- Email: [email protected]
On topological 2D string and intersection theory
27 July 1995
m=J . --
v;s
PHYSICS
LETTERS
B
iid ELSEVIER
Physics Letters B 355 (1995) 135-140
On topological 2D string and intersection theory * Yaron Oz l School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat Aviv, Tel-Aviv 49978, Israel
Received 22 May 1995 Editor: M. Dine
Abstract The topological description of 2D string theory at the self-dual radius is studied in the algebra-geometrical formulation of the Ak+l topological models at k = -3. Genus-zero correlators of tachyons and their gravitational descendants are computed as intersection numbers on moduli space and compared to 2D string results. The interpretation of negative momentum tachyons as gravitational descendants of the cosmological constant, as well as modifications of this, is shown to imply a disagreement between 2D string correlators and the associated intersection numbers.
2D string theory at the self-dual radius is described by a topological field theory with an infinite number of primary fields. A topological Landau-Ginzburg formulation of the theory has been constructed in [ 1,2], and a topological interpretation of the WI+~ recursion relations for the tachyon correlators has been given in [3,41. However, the topological structure of the theory is still not well understood. In particular, the identification of the BRST cohomology states as gravitational primaries and descendants, and the relation between the correlators and intersection numbers on the moduli space of Riemann surfaces has not been clarified yet. It has been suggested in [ 1,2] that while nonnegative momentum tachyons correspond to primary fields, the negative momentum tachyons correspond to gravitational descendants of TO, the zero momentum
*Work supported in part by the US-Israel Binational Science Foundation, the Israeli Academy of Science and GIF - the GcrmanIsraeli Foundation for Scientific Research. ’ Email: [email protected] Elsevier Science B.V. SSDI 0370-2693(95)00740-7
tachyon. 2D string theory, however, is parity invariant under the reflection of momenta k +-+-k, thus positive and negative momentum tachyons appear symmetrically. This might imply that all the tachyons should be considered as primaries, which is not in contradiction with the formulations of [ 1,3]. Such an interpretation is in agreement with intersection calculations of the four-point tachyon correlator [ 5,6] as well as the 1 -+ n tachyon amplitude and five-point function at various kinematical regions [ 71. There is a third possibility, namely that the negative momentum tachyons can be considered both as primaries and as descendants. This, if correct, will lead to beautiful relations between intersection numbers on the moduli space of Riemann surfaces, and should be intimately related to the way gravitational descendants are built from matter [ 81. Since the generating functional for tachyon correlators is a tau function of the Toda lattice hierarchy [ 91, one expects a generalization of the relation that exists between intersection numbers and the KdV hierarchy as implied by topological gravity [ 10,111.
136
Y. Oz/Physics
Letters B 355 (1995) 135-140
In this letter we make a step in the study of the intersection theory associated with the topological 2D string. In particular, we study the possibility of relating negative momentum tachyons to gravitational descendants from the viewpoint of intersection theory on the moduli space of Riemann surfaces. We compute tachyon correlators as intersection numbers and show that the identification of negative momentum tachyons as gravitational descendants, as well as modifications of this, is in disagreement with 2D string results. The algebro-geometrical set up is given by the analytical continuation of that, associated with the Ak+l theory [5,12], to k = -3. Define the “vector bundles” v and w over the compactified moduli space of genus-g Riemann surfaces with s punctures fig,$, whose fibers V and W respectively are2 : v = HOG,,,
2) 3
w = H’ G&S, 731 ,
Gj(Tki))g
=
j
fi(cl(&>
)%‘t~
8 m>
7(2)
i=l J%
i=l
where o;l( Tk) is the n th descendant of the momentum k tachyon, Cr is the top Chern class and Cr ( Li)n E H2”( fig,s) are the Mumford-MoritaMiller stable classes. Ci is a line bundle on the moduli space with fiber at Z being the cotangent space to the point xi E X. The subtraction of the vector bundles Y and w should be understood in K-theory sense. Using the Riemann-Roth theorem for a line bundle L dim@(Z,,
L) - dimH1(2,,
L) = deg L - g + 1 , (3)
and deg (K) = 2g-2, s (ni - ki) = 0 . c
Cg, ni = s - 3 and satisfying (4). Since deg (R) = -1 < 0, the Kodaira vanishing theorem implies that dim V = 0. Using the Riemann-Roth theorem (3) we get that dim W = 0. Thus, in this case the correlator is given by a simple integral
This result is in contradiction with T_, being defined as the rr th gravitational descendants of To, since the correlators of c = 1 tachyons satisfy
(1)
where R G K2 c3i 0( zi) lmki is a line bundle over );g,s, K being the canonical bundle. The sections of R are two-forms that can have zeros at the points zi with order of at least ki - 1. Then {fi
T-, as the n th gravitational descendants of TO, T-, = a,,(To). The first family of correlators we wish to evaluate is (n%, ffni (Tki))u subject to the constraint
we get the selection rule for (2)
(nTki)u=O
if
CIki]@(-ki)
=S-3
7
(6)
i=l
i=l
which is proved using the TO and Tl Ward identities of the theory. Note, however, that it is possible to overcome the disagreement between (5) and (6) by postulating an infinite factor between the tachyon T_, and the gravitational descendant gn (To), Consider next the genus-zero three-point function
where cat
(ni - ki) = 0. deg (R)
= - EE,
ki < 0,
thus dim V = 0 and dim W = Cfzl ki. The integral in (7) is evaluated via
In 3
J%,s
(CI(mmx-W)
id
(4)
i=l
Let us consider from the intersection theory viewpoint, the definition of negative momentum tachyons
z v and w are called the (higher) direct image sheaves of ‘FL and are not necessarily locally free, i.e. the dimension of the fiber may depend on the point on &Ig,s.
MO.3
Ho.3 (8)
Y. Oz/Physics
Letters B 355 (1995) 135-140
where L,, = G& Lyni, and we used a basic property of the top Chern class of a Whitney sum of vector bundles
Using (13) and ( 14) we have
ch ( C,, .
Cr(Y @ 0) = Cr(V)Cr(W) Eq. (8) is in agreement correlator
137
@~)=&G(l+Cl(Li))
(9)
with the three-point
i=l
tachyon -(f:ki-l+Cl(o)+...).
(15)
i=l
(Tk,Tk&)
=
akl+kz+k3,0
(10)
7
On the other hand, since Lgrav 8 w is a line bundle we
thus excluding the possibility of an infinite factor between the negative momentum tachyon and the gravitational descendant. Note that in contrast to (8)) in minimal topological models coupled to topological gravity three-point functions with gravitational descendants vanish, which is one of the basic ingredients of the topological recursion relations of [ 51. Consider now the genus-zero four-point function
get ch(L,,
8 0) =
Eqs. (ll),
1 + Cr(C,,,
(12),
&Gj(Tkj))o
.
(16)
(15) and (16) yield
=
i=l
8 a)
J cl(&ave~) MO.4
4 =
ClG)
ni ES
i=l MO.4
deg (77,) = - CL,
ki <
dim W = ‘j$ recast as
1. The integral
ki -
S’I
Mo.4
h20.4
(17)
Thus, in order to evaluate the genus-zero four-point function of tachyon’s gravitational descendants we have to compute the first Chern class of the vector bundle w. Recall now that the four-point tachyon correlator is
)niG(-W)
i=l
Jn J
Cl (0)
in ( 11) can be
4
=
J
0, thus dim V = 0 and
4
(Cl(G>
-
C&c~ni)C+W)
r.131
i=l
MO.4
=
W&l”
8
w>=
J
Cl (&a”
8
WI
9
(TkTklTk2Tks)= (k-
ki) 3
i=l
No.4
MO.4
I)-e(k+k,)b(-km
(18)
(12)
In order to evaluate ( 12) we will use the property that the Chern character of a difference of vector bundles satisfies ch(EeF)
=ch(E)
The Chern character Chern classes ch(E)
=rankE+Cr(E) -I-...
o
-ch(F)
.
has an expansion
+ i(Cr(E)*
(13) in terms of
with k > 0. As seen in (18) the 2 --+ 2 amplitude (T-k,T-k2Tk3Tk4)o iS non-trivial and depends on various kinematical regions. We will now compute this amplitude from intersection theory viewpoint with negative momentum tachyons considered as gravitational descendants of To. Using (17) we have (ak,
- 2C2(E))
(fibkz
(To)Tk3Tk4)o
= h
+ k2 -
J
CI (w)
.
Mot
(14)
(19)
138
Y. Oz/Physics
Letters B 355 (1995) 135-140
In order to compute the first Chern class of the vector bundle w, recall that Cl(w)
w) ,
z Ct(det
and consider the matrix Tij Tij 25 U’(Xj)
j=l
with det w being the determinant line bundle associated with o. The rank of w is r = k3 + k4 - 1 and its fiber is the vector space H’(Zo,4, K2 &t
O(zi)’
= Ho(Ce,4, Km1 &I
G&s O(zi)‘+) O(Zi)-’
@)% O(Zi)“-I)*
3 (21)
where we used Serre duality H’(X,,L)
2 @(2,,
K c3 L-l)*,
and * denotes the dual bundIe. A basis of SL( 2, @) invariant bundle to o is, for instance, s1 = dz
_t
i=O,...,k3-
(20)
(22) sections of the dual
(Z-z1)(z-22) 212
_t (z -zt)‘(z-z2)Zf;1 si = dz (z_z3)&1z;2
’
, z =27***7k3
i--k3+1(Xj) =u
i=k3,...,ks+k4-2,
(z-z4)i-1zi
u=1+
’
i=2,...,kq,
’
(E/U)
where (Y = zt4/234 and E = zts/z34. We have to consider the order of zero of Tij as a goes to zero. It is convenient to change a basis from z&,i = 0, . . . , k3 - 1 to (c/u) i. Thus, the order of zero of det (T) is xi,’ = ; k3 (k3 - 1) . Similarly we get the contributions the other degenerate surfaces. We find
where zij G zi - zj. The first Chern class of w is evaluated by counting with multiplicities the zeros minus the poles of a section det (S) of det w. S is the Y x r matrix : Sij Z Si( Xj). The zeros and poles of det (S) are located on the boundary of J.&d. We denote by Cij the degenerate sphere corresponding to the merge of the punctures i and j, which in the Deligne-Mumford compactification is represented by a splitting of a sphere. The sections si have a pole of order i on Zt2, and a zero of order i - 1 on 213, while si have a pole of order i on 212 and a zero of order i - 1 on Zt4, thus contributing k3 + k4 - 1 to the first Chern class of w, whereweusedthefactthatCt(w) =-Ct(w*>. We also have to calculate the zeros of the determinant due to linear dependencies among the sections on 2,. Define u(z)
z -
= z -
Zl z3’
u(z)
z - Zl = z-24’
.I
C,(w)
(23)
k3 + k4 - 1 .
CY
Altogether 12
(25)
The matrix zj is basically Sij, but we factored out overall nonrelevant terms. It encodes the linear dependencies among the sections, and we have to calculate the zeros of its determinant on the degenerate surfaces 8ij. Consider, for instance, the degenerate surface I; 13. From (24) it follows that
$3(k3-1)
=dz-
, . . . , k3 + k4 - 1
j=l,...,
p&4-1)
det (T) N z13 1 (z-zl)i~Z-z2)z;q1
1,
214
(k3--l)(b-1) 234
.
from
(27)
we have = -i(k3
+ k4)(k3 + k4 - 5) - 2 . (28)
I-
Mop
Using ( 19) and (28) we get
=kl+k2+2+;(k3+kq)(ky+k4-5),
(29)
where we have xi1 ki = 0. For the special case 0 )T 1T1) 0 only the section st in (23) is (gt(To)at(T relevant, and the value of the cot-relator is one, in agreement with (29). The result (29) is clearly different from ( 18), in particular is does not depend on kinematical regions and it is not parity invariant. Note, in contrast, that if the negative momentum tachyons are considered as primaries, then dimV = 1, dim W = 0, and the intersection calculation
Cfi&.,o =/ Cl(v) i=l
NO.4
agrees with (18)
[5].
3
(30)
Y. Oz/Physics
Letters B 355 (1995) 135-140
In [ 41 a definition of a negative momentum tachyon as a sum of gravitational descendants in 1 -+ n amplitudes has been proposed, it T_ II=
i
CrI (j k0
- n)c+iV-n+J .
(31)
j=l
This definition together with (2) is clearly compatible with the 1 -+ 3 tachyons correlator. However, for the 2 -+ 2 amplitude we get dim V = dim W = 0, thus (T--klT--kzT$kq)o
= (1 - k~)
Cl(G>
$ MD.4
f(l
-
C1(L2)
k2>
=2-kl
-kz,
(32)
139
It is straightforward to apply the same manipulations which we used in order to derive (8) and ( 17)) in order to get formulas for higher-point functions of tachyons and their gravitational descendants. These formulas involve higher Chern classes of the sheaf w. The latter is not locally free and we expect jumps in the dimension of the fiber at the boundary of the moduli space 3 , thus complicating the calculation of the intersection numbers. A proper application of the Grothendieck-Riemann-Roth theorem is likely to be helpful for these computations. Finally, we hope that the ideas used in the calculations that we made will be useful for the complete study of topological 2D string theory.
s
Mac
in disagreement with ( 18). The intersection theory computations presented above imply that the identification of negative momentum tachyons as gravitational descendants in the topological formulation of the 2D string leads to a disagreement between tachyon correlators and the associated intersection numbers. The identification of all the tachyons as gravitational primaries is consistent with all the intersection theory computations which have been done so far. These lead us to suggest that indeed the proper set up is to consider all the tachyons as gravitational primaries, and interpret accordingly the topological numbers associated with their correlators. It should be clear, however, that our computations do not exclude the possibility that an insertion of a tachyon operator in a correlator might be equivalent to an insertion of some combination of gravitational descendants. This would simply mean that there are topological Ward identities underlying the theory. The gravitational descendants of the tachyons, some of whose correlators have been calculated above, are likely to correspond to discrete states of 2D string theory. However, as discussed in [ 141, it is not clear what is the exact relation between gravitational descendants and discrete states since correlators of the latter have not been calculated successfully in the Liouville framework. Such a relation, when revealed, will evidently shed light on the relation between the physical 2D string theory and its topological formulation. Also, an underlying integrable structure generalizing the Toda lattice hierarchy is expected.
Acknowledgments We would like to thank J. Bernstein for helpful discussions.
and R. Plesser
References [II A. Hanany,
[21
[31
[41
151 t61
[71 [81
[91
Y. Oz and R. Plesser, Topological LandauGinzburg formulation and integrable structure of 2D string theory, hep-th/9401030, Nucl. Phys. B 425 (1994) 150. D. Ghoshal and S. Mukhi, Topological Landau-Ginzburg model of two-dimensional string theory, hep-th/9312189, Nucl. Phys. B 425 (1994) 173. Y. Lavi, Y. Oz and J. Sonnenschein, ( 1, q = - 1) model as a topological description of 2D string theory, hep-thi9406056, Nucl. Phys. B 431 (1994) 223. D. Ghoshal, C. Imbimbo and S. Mukhi, Topological 2D string theory: Higher-genus amplitudes and W-infinity identities, MRI-PHY/13/94, CERN-TH-7458194, TIFR/TH/39-94, hep&/9410034. E. Witten, The N matrix model and gauged WZW models, Nucl. Phys. B 371 (1992) 191. S. Mukhi and C. Vafa, Two dimensional black hole as a topological coset model of c = 1 string theory, hepth/9301083, Nucl. Phys. B 407 (1993) 667. R. Dijkgraaf, G. Moore and R. Plesser, unpublished. A. Lossev, Descendants constructed from matter fields in topological Landau-Ginzburg theories coupled to topological gravity, ITEP preprint PRINT-92-0563 (January 1993), hepth/9211089. R. Dijkgraaf, G. Moore and R. Plesser, The partition function of 2D string theory, hep-th19208031, Nucl. Phys. B 394 (1993) 356.
3 Such phenomena of [7].
appeared
in tachyons
five-point
computations
140
K Oz/Physics
Letters B 355 (1995) 135-140
[lo] E. Witten, Two dimensional gravity and intersection theory on moduli space, in: Surveys in Differential Geometry 243310, Proc. Cambridge 1990 [ 111 M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Aii function, Commun. Math. Phys. 147 (1992) I. ] 121 E. Witten, Algebraic geometry associated with matrix models of two dimensional gravity, IASSNS-HEP-91/74.
[ 131 P. di Francesco and D. Kutaaov, World she& and spacetime physics in two dimensional (super) string theory, bepth/9109005, Nucl. Phys. B 375 (1992) 119. [ 141 A. Hanany and Y. Oz, c = 1 discrete states correlators via Wlfoo constraints, hep-th/94101.57, Phys. Lett. B 347 ( 1995) 255.
m=J . --
v;s
PHYSICS
LETTERS
B
iid ELSEVIER
Physics Letters B 355 (1995) 135-140
On topological 2D string and intersection theory * Yaron Oz l School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat Aviv, Tel-Aviv 49978, Israel
Received 22 May 1995 Editor: M. Dine
Abstract The topological description of 2D string theory at the self-dual radius is studied in the algebra-geometrical formulation of the Ak+l topological models at k = -3. Genus-zero correlators of tachyons and their gravitational descendants are computed as intersection numbers on moduli space and compared to 2D string results. The interpretation of negative momentum tachyons as gravitational descendants of the cosmological constant, as well as modifications of this, is shown to imply a disagreement between 2D string correlators and the associated intersection numbers.
2D string theory at the self-dual radius is described by a topological field theory with an infinite number of primary fields. A topological Landau-Ginzburg formulation of the theory has been constructed in [ 1,2], and a topological interpretation of the WI+~ recursion relations for the tachyon correlators has been given in [3,41. However, the topological structure of the theory is still not well understood. In particular, the identification of the BRST cohomology states as gravitational primaries and descendants, and the relation between the correlators and intersection numbers on the moduli space of Riemann surfaces has not been clarified yet. It has been suggested in [ 1,2] that while nonnegative momentum tachyons correspond to primary fields, the negative momentum tachyons correspond to gravitational descendants of TO, the zero momentum
*Work supported in part by the US-Israel Binational Science Foundation, the Israeli Academy of Science and GIF - the GcrmanIsraeli Foundation for Scientific Research. ’ Email: [email protected] Elsevier Science B.V. SSDI 0370-2693(95)00740-7
tachyon. 2D string theory, however, is parity invariant under the reflection of momenta k +-+-k, thus positive and negative momentum tachyons appear symmetrically. This might imply that all the tachyons should be considered as primaries, which is not in contradiction with the formulations of [ 1,3]. Such an interpretation is in agreement with intersection calculations of the four-point tachyon correlator [ 5,6] as well as the 1 -+ n tachyon amplitude and five-point function at various kinematical regions [ 71. There is a third possibility, namely that the negative momentum tachyons can be considered both as primaries and as descendants. This, if correct, will lead to beautiful relations between intersection numbers on the moduli space of Riemann surfaces, and should be intimately related to the way gravitational descendants are built from matter [ 81. Since the generating functional for tachyon correlators is a tau function of the Toda lattice hierarchy [ 91, one expects a generalization of the relation that exists between intersection numbers and the KdV hierarchy as implied by topological gravity [ 10,111.
136
Y. Oz/Physics
Letters B 355 (1995) 135-140
In this letter we make a step in the study of the intersection theory associated with the topological 2D string. In particular, we study the possibility of relating negative momentum tachyons to gravitational descendants from the viewpoint of intersection theory on the moduli space of Riemann surfaces. We compute tachyon correlators as intersection numbers and show that the identification of negative momentum tachyons as gravitational descendants, as well as modifications of this, is in disagreement with 2D string results. The algebro-geometrical set up is given by the analytical continuation of that, associated with the Ak+l theory [5,12], to k = -3. Define the “vector bundles” v and w over the compactified moduli space of genus-g Riemann surfaces with s punctures fig,$, whose fibers V and W respectively are2 : v = HOG,,,
2) 3
w = H’ G&S, 731 ,
Gj(Tki))g
=
j
fi(cl(&>
)%‘t~
8 m>
7(2)
i=l J%
i=l
where o;l( Tk) is the n th descendant of the momentum k tachyon, Cr is the top Chern class and Cr ( Li)n E H2”( fig,s) are the Mumford-MoritaMiller stable classes. Ci is a line bundle on the moduli space with fiber at Z being the cotangent space to the point xi E X. The subtraction of the vector bundles Y and w should be understood in K-theory sense. Using the Riemann-Roth theorem for a line bundle L dim@(Z,,
L) - dimH1(2,,
L) = deg L - g + 1 , (3)
and deg (K) = 2g-2, s (ni - ki) = 0 . c
Cg, ni = s - 3 and satisfying (4). Since deg (R) = -1 < 0, the Kodaira vanishing theorem implies that dim V = 0. Using the Riemann-Roth theorem (3) we get that dim W = 0. Thus, in this case the correlator is given by a simple integral
This result is in contradiction with T_, being defined as the rr th gravitational descendants of To, since the correlators of c = 1 tachyons satisfy
(1)
where R G K2 c3i 0( zi) lmki is a line bundle over );g,s, K being the canonical bundle. The sections of R are two-forms that can have zeros at the points zi with order of at least ki - 1. Then {fi
T-, as the n th gravitational descendants of TO, T-, = a,,(To). The first family of correlators we wish to evaluate is (n%, ffni (Tki))u subject to the constraint
we get the selection rule for (2)
(nTki)u=O
if
CIki]@(-ki)
=S-3
7
(6)
i=l
i=l
which is proved using the TO and Tl Ward identities of the theory. Note, however, that it is possible to overcome the disagreement between (5) and (6) by postulating an infinite factor between the tachyon T_, and the gravitational descendant gn (To), Consider next the genus-zero three-point function
where cat
(ni - ki) = 0. deg (R)
= - EE,
ki < 0,
thus dim V = 0 and dim W = Cfzl ki. The integral in (7) is evaluated via
In 3
J%,s
(CI(mmx-W)
id
(4)
i=l
Let us consider from the intersection theory viewpoint, the definition of negative momentum tachyons
z v and w are called the (higher) direct image sheaves of ‘FL and are not necessarily locally free, i.e. the dimension of the fiber may depend on the point on &Ig,s.
MO.3
Ho.3 (8)
Y. Oz/Physics
Letters B 355 (1995) 135-140
where L,, = G& Lyni, and we used a basic property of the top Chern class of a Whitney sum of vector bundles
Using (13) and ( 14) we have
ch ( C,, .
Cr(Y @ 0) = Cr(V)Cr(W) Eq. (8) is in agreement correlator
137
@~)=&G(l+Cl(Li))
(9)
with the three-point
i=l
tachyon -(f:ki-l+Cl(o)+...).
(15)
i=l
(Tk,Tk&)
=
akl+kz+k3,0
(10)
7
On the other hand, since Lgrav 8 w is a line bundle we
thus excluding the possibility of an infinite factor between the negative momentum tachyon and the gravitational descendant. Note that in contrast to (8)) in minimal topological models coupled to topological gravity three-point functions with gravitational descendants vanish, which is one of the basic ingredients of the topological recursion relations of [ 51. Consider now the genus-zero four-point function
get ch(L,,
8 0) =
Eqs. (ll),
1 + Cr(C,,,
(12),
&Gj(Tkj))o
.
(16)
(15) and (16) yield
=
i=l
8 a)
J cl(&ave~) MO.4
4 =
ClG)
ni ES
i=l MO.4
deg (77,) = - CL,
ki <
dim W = ‘j$ recast as
1. The integral
ki -
S’I
Mo.4
h20.4
(17)
Thus, in order to evaluate the genus-zero four-point function of tachyon’s gravitational descendants we have to compute the first Chern class of the vector bundle w. Recall now that the four-point tachyon correlator is
)niG(-W)
i=l
Jn J
Cl (0)
in ( 11) can be
4
=
J
0, thus dim V = 0 and
4
(Cl(G>
-
C&c~ni)C+W)
r.131
i=l
MO.4
=
W&l”
8
w>=
J
Cl (&a”
8
WI
9
(TkTklTk2Tks)= (k-
ki) 3
i=l
No.4
MO.4
I)-e(k+k,)b(-km
(18)
(12)
In order to evaluate ( 12) we will use the property that the Chern character of a difference of vector bundles satisfies ch(EeF)
=ch(E)
The Chern character Chern classes ch(E)
=rankE+Cr(E) -I-...
o
-ch(F)
.
has an expansion
+ i(Cr(E)*
(13) in terms of
with k > 0. As seen in (18) the 2 --+ 2 amplitude (T-k,T-k2Tk3Tk4)o iS non-trivial and depends on various kinematical regions. We will now compute this amplitude from intersection theory viewpoint with negative momentum tachyons considered as gravitational descendants of To. Using (17) we have (ak,
- 2C2(E))
(fibkz
(To)Tk3Tk4)o
= h
+ k2 -
J
CI (w)
.
Mot
(14)
(19)
138
Y. Oz/Physics
Letters B 355 (1995) 135-140
In order to compute the first Chern class of the vector bundle w, recall that Cl(w)
w) ,
z Ct(det
and consider the matrix Tij Tij 25 U’(Xj)
j=l
with det w being the determinant line bundle associated with o. The rank of w is r = k3 + k4 - 1 and its fiber is the vector space H’(Zo,4, K2 &t
O(zi)’
= Ho(Ce,4, Km1 &I
G&s O(zi)‘+) O(Zi)-’
@)% O(Zi)“-I)*
3 (21)
where we used Serre duality H’(X,,L)
2 @(2,,
K c3 L-l)*,
and * denotes the dual bundIe. A basis of SL( 2, @) invariant bundle to o is, for instance, s1 = dz
_t
i=O,...,k3-
(20)
(22) sections of the dual
(Z-z1)(z-22) 212
_t (z -zt)‘(z-z2)Zf;1 si = dz (z_z3)&1z;2
’
, z =27***7k3
i--k3+1(Xj) =u
i=k3,...,ks+k4-2,
(z-z4)i-1zi
u=1+
’
i=2,...,kq,
’
(E/U)
where (Y = zt4/234 and E = zts/z34. We have to consider the order of zero of Tij as a goes to zero. It is convenient to change a basis from z&,i = 0, . . . , k3 - 1 to (c/u) i. Thus, the order of zero of det (T) is xi,’ = ; k3 (k3 - 1) . Similarly we get the contributions the other degenerate surfaces. We find
where zij G zi - zj. The first Chern class of w is evaluated by counting with multiplicities the zeros minus the poles of a section det (S) of det w. S is the Y x r matrix : Sij Z Si( Xj). The zeros and poles of det (S) are located on the boundary of J.&d. We denote by Cij the degenerate sphere corresponding to the merge of the punctures i and j, which in the Deligne-Mumford compactification is represented by a splitting of a sphere. The sections si have a pole of order i on Zt2, and a zero of order i - 1 on 213, while si have a pole of order i on 212 and a zero of order i - 1 on Zt4, thus contributing k3 + k4 - 1 to the first Chern class of w, whereweusedthefactthatCt(w) =-Ct(w*>. We also have to calculate the zeros of the determinant due to linear dependencies among the sections on 2,. Define u(z)
z -
= z -
Zl z3’
u(z)
z - Zl = z-24’
.I
C,(w)
(23)
k3 + k4 - 1 .
CY
Altogether 12
(25)
The matrix zj is basically Sij, but we factored out overall nonrelevant terms. It encodes the linear dependencies among the sections, and we have to calculate the zeros of its determinant on the degenerate surfaces 8ij. Consider, for instance, the degenerate surface I; 13. From (24) it follows that
$3(k3-1)
=dz-
, . . . , k3 + k4 - 1
j=l,...,
p&4-1)
det (T) N z13 1 (z-zl)i~Z-z2)z;q1
1,
214
(k3--l)(b-1) 234
.
from
(27)
we have = -i(k3
+ k4)(k3 + k4 - 5) - 2 . (28)
I-
Mop
Using ( 19) and (28) we get
=kl+k2+2+;(k3+kq)(ky+k4-5),
(29)
where we have xi1 ki = 0. For the special case 0 )T 1T1) 0 only the section st in (23) is (gt(To)at(T relevant, and the value of the cot-relator is one, in agreement with (29). The result (29) is clearly different from ( 18), in particular is does not depend on kinematical regions and it is not parity invariant. Note, in contrast, that if the negative momentum tachyons are considered as primaries, then dimV = 1, dim W = 0, and the intersection calculation
Cfi&.,o =/ Cl(v) i=l
NO.4
agrees with (18)
[5].
3
(30)
Y. Oz/Physics
Letters B 355 (1995) 135-140
In [ 41 a definition of a negative momentum tachyon as a sum of gravitational descendants in 1 -+ n amplitudes has been proposed, it T_ II=
i
CrI (j k0
- n)c+iV-n+J .
(31)
j=l
This definition together with (2) is clearly compatible with the 1 -+ 3 tachyons correlator. However, for the 2 -+ 2 amplitude we get dim V = dim W = 0, thus (T--klT--kzT$kq)o
= (1 - k~)
Cl(G>
$ MD.4
f(l
-
C1(L2)
k2>
=2-kl
-kz,
(32)
139
It is straightforward to apply the same manipulations which we used in order to derive (8) and ( 17)) in order to get formulas for higher-point functions of tachyons and their gravitational descendants. These formulas involve higher Chern classes of the sheaf w. The latter is not locally free and we expect jumps in the dimension of the fiber at the boundary of the moduli space 3 , thus complicating the calculation of the intersection numbers. A proper application of the Grothendieck-Riemann-Roth theorem is likely to be helpful for these computations. Finally, we hope that the ideas used in the calculations that we made will be useful for the complete study of topological 2D string theory.
s
Mac
in disagreement with ( 18). The intersection theory computations presented above imply that the identification of negative momentum tachyons as gravitational descendants in the topological formulation of the 2D string leads to a disagreement between tachyon correlators and the associated intersection numbers. The identification of all the tachyons as gravitational primaries is consistent with all the intersection theory computations which have been done so far. These lead us to suggest that indeed the proper set up is to consider all the tachyons as gravitational primaries, and interpret accordingly the topological numbers associated with their correlators. It should be clear, however, that our computations do not exclude the possibility that an insertion of a tachyon operator in a correlator might be equivalent to an insertion of some combination of gravitational descendants. This would simply mean that there are topological Ward identities underlying the theory. The gravitational descendants of the tachyons, some of whose correlators have been calculated above, are likely to correspond to discrete states of 2D string theory. However, as discussed in [ 141, it is not clear what is the exact relation between gravitational descendants and discrete states since correlators of the latter have not been calculated successfully in the Liouville framework. Such a relation, when revealed, will evidently shed light on the relation between the physical 2D string theory and its topological formulation. Also, an underlying integrable structure generalizing the Toda lattice hierarchy is expected.
Acknowledgments We would like to thank J. Bernstein for helpful discussions.
and R. Plesser
References [II A. Hanany,
[21
[31
[41
151 t61
[71 [81
[91
Y. Oz and R. Plesser, Topological LandauGinzburg formulation and integrable structure of 2D string theory, hep-th/9401030, Nucl. Phys. B 425 (1994) 150. D. Ghoshal and S. Mukhi, Topological Landau-Ginzburg model of two-dimensional string theory, hep-th/9312189, Nucl. Phys. B 425 (1994) 173. Y. Lavi, Y. Oz and J. Sonnenschein, ( 1, q = - 1) model as a topological description of 2D string theory, hep-thi9406056, Nucl. Phys. B 431 (1994) 223. D. Ghoshal, C. Imbimbo and S. Mukhi, Topological 2D string theory: Higher-genus amplitudes and W-infinity identities, MRI-PHY/13/94, CERN-TH-7458194, TIFR/TH/39-94, hep&/9410034. E. Witten, The N matrix model and gauged WZW models, Nucl. Phys. B 371 (1992) 191. S. Mukhi and C. Vafa, Two dimensional black hole as a topological coset model of c = 1 string theory, hepth/9301083, Nucl. Phys. B 407 (1993) 667. R. Dijkgraaf, G. Moore and R. Plesser, unpublished. A. Lossev, Descendants constructed from matter fields in topological Landau-Ginzburg theories coupled to topological gravity, ITEP preprint PRINT-92-0563 (January 1993), hepth/9211089. R. Dijkgraaf, G. Moore and R. Plesser, The partition function of 2D string theory, hep-th19208031, Nucl. Phys. B 394 (1993) 356.
3 Such phenomena of [7].
appeared
in tachyons
five-point
computations
140
K Oz/Physics
Letters B 355 (1995) 135-140
[lo] E. Witten, Two dimensional gravity and intersection theory on moduli space, in: Surveys in Differential Geometry 243310, Proc. Cambridge 1990 [ 111 M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Aii function, Commun. Math. Phys. 147 (1992) I. ] 121 E. Witten, Algebraic geometry associated with matrix models of two dimensional gravity, IASSNS-HEP-91/74.
[ 131 P. di Francesco and D. Kutaaov, World she& and spacetime physics in two dimensional (super) string theory, bepth/9109005, Nucl. Phys. B 375 (1992) 119. [ 141 A. Hanany and Y. Oz, c = 1 discrete states correlators via Wlfoo constraints, hep-th/94101.57, Phys. Lett. B 347 ( 1995) 255.