aa r X i v : . [ h e p - t h ] A p r ITEP/TH-21/08
NSR Superstring Measures Revisited
A.Morozov
ITEP, Moscow, Russia
ABSTRACT
We review the remarkable progress in evaluating the NSR superstring measures, originated by E.D’Hoker and D.Phong. Theserecent results are presented in the old-fashioned form, which allows us to highlight the options that have been overlooked in originalconsiderations in late 1980’s.
Contents θ -functions [75]-[80] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Hyperelliptic surfaces [75, 76, 89] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Relations between modular forms at particular genera . . . . . . . . . . . . . . . . . . . . . . . . 72.4.1 Genus one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Genus two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Genus three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.4 Genus four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 After the role of holomorphicity in 2 d conformal theories was fully realized and exploited in [1] it was naturalto look for the holomorphic factorization in the conformal-invariant first-quantized theories of critical strings[2]. The problem here was that the relevant quantities had to be meromorphic not only in z -variables, whichdefine positions of operators in operator-product expansions, but also in the moduli of Riemann surfaces.The relevant holomorphic anomalies in Polyakov’s combination of determinants, which define string measuresfor bosonic, super- and heterotic strings, were evaluated in [3] and shown to vanish together with conformalanomaly of [2]. This Belavin-Knizhnik theorem became a starting point for construction of perturbative stringand conformal field theories, reviewed, for example, in [4]-[8]. Without Belavin-Knizhnik theorem the Polyakovstring measures could be discussed in terms of either Shottky parametrization [9] or Selberg traces [10]. Withthis theorem the adequate language became that of the Mumford measure dµ on the moduli space of complex1urves (= Riemann surfaces) [11, 12]: the measure for bosonic string was proved in [3] to be | dµ | det(Im T ) , whilethat for the NSR superstring [13] had to contain an extra factor of (cid:16) det (Im T ) (cid:17) with dµ presumably multipliedby some modular form Ξ of the weight 8 (in the case of heterotic string [14] this NSR measure is multiplied bya complex conjugate of dµ times one of the two similarly different but actually coinciding weight-eight modularform, denoted by ξ and ξ below [3, 15, 16, 17]). The first big success on this way was explicit construction of dµ for the genera 2 , e , expressedthrough determinants in [2] and holomorphically factorized in [3], integrate away the ”supermoduli” and obtainthe relevant modification dµ [ e ] of the Mumford measure. This road looked straightforward [40]-[67], until it wasshown in [68]-[72] that naive integration over supermoduli does not work and its proper version requires a lotof work. This work was finally done by Eric D’Hoker and Duong Phong (DHP) in a series of impressive papers[18]-[26], but only 15 year later and only for genus 2 so far.The second equally obvious approach was to make educated guesses for NSR superstring measure, i.e. tofind the relevant weight-8 modular forms from their expected properties, at least for the first low genera, likeit was done in [15, 16, 17] for dµ itself. As explained in [43], the main obstacle on this way was modularnon-invariance of the Riemann identities – which are necessarily used for cancelation of tachionic divergenciesafter GSO projection (=sum over characteristics) [73]. After a series of attempts [74] – now known to be partlymisleading – this approach was temporarily abandoned. Now, after the DHP triumph it is used again andalready led to explicit construction of NSR measures at genera 3 [35], 4 [36, 37] and – somewhat less explicitly– for all higher genera [36]. The problem for g > dµ does not possess anynice representation in terms modular forms (only a far more transcendental formulas of [6, 68, 8] are currentlyavailable), but the result of [36] supports the original suggestion of [3, 16] that the ratio Ξ [ e ] = dµ [ e ] /dµ isa modular form (then it has modular weight 8) and this Ξ [ e ] is proposed in [36] in a simple and clear form.The only remaining problem with these suggestions at g ≥ , , , Theta-functions are special functions, associated with abelian varieties: g -dimensional tori, which are factorsof C g over relations z i ∼ z i + T ij z j , where symmetric period matrices T ij with positive definite imaginary part(Im T ) are points in the g ( g + 1) / modular )transformations T ∼ ( AT + B ) / ( CT + D ) from the group Sp( g, Z ).Bosonic and super-string measures on the moduli space of Riemann surfaces are defined in terms of theta-functions with semi-integer characteristics, this is taken into account in the following definition: θ (cid:20) ~δ~ε (cid:21) (cid:0) ~z | T (cid:1) = X ~n ∈ Z g exp (cid:26) iπ (cid:16) ~n + 12 ~δ (cid:17) T (cid:16) ~n + 12 ~δ (cid:17) + 2 πi (cid:16) ~n + 12 ~δ (cid:17)(cid:16) ~z + 12 ~ε (cid:17)(cid:27) (1)Sums are over all g vectors ~n with integer coordinates, each coordinate of characteristic vectors ~δ and ~ε can takevalues 0 or 1. Characteristic is called even or odd if scalar product ~δ~ε is even or odd respectively and associatedtheta-function is even or odd in ~z . The value of theta-function at ~z = 0 is called theta-constant, it automaticallyvanishes for odd characteristic. We often denote characteristics by e = { ~δ, ~ε } , in most cases these will be evencharacteristics, when we refer to some odd characteristic it is labeled by ∗ . There are N e = 2 g − (2 g + 1) even Note, however, that a lot of results from that period remain quite important and actually relevant for discussion of correlatorson the lines of [25]. N ∗ = 2 g − (2 g −
1) odd semi-integer characteristics: g N e N ∗ . . . With a pair of characteristics (not obligatory even) we associate a sign factor < e , e > = exp (cid:8) iπ ( ~δ ~ε − ~ε ~δ ) (cid:9) = ( ~δ ~ε − ~ε ~δ )mod 2 = < e , e > (2)which takes values ±
1. In particular, < e, e > = 1.
Functions of T , transforming multiplicatively under modular transformations, f ( T ) → (cid:16) det( CT + D ) (cid:17) − k f ( T ),are called modular forms of weight k . Theta-constants are not modular forms, they are not simply multi-plied by (cid:16) det( CT + D ) (cid:17) − / , but also acquire additional numerical factors proportional to e iπ/ and changecharacteristics.The simplest modular forms can be made from the 8-th powers of θ -constants, since modular transformationsact on them just by permuting their characteristics. In particular, for any integer k and gξ k ≡ N e X e θ ke (3)is a modular form of weight 4 k . Important for NSR measures are ξ = N e X e θ e and ξ = N e X e θ e (4)Also Π ≡ N e Y e θ e (5)of weight N e / g ≥
3, while roots of unity arise and Π should be raised to power 8 and 2at g = 1 and g = 2 respectively. This Π is the building block of Mumford measure at g = 1 , ,
3, see s.3 below.However, the set of modular forms is by no means exhausted by these trivial characters of the permutationgroup. Most important are other examples, having the same form for all g , like ξ k, l ≡ N e X e,e ′ < e, e ′ > θ ke θ le ′ = N e X e θ ke ξ l [ e ] (6)which has weight 4( k + l + 1). Modular invariance of ξ k, l implies that ξ l [ e ] ≡ N e X e ′ < e, e ′ > θ le ′ (7)transforms under modular transformations exactly like ξ e (we call such forms ”semi-modular”). The sign factors < e, e ′ > serve to restore modular invariance whenever θ e ′ appear instead of θ e ′ .As discovered in [18]-[26], [34]-[37] and formulated in a very clear and general form in [36], superstringmeasures are actually constructed from a wider family of modular forms of weight 8, of which ξ , and ξ and ξ , are just the first three members: ξ ( p )8 = N e X e ξ ( p )8 [ e ] (8)3here ξ (0)8 [ e ] = θ e , i . e . ξ (0)8 = ξ ,ξ (1)8 [ e ] = θ e N e X e θ e + e = θ e ξ , i . e . ξ (1)8 = ξ ,ξ (2)8 [ e ] = θ e N e X e ,e θ e + e θ e + e θ e + e + e ,ξ (3)8 [ e ] = θ e N e X e ,e ,e θ e + e θ e + e θ e + e θ e + e + e θ e + e + e θ e + e + e θ e + e + e + e ,. . . (9)and in general ξ ( p )8 [ e ] = N e X e ,...,e p θ e · p Y i θ e + e i ! · p Y i
1) instead of 2 p − G ( p ) = 0 for p > g .There is no a priori reason to prefer G ( p )8 over ξ ( p )8 , but in [36] it was demonstrated that NSR measures areactually ”more universal” (coefficients do not depend on g ) when expressed in terms of G ( p )8 , see s.4.2 below. There are no non-vanishing modular forms of weight 2 made from the 4-th powers of theta-constants, insteadthere is a set of Riemann identities R ∗ ≡ N e X e < e, ∗ > θ e = 0 (13)for all of the N ∗ odd characteristics ∗ . Of N ∗ = 2 g − (2 g −
1) Riemann identities there are (4 g −
1) = (2 g + 1)(2 g −
1) linearly independent, and they reduce the number of linearly -independent θ [ e ] from N e =2 g − (2 g + 1) to (2 g − g + 1). Other relations between theta-constants involve powers of θ . In naivesuperstring considerations an even stronger version of Riemann identity is commonly used, where up to threeof the four theta-constants are promoted to theta-functions: R ∗ ( ~z , ~z , ~z | T ) ≡ N e X e < e, ∗ > θ e ( ~ θ e ( ~z ) θ e ( ~z ) θ e ( ~z ) = 0 (14)for any three vectors ~z , ~z , ~z . Both (13) and (14) are corollaries of a general relation X all e < e, ∗ > θ e ( ~z ) θ e ( ~z ) θ e ( ~z ) θ e ( ~z ) == 2 g θ ∗ (cid:18) ~z + ~z + ~z + ~z (cid:19) θ ∗ (cid:18) ~z + ~z − ~z − ~z (cid:19) θ ∗ (cid:18) ~z − ~z + ~z − ~z (cid:19) θ ∗ (cid:18) ~z − ~z − ~z + ~z (cid:19) (15)If one needs a sum over even characteristics at the l.h.s. it is enough to add the same formula with ~z → − ~z to the r.h.s. (and divide by two). In particular, X e < e, ∗ > θ e ( ~ θ e ( ~z ) = 2 g θ ∗ (cid:18) ~z (cid:19) , (16)plays important role in superstring calculus. For block-diagonal matrices T = (cid:18) T T (cid:19) with g = g + g the theta-functions factorize into products θ e ( ~z | T ) = θ e ( ~z | T ) θ e ( ~z | T ). Above-mentioned modular forms behave as multiplicative characters under thisdecomposition: they also factorize, ξ k ( T ) = ξ k ( T ) ξ k ( T ) , ξ k, l ( T ) = ξ k, l ( T ) ξ k, l ( T ) ,ξ ( p )8 [ e ]( T ) = ξ ( p )8 [ e ]( T ) ξ ( p )8 [ e ]( T ) , R ∗ ( T ) = R ∗ ( T ) R ∗ ( T ) , (17)while Π in (5) vanishes, because some even characteristics e get decomposed into two odd, for example (cid:20) (cid:21) → (cid:20) (cid:21) ⊗ (cid:20) (cid:21) . 5 .2 Moduli space and Riemann θ -functions [75]-[80] Riemann theta-functions are associated with tori which are Jacobians of Riemann surfaces (complex curves).Then g is the genus of the curve and T ij is its period matrix. Period matrices define an embedding of moduli spaceof Riemann surfaces into Siegel semi-space, and moduli space has non-vanishing codimension g ( g +1) / − (3 g − g ≥
4. In terms of T matrices this embedding is defined by a set of transcendental Shottky relations . Todaythe best known formulation of these relations is that the corresponding theta-function is a τ -function of KP-hierarchy [81]-[85] or, in other words, satisfy the Wick theorem [8, 86, 87],det i,j θ e ( ~x i − ~y j ) E ( x i , y j ) θ e ( ~
0) = θ e ( P i ~x i − P i ~y i ) θ e ( ~ Q i
2. Non-vanishing are only theta-constants associatedwith even non-singular characteristic, k = 0, and these non-vanishing theta-constants are expressed throughramification points by Thomae formulas: θ [ e ] = ± (det σ ) g +1 Y i
2, while N nse = N e − g = 3 – so that exactly one even theta-constant vanishes and thus Π = 0 at codimension-onehyperelliptic locus in the moduli space at g = 3. The deviation from the hyperelliptic locus is measured by √ Π which has modular weight 9, and therefore the relations between modular forms of lower weights (includingthose of weight 8, which are relevant for NSR measures) can be exhaustively studied in hyperelliptic terms, i.e.pure algebraically. To be more precise, if two forms of weight ≤ g > g − − (2 g −
1) = g −
2. Of course, Π = 0 at all these loci, but additional g − a i , multiplied by appropriate power of det σ . This makes hyperellipticparametrization extremely convenient for study of relations between modular forms, at least for low genera andweights. Three theta-constants are related by Riemann identity θ = θ + θ ≡ b + c (24)The space of modular forms at genus one is generated by two Eisenstein series: E = ′ X m,n m + nτ ) ∼ ξ = X e =1 θ e = ( b + c ) + b + c = 2( b + bc + c ) (25)and E = ′ X m,n m + nτ ) ∼ θ (cid:20) (cid:21) − θ (cid:20) (cid:21) ! θ (cid:20) (cid:21) + θ (cid:20) (cid:21) ! θ (cid:20) (cid:21) + θ (cid:20) (cid:21) ! == ( b − c )(2 b + c )( b + 2 c ) (26)They are related to Dedekind function η = e iπτ/ Q ∞ n =1 (cid:0) − e πinτ (cid:1) by η = Π = (cid:16) θ θ θ (cid:17) = (cid:16) bc ( b + c ) (cid:17) = 11728 ( E − E ) (27)For any of the three even theta-characteristic e we have:2 θ e − θ e X e ′ θ e ′ = 2 < e, ∗ > θ e Y e ′ θ e ′ = 2 < e, ∗ > θ e η = 2 θ e Π ∗ (28)i.e. 2( b + c ) − ( b + c ) · b + bc + c ) = 2( b + c ) · bc ( b + c )2 b − b · b + bc + c ) = − b · bc ( b + c )2 c − c · b + bc + c ) = − c · bc ( b + c )Thus for g = 1 the two vanishing-relations (13) and (19) are actually the same. Note that we absorbed thesign-factor < e, ∗ > into the definition of Π ∗ . 7nder modular transformations τ → τ + 1 τ → − /τθ = b + c = a b − aθ = b a − cθ = c − c − b For g = 1 all our forms of weights 4 and 8 are expressed through θ e , and ξ = P e θ e : ξ [ e ] ≡ X e ′ < e, e ′ > θ e ′ = 2 θ e ,ξ , ≡ X e,e ′ θ e < e, e ′ > θ e ′ = 2 X e θ e = 2 ξ ,ξ [ e ] = X e ′ < e, e ′ > θ e ′ = − θ e + 32 θ e X e ′ θ e ′ (28) = θ e X e ′ θ e ′ − Π ∗ = ξ θ e − Π ∗ ,ξ , ≡ X e,e ′ θ e < e, e ′ > θ e ′ = 2 X e θ e = 2 ξ = ξ = X e θ e ! (29)For the set of the CDG-Grushevsky forms (9) and (10) we have: ξ ( p )8 [ e ] = α p θ e + β p θ e X e ′ θ e ′ = α p ξ (0)8 [ e ] + β p ξ (1)8 [ e ] (28) = w p θ e ξ + α p θ e Π ∗ , (30)where w p = α p + 2 β p . It follows that ξ ( p )8 ≡ X e ξ ( p )8 [ e ] = w p ξ = 2 p − ξ (31)Numerical coefficients α p , β p and w p are easily evaluated, if theta-constants are expressed through b and c : g = 1 : p α p β p w p − − −
14 15 16 . . .p − p − −
1) 2 p − p (32)In particular, it follows that ξ (2)8 [ e ] = 2 θ e ξ [ e ].In hyperelliptic parametrization θ = a a , θ = a a , θ = a a (33)and formulas look a little more involved than in terms of b and c , for example: ξ = X e θ e = a a + a a + a a = − s + 6 s s + 72 s − s s + 12 s , (34)8here s m = P k =1 a ki . Also, ξ = X e θ e = a a + a a + a a = 2 ξ R ∗ = X e < e, ∗ > θ e ∼ a a − a a − a a = 0 ξ , = a a ( a a + a a + a a ) + a a ( a a + a a − a a ) + a a ( a a − a a + a a )and ξ , = a a ( a a + a a + a a ) + a a ( a a + a a − a a ) + a a ( a a − a a + a a )Still, all the relations, including (30), can be easily derived in this parametrization, and such derivations arestraightforwardly generalized to g = 2 ,
3. The more economic b, c parametrization is also generalizable (it isrelated to expressions through theta-constants of doubled argument, θ (2 T ), which was actually used in [35]),but this is a slightly more involved technique, unnecessary for our presentation.Formula (22) looks as follows: S S ∪ U S ∩ U S ◦ U θ e ∅ ∅ ∼ + a a a a ∼ + a a
13 134 3 14 ∼ − a a a a ∼ − a a
14 134 4 13 ∼ − a a a a ∼ + a a
23 234 3 24 ∼ − a a a a ∼ + a a
24 234 4 23 ∼ − a a a a ∼ − a a ∼ + a a a a ∼ + a a
12 1234 ∅ ∅ U = { a , a } : this is the choice which reproduces (33). In the last two lines S ◦ U ) = g + 1 = 2, such sets S correspond to the odd characteristic with vanishing theta-constant. Of six (as many as there are odd characteristics *) Riemann identities (13) there are five linearly independent,and they express 10 a priori different θ e through 5 linearly independent ones. In addition there is one non-linearrelation: χ = 4 ξ − ξ = 0 , i . e . ξ (0)8 ≡ ξ = 14 ξ , ξ (1)8 = ξ (35)Further, ξ , = 4 ξ ,ξ , = 4 ξ = ξ (36)and ξ (2)8 [ e ] = 4 θ e ξ [ e ] , ξ (2)8 = X e ξ (2)8 [ e ] = 4 ξ , = 4 ξ (37)9 ( p )8 [ e ] = α p θ e + β p θ e X e ′ θ e ′ + γ p θ e X e ′ ,e ′′ θ e ′ θ e ′′ θ e + e ′ + e ′′ = α p ξ (0)8 [ e ] + β p ξ (1)8 [ e ] + γ p ξ (2)8 [ e ] (38)It follows that ξ ( p )8 ≡ X e ξ ( p )8 [ e ] = (cid:18) α p + β p + 4 γ p (cid:19) ξ = 14 w p ξ (39)where w p = α p + 4 β p + 16 γ p . Numerical coefficients α p , β p and γ p are easily evaluated if theta-constants areexpressed in hyperelliptic parametrization, where they become simple algebraic relations. g = 2 : p α p β p γ p w p −
14 7 644 56 −
90 35 256 . . .p p − − p − − − p − p − − (2 p − p − − p (40)The simplest way to prove this kind of identities is to use hyperelliptic parametrization, where they becomesimple algebraic relations. In the basis selected in [34] – it corresponds to taking U = { a , a , a } in (22) – wehave: odd characteristics : S
14 16 46 23 25 352356 2345 1235 1456 1346 1246 e ( S ) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) θ e even characteristics: S ∅
24 13 56 26 45 15 36 34 12123456 1356 2456 1234 1345 1236 2346 1245 1256 3456 e ( S ) h i h i h i h i h i h i h i h i h i h i θ e − a a a a a a − a a a a − a a a a − a a − a a − a a Note that there is no direct counterpart of the relation (28) already for g = 2: the form χ = 4 ξ − ξ is nota linear combination of Riemann identities (13). Moreover, one can easily check that it does not automaticallyvanish for arbitrary set of 5 linearly-independent θ e : from genus two χ = 0 is an additional relation betweentheta-constants, algebraically (not only linear) independent of Riemann identities. However, association of theta-characteristics – the map S → e ( S ) – in [34] does not look consistent with the rule (23), and wechoose another one in the second line of the table. .4.3 Genus three The number N ∗ of Riemann identities is now 28, of which g − = 21 are linearly independent and there are (2 g +1)(2 g − +1)3 = 36 −
21 = 15 linearly independent θ e . Again, there are additional non-linear relations, including χ = 8 ξ − ξ = 8 X e θ e − X e θ e ! = 0 (41)Hyperelliptic locus has codimension one in moduli space and is defined by Π = Q e θ e = 0. Still, hyperellipticparametrization can be used to prove formulas at genus 3 for modular functions of weights ≤
8, becausedeviations from hyperellipticity are proportional to √ Π which has weight 9.
As shown in [80], and widely used since [3, 16, 18], χ = 0 exactly at the moduli space, embedded as codimension-one subspace in the Siegel upper semi-space. Hyperelliptic locus now has codimension g − g < After a brief exposition of the theory of theta-constants – note that we do not need anything more than abovesimple statements – we are ready to switch to the string measures. As already mentioned in the Introduction,Belavin-Knizhnik theorem [3] expresses them through the holomorphic Mumford measure on the moduli spaceof complex curves, which has degree-2 poles at the boundaries: namely when one of the cycles (contractible ornon-contractible) gets shrinked. The degree of the pole is controlled by the negative mass squared of a tachyon,present in the spectrum of bosonic string. Residues at the poles are given by two-point a function in the caseof non-contractible cycle (when genus g curve degenerates into the one of g −
1) and a product of two one-pointfunctions in the case of contractible cycle (when the curve splits into two of genera g and g = g − g ). In factthe values of pole degrees are enough to determine the measure and above properties can be used to read offexpressions for one- and two-point functions. The most interesting object is the string measure on the universalmoduli space , unifying all genera and all the correlators (scattering amplitudes) [90]. n -point correlators canalso be promoted to stringy correlators by inclusion of Riemann surfaces with boundaries and/or non-oriented[91].In fact all these generalizations are rather straightforward once the structure of string measures for particulargenera is clarified – and we list here original expressions from [15, 16]. For somewhat less explicit expressionsfor all genera see [4]-[8]. Genus one: τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:16)Q e θ [ e ]( τ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i . e . dµ = dτ Π (42) Genus two: (cid:0) det (Im T ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dT dT dT (cid:16)Q e θ [ e ]( τ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i . e . dµ = Q i Genus four: This is the first time when the module space is smaller then Teichmuller one, it has complex codimension oneand is defined by the zero of a single Shottky condition χ = 0 (45)where χ is the weight-8 modular form on Teichmuller space, χ ( T ) = 16 X e θ [ e ] − X e θ [ e ] ! (46)Bosonic string measure is 1 (cid:0) det (Im T ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ij ≤ j dT ij χ ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (47)This wonderful formula, suggested in [3] and [16] never attracted attention that it deserves and was not investi-gated as carefully as its lower-genera counterparts. Note that instead of the holomorphic delta-function of χ in(47) one can put the sum of the NSR measures P e Ξ [ e ], which vanishes on the moduli space and is essentiallythe same as χ . Superstring possesses space-time supersymmetry in critical dimension d = 10. Two approaches are developedin order to describe it in the first quantization formalism, i.e. with the help of the two-dimensional actionson string world sheet. One approach (Green-Schwarz formalism [94]-[97]) is explicitly d = 10 supersymmetric,but the two-dimensional action is highly non-linear and possesses sophisticated κ -symmetry. Another, NSRapproach [13, 73] is based on the theory of fermionic string , defined as possessing the world-sheet, i.e. 2 d supersymmetry. On world sheets with non-trivial topologies one can impose a variety of boundary conditionson 2 d fermions, associated with different spin-structures or, what is the same, the theta-characteristics. Thecorresponding holomorphic NSR measures dµ [ e ] on the moduli space of Riemann surfaces also depend on theta-characteristics. Fermionic string does not have 10 d space-time supersymmetry, it has tachyon and divergencies,just as bosonic string. However, superstring Hilbert space is just a subspace in the Hilbert space of fermionicspace, and the relevant GSO projection [73] is provided simply by a sum of any holomorphic conformal blockover the spin-structures: D A E = Z (cid:16) det (Im T ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X e A [ e ] dµ [ e ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (48)where A [ e ] is a combination of holomorphic Green functions, associated with the multi-point observable A .In genus one the three NSR measures are well known [13]: dµ [ e ] = < e, ∗ > θ e dτη , (49)what means that they are expressed through Mumford measure dµ = dτη = dτ Π from (42): dµ e = < e, ∗ > θ e η dµ = θ e Π ∗ dµ (50)where ∗ is the only odd theta-characteristic at g = 1. (Of course, for genus one the measure includes the 6-thpower of Im τ instead of the 5-th one in for g > dµ [ e ] = Ξ [ e ] dµ, (51)12here Ξ [ e ] is a semi-modular form of weight 8. This is a non-trivial hypothesis for g ≥ 4, because there isno obvious reason why dµ [ e ] /dµ should have any nice continuation to entire Siegel space, beyond the modulispace. Still, if this hypothesis is true, for any correlator in superstring theory we have a simple representationin terms of an integral over moduli space: D A E = Z | dµ | (cid:16) det (Im T ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X e A [ e ] Ξ [ e ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (52)Under these assumptions the only unknown is the set of forms Ξ [ e ], which should satisfy two simple properties:factorization and the condition of vanishing cosmological constant, X e dµ [ e ] = 0 , i . e . X e Ξ [ e ] = 0 (53)For genus 1 eq.(53) for (50) is an immediate corollary of the Riemann identity (13), X e < e, ∗ > θ [ e ] = 0 (54)It seemed a natural generalization of conjecture (51) to extend this property to all genera [43, 44]:Ξ [ e ] ? = < e, ∗ > θ e K ∗ , (55)especially because (14) would then automatically guarantee the vanishing of all g ≥ , , ∗ , which would un-acceptedly show up in non-vanishing 4-point function and in higher correlators.Worse than that, an appropriate form K ∗ does not seem to exist.It was believed that the NSR measure can be derived , starting from explicitly 2 d -supersymmetric formalismfor fermionic string, based on the clever definition of super-Riemann surfaces, by integrating over odd super-moduli. However, naive simplified approaches of this kind (attempting to trivialize the supermoduli bundle overthe ordinary module space) failed, and accurate integration was performed only recently in [18]-[21] and onlyfor g = 2. The outcome was a confirmation of hypothesis (51) and a clear denunciation of (55): it appearedthat instead of continuing (54) from g = 1 to g > g = : Ξ [ e ] = X e < e, ∗ > θ [ e ] Π ∗ (28) = 2 X e θ e − X e θ e ! = χ = 2 ξ (0)8 − ξ (1)8 (56)and continue the r.h.s. (note that relation (28) does not survive at g ≥ 2, so that continuations of its two sidesdeviate from each other). Such continuation was derived in [18]-[21] for g = 2, reformulated and generalizedto g = 3 , g in [36]. SinceCPG-Grushevsky conjecture for g ≥ dµ [ e ] through ξ ( p )8 with p ≥ 3, it does not contain an explicit θ e factor, what makes puzzling the story about the 1 , , The natural generalization of the r.h.s. of (56) is any g : Ξ [ e ] = g X p =0 h p ξ ( p )8 [ e ] , (57)where CDG-Grushevsky forms at the r.h.s. are defined in (9) and (10) and coefficients h p are constrained byrequirements of factorization and vanishing of the cosmological constant.The latter one implies that N e X e Ξ [ e ] = g X p =0 h p ξ ( p )8 = 0 (58)13ince the l.h.s. is a modular form of weight 8, it has good chances to be proportional to ξ 24 (19) = 2 g ξ and thesame is actually true for all the terms in the sum: ξ ( p )8 = 12 W p ξ (59)Thus the requirement (58) simply states that g X p =0 h p W p = 0 (60)Coefficients W p can be evaluated by different methods, but the simplest one is to go to the high-codimensionsubset at the boundary of moduli space, when the curve degenerates into a set of tori and period matrix T becomes diagonal T = diag( τ , . . . , τ g ). Then ξ ( T ) → Q gi =1 ξ ( τ i ) = ξ ⊗ g and ξ ( p )8 ( T ) −→ g Y i =1 ξ ( p )8 ( τ i ) (31) = (cid:16) w p (cid:17) g g Y i =1 ξ ( τ i ) (61)so that W p = 2 (cid:16) w p (cid:17) g (32) = 2 g ( p − (62)Of course, (60) is an important but non-restrictive constraint on the coefficients h p . All the h p are determinedif the same reduction to genus one is made for the individual Ξ [ e ]: On one side,Ξ [ e ]( T ) → g Y i =1 Ξ [ e i ]( τ i ) (50) = g Y i =1 n θ e i Π ∗ ( τ i ) o (63)on another sideΞ [ e ]( T ) (58) = g X p =0 h p ξ ( p )8 [ e ] −→ g X p =0 h p ( g Y i =1 ξ ( p )8 [ e i ]( τ i ) ) (30) = g X p =0 h p ( g Y i =1 (cid:16) w p θ e i ξ + α p θ e i Π ∗ (cid:17) ( τ i ) ) (64)Comparing the two expressions we obtain a set of g + 1 linear equations for g + 1 coefficients h p : g X p =0 h p w kp (2 α p ) g − k = 2 g δ k, or g X p =0 ˜ h p λ kp = 2 g δ k, (65)with k = 0 , . . . , g , ˜ h p = (2 α p ) g ˜ h p and λ p = w p / α p , so that h p is the ratio of Van-der-Monde determinants:˜ h p = 2 g ∆ p ( λ )∆( λ ) = 2 g g Y i = p λ i λ i − λ p and h p = g Y i = p w i w i α p − w p α i (66) g h h h h h h . . . − − 12 112 − 13 112 − − 421 118 − − − − . . . (67)14t is easy to check, that the vanishing relations (60) and thus (58) are true with these values of h p .In Grushevsky’s basis [36] the coefficients are much nicer, moreover, they are actually independent of g .Indeed, substituting ξ ( p )8 in the form (11) and h p from the table into (57) we obtain: g = 1 Ξ [ e ] = (cid:16) G [ e ] − G (1)8 [ e ] (cid:17) g = 2 Ξ [ e ] = (cid:16) G [ e ] − G (1)8 [ e ] + G (2)8 [ e ] (cid:17) g = 3 Ξ [ e ] = (cid:16) G [ e ] − G (1)8 [ e ] + G (2)8 [ e ] − G (3)8 [ e ] (cid:17) g = 4 Ξ [ e ] = (cid:16) G [ e ] − G (1)8 [ e ] + G (2)8 [ e ] − G (3)8 [ e ] + G (4)8 [ e ] (cid:17) g = 5 Ξ [ e ] = (cid:16) G [ e ] − G (1)8 [ e ] + G (2)8 [ e ] − G (3)8 [ e ] + G (4)8 [ e ] − G (5)8 [ e ] (cid:17) . . . and finally dµ [ e ] = Ξ [ e ] dµ, Ξ [ e ] = 12 g g X p =0 ( − ) p Q pi =1 (2 i − G ( p )8 [ e ] (68)(the coefficient in the term with p = 0 is unity, by the usual rule Q = 1, like 0! = 1). Note that in [36] thenormalization of G ( p )8 was chosen differently, therefore the coefficients in (68) are also different. In addition to (61) one can consider reductions to lower-codimension components of the boundary, where, forexample, the curve degenerates into two of genera g and g with g + g = g . This is an important check, butthe result actually follows from above much simpler consideration.For example, the genus-threeΞ = 821 ξ (0)8 − ξ (1)8 + 112 ξ (2)8 − ξ (3)8 (69)decomposes into genus-one and genus-two quantitiesΞ −→ Ξ τ T T T T = 821 ξ (0)8 ( τ ) ⊗ ξ (0)8 (cid:18) T T T T (cid:19) − ξ (1)8 ( τ ) ⊗ ξ (1)8 (cid:18) T T T T (cid:19) ++ 112 ξ (2)8 ( τ ) ⊗ ξ (2)8 (cid:18) T T T T (cid:19) − ξ (3)8 ( τ ) ⊗ ξ (3)8 (cid:18) T T T T (cid:19) = (71) = (cid:18) ξ (0)8 − ξ (1)8 (cid:19) ( τ ) ⊗ (cid:18) ξ (0)8 − ξ (1)8 + 112 ξ (2)8 (cid:19) (cid:18) T T T T (cid:19) (57) = Ξ ( τ ) ⊗ Ξ (cid:18) T T T T (cid:19) (70)where we substituted the genus-one and genus-two relations: ξ (2)8 ( τ ) (30) = − ξ (0)8 ( τ ) + 3 ξ (1)8 ( τ ) ,ξ (3)8 ( τ ) (38) = − ξ (0)8 ( τ ) + 7 ξ (1)8 ( τ ) (71)and ξ (3)8 (cid:18) T T T T (cid:19) = 8 ξ (0)8 (cid:18) T T T T (cid:19) − ξ (1)8 (cid:18) T T T T (cid:19) + 7 ξ (2)8 (cid:18) T T T T (cid:19) (72)We omit characteristics labels in this section to simplify the formulas.Similarly, to check the decomposition with g = g + g ,Ξ = g X p =0 h p ξ ( p )8 −→ g X p =0 h p ξ ( p )8 ⊗ ξ ( p )8 = g X p =0 h p ξ ( p )8 ! ⊗ g X p =0 h p ξ ( p )8 ! = Ξ ⊗ Ξ (73)15ne needs to know the analogues of (30) and (38) to substitute into the underlined expression. After that thenext equality is just an algebraic identity for coefficients h p in (67). Remarkably, generalizations of (30) and (38)can be found for all genera by pure algebraic means: analyzing restrictions to hyperelliptic loci. Despite theseloci have high codimension g − 2, all the coefficients are unambiguously fixed in these restrictions. Eqs.(30) and(38) themselves are actually enough to validate decompositions g = m · n · m and n .To show just one more example, the decomposition 4 → H ξ (0) ⊗ ξ (0) + H ξ (1) ⊗ ξ (1) + H ξ (2) ⊗ ξ (2) + H ξ (3) ⊗ ξ (3) + H ξ (4) ⊗ ξ (4) == (cid:16) h ξ (0) + h ξ (1) + h ξ (2) (cid:17) ⊗ (cid:16) h ξ (0) + h ξ (1) + h ξ (2) (cid:17) (74)where H p correspond to genus 4 (the forth line in (67) while h p – to genus 2 (the second line in (67),– andgenus-two modular forms ξ ( p )8 [ e ] are related by (38): ξ (3) = 8 ξ (0) − ξ (1) + 7 ξ (2) ,ξ (4) = 56 ξ (0) − ξ (1) + 35 ξ (2) (75)Collecting the coefficients at different independent products of forms in (74), we obtain: ξ (0) ⊗ ξ (0) H + 8 H + 56 H = h 20 64315 − + = ξ (1) ⊗ ξ (1) H + 14 H + 90 H = h − − + = ξ (2) ⊗ ξ (2) H + 7 H + 35 H = h 22 118 − + = ξ (0) ⊗ ξ (1) − · H − · H = h h − · = − ξ (0) ⊗ ξ (2) · H + 56 · H = h h − + · = ξ (1) ⊗ ξ (2) − · H − · H = h h − · = − (76)Equalities in the last column obtained by substitution of the coefficients from (67) are indeed true. To conclude, we reviewed spectacular new development in perturbative superstring theory, caused by the ground-breaking papers [18]-[26] of Eric D’Hoker and Duong Phong and their direct continuation in [32]-[38]. The mainreason why these formulas have not been discovered in the first attack on NSR measures in 1980’s seems relatedto three prejudices.First, starting from [43], the vanishing of cosmological constant was attributed to Riemann identities, whilethe simple relation (28) at genus one allowed two kinds of generalizations: to (13) and to (19). It turned outthat the second choice is more appropriate.Second, NSR measure dµ e was believed to be proportional to θ e , so that expressions for to 1,2,3,4-pointfunctions would not contain θ e in denominators. Remarkably, this prejudice was still alive in [18] and was finallybroken only in [35], though it was actually based on the misleading overestimate of the role of the Riemannidentities (since they had a generalization (14), the vanishing of 1,2,3-point functions would automatically cometogether with that of the 0-function – if Riemann identities were the right thing to rely upon).Third, naive integration over odd supermoduli was associated with a correlator of the superghost β, γ -fields[68], which produced a non-trivial theta-function in denominator and summation over spin structures (theta-characteristics) looked hopeless. An artistic choice of odd moduli was then required in order to eliminate thistheta-function and perform the summation. Exact treatment of odd moduli in [18]-[26] confirmed that themeasure dµ e is simple and has nothing non-trivial in denominator (at least for genus two) and this opened theway for a new stage of guess-work, based on the search of the modular forms with given properties.Today all these problems seem to be largely resolved, the outcome – eqs.(57), (67) and (68) – is nearlyobvious (once you know it) and it deserves to be widely known . Our main goal in this text was to giveas simple presentation of the subject as possible, avoiding unnecessary details about supermoduli integration16nd modular-forms theory, relying instead only on widespread knowledge of elementary string theory. To avoidoverloading the text we did not include consideration of non-renormalization theorems for 1,2,3-point functions[40], in particular, the resolution of the θ e ”paradox”, and the most interesting expressions for 4-point functions(found and proved in above-cited references). Already at the level of 4-point functions the NSR string withGSO projection can be compared to Green-Schwarz superstring [94]-[96], where equally impressive progress isalso achieved in recent years due to the works of Nathan Berkovits [97] – and this is a separate issue of greatimportance to be addressed elsewhere. Acknowledgements I am grateful to my colleagues and friends, who taught me a lot about various aspects of Riemann surfaces andstring measures: A.Beilinson, A.Belavin, A.Gerasimov, E.D’Hoker, R.Iengo, R.Kallosh, I.Krichever, A.Levin,D.Lebedev, O.Lechtenfeld, Yu.Manin, G.Moore, P.Nelson, M.Olshanetsky, D.Phong, G.Shabat, A.Schwarz,T.Shiota, A.Turin, A.Voronov, Al.Zamolodchikov and especially to V.Knizhnik, A.Perelomov and A.Rosly.This work is partly supported by Russian Federal Nuclear Energy Agency and Russian Academy of Sciences,by the joint grant 06-01-92059-CE, by NWO project 047.011.2004.026, by INTAS grant 05-1000008-7865, byANR-05-BLAN-0029-01 project, by RFBR grant 07-02-00645 and by the Russian President’s Grant of Supportfor the Scientific Schools NSh-3035.2008.2 References [1] A.Belavin, A.Polyakov and A.Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional QuantumField Theory , Nucl.Phys. B241 (1984) 333[2] A.Polyakov, Quantum Geometry of Bosonic String , Phys.Lett. 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