function Z. Stating things as concisely as possible, this means the formal sum of formal integrals
g = 0 /,'g
Here S is the Polyakov energy, i.e., the Dirichlet energy of an arbitrary embedding of the propagating string into a background spacetime which, as a first approximation, can be taken to be the euclidean space of dimension d. The integration is with respect to the so-called Polyakov measure over the moduli space .~f/g of each genus g. The Polyakov measure exhibits conformal anomaly cancellation in the critical spacetime dimension d = 2 6 . Integration is followed by sum-
Volume 252, number 4
PHYSICS LETTERS B
m a t i o n over the genus but it is well k n o w n that the perturbative series (4) is infrared divergent. Perturbative string theory also suffers from the philosophical drawback that the topology and geometry of a background space need to be given. Merely the d i m e n s i o n arises as a constraint. Perhaps we are just scratching the surface of some underlying intrinsic geometric principle that would imply more stringent conditions on the global properties of spacetime. Our heuristic proposal is to replace the perturbarive series (4) by integration over the universal Teichmtiller space T ( 1 ) and to take the area of the quasidisk as the "universal Polyakov energy". Explicitly, we want to give a m e a n i n g to the formal integral Z=
f
e -A ,
T(I)
where A is given by (3). This proposal is hardly more than a shot in the dark as we are unable to provide any mathematical evidence for its justification. Notice, however, the formal resemblance of the quadratic expression in (3) to a "Dirichlet energy". Specifically, we do not know how to construct a "universal Polyakov measure" on T ( 1 ). In ref. [ 11 ] a formal measure on the l e a f M was discussed. As the curvature structure of M involves the critical n u m b e r 26, we suspect that the conformal a n o m a l y cancellation p h e n o m e n o n has its counterpart in the present universal set-up. Yet, so far we are unable to bring the dangling ends together.
27 December 1990
Acknowledgement These thoughts came to my m i n d during the special year 1989-1990 on quasiconformal mappings at Institut Mittag-Leffier, Sweden. I thank Professor Mark Bowick, Professor Clifford Earle, Professor Jouko Mickelsson, and Professor Robert Penner for discussions and the Academy of F i n l a n d for a scholarship.
References [ 1] D. Friedan and S. Shenker, Phys. Left. B 175 (1986) 287. [2] S. Nag, The complex analytic theory of Teichmiillerspaces (Wiley, New York, 1988). [3 ] O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane (Springer, Berlin, 1973). [4] P.L. Duren, Univalent functions (Springer, Berlin, 1983). [ 5 ] A.A. Kirillovand D.V. Yurev, Funct. Anal. Appl. 20 ( 1986) 322. [6 ] F.W. Gehring, Characteristicproperties ofquasidisks, S6m. Math. Sup. 84 (Presses Universit6 de Montr6al, Montreal, 1982). [ 7 ] R. Bowen, Publ. Math. IHES 50 (1979) 259. [ 8] M.J. Bowickand S.G. Lahiri,J. Math. Phys. 29 (1988) 1979. [9 ] S. Nag and A. Verjovsky,Commun. Math. Phys. 130 (1990) 123. [ 10] T. Ratiu and A. Todorov, An infinite-dimensionalpoint of view on Weil-Peterssonmetric, Max-PlanckInstitut preprint (1990). [ 11 ] H.-W. Wiesbrock, The mathematics of the string algebra, preprint DESY 90-003. [12] B. Zumino, The geometry of the Virasoro group for physicists, in: Proc. Carg6se Summer School on Particle physics (1987), ed. R. Gastmans (Plenum, New York). [ 13] E. D'Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917.
557