- Email: [email protected]
The BRST cohomology of the N=2 string
Nuclear Physics B 671 (2003) 51–66 www.elsevier.com/locate/npe
The BRST cohomology of the N = 2 string I. Kriz 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA Received 18 July 2003; accepted 27 August 2003
Abstract In this paper, we prove by rigorous calculation that BRST cohomology and old covariant quantization give the same spectra for the N = 2-supersymmetric string. This confirms an old conjecture, but the precise mechanism by which it happens is more subtle than assumed, and may shed new light on BRST at extended supersymmetry. 2003 Elsevier B.V. All rights reserved. PACS: 11.25.Hf; 11.25.Mj; 11.30.Pb; 04.65.+e
1. Introduction The N = 2 string (see, e.g., [1,2,9–12]) has 4 real bosons and fermions, coupled with N = 2 supergravity (1 graviton, 2 gravitinos and a U (1)-gauge field). Old covariant quantization (OCQ) for the N = 2 string was carried out, e.g., in [10–12] and gives a physical spectrum which has only one state for each momentum on the mass shell. The BRST complex for this theory was developed in [1,2,9]. However, the cohomology of that complex was never calculated directly. In this paper, we show that the N = 2 BRST cohomology gives the same spectrum as OCQ up to ghost degeneracy similarly as is the case of N = 0 and N = 1 strings, thus confirming a long standing conjecture. Despite the analogous nature of the results, computing the BRST cohomology of the N = 2 string is quite different from the N = 0 and N = 1 cases: at N = 0 and N = 1, (as shown by Polchinski [13, Section 4.4]), the ghost and antighost mode towers are matched by the BRST differential precisely with one tower of bosons (and fermions for N = 1) E-mail address: [email protected] (I. Kriz). 1 The author was supported by NSF grant DMS 0305853.
0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.08.035
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each, thus cancelling two towers of bosons (and fermions), which gives the lightcone gauge spectrum. When we try to argue the same way for the N = 2 string, however, we arrive at a puzzle. Just as in the N = 0 and N = 1 cases, Virasoro ghosts/antighosts cancel two bosonic towers, and the two pairs of supersymmetry ghosts/antighosts cancel the four fermionic towers. But then we are left with the two remaining bosonic towers and the ghost/antighost pair coming from the U (1)–R-symmetry. There is no way analogous to the N = 0 and N = 1 cases of cancelling the U (1)-ghosts/antighosts with bosons, since the R-symmetry acts trivially on bosons! This, in fact, originally led the author to erroneously believe that the BRST cohomology of the N = 2 string was intrinsically different from its OCQ spectrum, and that the BRST spectrum had more modes. It turns out, however, that the remaining bosons and U (1)-ghosts do cancel for a more subtle reason; consequently, the BRST and OCQ spectra of the N = 2 are the same up to 4 degenerate ghost 0-modes at each chirality, as expected. The cancellation mechanism involves a complicated interplay of counterterms which certainly does not seem to have been known before. In fact, the N = 2 BRST differential has 27 terms (23 after a transformation), and calculating general differentials directly seems almost impossible (for illustration, just a few explicit examples of lowest weight differentials are given in Appendix A). In fact, the author knows no way of rigorously computing the N = 2 BRST cohomology without a spectral sequence. Introducing this technique, and carefully showing how it applies to the N = 2 BRST case, is the main purpose of this paper. The present paper is organized as follows: in Section 2, we review the superconformal algebra of constraints KN and introduce a notation which is well suited for what follows. In Section 3, we write down explicitly the N = 2 BRST complex in our notation. In Section 4, we calculate, as a warm-up case, the BRST cohomology of the N = 0 and N = 1 strings (which is essentially only a slight simplification of Polchinski’s argument [13, Section 4.4]). Our main result, the calculation of the BRST cohomology of the N = 2 string, is given in Section 5.
Notation For our purposes, it is crucial to reconcile mathematical and physical notation. For a physical notation closest to our purposes, we refer the reader to [2]. We give here a short dictionary between the mathematical terms of the present paper and the language of [2]. The action constraints of [2, Eqs. (2.3a)–(2.3c)] are (super-versions of ) Euler equations of the corresponding variation problem; they determine the classical solution. In the quantized case, the constraints determine the energy-stress tensor, or, as one says in mathematics, the superconformal algebra action. In our case, the superconformal algebra is called K2 . We review the superconformal algebras KN , but see [5] for a more detailed mathematical discussion. In [2], the constraints in the present case are denoted by Θ, S i , T and later, in the same order, L, Gi , T . This notation coincides with the notation used today (cf. [6]), except there T is denoted by J . We use the notation G0 , G1 , G2 , G{12} (the exact translation is in the next section).
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The massive mode generating fields are, in [2], denoted by α i,a (bosons) and bi,a (fermions), i, a = 1, 2. In this paper, D = 4 (real) plays such a special role that we name these fields by individual letters x, y, z, t (bosons) and ξ , ψ, ζ , τ (fermions). The ghost generating fields are, in [2], denoted by λ, ξ 1 , ξ 2 , η and the antighost fields ¯ From our point of view, these fields are a part of a general are denoted by λ¯ , ξ¯ 1 , ξ¯ 2 , η. picture, and we call the ghost fields c12 , c1 , c2 , c0 and the antighost fields b12 , b 1 , b 2 , b 0 (in the same order).
2. A review of the N -superconformal Lie algebra KN We begin with a review of the N -superconformal algebra KN (see [4–7,11]). As a superLie algebra, KN can be described as the free complex vector super-space on basis GIr where I ⊂ {1, . . . , N}, r ≡ !I2 mod 1 in the NS sector and r ∈ Z in the Ramond sector (!I is the number of elements I ). It saves some sign conventions if we identify a sequence I = (i1 < · · · < ik ) with the set |I | = {i1 , . . . , ik }, and extend the definition of GIr to nonincreasing sequences by Gσr (I ) = sign(σ )GIr for a permutation σ on {1, . . . , N}. Now the super-Lie bracket on KN is defined as follows (we always write [ ] for the Lie bracket, regardless of parity): For I = (i1 , . . . , ik ), J = (j1 , . . . , j& ), |I | ∩ |J | = ∅, I J = (i1 , . . . , ik , j1 , . . . , j& ), I J J Gr , Gs = r(2 − &) − s(2 − k) GIr+s . For I , J as above, but with |I | ∩ |J | = {ik } = {j1 }, I ⊕ J = (i1 , . . . , ik−1 , j2 , . . . , j& ), I J I ⊕J Gr , Gs = Gr+s . When !(|I | ∩ |J |) > 1, I J Gr , Gs = 0. We will be mostly interested in the case N = 2, where these symbols relate to standard {i} {12} notation [6] by G∅r = 2Lr , Gr = Gir , Gr = iJr . However, even there we will see a benefit to having a systematic approach to the description of the Lie bracket. The superalgebra KN can be thought of as the finite energy part of the superalgebra of infinitesimal deformations of N -super-conformal structure on a boundary component of an N -superconformal surface (cf. [7]). We recall that an N -superconformal surface is an (1|N)-dimensional complex supermanifold which possesses an atlas consisting of coordinate patches (z|θ1 , . . . , θN ) satisfying the following equations: consider the N -tuple of differential operators D = (D1 , . . . , DN )T , where Dk =
∂ ∂ + θk , ∂θk ∂z
k = 1, . . . , N.
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Considering another set of local coordinates (˜z|θ˜1 , . . . θ˜N ), we have N N ∂ ˜ ˜ ˜ ˜ . θj Di θj (Di θj )Dj + Di z˜ − Di = ∂ z˜ j =1
j =1
The condition is that ˜ D = M D, where Mij = Di θ˜j , which leads to the equations Di z˜ −
N
θ˜j Di θ˜j = 0,
i = 1, . . . , N.
(1)
j =1
To find possible infinitesimal deformations of coordinates, we must consider the linearized equations (1), which are Di z˜ = θ˜i +
N
θj Di θ˜j .
(2)
j =1
Solutions of the linearized equations are (by definition) vector fields. One finds that a (topological) basis of the space of solutions, written in vector field notation, is of the form GI
n+ |I2| −1
= (|I | − 2) · zn θI
|I |−1 N ∂ ∂ ∂ + zn (−1)j θIk−j − nzn−1 θI θi , ∂z ∂θik−j ∂θi j =0
j =1
(3)
where θ(i1 ,...,ik ) = θ1 · · · θk and, for I = (i1 , . . . , ik ), Ij = (i1 , . . . , iˆj , . . . , ik ). These choices of GIr ’s obey the Lie relations stated above. Here the numbers n which occur vary with the boundary conditions imposed in the super-directions, although they are always of the form n ∈ Z + αI for some numbers αI . We shall not need a detailed analysis of the boundary conditions: in this paper, we will restrict attention to the R and NS sectors described in the beginning of this section. 3. The N = 2 BRST complex In this section, we shall describe the BRST complex Hp of the N = 2 free string at a given momentum p. Here and from now on we shall work in the chiral theory, which can be interpreted as calculating the spectrum of the free open N = 2 string. In the closed string case, both chiralities are present, which in our language means simply calculating the cohomology of a tensor product of two copies of Hp , which follows by the Künneth theorem.
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In the chiral case, Hp is a Z/2 × Z/2-graded commutative algebra. This means that two homogeneous elements u, v of bidegrees (d1 , d2 ) and (e1 , e2 ), di , ei ∈ Z/2, satisfy uv = (−1)d1 e1 +d2 e2 vu.
(4)
This is the Deligne–Morgan sign convention [3]. In fact, Hp , where p is a quadruple of numbers (X0 , Y0 , Z0 , T0 ), is a free (Z/2 × Z/2)-graded commutative algebra on generators xn , yn , zn , tn
of bidegree (0, 0), n < 0, n ∈ Z,
ξr , ψm , ζr τm ,
of bidegree (0, 1),
(5)
r 0,
m < 0, 1 in the NS sector and r, m ∈ Z in the R sector, r, m ∈ Z + 2 0 12 bn0 , cm , bn12, cm 1 2 br1 , cm , br2 , cm
of bidegree (1, 0), n < 0, m 0,
(6) (7)
of bidegree (1, 1),
m 0, 1 in the NS sector and r, m ∈ Z in the R sector. r, m ∈ Z + (8) 2 The generators (5), (6) are called matter generators and the generators (7), (8) are called Fadeev–Popov ghost generators. Additional numerical invariants we need to keep track of are the weight which on a monomial in the generators (5)–(8) is the sum of the negatives of the subscripts and the ghost number which is 0 on matter generators and −1, 1 on the b’s and c’s, respectively. Now to construct operators Hp → Hp , one introduces generating operators r < 0,
xn , yy , zn , tn , ξr , ψr , ζr , τr , bn0 , cn0 , br1 , cr1 , br2 , cr2 , bn12 , cn12 ,
(9)
in the NS sector and r ∈ Z in the R sector without where n ∈ Z, r ∈ Z + positivity restrictions. The operator generators whose indices satisfy the (non)-positivity requirements of (5)–(8) act on Hp by multiplication on the left (and thus can be naturally identified with the corresponding generators of Hp ). The remaining operator generators act by 0 on the vacuum 1 with the exception of x0 , y0 , z0 , t0 , which act as multiplication by numbers X0 , Y0 , Z0 , T0 . The action of operator generators on the entire Hp is then determined by the operator commutation relations 1 2
[xn , x−n ] = [yn , y−n ] = [zn , z−n ] = [tn , t−n ] = n, [ξr , ξ−r ] = [ψr , ψ−r ] = [ζr , ζ−r ] = [τr , τ−r ] = 1, I I cr , b−r = 1,
(10)
and all other commutators of the generators (9) which are not permutations of (10) are 0. Here the bidegrees of operator generators are specified by the same formulas as for the generators of Hp , and the commutator is defined on elements a, b of bidegrees (d1 , d2 )
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and (e1 , e2 ) as [a, b] = ab − (−1)d1 d2 +e1 e2 ba.
(11)
Physically, a structure of the kind just described naturally arises in quantization. Mathematically, the construction also has a better explanation, which will be given in Section 5 below. For now, it is important that the superconformal algebra K2 acts on Hp . It turns out helpful to write its operators as GIr = m GIr + gh GIr
(12)
where m stands for matter and gh stands for ghost. The formula for the BRST differential is then 1 I I I I Q = Qm + Qgh , Qm = (13) m Gr c−r , Qgh = gh Gr c−r . 2 I,r
We have 0 m Gr
=
I,r
:xn+r x−n : + :yn+r y−n : + :zn+r z−n : + :tn+r t−n :
n
1 m Gr
− n:ξn+r ξ−n : − n:ψn+r ψ−n : − n:ζn+r ζ−n : − n:τn+r τ−n :, = ξn+r x−n + ψn+r y−n + ζn+r z−n + τn+r t−n ,
(14) (15)
n
2 m Gr
=
−ψn+r x−n + ξn+r y−n − τn+r z−n + ζn+r t−n ,
(16)
n
12 m Gr
=
n
I gh Gr
=
:ψn+r ξ−n : + :τn+r ζ−n :,
J . GIr , bsJ c−s
(17) (18)
J,s
Here : : denotes the operation of normal ordering. A monomial is said to be in normal order if all its generators with negative subscripts are to the left of all generators with positive subscripts. Then : : is the operation of bringing a monomial to normal order using the relations (4), not (10). In (18), the symbol [GIr , bsJ ] is defined by computing [GIr , GJs ] and replacing G’s by b’s in the result. The formula (13) is then a semiinfinite version of Lie algebra cohomology, although it hides anomaly cancellation and mass-shell shift cancellation which one has to deal with in the quantum case (in particular, mass-shell shifts are trivial for the N = 2 string). In the present paper, the only thing we will use is that the resulting formula is consistent; for reasoning on the criticality of the D = 4 N = 2 string, and mass-shell shift cancellations in this case, we refer the reader to [11]. Now explicitly, (14)–(18) give 0 0 0 0 :xn+r x−n :c−r + :yn+r yn :c−r + :zn+r z−n :c−r + :tn+r t−n :c−r Qm = 0 0 0 0 − n:ξn+r ξ−n :c−r − n:ψn+r ψ−n :c−r − n:ζn+r ζ−n :c−r − n:τn+r τ−n :c−r 1 1 1 1 + ξn+r x−n c−r + ψn+r y−n c−r + ζn+r z−n c−r + τn+r t−n c−r
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2 2 2 2 − ψn+r x−n c−r + ξn+r y−n c−r − τn+r z−n c−r + ζn+r t−n c−r 12 12 + :ψn+r ξ−n :c−r + :τn+r ζ−n :c−r
(19)
and Qgh =
0 0 0 1 1 0 2 2 0 (r − n):br+n c−n c−r : + (r − 2n):br+n c−n c−r : + (r − 2n):br+n c−n c−r : 1 0 1 1 12 12 0 12 2 1 2 12 1 − 2n:br+n c−n c−r : + :br+n c−n c−r : + (r − n)br+n c−n c−r + br+n c−n c−r 2 1 0 2 2 1 12 2 + :br+n c−n c−r : − br+n c−n c−r . (20) 2
The operator Q satisfies QQ = 0, which can be checked by direct calculation. Thus, Hp is a cochain complex. The purpose of this paper is to calculate its cohomology, at least with some mild “genericity” restrictions on p. We will find it advantageous to introduce the following base change in the matter operators: 1 xn = √ (xn + iyn ), 2 1 zn = √ (zn + itn ), 2 1 ξn = √ (ξn + iψn ), 2 1 ζn = √ (ζn + iτn ), 2
1 yn = √ (xn − iyn ), 2 1 tn = √ (zn − itn ), 2 1 ψn = √ (ξn − iψn ), 2 1 τn = √ (ζn − iτn ). 2
(21)
This changes (12) to [xn , y−n ] = [zn , t−n ] = n,
[ξn , ψ−n ] = 1.
(22)
Again, all commutators of the new matter operator generators which are not permutations of (22) vanish. This transforms (19) to 0 0 Qm = 2:xn+r y−n :c−r + 2:zn+r t−n :c−r 0 0 − (2n + r):ξn+r ψ−n :c−r − (2n + r):ζn+r τ−n :c−r 1 1 1 1 + ξn+r y−n c−r + ψn+r x−n c−r + ζn+r t−n c−r + τn+r z−n c−r 2 2 2 2 + iξn+r y−n c−r − iψn+r x−n c−r + iζn+r t−n c−r − iτn+r z−n c−r 12 12 + i:ψn+r ξ−n :c−r + i:τn+r ζ−n :c−r .
This completes the definitions we need.
(23)
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4. A warm-up case: the N = 0 and N = 1 string BRST spectra In this section, we will introduce our main method, and will apply it to a simpler case: the N = 0 and N = 1 critical free strings. In these cases, the BRST cohomology has been calculated by Polchinski [13, Section 4.4]. The BRST complexes are as follows: In the N = 0 case, only K0 acts. Then Hp has a set of 26 sequences of generators (x1 )n , . . . , (x26)n , n ∈ Z, n < 0 and Qm =
26
0 :(xi )n+r (xi )−n :c−r .
i=1 n
The definition of Qgh changes by omitting all summands containing brI or crI with I = ∅, and adding the mass shell shift term −2ac00
(24)
with a = 1 (needed to get QQ = 0). To keep our notation, we shall put xn = (x25 )n , yn = (x26 )n , and define xn , yn by (21). In the N = 1 case, K1 acts. Then Hp has a set of 10 sequences of bosonic generators (x1 )n , . . . , (x10)n , n < 0, n ∈ Z, and 10 fermionic generators (ξ1 )r , . . . , (ξ10 )r , r < 0, r ∈ Z + 12 in the NS sector and r ∈ Z in the R sector, and one has Qm =
10
0 0 1 :(xi )n+r (xi )−n :c−r − n:(ξi )n+r (ξi )−n :c−r + (ξi )n+r (xi )−n c−r .
i=1 n
The definition of Qgh is got from (20) by omitting all summands containing brI or crI where I = ∅, {1} and adding the term (24) where a = 1/2 in the NS sector and a = 0 in the R sector. Again, to keep our notation, we put xn = (x9 )n , yn = (x10 )n , ξn = (ξ9 )n , ψn = (ξ10 )n and define xn , yn , ξn , ψn by (21). Now to calculate the BRST cohomology, we use a spectral sequence (see [8] for a general reference). To get a spectral sequence, we introduce a decreasing filtration F i on Hp (i.e., F i Hp ⊃ F i+1 Hp ). This is done by assigning a filtration degree to each generating operator as follows: |xn | = 1 |yn | = −1 0 b = −1, n
for n = 0, 0 c = 1 n
for n = 0, for all n,
and additionally in the N = 1 case: for all n, |ξn | = cn1 = 1 1 |ψn | = bn = −1 for all n. The other generating operators (xi )n , n 24 (N = 0 case) and (xi )n , (ξi )n , n 8 (N = 1 case) have filtration degrees 0. Now the filtration degree of a monomial in (xi )n , n 24, xn , yn (N = 0 case) and (xi )n , (ξi )n , i 8, xn , yn , ξn , ψn (N = 1 case) is the sum of filtration degrees of its factors,
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and the filtration degree of a linear combination of such monomials is the minimum of the filtration degrees of its summands. F i Hp is the submodule of all elements of filtration degree i. The main point now is that since the summands of Q have filtration degree 0, and also the relations (22) have filtration degree 0, so Q is compatible with filtration in the sense that QF i Hp ⊆ F i Hp . Now the E0 -term of the spectral sequence is E0 Hp , the associated graded object of Hp , which is isomorphic to Hp . The differential d0 is given by the summands of Q of filtration degree 0, which are 0 1 Q0 = (25) 2y0 xr c−r + x0 ψr c−r r=0
(omit the second summand in the N = 0 case). From this it follows that, assuming x0 , y0 = 0, up to scalar multiple we have 0 → x−r , d0 : b−r 1 d0 : b−r
(r = 0),
ψ−r → ,
0 d0 : y−r → c−r ,
d0 : ξ−r
(r = 0),
1 c−r →
(26)
(omit the last two differentials for the N = 0 case), and so d0 is a differentiation of algebras, i.e. d0 (uv) = d0 (u)v + (−1)d ud0 (v), where d is the cohomological degree of u. But (26) is the standard Koszul complex differential, and its cohomology cancels all the generators (26). Therefore, E1 Hp = Sym (xi )n , i = 1, . . . , 8 ⊗ Λ (ξi )n , i = 1, . . . , 8 ⊗ Λ c00 (27) in the N = 1 case and E1 Hp = Sym (xi )n , i = 1, . . . , 24 ⊗ Λ c00
(28)
in the N = 0 case (same restrictions on n as above). Now the differential d1 on E1 is given by terms of Q of filtration degree 1, which are 2:(xi )n (xi )−n :c00 Q1 = 2x0 y0 c00 − 2ac00 + iD−2 n 0 1 − 2n:(ξi )n (ξi )−n :c00 − 2n:bn0 c−n :c00 − 2n:bn1 c−n :c00 ,
where D = 10 for N = 1, D = 26 for N = 0, omit the ξ and effect of (29) is multiplication by D 2 xi c00 2w − 2a +
b1 -term
(29) for N = 0. Now the
(30)
i=1
where w is the weight of the monomial to which Q1 is being applied. Thus, E2 Hp is the summand of (27), (28) of weight w where 2w − 2a +
D i=1
xi2 = 0
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(the mass shell condition). Since E2 Hp is concentrated in filtration degree 0, 1, there is no room for higher differentials, and the spectral sequence collapses to E2 Hp , i.e., E2 Hp is the associated-graded object of the BRST cohomology. 5. The BRST cohomology of the N = 2 string To treat the case of N = 2, we will need to understand better the algebraic structure described in Section 3. Concretely, we shall consider the associative algebra A+ with generators (9) and relations (10). Even more mathematically precisely, A+ is a completion of this algebra in the sense that we allow infinite sums of monomials in normal order. Now the BRST complex at a given momentum p = (X0 , Y0 , Z0 , T0 ) is Hp = A+ /Ip where I = Ip is the completed left ideal (infinite sums of monomials in normal order allowed) spanned by brI , cnI , xn , yn , zn , tn , ξn , ψr , ζn , τr , r 0, n > 0, x0 − X0 , y0 − Y0 , z0 − Z0 , t0 − T0 . Our calculation will be performed under the assumption X0 , Y0 , Z0 , T0 = 0 (where these are defined from X0 , Y0 , Z0 , T0 by a formula analogous to (21)). For this reason, we will, instead of the algebra A+ , work with the algebra A = A+ (x0 )−1 , (y0 )−1 , (z0 )−1 , (t0 )−1 . Now the point of this discussion is that one can consider the BRST differential directly on A, where it has the form du = [Q, u],
u ∈ A.
This is a differential because [Q, Q] = 0, and additionally is a differentiation of algebras, i.e. d(u1 u2 ) = d(u1 )u2 + (−1)d1 u1 d(u2 )
(31)
where d1 is the cohomological degree (= ghost number) of u1 . The reason Q induces a compatible differential on A/I is that Q ∈ I . Now we shall introduce a complete decreasing filtration F i both on A and Hp = A/I which are compatible in the sense that the projection π : A → A/I
(32)
satisfies π(F i A) ⊆ F i A/I . Additionally, we will have Q ∈ F 0 A.
(33)
Now under these circumstances, one always has two accompanying spectral sequences on A and A/I , which are a spectral sequence of (non-commutative) differential graded (DG) algebras and differential graded modules (which means that formulas analogous to (31) in both the DG algebra and module sense will continue to hold for the higher terms Er and differentials dr of the spectral sequence). All this was true in the N = 0, N = 1 cases also,
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but in the N = 2 case, there are actual higher differentials to calculate, and the present technique is a tool for calculating differentials on non-linear monomias in Hp : we will see that such terms play a crucial role. The filtration is given both on A and A/I again by specifying filtration degrees of generators and then extending to all elements in an analogous manner as above. We put |x | = 1 (n = 0), 0n c = |ξ | = c1 = 1 for all n, n n n 0 b = |ψ | = b 1 = −1 for all n,
n n n |y | = −1 for n = 0, 2n 12 2 12 c = c = 2, b = b = −2 n n n n
for all n.
zn , tn , ζn , τn
The remaining matter generators have filtration degree 0. Simple verification shows that (33) indeed holds. To start out the calculation of the spectral sequence, the d0 differential is, again, given by [Q0 , ?] where Q0 is the sum of the terms of Q of filtration degree 0. We see from (20), (23) that (25) still holds, so d0 is still given by (26), and d0 acts as an algebra differential on monomials of normal order. Thus, both E0 (A), E0 (A/I ) are Koszul complexes. This shows that the cohomology E1 (A) is the algebra defined in the same way as A, except only on the generators zn , tn , ζn , τn , bn12 , cn12 , br2 , cr2 , c00 .
(34)
Also, E1 (A/I ) is the quotient of E1 (A) by the ideal defined in the same way as I , except on the generators (34) only. In other words, additive bases of E1 (A), E1 (A/I ) (in the completed sense in the case of E1 (A)) are monomials in normal order in the generators (34), with the same subscript restrictions as in A, A/I . As before, d1 is now given by commutator with the summands of Q of filtration degree 1, the non-zero terms of which are 0 0 c00 − 2nζn τ−n c00 − 2nbn0 c−n c0 Q1 = 2x0 y0 c00 + 2zn t−n 1 0 2 0 12 0 − 2nbn1 c−n c0 − 2nbn2 c−n c0 − 2nbn12c−n c0 .
The effect on an element u ∈ A, A/I is, again, left multiplication by 2w(u) + x02 + y02 + z02 + t02 c00 ,
(35)
where w is the weight. Therefore, E2 (A) is simply the sum of the kernel and cokernel of (35). In E2 (A/I ), one can say more explicitly that (35) leaves intact precisely all terms of E1 (A/I ) which have 1 2 X0 + Y02 + Z02 + T02 = 0 2 and kills all others; this is, again, the mass shell condition. Now to calculate d2 , we shall use the full force of our algebra machinery. First, let us recall how a d2 (or any higher dn ) is calculated in a spectral sequence associated with a decreasing filtration of a cochain complex C: we select an element u ∈ C whose pq representative in E0 C supports no differentials d0 , . . . , dn−1 , and hence survives to En C. w(u) +
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Then we find an element v ∈ F >p C such that d(u + v) ∈ F p+n C. Then the representative of d(u + v) in En C is dn u. The element v can be chosen to be a linear combination of elements whose representatives in En C have filtration degrees p + 1, . . . , p + n − 1 (i.e., only p + 1 for n = 2). Summands of dn (u) can come both from u or v; the element v will be called the counterterm. We claim that the d2 -differential on E2 (A) is given by 12 2 iτ−n cr+n − it−n cr+n , d2 τr = (36) 2 12 izs cr−s − iζs c−s+r , d2 ζr = (37) 12 2 2iζn+r t−n + i(r − n)br+n c−n , d2 br2 = (38) 2 2 i:τn+r ζ−n : − :ic−n br+n :, d2 br12 = (39) 2 irζn+r c−n , d2 zr = − (40) 2 irτn+r c−n , d2 tr = (41) d2 cr12 = d2 cr2 = 0.
(42)
Recall that (E2 (A), d2 ) is a (non-commutative) DGA, so this determines the value of d2 on any monomial. Further, since E2 (A/I ) is a quotient of E2 (A), the differential on E2 (A/I ) is induced. To prove (36)–(42), we use the procedure described above. To calculate d2 τr , consider commutators of τr with all summands Q2 of Q of filtration degree 2. The summands of Q2 which give non-trivial commutators are 0 1 2 12 (r − n)ζ−r τ−n cr+n + ζ−r t−n cr+n + iζ−r t−n cr+n + iτn ζ−r cn+r ,
which gives the commutator 0 1 2 12 −(r − n)τ−n cr+n − t−n cr+n − it−n cr+n + iτn cn+r .
(43)
The last two terms, of filtration degree 2, survive to (36). The first two terms of (43) are of filtration degree 1, and therefore must be cancelled by a counterterm. The counterterm is r −n 1 τ−n (44) xr+n − t−n ξn+r . 2x0 (r + n) x0 (Note that c00 is mass-shell related, so we may assume r + n = 0.) Now we must correct our candidate of d2 τr by adding the commutator of the sum Q1 of terms of Q of filtration −
degree 1 with (44). But one easily checks that every monomial thus obtained contains one of the matter generators ξn , xn as a factor, and thus vanishes in E2 (A). Thus, the counterterm contributes nothing in this case. (37) is analogous. In the case of (38), the summands of Q2 giving non-trivial commutators are 2 2 2 y−n c−r − iψn+r x−n c−r + iζn+r t−n c−r iξn+r
2 12 1 2 2 0 2 − iτn+r z−n c−r + (n − r)br+n c−n c−r − (n − 2r)br+n c−n c−r ,
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giving the commutator iξn+r y−n − iψn+r x−n + iζn+r t−n
12 1 2 0 − iτn+r z−n + (n − r)br+n c−n − (n − 2r)br+n c−n .
(45)
12 c 1 , b 2 c 0 require counterterms which The first term is in F0 A, and dies. The terms br+n −n r+n −n 12 ξ , b 2 x which, when commuted with Q , die similarly are scalar multiples of br+n 1 −n r+n −n x is in F A and hence dies, unless n = 0. In that case, as above. Also, the term −iψn+r 0 −n we have −iψr x0 of filtration degree −1, which requires adding the counterterm
ibr1 .
(46)
When computing the commutator of Q1 with (46), the terms giving a non-trivial commutator are 1 1 1 iξn+r y−n c−r + iψn+r x−n c−r + iζn+r t−n c−r
1 1 0 1 + iτn+r z−n c−r − i(n − 2r):br+n c−n c−r : 1 0 1 1 12 2 1 + i :br+n c−n c−r : + i(r − n)br+n c−n c−r . 2 From the resulting commutator with (46), the terms which survive to E2 (A) are 12 2 iζn+r t−n + iτn+r z−n + i(r − n)br+n c−n
which, when added to the third and fourth term of (45), gives (38). To get (39), the terms of Q2 which produce a non-trivial commutator with br12 are 12 12 12 12 0 2 12 1 i:ψn+r ξ−n :c−r + i:τn+r ζ−n :c−r − 2rbr+n c−r c−n + br+n c−r c−n ,
producing the commutator 12 0 2 1 ξ−n : + i:τn+r ζ−n : + 2rbr+n c−n − br+n c−n . i:ψn+r
(47)
The first summand of (47) dies, the second survives to E2 (A). The last two terms must be cancelled by counterterms (n = 0 in the third term to avoid the mass shell term). The counterterms are r 12 b x (n = 0) x0 n r+n −n 1 2 ξ−n . + br+n (48) x0 The first term of (48), when commuted with Q1 , dies in E2 A as above, while the second term has non-trivial commutator with the summands of Q1 1 2 ψn xs c−s−n − iψn xs c−n−s . The only cases which survive are s = 0 in the first summand (which we need for cancelling the relevant term of (47), and s = 0 in the second summand, which leads to 2 2 bn+r :. −i:c−n
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Adding this to the second term of (47) gives (39). To get (40), the terms of Q2 producing commutators with zr are 1 2 2zn+r t−r cn0 + ζn+r t−r c−n + iζn+r t−r c−n ,
leading to the commutators 0 1 2 2rzn+r c−n − rζn+r c−n − irζn+r c−n .
(49)
The last term survives to (40), the first term for n = 0 and second term need counterterms , ζ ξ , but both of these when commuted with Q which are multiples of zn+r x−n 1 n+r −n only produce terms which die in E2 (A) because of matter factors of the form x or ξ . are Concretely, the summands of Q1 producing a non-trivial commutator with zn+r x−n 1 2 + iξs yn c−s−n ξs yn c−n−s
leading to 1 2 zn+r − inξs c−s−n zn+r , −nξs c−n−s ξ are and summands of Q1 producing a non-trivial commutator with ζn+r −n 0 1 + ψn xs c−n−s , −(s − n)ξs ψn c−s−n
where −s − n = 0 in the first summand and s = 0 in the second summand. This produces 0 1 −(s − n)ξs c−s−n ζn+r + xs c−n−s ζn+r
which dies as promised. This proves (40). (41) is analogous. For (42), only ghost summands of Q give non-trivial commutators with cr2 , cr12 . Some counterterms are needed to cancel the commutators, but none produce non-zero terms in d2 . Now by looking at (36)–(42), it is not immediately obvious what the cohomology is. To settle that, we will introduce an additional filtration on (E2 (A), d2 ), (E2 (A/I ), d2 ) and use the resulting spectral sequence(!). To simplify notation, set δ := d2 . The filtration follows a now familiar pattern: it is a decreasing filtration Φ defined as above by specifying filtration degrees of generators. The formula is 2 12 b = b = |ζ | = −1, |zn | = −1 if n = 0, r r r 2 12 c = c = |τ | = 1, |t | = 1 if n = 0. r
r
r
n
δΦ i
⊆ Φi ,
One easily checks that so the filtration is compatible with the differential. As above, the differential δ0 on E0 E2 (A) is given by the degree 0 summands of δ. Comparing with (36)–(42), we see that δ0 τr = −t0 cr2 ,
δ0 br2 = 2it0 ζr ,
and all other δ0 ’s on generators are 0. Thus both (E0 E2 (A), δ0 ), (E0 E2 (A/I ), δ0 ) are Koszul complexes, and their cohomologies E1 E2 (A), E1 E2 (A/I ) have as additive bases the sets of normal-ordered monomials in zn , tn , bn12 , cn12 with the same values of the indices
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65
as in the definition of A, A/I . Moreover, E1 E2 (A) is an algebra (same relations as A), and E1 E2 (A/I ) is its factor by a left ideal. Now one sees that δ1 = 0, so E2 E2 (A) = E1 E2 (A), E2 E2 (A/I ) = E1 E2 (A/I ). Now let us calculate δ2 (zr ), by the same method as above. Thus, we apply δ first and get 2 c−n . −ir ζn+r But this is of filtration degree 0, so we must add the counterterm r :. − :ζn+r τ−n t0
(50)
Note that if and only if both n + r, −n have the same sign (are both non-positive or nonnegative, which happens for finitely many but more than 0 terms, which in addition have the same sign and hence cannot cancel), this will produce a non-zero multiple of cr12 by using commutators to bring the δ of (50) to normal order (see (36), (37)). Thus, we see that δ2 (zr ) = Cc12 ,
C = 0.
(51)
Now consider δ2 (br12 ). Applying δ first, we obtain 2 2 ζ−n : − :c−n br+n :. i :τn+r This is cancelled by a non-zero multiple of the counterterm 2 τ−n bn+r . b 2 ), by bringing But as above, we see that when n + r, −n have the same sign, δ(τ−n n+r terms to normal order, produces non-zero multiples (with the same sign) of tr . Thus, we see that δ2 br12 = Dtr , D = 0. (52)
Now (51), (52) show that (E2 E2 (A), δ2 ), (E2 E2 (A/I ), δ2 ) are again Koszul complexes, and that their cohomology is Λ c012 ⊗ Ker(d1 ) ⊕ Coker(d1 ) · c0 , (53) 12 0 2 2 2 2 Λ c0 , c0 if x0 + y0 + z0 + t0 = 0 and 0 else, (54) respectively. We see that the δ spectral sequences and also the d-spectral sequences must collapse to (53), (54), for the only remaining terms are in filtration degrees 0, 1, 2. This is what we claimed.
Appendix A. A few low weight explicit differentials It is quite difficult to write down explicitly all the N = 2 BRST differentials on Hp even in the lowest non-trivial weight w = 1 because of proliferation of counterterms. To illustrate the difficulty, and the great simplification the spectral sequence provides, we list, as an example, some (not all) of the differentials in weight 1 in the NS sector. We use the
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matter generators (9); the generators (21) were introduced to work better with the filtration, but the generators (9) give more symmetric explicit formulas. We have 1 0 1 2 2 = −2 b−1/2 c−1/2 + b−1/2 c−1/2 − x0 x−1 − y0 y−1 − z0 z−1 − t0 t−1 , Qb−1 1 1 2 2 0 c−1/2 − b−1/2 c−1/2 , Q x0 y−1 − y0 x−1 + z0 t−1 − t0 z−1 − b−1/2 = 4c−1 12 2 1 1 2 Qb−1 = ψ−1/2 ξ−1/2 + τ−1/2 ζ−1/2 + b−1/2 c−1/2 − b−1/2 c−1/2 , 12 Q(ξ−1/2 ψ−1/2 − x0 y−1 + y0 x−1 ) = Q(ζ−1/2 τ−1/2 − z0 t−1 + t0 z−1 ) = c−1 , 2 1 2 1 Q z0 b−1/2 ζ−1/2 + b−1/2τ−1/2 − t0 −b−1/2 τ−1/2 + b−1/2ζ−1/2 2 2 1 1 − x0 b−1/2 ξ−1/2 + b−1/2 ψ−1/2 + y0 −b−1/2 ψ−1/2 + b−1/2 ξ−1/2 12 1 2 2 2 − c012 b−1/2 − b−1/2 − z02 + t02 − x02 + y02 b−1
= 2(ξ−1/2 ψ−1/2 − ζ−1/2 τ−1/2 − x0 y−1 + y0 x−1 + z0 t−1 − t0 z−1 ).
References [1] A. Bilal, BRST approach to the N = 2 superconformal algebra, Phys. Lett. B 180 (3) (1986) 255. [2] R. Bogojevic, Z. Hlousek, The BRST quantization of the O(2) string, Phys. Lett. B 179 (1986) 69. [3] P. Deligne, J.W. Morgan, Notes on supersymmetry (following J. Bernstein), in: Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, Amer. Math. Society and the Inst. for Advanced Study, 1999, pp. 41– 98. [4] J. Distler, P. Nelson, Semirigid supergravity, Phys. Rev. Lett. 66 (5) (1991) 1955. [5] D. Fatori, V.G. Kac, Classification of finite simple Lie conformal superalgebras, J. Algebra, in press. [6] B.R. Greene, hep-th/9702155. [7] I. Kriz, On the N -superconformal algebra, in preparation. [8] J. McCleary, A User’s Guide Spectral Sequences, in: Cambridge Stud. Adv. Math., Vol. 58, Cambridge Univ. Press, Cambridge, 2001. [9] S. Mathur, S. Mukhi, Becchi–Rouet–Stora–Tyutin quantization of twisted extended fermionic string, Phys. Rev. D 36 (2) (1987) 465. [10] S. Mathur, S. Mukhi, The N = 2 fermionic string: path integral, spin structures and supermoduli on the torus, Nucl. Phys. B 302 (1988) 130–148. [11] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Cambridge Univ. Press, Cambridge, 1987. [12] H. Ooguri, C. Vafa, Geometry of N = 2 strings, Nucl. Phys. B 361 (1991) 469–518. [13] J. Polchinski, String Theory, Vol. 1, Cambridge Univ. Press, Cambridge, 1999.
The BRST cohomology of the N = 2 string I. Kriz 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA Received 18 July 2003; accepted 27 August 2003
Abstract In this paper, we prove by rigorous calculation that BRST cohomology and old covariant quantization give the same spectra for the N = 2-supersymmetric string. This confirms an old conjecture, but the precise mechanism by which it happens is more subtle than assumed, and may shed new light on BRST at extended supersymmetry. 2003 Elsevier B.V. All rights reserved. PACS: 11.25.Hf; 11.25.Mj; 11.30.Pb; 04.65.+e
1. Introduction The N = 2 string (see, e.g., [1,2,9–12]) has 4 real bosons and fermions, coupled with N = 2 supergravity (1 graviton, 2 gravitinos and a U (1)-gauge field). Old covariant quantization (OCQ) for the N = 2 string was carried out, e.g., in [10–12] and gives a physical spectrum which has only one state for each momentum on the mass shell. The BRST complex for this theory was developed in [1,2,9]. However, the cohomology of that complex was never calculated directly. In this paper, we show that the N = 2 BRST cohomology gives the same spectrum as OCQ up to ghost degeneracy similarly as is the case of N = 0 and N = 1 strings, thus confirming a long standing conjecture. Despite the analogous nature of the results, computing the BRST cohomology of the N = 2 string is quite different from the N = 0 and N = 1 cases: at N = 0 and N = 1, (as shown by Polchinski [13, Section 4.4]), the ghost and antighost mode towers are matched by the BRST differential precisely with one tower of bosons (and fermions for N = 1) E-mail address: [email protected] (I. Kriz). 1 The author was supported by NSF grant DMS 0305853.
0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.08.035
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each, thus cancelling two towers of bosons (and fermions), which gives the lightcone gauge spectrum. When we try to argue the same way for the N = 2 string, however, we arrive at a puzzle. Just as in the N = 0 and N = 1 cases, Virasoro ghosts/antighosts cancel two bosonic towers, and the two pairs of supersymmetry ghosts/antighosts cancel the four fermionic towers. But then we are left with the two remaining bosonic towers and the ghost/antighost pair coming from the U (1)–R-symmetry. There is no way analogous to the N = 0 and N = 1 cases of cancelling the U (1)-ghosts/antighosts with bosons, since the R-symmetry acts trivially on bosons! This, in fact, originally led the author to erroneously believe that the BRST cohomology of the N = 2 string was intrinsically different from its OCQ spectrum, and that the BRST spectrum had more modes. It turns out, however, that the remaining bosons and U (1)-ghosts do cancel for a more subtle reason; consequently, the BRST and OCQ spectra of the N = 2 are the same up to 4 degenerate ghost 0-modes at each chirality, as expected. The cancellation mechanism involves a complicated interplay of counterterms which certainly does not seem to have been known before. In fact, the N = 2 BRST differential has 27 terms (23 after a transformation), and calculating general differentials directly seems almost impossible (for illustration, just a few explicit examples of lowest weight differentials are given in Appendix A). In fact, the author knows no way of rigorously computing the N = 2 BRST cohomology without a spectral sequence. Introducing this technique, and carefully showing how it applies to the N = 2 BRST case, is the main purpose of this paper. The present paper is organized as follows: in Section 2, we review the superconformal algebra of constraints KN and introduce a notation which is well suited for what follows. In Section 3, we write down explicitly the N = 2 BRST complex in our notation. In Section 4, we calculate, as a warm-up case, the BRST cohomology of the N = 0 and N = 1 strings (which is essentially only a slight simplification of Polchinski’s argument [13, Section 4.4]). Our main result, the calculation of the BRST cohomology of the N = 2 string, is given in Section 5.
Notation For our purposes, it is crucial to reconcile mathematical and physical notation. For a physical notation closest to our purposes, we refer the reader to [2]. We give here a short dictionary between the mathematical terms of the present paper and the language of [2]. The action constraints of [2, Eqs. (2.3a)–(2.3c)] are (super-versions of ) Euler equations of the corresponding variation problem; they determine the classical solution. In the quantized case, the constraints determine the energy-stress tensor, or, as one says in mathematics, the superconformal algebra action. In our case, the superconformal algebra is called K2 . We review the superconformal algebras KN , but see [5] for a more detailed mathematical discussion. In [2], the constraints in the present case are denoted by Θ, S i , T and later, in the same order, L, Gi , T . This notation coincides with the notation used today (cf. [6]), except there T is denoted by J . We use the notation G0 , G1 , G2 , G{12} (the exact translation is in the next section).
I. Kriz / Nuclear Physics B 671 (2003) 51–66
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The massive mode generating fields are, in [2], denoted by α i,a (bosons) and bi,a (fermions), i, a = 1, 2. In this paper, D = 4 (real) plays such a special role that we name these fields by individual letters x, y, z, t (bosons) and ξ , ψ, ζ , τ (fermions). The ghost generating fields are, in [2], denoted by λ, ξ 1 , ξ 2 , η and the antighost fields ¯ From our point of view, these fields are a part of a general are denoted by λ¯ , ξ¯ 1 , ξ¯ 2 , η. picture, and we call the ghost fields c12 , c1 , c2 , c0 and the antighost fields b12 , b 1 , b 2 , b 0 (in the same order).
2. A review of the N -superconformal Lie algebra KN We begin with a review of the N -superconformal algebra KN (see [4–7,11]). As a superLie algebra, KN can be described as the free complex vector super-space on basis GIr where I ⊂ {1, . . . , N}, r ≡ !I2 mod 1 in the NS sector and r ∈ Z in the Ramond sector (!I is the number of elements I ). It saves some sign conventions if we identify a sequence I = (i1 < · · · < ik ) with the set |I | = {i1 , . . . , ik }, and extend the definition of GIr to nonincreasing sequences by Gσr (I ) = sign(σ )GIr for a permutation σ on {1, . . . , N}. Now the super-Lie bracket on KN is defined as follows (we always write [ ] for the Lie bracket, regardless of parity): For I = (i1 , . . . , ik ), J = (j1 , . . . , j& ), |I | ∩ |J | = ∅, I J = (i1 , . . . , ik , j1 , . . . , j& ), I J J Gr , Gs = r(2 − &) − s(2 − k) GIr+s . For I , J as above, but with |I | ∩ |J | = {ik } = {j1 }, I ⊕ J = (i1 , . . . , ik−1 , j2 , . . . , j& ), I J I ⊕J Gr , Gs = Gr+s . When !(|I | ∩ |J |) > 1, I J Gr , Gs = 0. We will be mostly interested in the case N = 2, where these symbols relate to standard {i} {12} notation [6] by G∅r = 2Lr , Gr = Gir , Gr = iJr . However, even there we will see a benefit to having a systematic approach to the description of the Lie bracket. The superalgebra KN can be thought of as the finite energy part of the superalgebra of infinitesimal deformations of N -super-conformal structure on a boundary component of an N -superconformal surface (cf. [7]). We recall that an N -superconformal surface is an (1|N)-dimensional complex supermanifold which possesses an atlas consisting of coordinate patches (z|θ1 , . . . , θN ) satisfying the following equations: consider the N -tuple of differential operators D = (D1 , . . . , DN )T , where Dk =
∂ ∂ + θk , ∂θk ∂z
k = 1, . . . , N.
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Considering another set of local coordinates (˜z|θ˜1 , . . . θ˜N ), we have N N ∂ ˜ ˜ ˜ ˜ . θj Di θj (Di θj )Dj + Di z˜ − Di = ∂ z˜ j =1
j =1
The condition is that ˜ D = M D, where Mij = Di θ˜j , which leads to the equations Di z˜ −
N
θ˜j Di θ˜j = 0,
i = 1, . . . , N.
(1)
j =1
To find possible infinitesimal deformations of coordinates, we must consider the linearized equations (1), which are Di z˜ = θ˜i +
N
θj Di θ˜j .
(2)
j =1
Solutions of the linearized equations are (by definition) vector fields. One finds that a (topological) basis of the space of solutions, written in vector field notation, is of the form GI
n+ |I2| −1
= (|I | − 2) · zn θI
|I |−1 N ∂ ∂ ∂ + zn (−1)j θIk−j − nzn−1 θI θi , ∂z ∂θik−j ∂θi j =0
j =1
(3)
where θ(i1 ,...,ik ) = θ1 · · · θk and, for I = (i1 , . . . , ik ), Ij = (i1 , . . . , iˆj , . . . , ik ). These choices of GIr ’s obey the Lie relations stated above. Here the numbers n which occur vary with the boundary conditions imposed in the super-directions, although they are always of the form n ∈ Z + αI for some numbers αI . We shall not need a detailed analysis of the boundary conditions: in this paper, we will restrict attention to the R and NS sectors described in the beginning of this section. 3. The N = 2 BRST complex In this section, we shall describe the BRST complex Hp of the N = 2 free string at a given momentum p. Here and from now on we shall work in the chiral theory, which can be interpreted as calculating the spectrum of the free open N = 2 string. In the closed string case, both chiralities are present, which in our language means simply calculating the cohomology of a tensor product of two copies of Hp , which follows by the Künneth theorem.
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In the chiral case, Hp is a Z/2 × Z/2-graded commutative algebra. This means that two homogeneous elements u, v of bidegrees (d1 , d2 ) and (e1 , e2 ), di , ei ∈ Z/2, satisfy uv = (−1)d1 e1 +d2 e2 vu.
(4)
This is the Deligne–Morgan sign convention [3]. In fact, Hp , where p is a quadruple of numbers (X0 , Y0 , Z0 , T0 ), is a free (Z/2 × Z/2)-graded commutative algebra on generators xn , yn , zn , tn
of bidegree (0, 0), n < 0, n ∈ Z,
ξr , ψm , ζr τm ,
of bidegree (0, 1),
(5)
r 0,
m < 0, 1 in the NS sector and r, m ∈ Z in the R sector, r, m ∈ Z + 2 0 12 bn0 , cm , bn12, cm 1 2 br1 , cm , br2 , cm
of bidegree (1, 0), n < 0, m 0,
(6) (7)
of bidegree (1, 1),
m 0, 1 in the NS sector and r, m ∈ Z in the R sector. r, m ∈ Z + (8) 2 The generators (5), (6) are called matter generators and the generators (7), (8) are called Fadeev–Popov ghost generators. Additional numerical invariants we need to keep track of are the weight which on a monomial in the generators (5)–(8) is the sum of the negatives of the subscripts and the ghost number which is 0 on matter generators and −1, 1 on the b’s and c’s, respectively. Now to construct operators Hp → Hp , one introduces generating operators r < 0,
xn , yy , zn , tn , ξr , ψr , ζr , τr , bn0 , cn0 , br1 , cr1 , br2 , cr2 , bn12 , cn12 ,
(9)
in the NS sector and r ∈ Z in the R sector without where n ∈ Z, r ∈ Z + positivity restrictions. The operator generators whose indices satisfy the (non)-positivity requirements of (5)–(8) act on Hp by multiplication on the left (and thus can be naturally identified with the corresponding generators of Hp ). The remaining operator generators act by 0 on the vacuum 1 with the exception of x0 , y0 , z0 , t0 , which act as multiplication by numbers X0 , Y0 , Z0 , T0 . The action of operator generators on the entire Hp is then determined by the operator commutation relations 1 2
[xn , x−n ] = [yn , y−n ] = [zn , z−n ] = [tn , t−n ] = n, [ξr , ξ−r ] = [ψr , ψ−r ] = [ζr , ζ−r ] = [τr , τ−r ] = 1, I I cr , b−r = 1,
(10)
and all other commutators of the generators (9) which are not permutations of (10) are 0. Here the bidegrees of operator generators are specified by the same formulas as for the generators of Hp , and the commutator is defined on elements a, b of bidegrees (d1 , d2 )
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and (e1 , e2 ) as [a, b] = ab − (−1)d1 d2 +e1 e2 ba.
(11)
Physically, a structure of the kind just described naturally arises in quantization. Mathematically, the construction also has a better explanation, which will be given in Section 5 below. For now, it is important that the superconformal algebra K2 acts on Hp . It turns out helpful to write its operators as GIr = m GIr + gh GIr
(12)
where m stands for matter and gh stands for ghost. The formula for the BRST differential is then 1 I I I I Q = Qm + Qgh , Qm = (13) m Gr c−r , Qgh = gh Gr c−r . 2 I,r
We have 0 m Gr
=
I,r
:xn+r x−n : + :yn+r y−n : + :zn+r z−n : + :tn+r t−n :
n
1 m Gr
− n:ξn+r ξ−n : − n:ψn+r ψ−n : − n:ζn+r ζ−n : − n:τn+r τ−n :, = ξn+r x−n + ψn+r y−n + ζn+r z−n + τn+r t−n ,
(14) (15)
n
2 m Gr
=
−ψn+r x−n + ξn+r y−n − τn+r z−n + ζn+r t−n ,
(16)
n
12 m Gr
=
n
I gh Gr
=
:ψn+r ξ−n : + :τn+r ζ−n :,
J . GIr , bsJ c−s
(17) (18)
J,s
Here : : denotes the operation of normal ordering. A monomial is said to be in normal order if all its generators with negative subscripts are to the left of all generators with positive subscripts. Then : : is the operation of bringing a monomial to normal order using the relations (4), not (10). In (18), the symbol [GIr , bsJ ] is defined by computing [GIr , GJs ] and replacing G’s by b’s in the result. The formula (13) is then a semiinfinite version of Lie algebra cohomology, although it hides anomaly cancellation and mass-shell shift cancellation which one has to deal with in the quantum case (in particular, mass-shell shifts are trivial for the N = 2 string). In the present paper, the only thing we will use is that the resulting formula is consistent; for reasoning on the criticality of the D = 4 N = 2 string, and mass-shell shift cancellations in this case, we refer the reader to [11]. Now explicitly, (14)–(18) give 0 0 0 0 :xn+r x−n :c−r + :yn+r yn :c−r + :zn+r z−n :c−r + :tn+r t−n :c−r Qm = 0 0 0 0 − n:ξn+r ξ−n :c−r − n:ψn+r ψ−n :c−r − n:ζn+r ζ−n :c−r − n:τn+r τ−n :c−r 1 1 1 1 + ξn+r x−n c−r + ψn+r y−n c−r + ζn+r z−n c−r + τn+r t−n c−r
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2 2 2 2 − ψn+r x−n c−r + ξn+r y−n c−r − τn+r z−n c−r + ζn+r t−n c−r 12 12 + :ψn+r ξ−n :c−r + :τn+r ζ−n :c−r
(19)
and Qgh =
0 0 0 1 1 0 2 2 0 (r − n):br+n c−n c−r : + (r − 2n):br+n c−n c−r : + (r − 2n):br+n c−n c−r : 1 0 1 1 12 12 0 12 2 1 2 12 1 − 2n:br+n c−n c−r : + :br+n c−n c−r : + (r − n)br+n c−n c−r + br+n c−n c−r 2 1 0 2 2 1 12 2 + :br+n c−n c−r : − br+n c−n c−r . (20) 2
The operator Q satisfies QQ = 0, which can be checked by direct calculation. Thus, Hp is a cochain complex. The purpose of this paper is to calculate its cohomology, at least with some mild “genericity” restrictions on p. We will find it advantageous to introduce the following base change in the matter operators: 1 xn = √ (xn + iyn ), 2 1 zn = √ (zn + itn ), 2 1 ξn = √ (ξn + iψn ), 2 1 ζn = √ (ζn + iτn ), 2
1 yn = √ (xn − iyn ), 2 1 tn = √ (zn − itn ), 2 1 ψn = √ (ξn − iψn ), 2 1 τn = √ (ζn − iτn ). 2
(21)
This changes (12) to [xn , y−n ] = [zn , t−n ] = n,
[ξn , ψ−n ] = 1.
(22)
Again, all commutators of the new matter operator generators which are not permutations of (22) vanish. This transforms (19) to 0 0 Qm = 2:xn+r y−n :c−r + 2:zn+r t−n :c−r 0 0 − (2n + r):ξn+r ψ−n :c−r − (2n + r):ζn+r τ−n :c−r 1 1 1 1 + ξn+r y−n c−r + ψn+r x−n c−r + ζn+r t−n c−r + τn+r z−n c−r 2 2 2 2 + iξn+r y−n c−r − iψn+r x−n c−r + iζn+r t−n c−r − iτn+r z−n c−r 12 12 + i:ψn+r ξ−n :c−r + i:τn+r ζ−n :c−r .
This completes the definitions we need.
(23)
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4. A warm-up case: the N = 0 and N = 1 string BRST spectra In this section, we will introduce our main method, and will apply it to a simpler case: the N = 0 and N = 1 critical free strings. In these cases, the BRST cohomology has been calculated by Polchinski [13, Section 4.4]. The BRST complexes are as follows: In the N = 0 case, only K0 acts. Then Hp has a set of 26 sequences of generators (x1 )n , . . . , (x26)n , n ∈ Z, n < 0 and Qm =
26
0 :(xi )n+r (xi )−n :c−r .
i=1 n
The definition of Qgh changes by omitting all summands containing brI or crI with I = ∅, and adding the mass shell shift term −2ac00
(24)
with a = 1 (needed to get QQ = 0). To keep our notation, we shall put xn = (x25 )n , yn = (x26 )n , and define xn , yn by (21). In the N = 1 case, K1 acts. Then Hp has a set of 10 sequences of bosonic generators (x1 )n , . . . , (x10)n , n < 0, n ∈ Z, and 10 fermionic generators (ξ1 )r , . . . , (ξ10 )r , r < 0, r ∈ Z + 12 in the NS sector and r ∈ Z in the R sector, and one has Qm =
10
0 0 1 :(xi )n+r (xi )−n :c−r − n:(ξi )n+r (ξi )−n :c−r + (ξi )n+r (xi )−n c−r .
i=1 n
The definition of Qgh is got from (20) by omitting all summands containing brI or crI where I = ∅, {1} and adding the term (24) where a = 1/2 in the NS sector and a = 0 in the R sector. Again, to keep our notation, we put xn = (x9 )n , yn = (x10 )n , ξn = (ξ9 )n , ψn = (ξ10 )n and define xn , yn , ξn , ψn by (21). Now to calculate the BRST cohomology, we use a spectral sequence (see [8] for a general reference). To get a spectral sequence, we introduce a decreasing filtration F i on Hp (i.e., F i Hp ⊃ F i+1 Hp ). This is done by assigning a filtration degree to each generating operator as follows: |xn | = 1 |yn | = −1 0 b = −1, n
for n = 0, 0 c = 1 n
for n = 0, for all n,
and additionally in the N = 1 case: for all n, |ξn | = cn1 = 1 1 |ψn | = bn = −1 for all n. The other generating operators (xi )n , n 24 (N = 0 case) and (xi )n , (ξi )n , n 8 (N = 1 case) have filtration degrees 0. Now the filtration degree of a monomial in (xi )n , n 24, xn , yn (N = 0 case) and (xi )n , (ξi )n , i 8, xn , yn , ξn , ψn (N = 1 case) is the sum of filtration degrees of its factors,
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and the filtration degree of a linear combination of such monomials is the minimum of the filtration degrees of its summands. F i Hp is the submodule of all elements of filtration degree i. The main point now is that since the summands of Q have filtration degree 0, and also the relations (22) have filtration degree 0, so Q is compatible with filtration in the sense that QF i Hp ⊆ F i Hp . Now the E0 -term of the spectral sequence is E0 Hp , the associated graded object of Hp , which is isomorphic to Hp . The differential d0 is given by the summands of Q of filtration degree 0, which are 0 1 Q0 = (25) 2y0 xr c−r + x0 ψr c−r r=0
(omit the second summand in the N = 0 case). From this it follows that, assuming x0 , y0 = 0, up to scalar multiple we have 0 → x−r , d0 : b−r 1 d0 : b−r
(r = 0),
ψ−r → ,
0 d0 : y−r → c−r ,
d0 : ξ−r
(r = 0),
1 c−r →
(26)
(omit the last two differentials for the N = 0 case), and so d0 is a differentiation of algebras, i.e. d0 (uv) = d0 (u)v + (−1)d ud0 (v), where d is the cohomological degree of u. But (26) is the standard Koszul complex differential, and its cohomology cancels all the generators (26). Therefore, E1 Hp = Sym (xi )n , i = 1, . . . , 8 ⊗ Λ (ξi )n , i = 1, . . . , 8 ⊗ Λ c00 (27) in the N = 1 case and E1 Hp = Sym (xi )n , i = 1, . . . , 24 ⊗ Λ c00
(28)
in the N = 0 case (same restrictions on n as above). Now the differential d1 on E1 is given by terms of Q of filtration degree 1, which are 2:(xi )n (xi )−n :c00 Q1 = 2x0 y0 c00 − 2ac00 + iD−2 n 0 1 − 2n:(ξi )n (ξi )−n :c00 − 2n:bn0 c−n :c00 − 2n:bn1 c−n :c00 ,
where D = 10 for N = 1, D = 26 for N = 0, omit the ξ and effect of (29) is multiplication by D 2 xi c00 2w − 2a +
b1 -term
(29) for N = 0. Now the
(30)
i=1
where w is the weight of the monomial to which Q1 is being applied. Thus, E2 Hp is the summand of (27), (28) of weight w where 2w − 2a +
D i=1
xi2 = 0
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(the mass shell condition). Since E2 Hp is concentrated in filtration degree 0, 1, there is no room for higher differentials, and the spectral sequence collapses to E2 Hp , i.e., E2 Hp is the associated-graded object of the BRST cohomology. 5. The BRST cohomology of the N = 2 string To treat the case of N = 2, we will need to understand better the algebraic structure described in Section 3. Concretely, we shall consider the associative algebra A+ with generators (9) and relations (10). Even more mathematically precisely, A+ is a completion of this algebra in the sense that we allow infinite sums of monomials in normal order. Now the BRST complex at a given momentum p = (X0 , Y0 , Z0 , T0 ) is Hp = A+ /Ip where I = Ip is the completed left ideal (infinite sums of monomials in normal order allowed) spanned by brI , cnI , xn , yn , zn , tn , ξn , ψr , ζn , τr , r 0, n > 0, x0 − X0 , y0 − Y0 , z0 − Z0 , t0 − T0 . Our calculation will be performed under the assumption X0 , Y0 , Z0 , T0 = 0 (where these are defined from X0 , Y0 , Z0 , T0 by a formula analogous to (21)). For this reason, we will, instead of the algebra A+ , work with the algebra A = A+ (x0 )−1 , (y0 )−1 , (z0 )−1 , (t0 )−1 . Now the point of this discussion is that one can consider the BRST differential directly on A, where it has the form du = [Q, u],
u ∈ A.
This is a differential because [Q, Q] = 0, and additionally is a differentiation of algebras, i.e. d(u1 u2 ) = d(u1 )u2 + (−1)d1 u1 d(u2 )
(31)
where d1 is the cohomological degree (= ghost number) of u1 . The reason Q induces a compatible differential on A/I is that Q ∈ I . Now we shall introduce a complete decreasing filtration F i both on A and Hp = A/I which are compatible in the sense that the projection π : A → A/I
(32)
satisfies π(F i A) ⊆ F i A/I . Additionally, we will have Q ∈ F 0 A.
(33)
Now under these circumstances, one always has two accompanying spectral sequences on A and A/I , which are a spectral sequence of (non-commutative) differential graded (DG) algebras and differential graded modules (which means that formulas analogous to (31) in both the DG algebra and module sense will continue to hold for the higher terms Er and differentials dr of the spectral sequence). All this was true in the N = 0, N = 1 cases also,
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but in the N = 2 case, there are actual higher differentials to calculate, and the present technique is a tool for calculating differentials on non-linear monomias in Hp : we will see that such terms play a crucial role. The filtration is given both on A and A/I again by specifying filtration degrees of generators and then extending to all elements in an analogous manner as above. We put |x | = 1 (n = 0), 0n c = |ξ | = c1 = 1 for all n, n n n 0 b = |ψ | = b 1 = −1 for all n,
n n n |y | = −1 for n = 0, 2n 12 2 12 c = c = 2, b = b = −2 n n n n
for all n.
zn , tn , ζn , τn
The remaining matter generators have filtration degree 0. Simple verification shows that (33) indeed holds. To start out the calculation of the spectral sequence, the d0 differential is, again, given by [Q0 , ?] where Q0 is the sum of the terms of Q of filtration degree 0. We see from (20), (23) that (25) still holds, so d0 is still given by (26), and d0 acts as an algebra differential on monomials of normal order. Thus, both E0 (A), E0 (A/I ) are Koszul complexes. This shows that the cohomology E1 (A) is the algebra defined in the same way as A, except only on the generators zn , tn , ζn , τn , bn12 , cn12 , br2 , cr2 , c00 .
(34)
Also, E1 (A/I ) is the quotient of E1 (A) by the ideal defined in the same way as I , except on the generators (34) only. In other words, additive bases of E1 (A), E1 (A/I ) (in the completed sense in the case of E1 (A)) are monomials in normal order in the generators (34), with the same subscript restrictions as in A, A/I . As before, d1 is now given by commutator with the summands of Q of filtration degree 1, the non-zero terms of which are 0 0 c00 − 2nζn τ−n c00 − 2nbn0 c−n c0 Q1 = 2x0 y0 c00 + 2zn t−n 1 0 2 0 12 0 − 2nbn1 c−n c0 − 2nbn2 c−n c0 − 2nbn12c−n c0 .
The effect on an element u ∈ A, A/I is, again, left multiplication by 2w(u) + x02 + y02 + z02 + t02 c00 ,
(35)
where w is the weight. Therefore, E2 (A) is simply the sum of the kernel and cokernel of (35). In E2 (A/I ), one can say more explicitly that (35) leaves intact precisely all terms of E1 (A/I ) which have 1 2 X0 + Y02 + Z02 + T02 = 0 2 and kills all others; this is, again, the mass shell condition. Now to calculate d2 , we shall use the full force of our algebra machinery. First, let us recall how a d2 (or any higher dn ) is calculated in a spectral sequence associated with a decreasing filtration of a cochain complex C: we select an element u ∈ C whose pq representative in E0 C supports no differentials d0 , . . . , dn−1 , and hence survives to En C. w(u) +
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Then we find an element v ∈ F >p C such that d(u + v) ∈ F p+n C. Then the representative of d(u + v) in En C is dn u. The element v can be chosen to be a linear combination of elements whose representatives in En C have filtration degrees p + 1, . . . , p + n − 1 (i.e., only p + 1 for n = 2). Summands of dn (u) can come both from u or v; the element v will be called the counterterm. We claim that the d2 -differential on E2 (A) is given by 12 2 iτ−n cr+n − it−n cr+n , d2 τr = (36) 2 12 izs cr−s − iζs c−s+r , d2 ζr = (37) 12 2 2iζn+r t−n + i(r − n)br+n c−n , d2 br2 = (38) 2 2 i:τn+r ζ−n : − :ic−n br+n :, d2 br12 = (39) 2 irζn+r c−n , d2 zr = − (40) 2 irτn+r c−n , d2 tr = (41) d2 cr12 = d2 cr2 = 0.
(42)
Recall that (E2 (A), d2 ) is a (non-commutative) DGA, so this determines the value of d2 on any monomial. Further, since E2 (A/I ) is a quotient of E2 (A), the differential on E2 (A/I ) is induced. To prove (36)–(42), we use the procedure described above. To calculate d2 τr , consider commutators of τr with all summands Q2 of Q of filtration degree 2. The summands of Q2 which give non-trivial commutators are 0 1 2 12 (r − n)ζ−r τ−n cr+n + ζ−r t−n cr+n + iζ−r t−n cr+n + iτn ζ−r cn+r ,
which gives the commutator 0 1 2 12 −(r − n)τ−n cr+n − t−n cr+n − it−n cr+n + iτn cn+r .
(43)
The last two terms, of filtration degree 2, survive to (36). The first two terms of (43) are of filtration degree 1, and therefore must be cancelled by a counterterm. The counterterm is r −n 1 τ−n (44) xr+n − t−n ξn+r . 2x0 (r + n) x0 (Note that c00 is mass-shell related, so we may assume r + n = 0.) Now we must correct our candidate of d2 τr by adding the commutator of the sum Q1 of terms of Q of filtration −
degree 1 with (44). But one easily checks that every monomial thus obtained contains one of the matter generators ξn , xn as a factor, and thus vanishes in E2 (A). Thus, the counterterm contributes nothing in this case. (37) is analogous. In the case of (38), the summands of Q2 giving non-trivial commutators are 2 2 2 y−n c−r − iψn+r x−n c−r + iζn+r t−n c−r iξn+r
2 12 1 2 2 0 2 − iτn+r z−n c−r + (n − r)br+n c−n c−r − (n − 2r)br+n c−n c−r ,
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giving the commutator iξn+r y−n − iψn+r x−n + iζn+r t−n
12 1 2 0 − iτn+r z−n + (n − r)br+n c−n − (n − 2r)br+n c−n .
(45)
12 c 1 , b 2 c 0 require counterterms which The first term is in F0 A, and dies. The terms br+n −n r+n −n 12 ξ , b 2 x which, when commuted with Q , die similarly are scalar multiples of br+n 1 −n r+n −n x is in F A and hence dies, unless n = 0. In that case, as above. Also, the term −iψn+r 0 −n we have −iψr x0 of filtration degree −1, which requires adding the counterterm
ibr1 .
(46)
When computing the commutator of Q1 with (46), the terms giving a non-trivial commutator are 1 1 1 iξn+r y−n c−r + iψn+r x−n c−r + iζn+r t−n c−r
1 1 0 1 + iτn+r z−n c−r − i(n − 2r):br+n c−n c−r : 1 0 1 1 12 2 1 + i :br+n c−n c−r : + i(r − n)br+n c−n c−r . 2 From the resulting commutator with (46), the terms which survive to E2 (A) are 12 2 iζn+r t−n + iτn+r z−n + i(r − n)br+n c−n
which, when added to the third and fourth term of (45), gives (38). To get (39), the terms of Q2 which produce a non-trivial commutator with br12 are 12 12 12 12 0 2 12 1 i:ψn+r ξ−n :c−r + i:τn+r ζ−n :c−r − 2rbr+n c−r c−n + br+n c−r c−n ,
producing the commutator 12 0 2 1 ξ−n : + i:τn+r ζ−n : + 2rbr+n c−n − br+n c−n . i:ψn+r
(47)
The first summand of (47) dies, the second survives to E2 (A). The last two terms must be cancelled by counterterms (n = 0 in the third term to avoid the mass shell term). The counterterms are r 12 b x (n = 0) x0 n r+n −n 1 2 ξ−n . + br+n (48) x0 The first term of (48), when commuted with Q1 , dies in E2 A as above, while the second term has non-trivial commutator with the summands of Q1 1 2 ψn xs c−s−n − iψn xs c−n−s . The only cases which survive are s = 0 in the first summand (which we need for cancelling the relevant term of (47), and s = 0 in the second summand, which leads to 2 2 bn+r :. −i:c−n
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Adding this to the second term of (47) gives (39). To get (40), the terms of Q2 producing commutators with zr are 1 2 2zn+r t−r cn0 + ζn+r t−r c−n + iζn+r t−r c−n ,
leading to the commutators 0 1 2 2rzn+r c−n − rζn+r c−n − irζn+r c−n .
(49)
The last term survives to (40), the first term for n = 0 and second term need counterterms , ζ ξ , but both of these when commuted with Q which are multiples of zn+r x−n 1 n+r −n only produce terms which die in E2 (A) because of matter factors of the form x or ξ . are Concretely, the summands of Q1 producing a non-trivial commutator with zn+r x−n 1 2 + iξs yn c−s−n ξs yn c−n−s
leading to 1 2 zn+r − inξs c−s−n zn+r , −nξs c−n−s ξ are and summands of Q1 producing a non-trivial commutator with ζn+r −n 0 1 + ψn xs c−n−s , −(s − n)ξs ψn c−s−n
where −s − n = 0 in the first summand and s = 0 in the second summand. This produces 0 1 −(s − n)ξs c−s−n ζn+r + xs c−n−s ζn+r
which dies as promised. This proves (40). (41) is analogous. For (42), only ghost summands of Q give non-trivial commutators with cr2 , cr12 . Some counterterms are needed to cancel the commutators, but none produce non-zero terms in d2 . Now by looking at (36)–(42), it is not immediately obvious what the cohomology is. To settle that, we will introduce an additional filtration on (E2 (A), d2 ), (E2 (A/I ), d2 ) and use the resulting spectral sequence(!). To simplify notation, set δ := d2 . The filtration follows a now familiar pattern: it is a decreasing filtration Φ defined as above by specifying filtration degrees of generators. The formula is 2 12 b = b = |ζ | = −1, |zn | = −1 if n = 0, r r r 2 12 c = c = |τ | = 1, |t | = 1 if n = 0. r
r
r
n
δΦ i
⊆ Φi ,
One easily checks that so the filtration is compatible with the differential. As above, the differential δ0 on E0 E2 (A) is given by the degree 0 summands of δ. Comparing with (36)–(42), we see that δ0 τr = −t0 cr2 ,
δ0 br2 = 2it0 ζr ,
and all other δ0 ’s on generators are 0. Thus both (E0 E2 (A), δ0 ), (E0 E2 (A/I ), δ0 ) are Koszul complexes, and their cohomologies E1 E2 (A), E1 E2 (A/I ) have as additive bases the sets of normal-ordered monomials in zn , tn , bn12 , cn12 with the same values of the indices
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65
as in the definition of A, A/I . Moreover, E1 E2 (A) is an algebra (same relations as A), and E1 E2 (A/I ) is its factor by a left ideal. Now one sees that δ1 = 0, so E2 E2 (A) = E1 E2 (A), E2 E2 (A/I ) = E1 E2 (A/I ). Now let us calculate δ2 (zr ), by the same method as above. Thus, we apply δ first and get 2 c−n . −ir ζn+r But this is of filtration degree 0, so we must add the counterterm r :. − :ζn+r τ−n t0
(50)
Note that if and only if both n + r, −n have the same sign (are both non-positive or nonnegative, which happens for finitely many but more than 0 terms, which in addition have the same sign and hence cannot cancel), this will produce a non-zero multiple of cr12 by using commutators to bring the δ of (50) to normal order (see (36), (37)). Thus, we see that δ2 (zr ) = Cc12 ,
C = 0.
(51)
Now consider δ2 (br12 ). Applying δ first, we obtain 2 2 ζ−n : − :c−n br+n :. i :τn+r This is cancelled by a non-zero multiple of the counterterm 2 τ−n bn+r . b 2 ), by bringing But as above, we see that when n + r, −n have the same sign, δ(τ−n n+r terms to normal order, produces non-zero multiples (with the same sign) of tr . Thus, we see that δ2 br12 = Dtr , D = 0. (52)
Now (51), (52) show that (E2 E2 (A), δ2 ), (E2 E2 (A/I ), δ2 ) are again Koszul complexes, and that their cohomology is Λ c012 ⊗ Ker(d1 ) ⊕ Coker(d1 ) · c0 , (53) 12 0 2 2 2 2 Λ c0 , c0 if x0 + y0 + z0 + t0 = 0 and 0 else, (54) respectively. We see that the δ spectral sequences and also the d-spectral sequences must collapse to (53), (54), for the only remaining terms are in filtration degrees 0, 1, 2. This is what we claimed.
Appendix A. A few low weight explicit differentials It is quite difficult to write down explicitly all the N = 2 BRST differentials on Hp even in the lowest non-trivial weight w = 1 because of proliferation of counterterms. To illustrate the difficulty, and the great simplification the spectral sequence provides, we list, as an example, some (not all) of the differentials in weight 1 in the NS sector. We use the
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matter generators (9); the generators (21) were introduced to work better with the filtration, but the generators (9) give more symmetric explicit formulas. We have 1 0 1 2 2 = −2 b−1/2 c−1/2 + b−1/2 c−1/2 − x0 x−1 − y0 y−1 − z0 z−1 − t0 t−1 , Qb−1 1 1 2 2 0 c−1/2 − b−1/2 c−1/2 , Q x0 y−1 − y0 x−1 + z0 t−1 − t0 z−1 − b−1/2 = 4c−1 12 2 1 1 2 Qb−1 = ψ−1/2 ξ−1/2 + τ−1/2 ζ−1/2 + b−1/2 c−1/2 − b−1/2 c−1/2 , 12 Q(ξ−1/2 ψ−1/2 − x0 y−1 + y0 x−1 ) = Q(ζ−1/2 τ−1/2 − z0 t−1 + t0 z−1 ) = c−1 , 2 1 2 1 Q z0 b−1/2 ζ−1/2 + b−1/2τ−1/2 − t0 −b−1/2 τ−1/2 + b−1/2ζ−1/2 2 2 1 1 − x0 b−1/2 ξ−1/2 + b−1/2 ψ−1/2 + y0 −b−1/2 ψ−1/2 + b−1/2 ξ−1/2 12 1 2 2 2 − c012 b−1/2 − b−1/2 − z02 + t02 − x02 + y02 b−1
= 2(ξ−1/2 ψ−1/2 − ζ−1/2 τ−1/2 − x0 y−1 + y0 x−1 + z0 t−1 − t0 z−1 ).
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