MARCO MATONE. Department of Physics \G. Galilei" - Istituto Nazionale di Fisica Nucleare ABSTRACT

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1 DFPD 95/TH/38 hep-th/95068 KOEBE /4-THEOREM AND INEQUALITIES IN N= SUPER-QCD y PostScript processed by the SLAC/DESY Libraries on 7 Jun 995. MARCO MATONE Department of Physics \G. Galilei" - Istituto Nazionale di Fisica Nucleare University of Padova Via Marzolo, Padova, Italy ABSTRACT The critical curve C on which Ima D =a = 0 determines hyperbolic domains. As a consequence of the Koebe /4-theorem and Schwarz lemma, the correlator hi and its dual satisfy inequalities such as h Im hihd i h. Such inequalities are strictly related to the structure of C. We describe this curve in a parametric form related to the Schwarzian equation. HEP-TH y Partly supported by the European Community Research Programme Gauge Theories, applied supersymmetry and quantum gravity, contract SC-CT9-0789

2 . The eective action of the low-energy limit of N = super Yang-Mills, solved exactly in [], is described in terms of the prepotential F [] S ef f = Z 4 Im d d i D i + Z d ij W i W j ; () where i i and F=@ j. Let us denote by a i h i iand a i D h i Di the vevs of the scalar component of the chiral supereld. For gauge group SU() the moduli space of quantum vacua, parameterized by u = htr i,is 3, the Riemann sphere with punctures at and (we will set = ). It turns out that [] a D = p Z u dx p p x u p x ; a = Z dx p x u p x ; () which satisfy the equation [3][4] ( u )@ u In [4] it has been shown that the function =0: (3) 4 G(a) =i(f(a) aa D =); (4) is modular invariant and satises the equation ( G )@ ag+ a 4 (@ ag) 3 =0: (5) It turns out that u = G(a), therefore F (hi) = i htr i + hih Di: (6) An important aspect considered in [] concerns the critical curve C on which Ima D =a =0. Ccan be seen as the curve on which the torus with modular parameter ^ = a D =a degenerates. Let us consider the mass of a dyon hypermultiplet M = p jn m a D + n e aj; (7) where n e and n m are the electric and magnetic charges respectively. M can be seen as eigenvalue of the Laplacian on this torus. More precisely we have the Schrodinger equation nmne = E nmne nmne ; (8)

3 where z is the Laplacian on the torus and nmn e = p cos (n m x n e y), z = x +^y. It is easy to check that E nmne = M jaj (Im) : (9) Notice that (8) admits the following interpretation. One can consider the theory N = in D= 6. Compactifying two dimensions on the torus ^, one has Z P 5 + ip 6. Therefore in the massless sector P = 0 one has 6 =0=) 4 = =jzj ; = jaj (Im^) (0) In crossing the curve C a BPS-saturated particle of given charges can appear or disappear. Eq.(8) and (9) show that the tori ^ Ccorrespond to critical points for the structure of the energy eigenvalues. It has been shown in [5][6] that inside C we haveima D =a < 0. Let us denote by D such a domain, so that C Let us describe C in the parametric form. In order to do this we nd a dierential equation for G = G(a(z)), z = a D =a. From the chain rule for the Schwarzian derivative it follows that fg;zg= (@ z G) fz;gg; z = a D a : () Using (3) to evaluate fz;gg (note that a u a ( G ) 4 G000 3 G 0 a@ u a D is a constant) we get! 3 G 00 5 = G 0 ; () G 0 where z. Since z( ) = and z() = 0 and u = G, it follows that the initial conditions of the third-order equation () are G(a( )) = G(a()) = ; G(a(0)) =. Therefore the critical curve is described by C = fu = G(a(z))jz [ ; ]g : (3) Observe that by z G = i a : (4) The solutions of the dierential equations (5) and () should be related to the }-function. This is suggested by the fact that J H a F, where JH is the inverse of the uniformizing

4 map J H : H! 3 (H is the upper half plane). In particular the uniformizing equation (see u + 3+u 4( u ) implies that G as function of w H satises the equation whose solution is [7] G = } + ;; } } ;; =0; (5) ( G ) fg;wg=3+g ; (6) } ;; ;; ; J H : (7) ;; }. In [] it has been emphasized that the properties of the metric ds = jdaj = e '= j u j jduj ; (8) are at heart of the physics. Actually, the natural framework to investigate these properties is uniformization theory. An interesting aspect of the theory is that the classical moduli space is the Riemann sphere with a puncture whereas in the quantum case one has the Riemann sphere with three punctures. Thus, by Gauss-Bonnet formula, there is a \transition" from positively (classical moduli) to negatively (quantum moduli) curved space. This transition makes it evident that quantum aspects are related to deep aspects concerning uniformization theory. In particular one can apply basic inequalities, such as the Koebe /4-theorem [7], which are at heart of the theory of univalent functions (i.e. uniformization, Teichmuller spaces etc.). By means of the prepotential F it is possible to construct the positive denite metric ds P = j@3 F=@a 3 j (Im@ F=@a ) jdaj : (9) F=@a = J H (u), this metric corresponds to the Poincare metric on the u-moduli space ds P = j@3 F=@u@a@aj (Im@ F=@a ) jduj = e ' jduj ; (0) Note that e '= is a \non-chiral" solution of the uniformizing equation (5) (see [8]). 3

5 where ' satises the Liouville equation ' uu = e ' =. Metric (0) suggests to consider the Poincare metric also in the domain D. This metric can be explicitly constructed using the function z = a D =a. In fact, as shown in [6], z is the inverse of the map from a fundamental domain (identied in [6]) to 3. In particular by [6] it follows that the equality a D (u )=a(u )=a D (u )=a(u ) implies u = u. This property, i.e. univalence, is the crucial point. Since the Poincare metric on the fundamental domain is (Imw) jdwj (which isdivergent onr), we have that the Poincare metric on D is (here u ) ds = 4 jaa0 D a D a 0 j (a D a a D a) jduj = 6 = e ' D(u;u) jduj ; () (a D a a D a) jduj where the fact that aa 0 D a D a 0 =i= [4] has been used. Let us denote by D uthe Euclidean distance from u D to the (i.e. the critical curve C). As a consequence of Schwarz lemma and Koebe =4-theorem (see for example [7]) we have e ' D(u;u) ( D u) 4: () We note that a similar geometrical uncertainty relation appears in the description of the cuto (z min ) in D quantum gravity [8]. Up to now we considered unit where h =. However, the structure of () suggests to perform a dimensional analysis. Being a D a F and noticing that F has the dimensions of h, it follows that htr i(aa 0 D a D a 0 ) has the dimensions of h. By Eqs.()() and using the fact that Ima D =a < 0onD,wehave h D htr iim hihd i h D htr i (3) Let us denote by C d the curve indon which D htr i = d. For d ==wehave h Im hihd i h: (4) Similar inequalities arise on the complementary disk f D = b CnD. In particular one has h D htr iim hihd i h D htr i; u D: f (5) We note that () (3) and (4) should be useful in order to investigate the structure of the Euclidean distance D htr i. This implies that the dierential equation () describes the general structure of (3) (5). 4

6 Another interesting inequality is the Nehari theorem [9]. It states that a sucient condition for the univalence of a function g dened on the Poincare disk = fzjjzj < g is e ' jfg; zgj ; (6) whereas the necessary condition is e ' jfg; zgj 6; (7) where e ' =( jzj ) (the Poincare metric on ). It can be shown that the constant in (6) cannot be replaced by any larger one. An interesting question is to nd the sharp inequality for the case at hand. In conclusion we note that our investigation is related with the theory of quasidisks. They have interesting structures. For example a generic quasidisk has a fractal boundary [0]. Quasidisks also appear in some non-perturbative aspects of string theory []. It is a pleasure to thank G. Bonelli, P.A. Marchetti and M. Tonin for useful discussions. References [] N. Seiberg and E. Witten, Nucl. Phys. B46 (994) 9. [] N. Seiberg, Phys. Lett. 06B (988) 75. [3] A. Klemm, W. Lerche and S. Theisen, Nonperturbative Eective Actions of N= Supersymmetric Gauge Theories, CERN-TH/95-04, LMU-TPW 95-7, hep-th/ [4] M. Matone, Instantons and Recursion Relations in N =Susy Gauge Theories, DFPD 95/TH/5, hep-th/ [5] A. Fayyazuddin, Some comments on N= Supersymmetric Yang-Mills, Nordita 95/, hep-th/ [6] P. Argyres, A. Faraggi and A. Shapere, Curves of marginal stability in N= super-qcd, IASSNS-HEP-94/03, UK-HEP/95-07, hep-th/

7 [7] L. Ford, Automorphic Functions, Chelsea, 95. Z. Nehari, Conformal Mapping, Dover, 975. [8] M. Matone, Int. J. Mod. Phys. A0 (995) 89. [9] Z. Nehari, Bull. Am. Math. Soc. 55 (949) 545; E. Hille, Bull. Am. Math. Soc. 55 (949) 55. [0] R. Bowen, Hausdor dimension of quasi-circles, IHES 50 (979) [] O. Pekonen, Phys. Lett. 5B (990) 555; J. Geom. Phys. 5 (995). 6