Volume !75, number 3 PHYS:-'CSLETTERS B 7 August 1986 THE INTEGRABLE ANALYTIC GEOMETRY OF QUANTUM STRING ~ Daniei FRIEDAN and Stephen SHENKER Enrico Fermi and J~mes J~>anck L,~stitutes, and Department of Physics, Ui#versi.~y of Chicago, Chicago, IL 60637, USA Received 13 May 1986 The quantem theory of c~osed bosonic strings is formuNted as integrabie anMytic geometry on the universM moduli space of Riemann surfaces. Solztions of the equalion of motion of quantum strings are flat hermifian metrics in hoio- morphic vector bundles ever universal moduti space. In this 1otter we formulate quantum string theory two-dimensional conformabboots~rap equation *a as integrabie analytic geometry on the ~aniversal modu- The perturbation expansion of the strir~.g S-matrix is space of Riemann surfaces, based on the analytic written as a sum over world s~;rfaces in spacetime, formulation of two-dimensional conformal field which is the funcional integral version of a two-di- theory [ i ]. menNonal quantum field theory, The couplings of String theory currently provides the onty viable the twoodimensiona! qum~_tum field theory express candidate for a %ndamentaI theory of nature ~'~" , Any the background in which strings propagate. The per- such potentially fundamentai theory should have a. turbation expansion describes strings as the fluctua- completely nat~;ral mathematical formulation, The tions around the ground state° The gro~-,d state itself gauge symmetries of string theory should be the trans- is encoded as the two-dlmensional field theory ex- formations between mathematically equivalent descrip- pressing the background. tions of one underlying mathematical object° in par- A ground state era string is a background in which ticuiar, it is unnat~al to assume a particular space- the S-matrix of propagating strings is unitary, tn time as a setting for string theory, since string theory bosenic string theory, unitarity requires conformal is a quantization of general retati~ty. The structure of invafiance of the world surface, ConformaI invariance spacetime should be a property of the ground state. is expressed by the action of the Virasore algebra in One of our goals is a formulation of string theory the HiNert space of the two-dimensional field theory which is sufficiently abstract to allow discussion of of the cylindrical world surface swept out by a closed the nature of spacetime, A second goal is a formula- string. The central charge must have the particuIar tion in which effective calcw!ation might be possible. vMue c = 26, To solve the bosonic string equation of Even a potentially complete theory is unlikely to be motion is to construct two,dimenslonal conformM useful, or even testable, unless effective nonperturba., field theories with c = 26. Understanding co~fermM tire calculations are possible. ~is argues for an ~a- field theory on aH Riemann surfaces is the key to un- lyric or even aigebraic setting in which to do string derstanding quantum string theory. theory. Two-dimensions! confo~al field theory can be The equation of motion of strings is essentially the formulated as analytic geometry on the universal mo- duli space of Riemann surfaces [ 1], In this letter we adapt the generM ana!ytic geometry of conforms1 ~etd * This work was supported in part by DOE grant DE-FG02- 84ER-45t44, NSK grant PHY-8451285 and ~he Slosh Foundation. *~ For background references in string theory and two-dimen- *i For re%fences see refo [2 t. sional ccnformai field theory see refs. [I ,2 ]. 0370-2693/86/$ 03.50 © Eisevier Science Publishers B.V. 287 (North-Holland Physics Pub!isNng Division} Votume 175, number 3 PHYStCS LETTERS 13 7 August I986 theory to the speciai case of string ~heory, where its of compact, connected, stable Riemann surfaces of applbation is especially naturai. As the setting for the gents g. The stabb surfaces include the surfaces with analytic formulation of both two-dimensional con- nodes, c-~g is a compact, compbx analytic orbifold forma[ fie[d theory and string theory, we define the or V-manifold, of compiex dimension n = 3g - 3 for universal anaiytic moduli space of Riemann surfaces, g>i,n = !forg=iandn=0forg=0. T,.e~ s,~ace which apparendy has not been described previously. of surfaces with nodes is the compactification divisor The fundamental object of quantum string theory is @g = c-fy.g _ CyEg"Tb.e generic surface in @g has a hermitian metric h in a holomorp~c, infinite-dimen- exactty one node. The muItiple self-intersections of sional vector bundb W over universa! moduIi space. c3g, the subcemponents of q)g, are the spaces of The quantum equatio.,x of re.orion of strings is the flat- surfaces wit ~ mukipb r..odes. For every subcompo- ness equation o'n the metric h. Flatness equations are nent of C~,g, for every configuration of k nodes, there integrabie. Here, the solutions are exactly the unitary is a colbction of k independent analytic coordinates representations of the universa! modular group. String qi, ~ear qi = 0, defining the subcomponent by the k theory is thus reduced, in principie, to the represer.ta- independent equations q* = 0. ~n more abstract lan- tion theory of tl~e ~mlversal modular group. guage, a!t the se;]-mtersectlons.... of @g are transversal. in fermionic string theory, unitarity is equivabn~ to The generating f~.nctional of the connected part se.~percouformal invafiance of the world surface. The of the string S°matrix is fermionic theories are the on!y known string theories with consistent perUdrbative expansions. The equation = of motion of th_e fermionic string is esse~tia!ly the c] c°nn g={} C~g two-dimensional superconformal bootstrap equation. We concentrate in this letter on the closed bosonic where Z(N, m) is the string partition function on the string The superanalytic formuiation of the fermi,.mic modu.li spaces O# g. The string coup!ing constant 3, string paraliels the bosonic ~heory~ The fermionic string aepears in a factor )v2g-2 which is absorbed into Z. and two-dimensionai superconformal fie~d theory are CSconn is a .......... ~ ~ ,...... string background. The set in the super analytic geometry of the universal scattering amplitudes of strings are the coefficients of super moduti space, based on the superc,'mformal ten- the mfinitesima[ variations of the background i~) the sor ca{caius on super Riemann surfaces~ expansion of d corm around the string ground state. This work was motivated by .:he use of mod~atar The generating functional for the full S-matrix is invariance as a crucia~ constraint in string theory [3,4I the integral of the partition f~a~.ction over the moduli and ir~ two-dimensional conformal field theory [5 ]. space of all compact, smooth Pdemann surfaces, not and by some of the recent discussions of the fo,anda° necessarily cor._nected : tions of string theory and string field theory [6-9]~ Wiaib engaged it.: the presem work, we were encourag- F /- ~-- xmcSconn ) • Z,,m,m). (2) ed the paper Belav> and Knizhnik on the e I [ "=3 by of [10] OR partitio~:~ function of strings in flat spacetime, which also ~bcuses on the compbx analytic structure of ~ is the ~aniversai moduli space moduii space, and by other recent studies of determi- co nants of e!Iiptic operators on surfaces and their factori- ~= ~ S , (~) zation properties, on t:::e modu~J spaces of Riemann 2°(= k=0° g surfaces [ 11 l- We begin by establishing some notation and defin-. where Sym k is the k-fold symmetric product. The b.g the uriversa! modui! space [ t ]o Let cfg a be the symme~.rization is due ~o the indistinguishability of modu!i space of compact, connected, smooth, germs conformalIy equivalent Riema~n surfaces. Eq. (2) is g Riemann surfaces *a Let ~g be the moduii space a direct rewriting of eq. (!), given that the partition function of a disconnected surface is the product of ,a For expianations of moduli space see ref. ~t 2!; for analytic the partition functions of the con~.eeted components. geome*ry see ref. {13 ]. The factors t/k! are provided by the symmetric pmduc~s. 288 Volume 175,number 3 PHYSICS LETTERS B 7 August 1986 The space 9e is highly disconnected in its naive The moduli space near @ is ,~aramet.~zed oy (m @ ,q) topology as a union of products of the individual = (m, xl, x 2 , q), with q = 0 being the s~rface with moduii spaces cy~g. But tP..e" mfinue,:yi "÷ I many disconnect- node m O. ~n the limit q -~ 0, z becomes a coordinate ed components of c'd are connected by the formation on the punctured comDex p~ane, wmc, is conforma!o of nodes. Any two surfaces in q~ can be obtained .ly equivalent to the infinite cylinder. To study the from a single connected surface of higher genus by ana- behavior of the partition function near @, we only !y::,c aeformat,ons wm.cn mcmde formatio~ and re- need to know the properties o,.;" ~o,:.,.ormat "~ field theory moval of nodes. The pa.,t.,Ao.; fm.~don extends to on the annular pIumbLqg joint in the limit q -~ 0, fa~eao,, - wb:h'* nodes, so it lives on the u:~.wersm"" ' moduli wNch is just conformal field theory on the pla=e. space of compact, stabie Riemann surfaces The conforms! fie~a;' ~"m,o,y ~ "" on the plane is radiaIiy quant~zee, w~:h Virasoro algebra L m L m . Near @ the partition function of the surface (m@, q} can. be =ou syrup(%) (4) .oictured as an ex~ectation~ vaiue m_ the ~au~a:-~" : qu~n:>-*~ zation: The compactification divisor @- = c~ _ c)e is the space Z = tr(q z° ~Lo p(m @)) (6) of compact Riemann surfaces with nodes.