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Riemann Surfaces, Conformal Fields and Strings I he ru k uf Rtc m:rnn s url acc~ in modern pa rticle r h y~ i cs I\ discussed. M<i thc111at- 1r :1!1 \. q11 :1111urn ri cld th eories c:1 11 he tlclinccl 011 the" : mani fo lds if the <1rc co11 - lmm11lll' in ari 11 n1. I hysic•tll y. Ri 1:t11 nnn surl:i tcs provide n 111 od1: I for th • wqrlJ , h~ c l ~ 'vc p t out IJ n prop:1gating rcla1i vistk stri ng. Thuh Ricmu nn surl uccs nrc th e natural se ll ing for confom1al fiel d theory. and. both these co n cc pl ~ together pruvidc .1 1i1r11tul .1 t1 011 of smng theory. I\~\ Words: string theory, conformal symmetry , 11•orld sh eets, Riemann surfaces I. FIELD THEORY AND RIEMANN SURFACES The space- tim e on whi ch th e particl ph ysici st writes a Lagrnngian field theory and deduces equations ol moiion, c b ss i c~ d solutions anJ quantum scaucring amplitud s is usually taken to be a Ri e­ nian111a11 manifold . 1 T he. ii.l ea is cent rn l l ' ge neral relati vi ty and grav1tat1 n.2 and has becom · mor imporlanl in pnrticlc phys ics w11h increasi ng cmphasi on unified th eo ries o f all for es including grav itation . R1cmannian gcomotr is usually incorpo.ratctl as a se ri es of pre­ Knpti ons. starting with the introduction of a i..:c ond -r:rnk ~y m ­ mctril: tcn·or fi ' ld, the metri c g1.,,(x). anu combinations o[ its de­ r1 \tativc, ~u c h a ~ th e hri stoff I conn ·cti on 1· ~... (,r) unJ the Ricniann curvature tensor R~v 11 (x). These are then coupled invariantly to lhcnisclvcs and to other classical fields such as scalars <P(x) and (~,111 \11d /,au Phv.r .. ~ « I 9HX Gordon and B rcadi. 1'119 . \ 0 1 I~ . :-.; o. .1. pp. 133 - 155 Science l'u hti shc1s . Inc . ~-h •1;id;ihk tlircctly from the publisher Pt intcd in G1ca t B1 itain "-"P'" ' ~ permitted hy li ce nse only 133 vector gauge fields A~(x) . Some typical field-theory actions on Riemannian manifolds are: 2 4 (i) (114K ) f d xygR: the Einstein-Hilbert action (K is the grav­ itational constant); (ii) (1/2) f d 4xVggµv aµ<1>av<1>: the scalar field coupled to gravity; (iii) - (1/4) f d 4xVggµAg vp tr(FµvFAP): the Yang-Mills field cou- pled to gravity. (1) From the mathematician's point of view, a Riemannian manifold is a somewhat specialized structure. To define it, one starts with a topological space and introduces a differentiable structure (a set of coordinate charts such that coordinate changes across overlaps are given by differentiable functions). This gives rise to a differ­ entiable manifold. At the next stage a Riemannian metric is in­ troduced, and one gets a Riemannian manifold. For the particle physicist it has generally been this last attribute which has played a major role; the topology and differentiable structure have (until recently) played a relatively minor part. An interesting specialization of the notion of a differentiahle structure is that of a complex structure. 3 The coordinate charts are chosen to be complex, and coordinate changes across overlaps are given by complex analytic functions. In this way one gets a complex manifold. This can in turn be endowed with a metric. The various structures we have discussed can be listed in order of complexit~· (Fig. 1). Riemann surfaces belong in the lower branch of this flow chart. They are, quite simply, one-complex-dimensional (connected I manifolds. 4 They need not be thought of as metric spaces at all. An important fact about them is that they are relatively simple Rier.tannian .______,manifold .---------,~ topological differentiable space manifold ~ r-c-o-n1P-l"--X-.. comp le~+- rr.a~: - f ld ~ fold \H .h 1nan i ,o metr1c FIGURE 1 134 and rather well-understood objects. Their topologies and complex structures are completely classified. In sharp contrast, differenti­ able manifolds in four dimensions, and possible Riemannian met­ rics on them, are far from being classified. From the physicist's point of view, Riemann surfaces ar s arccly interesting as models of space-time! • r one thing th y are tw - dimensional as real manifolds. But their mathematical implicily makes it very tempting to try and c n tru t fi eld-th eory model on them. IL STRING DYNAMICS AND RIEMANN SURFACES Had physicists two de ad s ago reasoned things out in the manner di ·cussed above, they w uld probably have come to a halt at this point. orrunately for u , things worked Ltt quite di ffe rentl y. Two­ t.limensi nal 'space-times" entered particl physi in the descrip­ tion of the "world sheet" swept out by a pr pagating relativistic string5 (Fig. 2). It is a theorem4 that a two-real-dimensional manifold can be described as a one-complex-dimensional manifold if it is orientable (which means that one can define a unique normal vector smoothly everywhere). For closed strings, this will be true of their world sheets if the strings themselves are assigned an orientation which 1s preserved in interactions. Thus closed oriented strings sweep out Riemann surfaces. It does not (yet) follow that, in describing the dynamics of such a string, it will be sufficient to deal with Riemann surfaces, in­ dependent of a metric. In fact, as presently formulated, the basic FIGURE 2 X (cr,ll FIGURE 3 postulate of slrin, th eory does requir a lw< -d imensional metric in the Riemann surface, as we wi ll see. but the I p ndcncc 011 them Irie ultimat ·ly ancels out in ph sical qu an tities. In n some­ what r unclaboul way, then , string th eories are fi Id lh cori s on Riemann surfaces. Thi come · ab< ul in th e l'o ll owing way. Introduce a string o­ < rdinatc ; JJ.(a J . wh re 1' labels points in a Euclid 'an ·pace­ tim and (cr,r ure local real oordinates n an oriented two­ dimensional ·urfacc. A string confi uration may be thought of i. a m<:1p from rhe Riemann surfac' int phy!iical space-Lime (Fi,. ). h osing a metri g,, 1,(a.r) on the tw -dimensi nnl surfa e, we 6 7 postulate an action · (2) where the integral is performed over the surface. The motivation for this cla · ica I action may b • seen from the fact that the classical equation f motion f r th e metric (varied as an independent field) is 0 (3) This is solved by where <l>(a,t) is an arbitrary function. Substituting this into Eq. (2) 136 r one gets the classical action (4) This is just the area of the world sheet swept out by the string in n-dime nsi onal space. ~ 1 Note that the <!>-dependence has dropped out. as a consequence of Wey! invariance (invariance of Eq. (2) unde r I «ti rescalings of the metric: 8ab(rr,t) ~ e<l> 1"">gab(a,t) for nrhi lra r (!>(er .t)). The connection between this action principle and Riemann sur­ faces as one-complex-dimensional manifolds comes about through 4 three important theorems : (i) Every orientable two-real-dimensional manifold admits a com­ plex structure, hence can be made into a Riemann surface. (ii) Every Riemann surface admits a unique metric of constant curvature, denoted kab(rr,t) . (iii) All possible metrics on a given Riemann surface are confor­ mally related: if 8ab( a ,t) and g~b( a ,t) are two metrics, then (5) for some function <!>(a,t). Fr Ill (i) irnd (iii) , every metric is conformally related to a unique one of constant curvature. Thus in the action (2), we can substitute any arbitra r 1 g 11 1> by e<l>gab for a suitable <!>(CT ,t). The <!>-dependence now drops ut. a nd we are left with (6) But now theorem (ii) tells us that each gab is uniquely associated to a given complex structure, that is, to a given Riemann surface. Thus the string action is (classically) an action on a Riemann sur­ face. 137 III. THE CLASSIFICATION OF COMPACT RIEMANN SURFACES The que ticm of how nrnn y inequi va lenl 811ti 's th ere are reduces lo th at of classifyin g incqui alcnt Riemann surfaces. Let us mak s mewhat more precise wh al i meant by a c mpl ex ·tructure. Th c ncept involved in definin g a manifold ca n be illusLrated by a di agram (Fig. 4 . µ. 1 and µ. 2 are two overlapping patches on the manifold . c!> 1 a11d <1> 2 as~oc i a t e subsets of u lide n pace, v1 and v2 • to th ir re ·pe ti vc patch . These are the coordinace maps. The " transition fun ti on" j' d esc rib e~ h w th e shaded regions on lht: I 'ft and right arc rel <tl ·J. by virtue ol' bt:i ng different co rc.li - natization of th ame haded region in M . l For a Riemann sm face lh e c ordinate patche. v1 and v2 li e in the complex plane. If the compiex coordinate on v 1 is z and on v2 , w, then the transition function is f: z ~ w = w(z) (7) The fact that w is an analytic fW1 ction of z, on every overl ap of co rtlinate patches, is wh at define r1 complex structure.

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