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Volume !75, number 3 PHYS:-'CSLETTERS B 7 August 1986

THE INTEGRABLE ANALYTIC OF QUANTUM STRING ~

Daniei FRIEDAN and Stephen SHENKER Enrico Fermi and J~mes J~>anck L,~stitutes, and Department of , 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 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 *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 , 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 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 , ConformaI invariance spacetime should be a property of the ground state. is expressed by the action of the Virasore 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 - 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-! 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 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 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.

288 175,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) 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 . 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 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. But nodes are invisible to a two-dimeasionat conformai field where p(m O ) is the density matrix of states on the theory. ~ no. is, the partition function of a surface boundary izil = I wNch is prepared by the withonodes is exactly equal to the partition function conformal field theory on the surface outside the re- of the smooth, possibly discormected~ surface which is gion of the _,node izii < i. The density matrix p is in- made by removing the nodes and erasing the punctures dependent ofq. The trace in eqo (6) can be perform- which, are left behind. T:ao~'~ mo.dw,e*" ~' s defining an anao ed by summing over a complete set of states in the lyric st ...... *u , ~~e on c.~ for which the removal of nodes is radial quamizationo Use the onedo~one correspon- analytic. We write mcb ~ w(mq) ) for the operaL.on*~ of dence between the scaling fields ~o(x) of the con%final removing the nodes from a surface m O ~ O and eras- field theory a.~d t:.ae"_ e~ge:.~a,~es" ne * t@_ = ~,0)i0), with L o lag the punctures, leaving £he smooth surface ~r(m0 ). exgen-vame h® and L 0 eigenvaiue h,;, to write the ex- We ~:.- see that °£ is an effectively connected, con:- pansion in terms of two- functions of the fields pact a:aG.~.~*"~ ~'- space on which...... q-,~ pa~'.-t:on~q function is ~(x) on the smooth surface m = rr(m ~ ): real analytic. Let us now describe explicitly the partition function Z = ~ qh~h~o(~{ p(m ,,c79 ){¢) ~0 near q) o Take a particular coordinate neighborhood of a node, consisting of two disks {z i : izfl < i} attached q vq ~(~Vc'l)~23)rn' (7) together at z~ = z 2 = 0 to form the node. A surface ~o with node is earametnzed oy m c~ = (m, x ~, x 2), where m -- ~r(mq))is the surface"wh,~:, ~;'~ r~m~,_.s= -~ waen" the ., ,e correiat!on functions of the fields can be recon° node is removed .-..rodthe punctures erased, and (x ! , x2) s~_u~;eca*r -" " from..... this expansion [I] The .:~admg ~'-" is the unardered pair of punctures on the surface m bution i-.~,... eq. (7) come~" ~-:,o.: ~" the ground state 'i'3/, ~ which .esut.. from remova~ of the aode. The opening which has h 0 = h0 = 0. x,:e unique SL2(C)-invariant of the node is parametrized by a single complex vari. ground state I0) always corresponds to the identity able g. First remove a sma~! neighborhood of the node, operator, which is indifferent to its location, so the ['@ < ~ai !/~ in both disks. ~be,'~_ ,~ ~,.~.~o*~o-~+~h~ ,a aemular co- partition function on @ itself is ordinate ,.~elgnbornood .,, , iq IV2 < tZl < !ql -~/2 to the Z(r~O,mo)=z(~,m), Z=zo ~. (8) remaining surface by ,:he identifications From the point of view of conforma! field theory, as z=. aU~/z2 if}q il/2

289 Voi.'ume 175, number 3 PHYSICS LETTERS B 7 August 1986

ture consistent wkh the naive analytic structure on The ghost sector is-::,o.. + eAac~,y~ +~" a conformai field c~, and for which the removal of nodes is an analytic theory.__The ghost partition function is Zgh operatiom The s~.,o_.~g~s~...... a:,.a:y,.~c: ~' structure is the one = hgh(~gh, ~gh) where w~:n fewest tocal holomorphic fmuctions. Write t.,~b~ diagram U3-+ ~ (9) is the chiral fermionb ghost functio~m! integral. !at®to preted as a ..u~c,.~on on moduE space, ~gh(m) wo:Ad in which @ -+ c'~ and ~ -+ qg are the mc..'us~.on maps, be identica!iy zero, because of the n = 3g - 3 zero and rr: @ _>c~ is the map which removes the nodes .'.nodes ofb(z)(dz) 2 i~ genusg > I, or the two con.- and erases the punctures. The ana:.ydc structure on stunt zero **nodes b 0 and e 0 in genusg = 1. We whou!d is +h~.... strongest for wmc,.~' " ~ (9) is a commuting dia- instead_interpret ~g>,(m) as a holomorphic halGdensio gram of anaiytic maps. A function for,,: ~ is hoIo- ty on '~, a holomorphic section of*" i - h; morphic if and only if it is ho!omorphic on the smooth Erie bundle K = Amax(T* ~I'0). Eq. (1 1) should be surfaces c~, and satisfies/}@ =fo ~r on the compacti- written fication divisor~ The ::opo~ogy on q£ associated wit~: this anaiytic structure is stronger the. the naive ~gh(m) = f db dee S(b'c) topology. In earticuiar, it is simple to show by induc- tion in ,*he,~. genus t:~a:~-* a giobal!y hobmorphic function on c~ is constant. ~n *bSs...... ~n se, •c~ is connected a~d X f d 2z!~!izl,zi)b(z~)dm 1,-- I °.o compact, in such a rigid analytic setting, the globaI holomorphic objects are highty co,strained and can be manipulated by ana~.ytic techniques with complete ~,~,~:,,:. As far as we K>OW, the universal modu!i space WI has not previously been descr::bed. A universai Teich- wo.,ere {din k} is a basis of one..forms on °Tg at m. corn m~aiier space is known, but the corresponding moduli ]ugate to a basis {~k} of Beitrami differentials on the ssace, is a single >om~"~ + ,~127.'., surface m. To be more precise, again in the iang~age The partition funtlon for strings in spacetime has of ref. [! ], ~gh(ra) is a hoiomorpbJc section of the form Z = Zg:=Zst , where Zg is the contribution K ® E 26 ® Wgh, where Wgh is a !ine bundle on c~ from the two-dimensional conformat field theory of with projectively fiat hermifian metric hgh~ s~arfaces in d.-dimensional spacetime, and Zg h is the :_'n the critical dimension d = 26 the depe,~dence partition function of the cNrat conformaI ghost sys- on surface geometry disappears in the combined te~ o~" 4" *'me worm* " surface functional integral fixed im spacetime and ghost system.. The cu.wature forms of the naturai eonformal gauge [2]. the projectively flat metrics h a and h ~1" cancel. The The spacefime sector is ~n ordinary two-dimem product h = hSth gh is a fia~~:erm,~mn~*~ metric i~ W sional cor._formal fie~d theory with central charge c = Wst ® Wgh~ The fibers of W are in general infinite = d~ Formulated in the im~guage of tel [1 ~, the par° dimensional in the critical dimension, E d ® E_26 .=,t,On , ~,z~ct,o~ Z ~S^a s~cLon of E d Ed,W ..... , = E 0 is the mv~a~ .:me bundle on%??, so ¢(m) ~.~nemi., ~c'~ = ,iX->:~w/z, (XH) be~-~g the Hedge =/go(m) V)st(m,) is a ho!omorp, hic section of V' bu~,dle. E c is the projective 1in® bundb on c72 which = K ® W. The string partition function is encodes the reIationship between the two-dimension- al conformat fieid theory and the surface geometry. z.,m, m~ - h(~,, ~) : ~(m)h~b.~b(m), 03) The spat®time partition function is written in terms which, as a section of K ® K is a density on ~ which of a projective]y fiat henn,~,a~*-: ...... -~" et f _.c h st in a projec- can be integrated, formally, over c'~ to give the S- tire vector bundb War over ~, and a hotomorp~c matrix. Thus eq. (2) for the S.matrix is mathematical- sectloz:} ~st(m) of the vector bundle Vst = Ed ® Wst iy natural. This is the formal orig}n of gauge symme- over c'te : try L~ the anaiytic qua:p.,tum string theory. To go be-

290 Volume 175, number 3 PHYSPCS LETT,~RS B 7 August 1986

yond form.at gauge invariance we wi!i need the super- Modular invariance of the partition function becomes e.na!ytic formulation of fermionic strings, since we do crossing of correiation functions on the not know of a bosonic string ground state for which world surface, when the corre[at:.'on ~5~,a~,.~,,_,:~'~-are re-

~he.. integraI in eq. (2)"" is ~LAte.%" constructed from the behaqor of the partition funco The hermitian connection D m W is written ha tion near @. Crossing symm~h, on ::he world sur- components face is duality ~m stnng theory. The analytic formui'ao tion of quantum strings *~~.,~us • ~,,,uar_me~s • • ~ * ~ t:_.ea two crucla:.. • D=8, D=~+A, properties of locality and duality. The fiat herm:~an vector bundle W an~ the unitary Dhab = 8h~ - n~.cAb C = O. (! 4) representation ~ of P are equivalent. W is reconstruct- A is the connection form of D, The quantum equa- ed )ore iE as the quotient of N X 9~ by P acting on tion of motion of bosonic strings is the :..'ames.'~ * equa- both spaces simult~eousiy, W = N ×r ~ . We say

tion that the analytic geometry of quantum strings ~s. ~xn~e- + grable in the sense that this equivalence should even- F=~A =O. 05) tuaily allow a purety group theoretic treatment of The connection in W can be thought of as the propa- string theory. gator of information across universal moduIi space. The fiber of the universai analytic covering N .+c£ r;.aness, the local equation of motion of quantum at a generic smooth point Ln c~ is isomorphic to P ito strings, is the absence of ambiguity in the propagation se!L At a singular point in c~£, the fiber eegenerat~s~

of information. to a homogeneous space P/P0, where P0 is the :'**:al,c.~e Near a smooth, non-orbifold point m c~, a basis group at the singular point. At a generic smooth poma+~, (wa} of covariant constant local hoiomorphic sec- ~e fiber of W is modelled on gg. At a singuIar point, tions can be chosen for W. In this local frame, A = 0 the fiber is W(0), modelled on the subspace ~(O) con- and h is the ~nstant hermitian matt! x ha~ sisting of the vectors Ln ~ which are inva~ant ~der = h(~as~-, wa). In components, Z = ,#a(m)h& cb(m), the iittle group P0' wmc. is manifestly real analytic ha m. Real analyticity The little group for a surface with nodes is the of the p~rtition function on c~ is equivaient to locali- group generated by the Dd.~n twists 7k :qk .+ e2~riqx on ,.~..~wo~,x: surface, and there%re to locality k~ arouna' t:.ea nodes, tt is enough to describe the generic spacetime._ The rent ana..y~_c~,.y~ +{ ~+ o~e, ,h~ partition func.- case of one node. '.in a basis of cova/ant constant sec- tion foIiows from the local flatness of h and #m~ ~ local tions near cD, the twist 7 can. be diagona!ized, with ana!yticity of ~b. eigenvaiues e 2rAh . W(h) is the ~ocatly defined sub- Because h and ¢ are globally defined on ~, Z bundle on which 7 has eigenva:me e2~rih 0W degenerates is single valued under analytic continuation aro'~md to W(0) at q = O. the singular poLnts in 9~, the orbifo!d po~.ts and the The definition W as a holomorphic vector bundie compactification divisor @. To l~e single valued on over c.~ includes a transition map rr* correspondfi',g to c~ can be interpreted as modular invariance. Let N the removal of nodes map ~ in diagram (9). In + * ° be the universal analytic covering space of c7g£~°Define of the Ioeal hotomozphic sections of W over q~, qg+, the universal modular group P to be the group of Deck and @, written c~x2(~), O~Y(c~), o/2 (@), the transi- or covering tr~sformations of N, ~ = N/P. P is es- tion map ~r* g,ves~" *~',.an commuting diagram sentially the fundamental group of ~ minus the singular points. ParaI:e,_ transport of the covariant con° st ant se~ons,'*~ w a aromad nontrivial closed in cw(c~) c~, avoiding the singular points, gives a representation

of the universal modular group P: T~ = "~wa. .oca s~c,~o, of W must satisfy lr*w(m ® ) = is ~itary because the metric h~b is finvariant under = w(~(m c~)) on cD. parattei transport. Therefore the string pa~x~1on~,. !~nc-e,. More concretely, let mq) be a surface with nodes, tion is single valued: and write m = ~(mcD ) = m I U m 2 ~.J _. Umk, the unim~ of con,nected comixme~'As.mi. Given sections Wa~ of v~ ~ (m)~a¢ ~m). 06) 291 Vohm~e 175, number 3 PHYSICS LETTERS g 7 August i986

W at m i, the symmetric oroduct w a w a ... Wa, is a we get the 1ending behavior sev.,o.:04-4 ~ of W at m. The* transmon* . . mapI ~r*~ ~s, gwen./'2 by the factorization structure tensors ~ ~h(rn ~ ) r*(w>, w.~ .°o v"bk) = Fg; b~...bkW~ • (18) • f z, : Note that a is an index for W/cD = W(O). Note also q~'0 Ogh(m)~ ~ f dr0 " 1 ~hat ob,...blc :.s sy..'nm~nc ~o_ 6r~dee. ~ymme:nc be- cause o~ the ghosts) in the radices be. The factofiza- :~o:, tensors provide tlae data which is used to build the ×

.,c~ d~d 2 =hc,d~ hg2d 2" ~(m, x~, x 2, q) q'2o cl-2dx~ dv dx2~r* ~(m), Eq. (21) makes h bear a strong formal resembh~,.c~ ~a(m !, m2,x i,x 2,q) to the metric of a Fock space, where Fg c is the sym- metric tensor product. q~0 q-2dx? dx2 ;'~F~ee/b(ml)~e(m2)" (24) To ~.derstand the behavior of ~) near @, we onIy need some basic facts about the behavior of the con- The factorization conditions (21) on hab and (24) form~ ghost fieids on the complex plane [2}. The on * give the factorizatinn of Z: Fourier components of the ghost fields satisfy Z(~,Zl,x2,q,m,x!,x2,q)

SL 2 ;mva:.a,,'~"* state .~s written [0% Because ",f.. the q~'O q-2q-21dql2fdxl dx2i2Z(g~' m). (25) background charge of tan ghost syst.,m, <010> - O, and the state conm~at"'7 e"~0 I0>, satisfying <0'D> = 1, is i0'> Belavin and ICnizhnik [i0] or!gina!iy found the = c_icoc I iO>, Note that ' 0,,x is not $L 2 invarieaat. g- 2q-2 singularity of the genus g partition %nction nser~ a sum over ~tates on each side of the node, hn fiat spacefime. at .<"~ circle !a;i -~ 1 and also at the circle Iz2i = I. We have now described a quantum ground state The !eadh'~g singu!afity comes from the insertion of the string as a flat hermAian metric in a holo. 10><0'l -. {0 , morphic vector bundle W over~, or equivalently a facLqg the rest of ...._e sa.face.o From eq. ~,:2), using ex- unitary mpresentafion ~ of P, along with a hoto- plic:;t Bekrami differentials dual to dx~, dx 2 and dq, morphic section ~ of V = K ® W. The fundamental

292 Volume 175, number 3 PHYSICS LETTERS B 7 August 1986 pr!ncipIes of string theory have been trans!ated into ycctor bm'~d!e f2 of differential forms on c~ with co- flatness and analyticity on c~. But tNs formu}ation efficients in W: of string theory is not concrete without a definite max prescrlptior~, for constructing ~. We wiIi discuss pos- a= ~ (^;,qN)®w, sible criteria for choosing ~, but our understanding p,q =0 of this probbm is still very incomplete. In particular, we lack a generM abstract formulation of Wick rota- AP, q'O~ = ,,.P(T*~!,0~ ,~ ,~ ,® ^q(T*N°,i), tion as analytic continuation of the representation ~. max Unto now we have implicitly been working in *be ab- a = E G, ap-- ® v stract analog of "e,aclidean" spacetime, where the p,q=O ~q spacetime system is a unitary conformal field theory with straightfo~vard factorization properties. ~n the (26) Wl,.~. rotated system we expect there to be a mani. fo!d of solutions of the kqfL~itesimal flatness eqaao The metric h on W, combined with the wedge product tion (1 ~), corresponding to physical fluctuations of the on forms, induces an inner product on sections ~ of string. The crucial condition on ~ wili be the positi~- ~, h~ V) = ha-5~a ~bb, whose value is a differential ty of the metric on physical states, after Wick rotation. form on c~ Integrating formally the volume e!ement To deal with these issues it wiIi surely be necessary compor,.ent in hff ff gives an i~deFmite metric on the to understand the gauge symmetries of the quantum space of sections of ~: string. The gauge symmetries of the c!assical string have been described in the context of classical string field theory [6,8]. Gauge symmetry in the quantum bosonic string is difficuk to disc~,~s because the par- tition function Z = h(~, ,~) cannot be integrated at The connection D m W combines with the exterior q = 0. The integral could only be finite at q = 0 if the g~ @on forms to give exterior derivatives residue ~*~ at the double pole, and the sLngle pole 5, D on sections of ~. 8 is the ordinary antiholo. residue as weli, were in the nu!l space of h. But Z morpNc exterior , ~ C f2~+l, and D is factorizes on the doubb poie, so the partition func- the covariant divergence operator DI2-~ C g2P -I The tion itself would then be identically zero. It is in- r,~al exterio~ derivative wkh coe:ficients in ~¢qis Q tr!gukng to specu!ate that an argument of this type, = 8 + D. D being a metric connection is equivaIent to for Z = O, could be used to show the vanisNng of the D 2 = 0. D being flat is equivMent to [g, Dt+ = 0. The cosmological constant, in the super anMytic formula- compatibility of the metric h with the connection I3 tion of fermionic strings, there must be an analogous is the formal self-adjointness condition D ? = ~ Thus factofization pole, but the k.ntegra! over universal super the quar..tum equation of motion of the string can be modu~ space c~ever,,y~ suppresses the factorization written in the gauge invariant form singularity by the GOS projection, and suppresses the Q2 = O, Qt = Q. (28) s~;bleading singularity by a remarkable canceIlation mechanism, wbhch should in principle fo!~ow simply if differential forms on c~ are written i,~ terms of f tom smooWmess of h ar.d analydcity of ~, or._ the ghost zero modes, the operator Q can be recognized super compactification dfvisor of super moduli space° as a quantum BRS operator° Thus eq. (28) is formai~ We should wait to formulate tl~e analytic geometry of iy analogous to the classical field equatio~ for open the ferrnionic string before discussing gauge symme~ strings [8~° TEe gauge symmetries of the two equations tries of the string S-mat~x. Bet we proceed anyway, have essentially the same structure. We , however, as practice for the fermionic strh'-~g. that eq. (28) is the quantum equation of motion of The obvious gauge symmetries are the unitary the cJesed string° gauge transformations in the hermitian bundle W. But The fiat'nermman ' ~ geometry, or equiva!entiy +~,e~_~ mere is a larger class of gauge transformations. The "~....tary .~o~es~nta..~on ~ of P, shouId be regarded as recto, bundte V is contained as a sub-b~mdle in the the pvlmary object in the strD.~g theory. For each such

293 Volume !75, number 3 PHYSICS LETTERS B 7 A~g~st t986

representafio~ gg ....v~e.ze is an associated com~.~p..~l~× choice of a representative ~ in ~:e co~:omo,.~,gy class ~2(~ )., The cohomoiogy spaces of this complex are is a choice of gauge. H(gt ) = H(°--~d,W) = Ep,qH~(g(). The cohometogy We need to understand which unitary represents° spaces H(~) form a vector bundle over .*,,he space of tions g( are a]lowed ground states of the string, k is representations of P. For the trivial representation 0, ,emp.~mg to posit the uniqueness of [~] (~) as a H(0) is the cohomology ring H(C~), which acts by basic condition on g(, since otherwise the ground m~pl.~t:o..~, of forms as an a~gebra oe, ~.X~)° As a state is not uniqueiy determined by the hermitian section of g20, ff satisfies 3-~ = 0 because ff is hoie- geometry a!one. This condition wouId be dim H p~''qv~

Spacetime geometry is associated w_..,~"~'0 pe~~u_oa~ive~+~-r ~ *~ ment, so that the string states in the ena!ytic .forma- string theory, and presumably with the choice of gauge lism can be interpreted physicMly. k.n ~ ~o The choice of gauge determ~nes the residues We should note that the strategy of the present of ¢ at t;~e single poles, and thus determ:.nes the mass° work can be traced in several ways to the original S- less physical particle spectrum. We do not know the matrix bootstrap program [!4 t . it is applied here in details o~~ *~~e association between geomet~ a-~d gauge the highly constrah'..ed context of string theoryu choice because we do not have a theory of measure. Anaiyticity, cross'ms symmetry, and unit arity o f the monte Measurement is the basic concep~ua.: promem S-matrix are expressed as flatness ofh and analyticity to be resolved in turning the formai structure of ana- of ¢ on universal mod~!i space. Analyticity, crossing ~y~..~ ~aan~um string theory into a theory of physics. symmetry, and unitarity are abstracted away from There should be no essential prebiem extending spacetime, and adopted as dynamical principles of the analytic construction to femnionic string. The key string theory, in the hope that the ground state of the ~',,gredients are the local superconformal tensor caicu- strw.g will resemble spacetime. !us on super RJemann surfaces, and, because a node Iooks ~,~e the complex plane, the structure of the We thank Dan.: Freed, Emil Martinec, Edward Witten superconformai ghost system on the complex plane and Scott Woipert for helpful conversations. We es- [z]~ In me hetero~ic string, the partition function Z peclai!y acknowledge discussions with Tom Banks on = h~ord~su p is an ordinary halfodensity in the anti° string field theory and on the foundations of string hoiomorphic variables and a super h alf..density in the theory. A preliminary description of t>is work was ho!omerphic variableso The spaces W and W are not given at the Workshop on Strings and _.Riemapm~ Sur- nece~arily comptex conjugate to each other, tn faces, MSRL Berkeiey, March 10-12, 1986o fermionic string theory, the fimteness of the generat- ing m:,c~ona, comes from cancellations of poles due efe~'e~ce$ to the sum over spin structures and ,~h~ GOS projec- tion, i.n the node viewed as a superconforma! puncture [ I ] D. Friedan and S.H. Shenker, The a~Mytic geometry of ed p!ane. conforms! field theory, EF~ preprint 86oi 8A. ~n• this work, we have defined the universal moduti [2 i E.g.M. Jacob, ed., Dual theory (North-.Ho!land, Amsterdam, 1974) ; space of Riemann surfaces, and we have written the J. Schwarz, ed., SuperstrL*kgs, the first fifteen years quantum equation of motion of the bosonic string as (World Scientific, Singapore, i 985); the flatness equation (I5). This equation can be irte- M..Green and D. Gross, eds., Prec. Workshop on Uni- gra..ee, ~,,. principle. The somdons' +" are the unitary fied string theories (~nstitute for , representations g{. of the universal modular group. Santa Barbara, CA, Juiy 29-A~.~g~st I6, 1985) (Worn ScientifiG Singapore, 1986), to be published; We have given the gauge ~_variant form of the quan. D. Friedan, E. Martinee and S.H. Shenker, Con%rma! turn equauo,~ of tues.,on and have speculated on the invar~anc:e, supersymmetry, and strifes theory, Chicago/ relevance ,~,~ ,~,,.,.,,~:'~n-;'ogenus Riema~.n surfaces to the Princeton preprint (1985), Nucl. Phys. B, to be publish- uniqueness of the ground state. Among the promising ed. aspects of ..b& anayt~c* forma~sm is the possibility of [3] J. Shapko, Phys. Rev. D5 (1972) 1945. [4! D. Gross, J. Harvey, E. M~tinec and R. Robin, Phys. precise control over string theory, so that certain key Roy. Letto 54 (i985) 502; Nucl. Phys. B256 (1985) 253; numbers I£~:e the cosmological constant might 5e de- B267 (1986) 75° termined exactly, even though it witl probably remain [5 ] J. Cardy, UCSB preprint (1985), NucL Physo B, to be impractical to describe the ground state explicitly and published. [6~ W. Siegel, Phys. Let*,. B 149 (i984) 157,162; B 151 ~o ..... e~,y ia every detail. The remaining problems of (t985) 39i, 396; .t.nmediate concern are: extending the analytic forma- W. Siegel and B. Zwiebach, NucI. Phys. B263 (1986) i05; lism to fermionic strings; giving a concrete description To Ba~_ks and M.E. Peskin, NucL Physo B264 (1986) 573° of the universal covering space N and the umve_sa [7] D. Friedan, EFI preprint 85-27, NucL Physo B, to be modular group; fimdkng an abstract version of Wick pubiished; Phys~ LetL B t62 (i985) 102. rotation and specifying the physical positivky con- [8~ E. Witten, Nuc!. Phys. B268 (t986) 253. [9~ T. Banks, private communication. straints on g~ ; and construct/ng a theory of measure-

295 Voiume !75~ number 3 PHYSICS LETTERS B 7 August 1986

A.A. BeNvin and VXL Knizhnik, Landau Institute pro- [12j E.g.L. Bets, Bull London Math. Soco 4 (1972) 257; print (! 986). S. Wolpert, Ann. Math° it8 (! 983) 491 ; On obtaLning J-M. Bismut and D. Freed, mare:scrip! (i985); a positive line bundle from t~e Weii-Petersscn class, R. Catenacci, M. ConMba, M. MarteH~ni and C. Reins, University of Maryland Mathematics Department pro- Pa-~a preprint (1986); print (I983). J.B= Bost and T. Jolicoeur, Sack:y prcprint (~ 986); ~3} E.g.P. Grfffiths and J. Hm'ris, Principles of algebraic N. So,berg and E. Witten, Princeton preprint (1986); geometry (Wiley, New York, I978)o L. Dixon and ~o Harvey, Princeton preprint (!988); [14] J.A. Whee~er, Phys. Revo 52 (t937) 1107; L. AtvarezK~aum~a, P. Ginsparg, G. Moore and C. Vafa, W. Heisenberg, Z. Phys. !20 (I943) 513,673; Harvard preprint (i986); G. Chew, S-matrix theory of the strong interactions Lo AIvarez~C,aum4, G. Moore and C. Vafa, Harvard Fro- (Benjamin, Reading, MA, i961); print (1986); A.B. Zaraotodchikov and AI.B. ZamoIodehNov, Ann. A. Cohen, G. Moore, P. Nelson and J. Poichinski, Phys. !20 (1979) 253. Harvard preprint (1986); C. Gomez, 1AS preprint (I986).

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