sign in sign up
<<
Home , Bosonic string theory, BRST quantization

4 BRST of the bosonic

4.1 BRST quantization

The BRST formalism is a consistent quantization prescription for a at the level of path integral, assuming that the gauge algebra is not field dependent. To begin with, consider an S[], where collectively denotes all fields in the theory, that is invariant with respect to gauge symmetries of the form

= dµ ✏↵ ✏↵ . (4.1) ↵ ↵ ⌘ ↵ Z Here ↵ is a continuous label that includes the spacetime coordinates. dµ↵ is a suitable measure on ↵-space, whereas ✏↵ is a function in ↵. When there is no room for confusion, we will omit the integration measure in our notation. The gauge variation ↵ acts on composite fields by Leibniz’s rule as a di↵erentiation, and obey an (infinite dimensional)

[↵,]=f↵ , (4.2)

7 where the structure constants f↵ by assumption are field independent. Formally, the partition function can be expressed as the Euclidean path integral

S[] Z = [D] e , (4.6) Z up to some unspecified normalization of the integration measure [D]. To define the latter properly, we can fix the gauge by replacing (4.6) with

A A S[] Z = [D] (F []) det(↵F [])e , (4.7) Z 7As an example, suppose the gauge symmetry of interest is di↵eomorphism. In this case, ↵ will be a ↵ ⇢ d spacetime coordinate y together with a vector index ⇢, ✏ will be a vector field " (y), with dµ↵ = d y.A scalar field ' transforms as '(x)= d(x y)@ '(x). (4.3) y,⇢ ⇢ (4.1) then reproduces the usual infinitesimal di↵eomorphism '(x)= "⇢(x)@ '(x). From the action of the ⇢ commutator of a pair of di↵eomorphisms on ',

d d [ , ]'(x)= (x z)@ (x y)@ '(x) (y, ⇢ z,) y,⇢ z, x ⇢ $ (4.4) = ddw ⌫ ⇥d(w z)@ d(x ⇤y)+⌫ d(w y)@ d(x z) '(x), ⇢ ⇢ w,⌫ Z ⇥ ⇤ we deduce the structure constant of di↵eomorphism,

f (w,⌫) = ⌫ d(w z)@ d(x y)+⌫ d(w y)@ d(x z). (4.5) (y,⇢)(z,) ⇢ ⇢

58 where A is an index that labels components of the gauge condition F A[]=0,suchthat A ↵ the Jacobian ↵F [] is non-degenerate. We can then introduce the ghosts bA, c ,anda Lagrangian multiplier field BA,sothat(4.7)canbewrittenas

Z = [DDB Db Dc↵]exp S[]+iB F A[] b c↵ F A[] . (4.8) A A A A ↵ Z In the full gauge fixed action with the ghosts included, the gauge symmetry is replaced by afermionicglobalsymmetryB,knownasBRSTsymmetry,thatactsonthefieldsas

= ic↵ , B ↵ BBA =0, (4.9) BbA = BA, i c↵ = f ↵cc. B 2

2 The definition is such that B =0,and

iB F A[] b c↵ F A[]= ib F A[] . (4.10) A A ↵ B A It follows that B is a symmetry of the action appearing in (4.8).

One may also construct a Noether current jB associated with BRST symmetry, and define a corresponding BRST charge Q ,suchthat = i Q , ,where , stands for B B { B ·}cl {· ·}cl the Poisson bracket. If the path integral measure is also BRST invariant, then QB can be 2 promoted to a Hermitian operator, such that QB =0. Aphysicalstate will be defined to be one that is BRST-invariant, or BRST-closed, | i Q =0.Intermsofthewavefunctional [], this condition is Q [] i []= B| i B ⌘ B 0. It follows that the transition amplitude between two physical states is invariant under deformations of the gauge condition F A[]=0.Toseethis,considerachangeF A A A ! F + 0F .Wehave

A ↵ f S[]+B (bAF ) A 0 f Ufi i = [DDBADbADc ] ⇤ [f ]e B(bA0F ) i[i]=0, (4.11) h | | i i f Z where Ufi is the operator defined by the path integral with boundary condi- tions = i at the initial time and = f at the final time. Furthermore, shifting a physical state by a BRST-exact state Q for some | i B| i | i does not a↵ect transition amplitudes between and any other physical states. Thus we | i may view as being equivalent to + Q , and identify the of physical | i | i B| i states with the of QB.

59 4.2 BRST on the

As already seen in section (2.4), the gauge fixing of the worldsheet theory of the critical bosonic string with the gauge condition gab =ˆgab leads to the action

µ a µ a S[X ,bab,c ]=SP [ˆgab,X ]+Sgh[ˆgab,bab,c ], (4.12)

In the language of BRST quantization, we have already integrated out the Lagrangian mul- tiplier field Bab as well as the dynamical metric gab.TheBRSTtransformationofbab, Bbab = Bab,isnowreplacedanoperatorthatisequivalenttoBab by the equation of motion, µ a which is the stress- tensor of the (X ,bab,c )system,namely

X gh Tab = Tab + Tab . (4.13)

The classical BRST transformation of the worldsheet fields, up to equations of motion, are

µ a µ BX = ic @aX ,

Bbab = iTab, (4.14) ca = icb ˆ ca. B rb B In the quantum theory, it is more precise to express this in terms of the BRST current ja as a local operator. In the conformal gaugeg ˆab = ab,workingincomplexcoordinates(z,z¯), the fields can be written as b b , b b , and c cz, c cz¯,withthefollowing ⌘ zz ⌘ z¯z¯ ⌘ ⌘ singular OPE among holomorphic ghost fields e e 1 1 b(z)c(0) ,c(z)b(0) . (4.15) ⇠ z ⇠ z The bc system by itself is a CFT with holomorphic stress-energy tensor

T gh = (@b)c 2b@c. (4.16) As usual, an implicit normal ordering is defined by subtracting o↵singular terms in the coincidence limit, and this form of the ghost stress-energy tensor is unambiguously fixed by the requirement that b is a primary of conformal weight (2, 0) and whereas c is one of weight ( 1, 0). One can indeed verify that the OPE of T gh with itself takes the form of the Virasoro algebra with central charge cgh = 26. Now we will denote the holomorphic and anti-holomorphic components of the BRST current by j j , j j ,andwritetheBRSTchargeas Bz ⌘ B Bz¯ ⌘ B dz dz¯ eQ = j (z) j (¯z). (4.17) B 2⇡i B 2⇡i B I I e 60 Note that the holomorphic and anti-holomorphic BRST currents are independently con- served. The holomorphic BRST current is 3 j = cT X + bc@c + @2c, (4.18) B 2

X 1 µ µ where T = @X @Xµ is the stress-energy tensor of the X CFT, of central charge ↵0 2 B D.Thelasttermproportionalto@ c does not a↵ect QB,andisincludedsothatj can transform as a conformal primary. In fact, jB(z)isaconformalprimaryonlywhenD =26.

QB obeys Q ,b = T = T X + T gh. (4.19) { B } The nilpotency of QB implies that

T (z)T (0) = T (z) Q ,b(0) = Q ,T(z)b(0) (4.20) { B } { B } has only 1 singularity, and thus the total central charge c = D 26 must be zero. We z2 will see later that this is also the condition for the cancelation of Weyl on a curved worldsheet. For computing BRST cohomology, it is useful to work with the oscillator representation of the BRST charge,

X m n QB = cnL n + : cmcnb m n : c0 +(anti holomorphic), (4.21) 2 n Z m,n Z X2 X2 where the normal ordering prescription is defined by moving all the positively graded os- cillators to the right of the negatively graded ones, including the appropriate signs when a pair of Grassmannian fields are exchanged.8 One can also directly verify from (4.21) that Q ,b = LX + Lgh. { B n} n n Let us comment on the space of states in the bc ghost system. The space of local vertex operators are generated by the OPE of b, c, and their derivatives. However, the lowest weight operators are not the identity operator, but c and c@c, both of which has conformal weight 1. In the description, they correspond to two degenerate ground states and |#i ,thatobey |"i b0 = c0 =0, |#i |"i (4.22) c = ,b = . 0|#i |"i 0|"i |#i Indeed, under the state/operator mapping, corresponds to the operator c(0), whereas |#i b 1 is mapped to the identity operator. Combing left and right movers, we will write e.g. |#i , for the state that corresponds to the operator cc(0). |# #i 8Note that this normal ordering on the Fourier modes is a priori unrelated to the normal ordering prescription on the products of local field operators. e

61 The bc system has a U(1) ghost number symmetry that assigns (left-moving) ghost num- ber 1 to c and 1tob. The corresponding Noether current is j = bc. (4.23) gh

While jgh is a conserved current of weight (1, 0), it is not a Virasoro primary. Indeed, the gh OPE of T with jgh takes the form 3 1 1 T gh(z)j (0) + j (0) + @j (0). (4.24) gh ⇠z3 z2 gh z gh This leads to an anomaly in the ghost number symmetry on a curved worldsheet, namely the operator jgh has a nonzero divergence 3 ja = R(g), (4.25) ra gh 4 where R(g)istheRicciscalaroftheworldsheetmetricgab.Consequently,thecorrelation functions of the bc CFT on a genus g Riemann surface must violate ghost number by 3 3g.Thereisasimilarviolationoftheghostnumberintheanti-holomorphicbc CFT. In particular, the sphere correlation function of the identity operator is zero, unlike in a unitary CFT. On the other hand, the following nonzero correlation function9 ee

3 c(z )c(¯z ) = z z z 2 (4.26) i i | 12 13 23| *i=1 + Y leads to the inner product e

, c c , = , , =1. (4.27) hh# #| 0 0|# #i hh# #|" "i e 4.3 Siegel constraint and BRST cohomology

The BRST prescription suggests that the physical single string states are given by the coho- mology of Q on the space of states of the Xµ CFT together with the b ,ca ghost CFT. B H ab However, a careful inspection of the perturbative amplitudes of string states, as we will formulate precisely in section 4.8, shows that certain states in decouple, and a H unitary S-matrix would be obtained if we make a further truncation on the physical states, namely we demand a physical to also obey | i b = b =0. (4.28) 0| i 0| i 9The sign on the RHS of (4.26) is due to our convention in which c(z )c(z )c(z ) = z z z in the e h 1 2 3 i 12 23 13 holomorphic bc CFT whereas the analogous correlator in the bc CFT is given by the complex conjugate expression, and that c and c anti-commute with one another. ee e 62 This condition will have a natural explanation in string field theory (chapter 8), as gauge constraint on the Batalin-Vilkovisky functional of string fields. In this context, (4.28) is known as the Siegel gauge condition. For now, we will assume (4.28), and proceed to analyze the string spectrum. The BRST closure of and (4.28) immediately imply that | i L = L =0. (4.29) 0| i 0| i

Clearly, QB commutes with L0 and L0. Fore the analysis of BRST cohomology, it suces to restrict to the space of states that obey (4.29) to begin with. Acting on such states, QB commutes with c b and c b ,andthusforaBRST-exactstatee Q of this form to obey 0 0 0 0 B| i the constraint (4.28), itself must obey (4.28) as well. Therefore, we can analyze the | i BRST cohomology withine thee space of states that obey both (4.28) and (4.29). H Thus, if we take to have spacetime momentum charge kµ,andtotalleftandright | i b oscillator levels N and N,definedby

e 1 1 µ N = ↵ m↵mµ + b mcm + c mbm , m m=1 X ✓ ◆ (4.30) 1 1 µ N = ↵ m↵mµ + b mcm + c mbm , m m=1 X ✓ ◆ e e e e e e e then (4.29) demands N = N,relatedtokµ by

↵ e 0 k2 + N 1=0. (4.31) 4 The mass of the corresponding string mode in spacetime would be 4 m2 = (N 1). (4.32) ↵0 This formula for the mass spectrum looks like what we found from quantization in the light cone gauge, but we have yet to fully take into account the BRST closure condition as well as identifying states that di↵er by BRST exact states. Acting on the space of states in with spacetime momentum kµ,whichwedenoteby H ,theBRSTchargeQ can be split into a part involving left oscillators only and a part Hk B involving right oscillators only, b b L R QB = QB + QB. (4.33) Clearly, QL and QR anti-commute, and are separately . Likewise can be con- B B Hk structed by acting on the oscillator ground state with left and right oscillators independently,

63 and takes the form = L R. Thus, it suces to analyze the BRST cohomology of Hk Hk ⌦Hk QL,R on L,R independently, and the full BRST cohomology for string states of momentum B Hk kµ is the tensor product of the cohomology of QL and QR on L and R. B B Hk Hk To illustrate this, let us begin with the level 0 example. After imposing (4.28) only the states k, , (4.34) | # #i are allowed at level 0, and BRST closure which implies (4.29) leads to k2 = m2 = 4 . Note ↵0 that there are no BRST exact states at level 0 in . Thus, we find a single spacetime particle H at level 0 that is a . The existence of the tachyon spoils the quantum consistency of critical bosonic closed string perturbation theory.b Nonetheless, the formulation of classical scattering amplitudes as well as the classical string field theory will still be consistent in this setup. Next, let us consider the BRST cohomology at level 1, where kµ is constrained by (4.31) L to be a null momentum. A general state of level 1 in k can have ghost number 1, 0, 1. L H QB raises ghost number by 1, and thus must annihilate the state of ghost number 1, which L b is of the form c 1 k, . On the other hand, QB acts on the ghost number -1 and 0 states as | #i

L ↵0 µ QBb 1 k, = kµ↵ 1 k, , | #i r 2 | #i (4.35) L µ ↵0 µ QB↵ 1 k, = k c 1 k, . | #i r 2 | #i It follows that, assuming k is nonzero, a general BRST closed state at level 1 in L takes Hk the form µ µ b (eµ↵ 1 + c 1) k, ,keµ =0, (4.36) | #i whereas a shift of (4.36) by a BRST exact state identifies

eµ eµ + ⇣kµ, ⇣, ⇠ 8 (4.37) 0. ⇠ L L µ We see that the cohomology of QB on k is represented by states of the form eµ↵ 1 k, H | #i subject to the transversality condition k e =0,andtheidentificatione e + ⇣k .This · µ ⇠ µ µ is the representation content of a single massless vector in spacetime. Combining left and right, we conclude that the BRST cohomology at level 1 on is representation by states H of the form µ ⌫ 2 b eµ⌫↵ 1↵ 1 k, , ,k= k e =0, (4.38) | # #i · subject to the identification e e e + ⇣ k + k ⇣ . (4.39) µ⌫ ⇠ µ⌫ µ ⌫ µ ⌫ 64 e To analyze the representation content of the level 1 states with respect to the spacetime Poincar´egroup, we can work in a frame in which the momentum points in the x1 direction, 1 0 1 so that k (k k )=0,andk =0,i =2, ,D 1=25.Thetransversality ⌘ p2 i ··· condition on eµ⌫ then implies

e+µ = eµ+ =0, (4.40) and the identification (4.39) can be used to set e µ = eµ =0onarepresentativestate of any given level 1 BRST cohomology class. We are then left with the BRST cohomology representatives 25 i j + i eij↵ 1↵ 1 k ,k = k =0, , . (4.41) | # #i i,j=2 X We can further decompose these statese into irreducible representations of the little group SO(24), by taking eij to be a symmetric traceless tensor, an anti-symmetric tensor, or ij. The corresponding multiplets of string states have the interpretation of the , the anti-symmetric 2-form potential B-field, and a massless scalar known as the , in spacetime. A important fact that is far from obvious is that the expectation value of the dilaton field controls the e↵ective string coupling, as will be explained in section 7.1.

4.4 Equivalence to light cone gauge

One can carry out the above analysis of BRST cohomology to higher levels, corresponding to massive string states, but this is a rather tedious and cumbersome. As already anticipated from the light cone gauge, there should be a simple characterization of the entire BRST cohomology. To see this, let us define the left-moving “light cone oscillator number”

1 lc 1 + + N ↵ m↵m ↵m↵m , (4.42) ⌘ m m=1 X where ↵ are the oscillators associated with X 1 (X0 X1), obeying [↵+ ,↵ ]= m± ± p2 m n lc ⌘ ± mm, n. N counts the number of ↵ excitations (rather than level) minus the number of ↵+ excitations. Without loss of generality, suppose the light cone momentum component + L k is nonzero. The left part of the BRST charge QB can be decomposed according to the grading by N lc quantum number 1, 0, or 1, L QB = Q1 + Q0 + Q 1. (4.43) L The nilpotency of QB implies that Q1 is nilpotent as well. Q1 admits a very simple expression,

↵0 + Q1 = k ↵mcm. (4.44) 2 r m=0 X6 65 Let us define a sort of “conjugate” operator to Q1,

2 1 + R 1 = ↵ mbm, (4.45) ↵ k+ r 0 m=0 X6 which carries N lc quantum number 1, as indicated by its subscript. Further, define

S0 = Q1,R 1 , { } U 1 = Q0,R 1 , (4.46) { } V 2 = Q 1,R 1 . { }

The operator S0 counts the total light cone plus ghost oscillator level,

1 + + S0 = ↵ m↵m ↵m↵m + mb mcm + mc mbm . (4.47) m=1 X We claim that

(1) The cohomology of Q1 is isomorphic to the kernel of S0.

(2) The kernel of S0 is isomorphic to the kernel of S0 + U 1 + V 2. L (3) The cohomology of QB is isomorphic to the kernel of S0 + U 1 + V 2. To prove claim (1), first note that Q commutes with S .Supposeastate is in the 1 0 | i kernel of S ,thenithasnoghostexcitation.DuetoSiegelconstraintb =0, must 0 0| i | i be proportional to in the ghost sector, which has ghost number 1. It follows that Q |#i 1| i has ghost number 2, and is also annihilated by S0,aswellasbyb0.Buttherearenosuch states, thus Q =0.ThisgivesamapfromthekernelofS to the cohomology of Q . 1| i 0 1 To go the other way, suppose Q =0.Wecandecompose into eigenstates of S , 1| i | i 0 and without loss of generality, assume that is an eigenstate of S with eigenvalue s.If | i 0 s =0, would be in the kernel of S .Ifs =0,wecanwrite | i 0 6 1 1 1 = S0 = Q1,R 1 = Q1R 1 , (4.48) | i s | i s{ }| i s | i and is therefore Q -exact. This shows that the map Ker(S ) Coh(Q )isanisomorphism. 1 0 ! 1 The isomorphism of claim (2) can be constructed explicitly as

Ker(S0) Ker(S0 + U 1 + V 2) ! 1 1 n 1 n (4.49) 1 ( ) (S0 (U 1 + V 2)) | i 7! | i⌘ | i 1+S0 (U 1 + V 2) n=0 X This is well defined because S0 is invertible when acting on the image of U 1 + V 2. 66 L To prove (3), first let us establish that the kernel of S0 +U 1 +V 2 is QB-closed. Suppose is annihilated by S0 +U 1 +V 2.SinceS0 +U 1 +V 2 commutes with the ghost number, | i without loss of generality, we can assume that has a definite ghost number. We can | i decompose according to the N lc quantum number, | i = . | i | ni (4.50) n n0 X The maximal N lc component must be annihilated by S , which implies that n =0, | n0 i 0 0 and the Siegel constraint implies that 0 ,therefore ,hasghostnumber1.Itfollowsthat L | i | i lc QB has ghost number 2, and is annihilated by S0+U 1+V 2.ThehigherN component of L | i QB must then be annihilate by S0,aswellasbyb0,buttherearenosuchstateswithghost | i L L number 2. Thus QB =0,andonceagainwehaveamapKer(S0 +U 1 +V 2) Coh(QB). | i ! Now suppose QL = 0. With the decomposition (4.50), we have Q =0,andso B| i 1| n0 i we can write = + Q , (4.51) | n0 i | i 1| i where is annihilated by S and has the same N lc eigenvalue n as ,and has N lc | i 0 0 | n0 i | i eigenvalue n 1. The state 0 1 0 1 (4.52) | i⌘1+S0 (U 1 + V 2)| i L is annihilated by S0 + U 1 + V 2 and thus by QB.Thedi↵erence L 0 Q | i| i B| i 1 n 1 1 n (4.53) = ( ) (S0 (U 1 + V 2)) (Q0 + Q 1) + n | i | i | i n=1 n n0 1 X X L lc is QB-closed and has maximal N quantum number n0 1. In other words, is equal L L | i to 0 Ker(S0 + U 1 + V 2), up to a QB-exact state, and up to a QB-closed remainder | i2 (4.53) that has lower N lc quantum number. Repeating the same procedure to the remainder L recursively, we see that can be put in Ker(S0 + U 1 + V 2)uptoaQB-exact state, thus | i completing the proof. L It follow from claims (2) and (3) that the cohomology classes of QB are in 1-1 corre- i spondence with states created by acting transverse oscillators ↵ m, m 1, on k, ,which | #i are precisely the states we obtained in light cone quantization. The above isomorphism Ker(S ) Coh(QL ) also gives an explicit construction of a general BRST cohomology 0 ! B representative from the state space of the transverse oscillator excitations. A corollary of the above discussion is that, the norm on the BRST cohomology (subject to Siegel constraint) defined by 2 = c c is positive definite. || || hh | 0 0| i e67 4.5 Equivalence to old covariant quantization

Every weight (1, 1) matter CFT primary V corresponds to a BRST closed operator ccV .In this section we will show that every BRST cohomology class (here always assumed to obey Siegel constraint) admits a representative of this form. In fact, we will prove a strongere result that goes under the name of “old covariant quantization” (OCQ), as follows. In the matter CFT, a primary state V is orthogonal to any descendant of the form X X | i = n,m 1 L nL m n,m .Adescendant can also be a primary itself, in which case it | i | i | i must be null, namely = 0. Such null states are abundant in the matter CFT because P h | i the latter is not quitee unitary, due to the timelike boson X0.Forinstance,takethematter µ oscillator ground state k for a null spacetime momentum k ,thenL 1L 1 k is a null | i | i primary state of weight (1, 1). e The OCQ Hilbert space is defined to be the space of weight (1, 1) primaries modulo HOCQ those that are also descendants. We claim that is in fact isomorphic to the BRST HOCQ cohomology, via the map V ccV = V , . (4.54) | i 7! | i | i⌦|# #i To show this, it suces to focus on the holomorphic sector, and consider the map e V cV = V (4.55) | i 7! | i | i⌦|#i from L , the space of weight 1 primaries modulo descendants with respect to the holo- HOCQ morphic , to the cohomology of the holomorphic part of BRST operaor, L QB. First, suppose the matter primary V is a null descendant .Thestate is L | i L | i | i⌦|#i QB-closed and thus is orthogonal to every QB-exact state. It is also null, and by the positive L definiteness of the inner product on QB-cohomology defined at the end of the previous section, we conclude that must be QL -exact. Thus, the map (4.55) is well defined | i⌦|#i B on L . HOCQ Let us define the operators

Nb b ncn,Nc c nbn, ⌘ ⌘ (4.56) n 1 n 1 X X that count the number of b and c oscillators in a state, respectively. If V is QL -exact, | i⌦|#i B namely V = QL for some , must be of the form | i⌦|#i B| i | i | i

= n b n + , | i | i⌦ |#i ··· (4.57) n 1 X

68 where are a set of matter CFT states, and involves states with N 2, N 1. It | ni ··· b c follows that L X QB = L n n + , | i | i⌦|#i ··· (4.58) n 1 X where on the RHS involves states with Nb 1, Nc 1, but by assumption such terms ··· L X L must be absent in QB . Thus we learn that V = n 1 L n n ,andistrivialin OCQ. | i | i | i H This proves that the map (4.55) is injective. P Now suppose we have a state that is QL -closed, obeys the Siegel constraint b =0, | i B 0| i and is orthogonal to all states of the form V where V is a weight 1 primary of the | i⌦|#i matter CFT. We can write = + 0 , (4.59) | i | i⌦|#i | i where 0 is a linear combination of states with N + N 1, and is a matter CFT | i b c | i state of weight 1 that is orthogonal to all matter primaries. We have already seen that L the latter condition implies that is QB-exact. It remains to show that 0 is also L | i⌦|#i | i QB-exact. We can recycle the same argument as in the previous section from (4.51) to (4.53) L to express 0 as the sum of a Q -exact state and another state 00 that lies in the kernel | i B | i of S0 + U 1 + V 2 and still obeys Nb + Nc 1. But S0 + U 1 + V 2 is invertible on the space of states that obey N + N 1, and thus we conclude that 00 =0.Thisprovesthatthe b c | i map (4.55) is surjective, hence the equivalence of with the BRST cohomology. HOCQ Note that in making the above argument, we only need to assume that the matter CFT contains a pair free boson X± corresponding to null directions in spacetime, and that the + physical state has nonzero charge k with respect to the translation symmetry in X.This allows for generalization of the equivalence between the BRST cohomology and to HOCQ strings propagating in more general backgrounds, such as AdS3 spacetime.

4.6 DDF operators

The explicit map between the Hilbert space of string states in lightcone gauge (namely

KerS0 of section 4.4) and the BRST cohomology representatives in the OCQ form (the RHS of (4.55)) can be constructed using the Del Giudice-Di Vecchia-Fubini (DDF) operators, defined as dz 2 2in + i i + XL (z) A = @X (z)e ↵0k . (4.60) n 2⇡ ↵ I r 0 + + i Here XL (z)istheholomorphicpartofX (z,z¯), and Aq is understood to act on an vertex + i operator at the origin, that carries spacetime momentum (k ,k,k )inthelightconeframe. 2in + + XL (z) i The operator e ↵0k is single-valued for integer n.Inthiscase,An is the conserved

69 charge associated with a weight (1, 0) current and commutes with the Virasoro algebra. i Furthermore, An obey the commutation relation

i j ij [An,Am]= nn, m, (4.61) i i which takes the same form as that of of free boson oscillators ↵n. However, unlike ↵n which i lowers the conformal weight by n, An takes a Virasoro primary to another Virasoro primary of the same weight.

i1 j1 Given a string state ↵ n1 ↵ m1 0; k in the lightcone gauge, we can produce a ··· ···| i BRST cohomology representative (up to normalization) e i1 j1 ik0 X (4.62) ccA n1 A m1 e · , ··· ··· where k is related to k by 0 e e

2 + + i i 0 0 0 (4.63) k = k + N, k = k ,k= k . ↵0k Here N = n + = m + is the total oscillator level of the original lightcone gauge state, 1 ··· 1 ··· either on the left or on the right (they are the same due to the level matching condition). 2 4 ik X Indeed, when k is on-shell, namely k = N, e 0· is a weight (1, 1) primary and (4.62) ↵0 is a BRST closed vertex operator of the OCQ form.

4.7 Riemann surfaces

We have so far assumed that the worldsheet di↵eomorphism and Weyl gauge symmetry can be used to fix the dynamical metric gab to an arbitrary fiducial metricg ˆab.Thisingeneral is only possible locally on the worldsheet. Globally, there may be obstructions, in that it is k only possible to fix gab to a finitely-many-parameter family of fiducial metricsg ˆab(t ), where k k t are a set of “moduli parameters”. Further, after fixing gab =ˆgab(t ), there may be residual unfixed gauge redundancies, that generate the “conformal Killing group”. Infinitesimally, the conformal Killing group is generated by di↵eomorphism parameters va and Weyl parameters !, such that gˆ = ˆ v ˆ v +2!gˆ =0. (4.64) ab ra b rb a ab Evidently, ! is then determined by va.Avectorfieldva that obeys (4.64) is called a conformal Killing vector (field). Since we will be formulating the string scattering amplitudes based on the Euclidean path integral, we shall restrict our attention to compact, Euclidean . As already seen in section 3.10, the of inequivalent fiducial metrics can be identified with the moduli space of Riemann surfaces. A Riemann surface ⌃is covered by holomorphic

70 coordinate charts Ui parameterized by zi,withholomorphictransitionfunctionszi = fij(zj) on their overlaps. The moduli parameters tk can be understood as parameters of the tran- sition maps fij modulo the equivalence relation due to holomorphic coordinate redefinitions of each zi.

4.7.1 The sphere

We begin with the genus g = 0 case, in which case the Riemann surface is topologically a sphere. The Riemann sphere has no moduli, and every metric on the sphere can be put to the form of the Euclidean metric ds2 = dzdz¯ (4.65) by a Weyl transformation. More precisely, the z coordinate chart U covers the sphere minus one point. There is another coordinate chart V ,parameterizedbythecomplexcoordinate w,suchthatontheoverlapU V ,thetwocoordinatesarerelatedbyw =1/z. \ The conformal Killing vector is simply a linear combination of a (globally defined) holo- morphic vector field and an anti-holomorphic vector field,

z z¯ v (z)@z + v (¯z)@z¯. (4.66)

w On a di↵erent coordinate chart parameterized by w,thevectorfieldisexpressedasv (w)@w + w¯ w z e v (¯w)@w¯,wherev (w)isrelatedtov (z)ontheoverlapofthecoordinatechartsby @w vw(w)=vz(z) . (4.67) e @z z On the Riemann sphere, a holomorphic vector field of the form v (z)@z on the chart U would be written as w2vz(1/w)@ (4.68) w on the chart V . vz(z)shouldbeholomorphicontheentirecomplexz-plane, as is w2vz(w) on the entire complex w-plane. This restricts vz(z)tobeaquadraticpolynomial,

z 2 v (z)=a0 + a1z + a2z . (4.69)

These CKVs generate a finite conformal Killing group transformation of the form ↵z + z z0 = , (4.70) 7! z + where ↵,,, can be taken to be complex numbers with ↵ = 1. Further, flipping the signs of ↵,,, simultaneously does not change the transformation (4.70), and preserves the condition ↵ =1.Therefore,theconformalKillinggroupoftheRiemannsphere is isomorphic to PSL(2, C).

71 4.7.2 The

Next consider the genus g = 1 case, where the Riemann surface is topologically a torus. Up to a Weyl transformation, the metric on the torus can always be put to the Euclidean form ds2 = dzdz¯,butwiththecoordinatez subject to the periodic identification

z z +2⇡ z +2⇡⌧, (4.71) ⇠ ⇠ where ⌧ = ⌧1 + i⌧2 for ⌧1,⌧2 R, ⌧2 > 0. We could equivalently describe the torus via coor- 2 dinate charts, but that would not be necessary. An alternative (1,2), where 1 and 2 are of periodicity 2⇡,isrelatedby

z = 1 + ⌧2, z¯ = 1 +¯⌧2, (4.72) so that the metric can be written as

ds2 = d1 + ⌧d2 2. (4.73) | | ⌧ is a complex parameter of the moduli space of the torus as a Riemann surface. M1 The only di↵eomorphism that preserves the Euclidean metric up to a Weyl transformation with an identification of the form (4.71) for some ⌧ is a linear holomorphic map, of the form z z z0 = , (4.74) 7! c⌧ + d where c and d are a pair of coprime integers. Note that z0 is identified with z0 +2⇡,andis also identified with z0 +2⇡⌧0,for a⌧ + b ⌧ 0 = , (4.75) c⌧ + d for another pair of integers a, b such that ad bc =1.Indeed,thetoruswithmodulus⌧ and ⌧ 0 are equivalent under a “large” di↵eomorphism and Weyl transformation. In terms of the a 1 2 1 2 coordinates, it can be expressed as a map ( , ) (0 ,0 ), with 7! 1 1 0 a b 2 = 2 , (4.76) 0 cd ✓ ◆ ✓ ◆✓ ◆ which is a di↵eomorphism for integers a, b, c, d that obey ad bc = 1. The metric (4.73) can a be written in 0 coordinates as

2 2 1 2 2 ds = c⌧ + d d0 + ⌧ 0d0 . (4.77) | | | | Thus, we learn that the modulus ⌧ should be identified up to PSL(2, Z)transformationsof the form (4.75). The moduli space of the torus is then

1 ⌧ C Im⌧>0 /P SL(2, Z). (4.78) M '{ 2 | } 72 can be parameterized by ⌧ in the fundamental domain M1 1 1 : ⌧ , ⌧ 1, (4.79) F 2  1  2 | | with the boundary @ identified using either ⌧ ⌧ +1or⌧ 1/⌧.Theresultingspace F ! ! may be viewed topologically as a sphere with a puncture at ⌧ = i ,correspondingto M1 1 a singular limit of the torus modulus, and two special points ⌧ = i and ⌧ = e2⇡i/3 where the torus develops a Z4 and Z6 discrete rotation symmetry respectively. The conformal Killing vector fields on the torus are easy to describe: they are simply 2 constant vector fields. The conformal Killing group of the torus with generic ⌧ is T oZ2, 2 where the T is the group of translations along the torus, and the Z2 is the orientation- preserving reflection symmetry z z. 7!

4.7.3 g 2 AgeneralRiemannsurfaceofarbitrarygenusmaybeconstructedbygluingtogetherRie- mann surfaces of lower genera using the “plumbing fixture”. Given two Riemann surfaces ⌃ and ⌃ ,ofgenusg and g respectively, we can choose a coordinate chart U ⌃ param- 1 2 1 2 ⇢ 1 eterized by z on the unit disc, and another coordinate chart V ⌃ parameterized by w on ⇢ 2 the unit disc, and identify a pair of annulus regions on the two discs using the map w = q/z, (4.80) for a nonzero complex parameter q with q 1. The identification produces a new Riemann | | surface ⌃of genus g = g1 + . Similarly, one may start with a single Riemann surface ⌃of genus g, pick two coordinate charts U and V ,andcutoutaholeoneachandidentifythemalongapairofannulusregions using the map (4.80), thereby producing a new Riemann surface ⌃0 of genus g0 = g +1. We can give a rough count of the dimension µ of the moduli space of the genus g g Mg Riemann surface based on the plumbing construction. We can choose the center of the pair of discs U and V on ⌃1 and ⌃2,andthecomplexparameterq,givingatotalof3complex parameters, or 6 real parameters. Thus, we expect µg1+g2 = µg1 + µg2 + 6, which leads to µ =6g 6. This answer is obviously incorrect for g =0or1,butisinfactthecorrect g number of real moduli for g 2. Amoreexplicitdescriptionofthegenusg Riemann surface, obtained by plumbing to- gether Riemann spheres, is as follows. We begin with the Riemann sphere C ,and [{1} consider a set of g loxodromic PSL(2, C)elements1, ,g.Eachi acts on the complex ··· coordinate z by aiz + bi z i(z)= , (4.81) 7! ciz + di

73 ai bi with aidi bici =1,suchthattheSL(2, C) matrix has a pair of distinct eigenvalues ci di 1 ✓ ◆ q ,q with q < 1. Generically, there are no group relations among , , .Together i i | i| 1 ··· g they generate a free group , known as the Schottky group, that acts on the Riemann sphere. The set ⇤of accumulation points of on the Riemann sphere is a certain Cantor set of measure zero. The quotient

⌃=(C ⇤)/ (4.82) [{1} is a Riemann surface of genus g.TheSchottkygroupinvolves3g complex parameters, but an overall conjugation by PSL(2, C), the conformal Killing group of the Riemann sphere, produces an equivalent Schottky . Thus the moduli space of ⌃is parameterized by 3g 3independentcomplexparameters.Infact,eachpointofthegenusg Riemann surface moduli space can be realized by a Schottky group through (4.82). However, Mg each Schottky group not only determines a Riemann surface, but also specifies a particular pair-of-pants decomposition of that Riemann surface. A given Riemann surface admits many di↵erent pair-of-pants decompositions, leading to discrete identifications of the Schottky parameters. One must also be careful with the boundary of the moduli space where the Riemann surface degenerates. As a result, the full geometry of the moduli space is Mg generally rather complicated for g 2. Another useful characterization of the moduli of a genus g Riemann surface ⌃is through I the period matrix. Pick a basis of 1-cycles ↵ , J , such that their intersection numbers are

↵I ↵J = =0,↵I = ↵I = I . (4.83) · I · J · J J · J

There is a basis of holomorphic 1-forms !I on ⌃that obey

I !J = J . (4.84) I Z↵ The period matrix ⌦is a g g complex matrix whose entries are ⇥

⌦IJ = !J . (4.85) ZI J J The cohomology class of !I can be written as [!I ]=ˇ↵I +⌦IJˇ , where (ˇ↵I , ˇ )arethe basis of integral cohomology classes dual to (↵I , ). It follows from ! ! =0that⌦ I ⌃ I ^ J IJ is symmetric, and from the positive definiteness of i ! ! that Im(⌦ )isapositive ⌃ I ^ J R IJ definite matrix. R I ⌦is defined up to the ambiguity of a symplectic change of basis of the 1-cycles (↵ ,J ). Namely, with respect to the basis

I I J IJ J J ↵0 = D J ↵ + C J ,I0 = BIJ↵ + AI J , (4.86)

74 where A, B, C, D are matrices with integer entries that obey

DAT CBT = I,BAT = ABT ,DCT = CDT , (4.87) parameterizing an element of Sp(2g, Z), the new period matrix is

1 ⌦0 =(A⌦+B)(C⌦+D) . (4.88)

g(g+1) Note that ⌦has a priori 2 independent complex entries, whose number coincides with the dimension of the moduli space for g 3, but exceeds dim( )forg 4. In the Mg  Mg latter case, there are nontrivial constraints among ⌦IJ. An explicit formula that expresses the period matrix in terms of the Schottky parameters is as follows. The PSL(2, C)transformation(4.81)canbeexpressedas (z) ⌘ z ⌘ z (z), i i = q i . (4.89) 7! i (z) ⇠ i z ⇠ i i i The components of the period matrix ⌦IJ in a corresponding symplectic basis is then given by ⌘ ⌘ ⇠ ⇠ 2⇡i⌦IJ IJ ( ( I ) J )( ( I ) J ) e = qI , (4.90) ((⌘I ) ⇠J )((⇠I ) ⌘J ) Y2 IJ where IJ is defined to be the subset of the Schottky group = 1, ,g that consists 1 h ··· 1i of group elements that do not end with I± on the right of its word, nor J± on the left of its word. In the case I = J,theidentitygroupelementisalsoexcludedfromIJ (but is included when I = J). 6

4.8 BRST formalism for string scattering

4.8.1 Gauge fixing the worldsheet path integral

We now give a heuristic derivation of the genus g, n-point bosonic string scattering amplitude by carefully gauge fixing the di↵eomorphism and Weyl symmetry in the worldsheet path µ integral based on the SP[gab,X ] defined on a genus g surface ⌃, with the asymptotic string states represented by vertex operators V , ,V ,startingfromthenaive 1 ··· n expression n µ SP 2 [DgabDX ]e d i g(i) Vi(i). (4.91) i=1 Z Y Z p Vi should transform as a scalar field under di↵eomorphism, and of weight 2 under Weyl trans- k formation. We will fix gab to a fiducial metric that depends on moduli t that parameterize , Mg gab =ˆgab(t), (4.92)

75 and fix the conformal Killing group by choosing

a =ˆa, (i, a) f, (4.93) i i 2 for an index set f whose order is equal to the dimension of the CKG. Since the order of f cannot exceed 2n,theformalismmakessenseonlyifthedimensionoftheCKGisnomore than 2n.Thisissatisfiedforg =0,n 3, for g =1,n 1, and for any g 2, which suces for formulating scattering amplitudes. Under an infinitesimal di↵eomorphism and Weyl transformation as in (2.19), as well as variation of the moduli tk,thegaugefixingconditions(4.92)and(4.93)varyby @gˆ (t) (g gˆ (t)) = v v +2!g tk ab , ab ab ra b rb a ab @tk (4.94) (a ˆa)=va( ). i i i In applying the Faddeev-Popov ansatz, we can treat the moduli tk on equal footing as the gauge parameters, provided that we integrate tk over the moduli space in the end. Thus, Mg we can make the following replacement of the path integral measure,

[Dg DXµ] da dtk da [ˆg , ˆa], ab i 7! i FP ab i (4.95) (i,a) g (i,a) f Z Y ZM Y62 where the FP determinant FP is given by

a FP[ˆgab, ˆi ]

a k a 1 2 ab k a a = [Db Dc ]d⇠ d⌘ exp S [ˆg, b, c] d gbˆ @ k gˆ (t)⇠ ⌘ c (ˆ ) . ab i 8 gh 4⇡ t ab i i 9 Z Z (i,a) f < p X2 = (4.96) k a : k ; Here ⇠ and ⌘i are Grassmann parameters; ⇠ plays the role of c ghost for the moduli k a variation t ,and⌘i plays the role of b ghost for the gauge fixing condition (4.93). The k a Grassmannian integration over ⇠ and ⌘i can be performed easily, giving 1 a a Sgh[ˆg,b,c] 2 ab a [ˆg , ˆ ]= [Db Dc ]e d gbˆ @ k gˆ c (ˆ ). FP ab i ab 4⇡ t ab i (4.97) Z k ✓ Z ◆ (i,a) f Y p Y2 We can include in the Polyakov action a topological term 0 d2pgR(g)= , (4.98) 4⇡ 0 Z where is the Euler characteristic of the worldsheet, which amounts to multiplying (4.91) 0 0 2g 2 by the factor e . Defining gs = e ,wemayalsowritethisfactorasgs ,andinterpret gs as the string that plays the role of the genus expansion parameter.

76 Putting these together, we arrive at a prescription for the genus g, n-point string scat- tering amplitude,

[V , ,V ] Ag 1 ··· n n 2g 2 1 2 ab k a a = gs d gbˆ @tk gˆabdt c (ˆi) di gˆ(i)Vi(i) , g 4⇡ ZM * k ✓ Z ◆ (i,a) f (i,a) f i=1 + Y p Y2 Y62 Y p ⌃,gˆ(t) (4.99) where represents a correlation function of vertex operators in the (Xµ,b ,ca) h···i⌃,gˆ(t) ab CFT on the surface ⌃equipped with the fiducial metricg ˆab(t). In the product over n vertex operators, it is understood that a =ˆa for (i, a) f. i i 2

4.8.2 The ghost correlator

Let us first examine the correlation function of the bc ghost CFT. As already shown in section 4.2, on a genus g Riemann surface ⌃, due to the ghost number anomaly, a correlation function in the ghost CFT can be nonvanishing only if it violates the total (holomorphic and anti- holomorphic) ghost number by 6g 6. This is indeed the case for the correlator appearing on the RHS of (4.99), as the number of b insertions is equal to dim( ), while the number of Mg c insertions is equal to dim(CKG); as we have seen in section 4.7, dim( ) dim(CKG) = Mg 6g 6holdsforeverygenusg. Since the bc CFT is free, we can explicitly evaluate its correlation functions through a Gaussian functional integral. Let us define the di↵erential operator that takes a vector P field on ⌃to a symmetric traceless tensor, 1 ( c) c + c g cd . (4.100) P ab ⌘ 2 ra b rb a abrd With respect to the pairing of tensor fields defined by

2 ab (f,f0) d pgf f 0 , (4.101) ⌘ ab Z the transpose of acts on a symmetric traceless tensor according to P ( T b)a = bb . (4.102) P r ab The bc ghost action can be written as 1 1 S = (b, c)= ( T b, c). (4.103) gh 2⇡ P 2⇡ P T is also the same as the adjoint of with respect to a Hermitian inner product defined P P analogously to (4.101) but with complex conjugation on one of the tensor fields. In particular,

77 T and T are positive semidefinite Hermitian operators. We can find an orthonormal P P PP basis v for vector fields on ⌃that diagonalizes T , n P P T v = v , (4.104) P P n n n (0) where the eigenvalues n are non-negative. The zero eigen-vector-fields will be denoted vj . If n is nonzero, we can define 1 u =( ) 2 v . (4.105) n n P n un are a set of orthonormal symmetric traceless tensor fields that obey

T 1 T u = u ,v=( ) 2 u . (4.106) PP n n n n n P n

We can complete un into a basis of symmetric traceless tensor fields including also the zero modes u(0). Note that while the nonzero eigenvectors of T and T come in pairs, the k P P PP zero eigenvectors do not. Expanding

a a bab()= bn(un)ab(),c()= cn(vn) (), (4.107) n n X X we can write the functional integral of the bc system as

S 1 [DbDc]e gh = db dc exp b c (4.108) ··· n m 2⇡ n n n ··· n m n ! Z Z Y Y X p a where represents general operator insertions. The zero modes of bab and c ,i.e.the ··· (0) (0) coecients of uk and vj in the expansion (4.107), do not appear in the ghost action Sgh.Thus,thefunctionalintegral(4.108)vanishesunlesstheoperatorinsertion includes a ··· precisely the zero modes of bab and c . (0) The condition vj = 0 takes the identical form to the conformal Killing equation (4.64), (0)) P and so vj is a basis of conformal Killing vectors. They are in 1-1 correspondence with the a (0) zero modes of c . uk ,ontheotherhand,representmetricdeformationsgab that are orthogonal to all di↵eomorphism and Weyl variations (2.19). Hence, the zero modes of bab are in correspondence with deformations of the moduli, tk. Now we see in that the b and c insertions in the correlation function appearing on the RHS of (4.99) precisely match the number of zero modes. It is conventional to define the Beltrami di↵erential 1 (µ ) @ k gˆ (t), (4.109) k ab ⌘ 2 t ab so that we can write 1 2 ab 1 d gbˆ @ k gˆ (b, µ ). (4.110) 4⇡ t ab ⌘ 2⇡ k Z p 78 The bc CFT correlator appearing in (4.99) can be evaluated as

1 T 2 (0) 1 a (uk ,µ`) a (b, µk) c (ˆi) = det 0 P P det det vj (ˆi), (4.111) 2⇡ 2⇡ 2⇡ j,(i,a) f * k (i,a) f +  2 Y Y2 ⌃ where det0 is the taken over nonzero modes of . P

4.8.3 BRST invariance

The prescription for the string amplitude based on gauge fixing the worldsheet path integral is only expected to be consistent when the integrated vertex operators

2 d g() Vi()(4.112) Z p are BRST invariant. In other words, Vi()shouldbeBRSTinvariantuptoatotalderivative. This indeed holds provided that Vi is a matter CFT primary with conformal weight (1, 1), in which case dz X dz¯ X QB Vi(0) = c(z)T (z) c(¯z)T (¯z) Vi(0) · 2⇡i 2⇡i (4.113) I  = @(cV )(0) + @¯(cV )(0) = @ (caV )(0). i i ea e i a If we choose to fix the CKG by fixing i for some i,thenthecorrespondingstringstate e appears in (4.99) through the “fixed vertex operator” ccVi,whichisBRSTclosed.Recall that, as we have already seen in section 4.5, every BRST cohomology class admits a repre- sentative of the form ccV ,whereV is a matter CFT primarye of weight (1, 1). Thus, (4.99) can capture the scattering amplitude of all string states, represented through matter CFT vertex operators Vi. e The b ghost insertions on the RHS of (4.99) are, however, not BRST closed. Indeed, their BRST transformations can be computed as

2 ab Q (b, µ )=(T,µ ) d gTˆ @ k gˆ (t). (4.114) B · k k ⌘ t ab Z p Inserting this into a correlator on ⌃, we have

2 ab QB (b, µk) = d gˆ @tk gˆab(t) T = 4⇡@tk . (4.115) · ··· ⌃,gˆ(t) ··· ⌃,gˆ(t) ··· ⌃,gˆ(t) D E Z p D E D E We see that while the correlator appearing on the RHS of (4.99) is not quite BRST invariant in the case of genus g 1, its BRST variation is a total derivative on the moduli space . Mg Upon integration over ,withthemeasure dtk,(4.99)isBRSTinvariantuptopossible Mg boundary terms. Q 79 In fact, we should also have worried about boundary terms in integrating correlators of

(4.113). After all, the coordinates of integrated vertex operators Vi can be treated on equal footing as the moduli parameters tk,andtogethertheycanbeviewedasmodulispace Mg,n of a genus g Riemann surface with n (ordered) punctures. What we have seen so far is that the genus g, n-point string amplitude is BRST invariant up to a boundary integral over @ . Mg,n

4.8.4 Boundary of the moduli space

There are two kinds of boundary components of @ :(1)theRiemannsurface⌃ pinches Mg,n g,n into a pair of surfaces ⌃g1,n1 and ⌃g2,n2 ,withg1 + g2 = g, n1 + n2 = n +2,joinedatapair of punctures, and (2) A handle pinches o↵, so that ⌃g,n degenerates into a surface ⌃g0 1,n+2 with a pair of punctures joined together.

Aspecialcaseof(1)iswheng1 =0,n1 =3.Thisisconformallyequivalenttoapair of punctures on ⌃colliding, say those that correspond to vertex operators V1 and V2.Bya Weyl transformation, we can assume that gab takes the Euclidean form ab on a coordinate patch that contains both V1 and V2.Wecanthenpasstocomplexcoordinateonthispatch, and represent the positions of V1 and V2 asbz1 and z2 respectively, and write the OPE

hn 2 hn 2 V (z , z¯ )V (z , z¯ )= z z¯ (z , z¯ ), 1 1 1 2 2 2 12 12 On 2 2 (4.116) n X e where is a basis of matter CFT operators of weight (h , h ). A priori, if there are operators On n n appearing on the RHS of (4.116) with h = h 1, the integral over z diverges near On n n  1 z1 = z2.Ifwecuto↵thez1 integral by excluding a smalle disc of radius ✏ centered at z2,we 2(hn 1) encounter a power divergence of order ✏ for ehn = hn < 1, and a logarithmic divergence of order log(1/✏)ifhn = hn =1.Thepowerdivergencecanbecanceledbyalocalcounter term and is unphysical, whereas the log divergence generallye has physical interpretations. e µ Suppose Vi represents a string state of spacetime momentum ki ,thatobeysmass-shell 2 2 2 4 condition ki = mi ,wheremi is times an integer, then the OPE of free boson CFT is ↵0 such that carries spacetime momentum charge kµ + kµ.Theconformalweightsh , h On 1 2 n n of are ↵0 (k + k )2 shifted by a non-negative integer. Generally, the connected part of a On 4 1 2 scattering amplitude in Minkowskian spacetime is expected to be an analytic function in thee µ momenta ki ,subjecttomass-shellconditionandmomentumconversation,awayfrompoles that correspond to intermediate on-shell particles or resonances10 and away from branch cuts that have to do with particle production. In the present situation, we can start with

10Resonance poles are generally expected to occur on the “second sheet” of complex s = (k + k )2 12 1 2 plane, after analytic continuing past the branch cut due to particle production.

80 on-shell momenta k1 and k2 that obey momentum conservation in a complex domain where ↵0 2 Re 4 (k1 + k2) is suciently large and positive, so that the integral of (4.116) converges, and analytically continue to generic values of k1,k2.Thisgivesthesameresultasthe ⇥ ⇤ 11 prescription of throwing away power divergences for generic k1,k2.

Near the momenta values where a log divergence in the z1-integral occurs, the divergent term takes the form of a pole in momenta, 1 2 2 (4.119) (k1 + k2) + M

2 4 for some M = (N 1), N Z 0.Thisisexpectedofascatteringamplitude,whenan ↵0 intermediate particle of mass M becomes on-shell. At the moment, it is not evident that all divergences of this type are accounted for by intermediate on-shell particles, since most of the operators appearing in the OPE (4.116) do not correspond to any BRST cohomology On class or on-shell string states. Nonetheless, we will show in section 5 that such divergences are precisely what is required of the unitary of the S-matrix.

Asimilarargumentappliestothesituationwhere⌃g,n pinches into ⌃g1,n1 and ⌃g2,n2 . Up to a conformal transformation, the pinching limit amounts to having a long cylinder of circumference 2⇡ and length 2⇡t connecting ⌃g1,n1 and ⌃g2,n2 ,witht .States µ n1 1 µ !1 n µ propagating through the cylinder has spacetime momentum P = k = k , i=1 i i=n1 i where k1, ,kn1 1 are the momenta of vertex operators on ⌃g1,n1 ,andkn1 , ,kn are the ··· P ··· P momenta of vertex operators on ⌃g2,n2 .Thecorrelationfunctionofthevertexoperatorsis suppressed by the propagator along the cylinder,

↵ exp 2⇡t(L + L ) =exp 2⇡t 0 P 2 + N + N 2 , (4.120) 0 0 2 h i  ✓ ◆ e e where N and N are the holomorphic and anti-holomorphic oscillator levels. Provided that n ,n 3, we can always define the amplitude by analytic continuation from a domain 1 2 in which Re(Pe2)issucientlylargeandpositive,wherethemoduliintegraloverthelarge t region converges. When the momenta are arranged such that P 2 approaches M 2 for 11As a toy example, consider the integral

2 x 2 2⇡ d z z = , Re(x) > 0. (4.117) z <1 | | x Z| | Its analytic continuation to Re(x) < 0is

2⇡ 2 x 2 2⇡ x =lim d z z ✏ , (4.118) x ✏ 0 ✏< z <1 | | x ! "Z | | # which is equivalent to throwing away power divergences.

81 some mass value M of a string state, the analytic continuation results in a pole of the form 2 2 1 (P + M ) ,aswillbeanalyzedinmoredetailinsection5.

Two exceptional case are (a) g 1, n1 =2,and(b)g 1, n1 =1.Inthecase(a), µ µ the momentum P = k1 flowing through the cylinder is always on-shell, and potentially leads to a divergence in the t-integral. Such a divergence, if present, should be canceled by a mass of the string state in question. This will be addressed in section 8.1. In the case (b), the momentum P µ is always zero; a potential divergence that results from the t-integral amounts to a tadpole diagram that ought to be canceled by a shift of the background string fields in spacetime. Near the boundary component of where a handle of ⌃ pinches o↵, the CFT Mg,n g,n correlation function contains a propagator of the form (4.120), summed over all intermediate states, including an integration over P µ over the Euclidean momentum space. In the critical bosonic , there will be a divergence in the t-integral having to do the tachyon. We will see in chapter 7.4 that such a divergence is absent in the tachyon-free bosonic “c =1” string theory in 1+1 dimensional spacetime, and in chapter 10 that such divergence is absent in the critical in 10 dimensional Minkowskian spacetime. One may further worry about the higher codimension locus in where di↵erent Mg,n boundary components meet. An example of this is when ⌃g,n degenerates into a torus joined to a genus g 1surface,andsimultaneouslypinchingo↵thehandleofthetorus.This amounts to the insertion of a local vertex operator, whose coecient may be logarithmically divergent due to the moduli integral. The interpretation of such a log divergence is a string- loop contribution to the Weyl anomaly on the worldsheet, which should be canceled by an adjustment of the spacetime background field due to the string-loop corrections to the spacetime equation of motion.

4.9 Reformulation in terms of Riemann surfaces

The formulation of perturbative string amplitude (4.99) makes explicit use of the fiducial metricg ˆab(t). This is somewhat redundant, as the worldsheet theory is insensitive to the choice of metric within a given di↵eomorphism and Weyl equivalence class. Indeed, it is possible to rewrite (4.99) in a way that only makes reference to the complex structure of the Riemann surface rather than the metric. Let us begin with the data of the Riemann surface ⌃(t), namely holomorphic coordinate charts Ui and transition maps on their overlaps, of the form zi = fij(zj; t). Here the moduli k dependence is characterized by the dependence of the transition functions fij on t .Inorder to directly compare Hermitian metrics on Riemann surfaces with moduli t and t0,weneed to specify a (non-holomorphic) di↵eomorphism from ⌃(t)to⌃(t0). Under an infinitesimal

82 k k k k deformation of the moduli t t0 = t + t ,thereisadi↵eomorphismthatmapsz to 7! i

k zi zi0 = zi + t vk,i(zi, z¯i), (4.121)

zi where the subscript i of vk,i indicates that the latter is defined on the patch Ui,andthe zi superscript zi of vk,i indicates that it is vector field written in the zi coordinate system. (4.121) is such that

zi0 = fij(zj0 ; t0) (4.122) is obeyed on U U .Expandingbothsidesintk, we have i \ j

zi zj vk,i = vk,j@zfij(zj,t)+@tk fij(zj,t) @z (4.123) = vzi + i , k,j @tk zj zi where by vk,j, we mean the vector field vk,j written in the zi coordinate system. 2 2 Now we can compare the metric ds = dzidz¯i on Ui with ds = dzi0dz¯i0.Thelatteris

2 2 k zi k ¯ zi ds = dzi(1 + t @zi vk,i)+dz¯it @z¯i vk,i (4.124) = e2! dz dz¯ + tk @ vzi dz2 + @¯ vzi dz¯2 , i i zi k,i i z¯i k,i i h ⇣ ⌘i for some !.Thus,theinfinitesimalmetricvariationis

k zi k ¯ zi gziz¯i = !,gzizi = t @zi vk,i,gz¯iz¯i = t @z¯i vk,i. (4.125)

In terms of the Weyl invariant form of the Beltrami di↵erential

b 1 bc (µ ) = gˆ @ k gˆ , (4.126) k a 2 t ac we have z ¯ zi z¯ zi (µk)z¯ = @z¯i vk,i, (µk)z = @zi vk,i. (4.127) The b ghost insertions appearing in (4.99) can be written as

(b, µ ) 1 k = d2z b (µ ) z + b (µ ) z¯ Bk ⌘ 2⇡ 2⇡ zz k z¯ z¯z¯ k z Z (4.128) 1 ⇥ z z ⇤ = d2z b @¯ v j + b @ v j . 2⇡ j zj zj z¯j k,j z¯j z¯j zj k,j j ZDj X In the last line, we have divided the Riemann surface into the union of domains D U , j ⇢ j and expressed the integration over Dj explicitly in the coordinate system (zj, z¯j). Using the

83 holomorphy of bzz,wecanturntheintegraloverDj into a contour integral over its boundary @D C ,andwrite j ⌘ j dz dz¯ = j b vzj j b vzj . Bk 2⇡i zj zj k,j 2⇡i z¯j z¯j k,j (4.129) j Cj X I ✓ ◆ Let C be the contour segment along the intersection C C ,orientedaccordingtothe ij i \ j direction of C .Bydefinition,C = C is the oppositely oriented contour segment. Using i ji ij (4.123), we can finally put in a form that only makes reference to the holomorphic Bk transition maps,

dzj @zj dz¯j @z¯j k = bz z bz¯ z¯ , (4.130) B 2⇡i j j @tk 2⇡i j j @tk (j`) ZCj` z` z¯` ! X where the sum runs over each unordered pair (ij)onlyonce.Ifwechoosethedomains Dj to be a pair-of-pants decomposition of ⌃, then Cij are circular contours along the intersection D D . i \ j As already mentioned, the positions of the integrated vertex operators in (4.99) can be combined with tk to parameterize the moduli space of n-punctured genus g Riemann Mg,n surface. This can be made more explicit by writing the matter vertex operator as

Vi = b 1b 1 (ccVi), (4.131) · and represent b 1, b 1 as contour integrals e e dz dz¯ e b 1 = b(z), b 1 = b(¯z). (4.132) 2⇡i 2⇡i ZCi ZCi Here Ci is a clockwise contour that encircles Vei and no other vertexe operators. We can view C as the boundary of a disc Di with holomorphic coordinate wi,suchthatVi is located at wi =0.SupposeDi lies within a coordinate chart U.WecannowrefineU into a chart Ui that contains D ,andanotherchartU 0 that contains U D .ThepositionofV on ⌃may i i i be viewed as parameter zi that appears in the transition map between Ui and U 0,

w = z0 z , (4.133) i i where z0 is the holomorphic coordinate on U 0.(4.133)istheonlytransitionfunctionthat depends on the modulus zi.Thus,b 1 and b 1 in (4.132) are the same as z and z . i i 2 k B B Grouping the integration measure (i,a) f d zi with dt ,anddenotethemcollectivelyas 6g 6+2n 62 d t for the 6g 6+2n moduli the puncturede Riemann surface, we can rewrite (4.99) Q as 6g 6+2n n 2g 2 6g 6+2n [V , ,V ]=g d t ccV , (4.134) Ag 1 ··· n s Bk i g,n * i=1 + ZM kY=1 Y where all vertex operators are now treated on equal footing. e

84