Physics Letters B 278 (1992) 97-100 PHYSICS LETTERS B North-Holland

Topological phases of the heterotic string* Jack Morse and Rolf Schimmrigk 1

Institut ffir Theoretische Physik, Universitdt Heidelberg, Philosophenweg 16, w-6900 Heidelberg, FRG and Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Received 5 January 1992

We analyse the phase structure of topological Calabi-Yau manifolds defined on the moduli space of instantons. We show in this framework that topological vacua describe new phases of the heterotic string theory in which the fiat directions corresponding to complex deformations are lifted. We also briefly discuss the phase structure of non-K~ihler manifolds.

1. Introduction

It is bel ieved that a topological a -model coupled to 2D topological gravity may represent a new phase o f an underlying string theory [ 1 ]. We investigate the na- ture of this phase by computing the scaling behaviour of the complete set of operators associated to the coho- mology groups of a general Ca lab i -Yau manifold. In the a -mode l corresponding to a string theory compact- ified on a Ca lab i -Yau manifold these fields become marginal, giving rise to degenerate string vacua. We show that in the topological phase of these theories the complex deformat ions and their gravitat ional de- scendents are no longer marginal at any critical level. Hence the corresponding degeneracy of the vacua is lifted. Thus a picture reminiscent of the Higgs phase transi t ion emerges: in the "high temperature" sym- metric topological phase of string theory certain com- ponents of the order parameter vanish. In the critical, "broken" phase these components freeze out and ac- quire nonvanishing vacuum expectat ion values.

Mot ivated by the issue of mir ror symmetry we also briefly analyse the phase structure of non-K~ihler man- ifolds. It is shown that the operators corresponding to the third cohomology of such spaces are marginal at every critical level.

* This work was supported in part by the National Science Foundation under Grant No. PHY-89-04035.

l Bitnet address: q25@dhdurzl.bitnet.

2. Topological Calabi-Yau manifolds via intersection theory

The configuration space of string theory is the space of maps .A4 = {~ : .Sg ~ M } defining the embed- ding of the worldsheet .Sg into the target manifold M. The idea of ref. [2] is to define the topological phase of string theory via intersection theory on the moduli space A4. To this end recall that associated to a form top c H p ( M ) on an n-dimensional manifold M is its Poincar6 dual Co o E Hn_p (M) . top has a delta function property with respect to the cycle Co,p, it essentially vanishes outside Cop. Hence the 2D op- erator that results from a pullback of the p-form of the manifold vanishes unless the embedding map maps the location of the operator into the Poincar6 dual of the form. This immediate ly leads to a dimen- sion count that nonvanishing correlation functions #l have to satisfy: every insertion of an operator intro- duces two real moduli but for a p-form there are also p constraints on the instanton maps. Hence one can associate a "ghost number" ( 2 - p ) to the p-form top.

The operator ring on a Ca lab i -Yau vacuum is rel- atively simple; since there are no one-forms the only nonvanishing Betti numbers are bo, b2, b3, b4, b6. We denote by T' the puncture operator, associated to

#1 Meaning intersections of hypersurfaces of the moduli space defined by the constraint on the embedding maps just mentioned.

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the connectivity of the manifold. ](~m, ]~m denote the K~ihler forms and their duals, 73~ denote the elements of the third cohomology and 7~ is the dual of the puncture operator. Since the dimension of the mod- uli space of instantons embedded in a three-complex- dimensional Calabi-Yau manifold is actually zero it follows from the ghost numbers of the operators that the only nonvanishing three-point functions are

('P]~.m~.n) = ?Iron,

( 'pT~z~j) = rlq,

qC. lCql¢ ) = G q - (1)

Assuming that these correlation functions are associ- ated to a path integral in which deformations along the operators are introduced one can integrate these equations on the small phase space parametrized by to,~, to,~:~, t0,~, to,v~, t0,~ ~2. The result

and hence the two-point function becomes

(ICE) = Ctox de y ~ c2e catO'tc. (5) a

These two-point functions are not independent but satisfy the constitutive equations

(v,z~j) = ,~ApTz),

(lclc) = c(~'~;) + C2aeCa($'~). (6) a

From the last equation it follows that (79/C) ,,, 1 and (E/C) ~ 1 and hence

Yg: = Ystr = 271c (7)

because

2--Tstr + ~ T M (70i--1) (Ol " " Or) ~ x .--.~=

(PP) = t0,n,

(P/G.) = t/mnt0,~.,

(Epic.q) = Cq~.to,pc.

('PDi) = qijto,vj,

(PR) = t0,~,

('P]Cm) = rlmnto, pc.,

(l~m~,n) = tlmnto,7~,

(DiD j) = rlijto,'p,

is corrected by instantons. The instantons (if isolated ) will only generate corrections to the correlation func- tions of the second cohomology group,

(]Cpl~.q]~r) ~-- Cpqr de 2 Epqr' (2) a

where the sum denotes the contributions of the various instantons,

a a a

Assuming for simplicity that there is only one (1,1) form one finds for the multipoint correlation functions of the ( 1,1 ) form in this case

Z n (If. n) = Cdn,3 + Ca, (4) a

n2 We use the notation of ref. [3].

With (E/C) ~ x -ystr+y~: +rt the constitutive equations then lead to the remaining relations

Yze = 3yx: = 2Yvi. (8)

The upshot of these relations is that the scaling dimensions of all the operators in the theory are de- termined in terms of the single quantity ~'str:

7(OoJv) = ¼PYstr. (9)

In addition, by use of the genus recursion relations

( a k ( O ~ ) O ~ O y )

= k(ak_~ (O~)O~)r/~(O~OpOy) (10)

we can further determine the scaling dimensions of the gravitational descendents an (O~) in terms of Ystr:

y(an(O~,v)) = ¼(p + 2n)ystr. (11)

The anomalous string dimension itself is deter- mined by the choice of the coupling constants tk,~ in the massive planar string equation

(P 0~) = q~pto,~ ~2C

+ Z ~ ' ~ k t k . a ( a k _ , ( 0 ~ ) 0 ~ ) . (12) k=l

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This equation can be reduced, via the recursion re- lations, to an equation purely in terms of the two- point functions of the order parameters of the theory. Choosing the coupling constants as - directions of order parameters:

t 0 , p - - x , t 0 , ~ - t ¢ , t 0 , g : - k ,

to,z~ i ~ 6~i, tom = 2, (13)

Combining this result with our previous result for the dimensions of the gravitat ional descendents we f ind

yk(e , , (O,op) ) = (p + 2 n ) / 2 k , (17)

from which we conclude that in the topological phase there are no marginal operators associated to the com- plex deformat ion of the Ca lab i -Yau manifold.

- directions of deformations:

t l ,v - 1, tk,v - - 1 / k , all others zero, (14)

leads to the general planar string equations at level k of a d i m c = 3 Ca lab i -Yau manifold with b2 = 1 #3

X ~

K =

k =

(1/k)(79R) k,

(79£)(p~)*-~, ( 7 9 ' D i ) ( 7 9 T ~ ) k - I ,

(mc)(79~) *-~

+ ( k - 1)(79~) k-2

X (Za (Ca(79~)-- 1)caeCa(7:'£) ) ,

2 = (7979)(79R) k - l + ( k - 1)(79R) k-2

× ((mc)(79£) + l(79v;)~iJ(z)jp))

+ (k - 1 ) (k - 2)(79R) k-3

+ Zco ×

k a

+ ~ (c~(79/~) - 2) e ~ ( ~ ¢ / / o

a / (15)

From these equations one readily derives (79;0) x -2/~ and hence the anomalous string dimension at level k is

3. Topological phases on non-Kiihler manifolds

In this section we analyse the topological behaviour of manifolds with no K~ihler deformations but with vanishing first Chern class. Because of the existence of Ca lab i -Yau manifolds with no complex deformations these spaces are of obvious interest in the context of a possible understanding of mirror symmetry in terms of intersection theory. We assume that the modul i space of instantons is again zero-dimensional and we wish to investigate the nature of the critical points in such a framework. Consider the operators 79, Di, R, corresponding to the puncture operator, the third co- homology and the dual of the puncture operator.

Since there are no K~ihler deformations it is obvious that the behaviour of these theories should resem- ble that of a two-complex-dimensional Ca lab i -Yau manifold, the K3 surface. The constitutive equations collapse to the one equation

(DiD j) = qij(79R), (18)

and the string equations simplify to

x = (1/k)(79Tg) k,

ei = (PZ)i)(797~) k-~,

,l = (p~e ) k-2 [ ( p p ) ( 7 9 r e )

+ ½ (k - 1)(pz) , )~ 'J(vjp)] . (19)

F rom this it follows that Y~ = Ystr + I l k , y~ = 27vi and

Ystr = 2 / k . (16)

#3 Terms with negative exponents at low k should be set to zero.

~'str = 2 - l / k , (20)

and hence the order parameters corresponding to the complex deformat ion are marginal at every level k. Thus it might be possible to establish an explicit map

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between these theories and the ones discussed in the previous section, if perhaps only for the marginal sec- tor.

4. Conclusion

phase structure of non-K~ihler manifolds with vanish- ing first Chern class. The anomalous dimensions of these theories turn out to be different from the K~ihler manifold and the order parameters corresponding to the primary operators of the complex deformation are always marginal for these theories.

We have shown that topological a-models on Calabi-Yau manifolds formulated via intersection theory on the moduli space of instantons define a new phase of string theory. In this new phase the space of order parameters of marginal operators collapses to a subspace. In the critical phase the order parameters corresponding to the complex deformations acquire nonvanishing vevs and the degeneracy of the various vacua is enlarged. All of this of course is rather rem- iniscent of the Higgs phase transitions, in which the order parameter acquires nonvanishing vevs as well.

Motivated by the issue of mirror symmetry in this topological framework we have also analysed the

Acknowledgement

We thank R. Dijkgraafand E. Witten for discussions and correspondence.

References

[1] E. Witten, Commun. Math. Phys. 118 (1988) 411. [2] E. Witten, Nucl. Phys. B 340 (1990) 281. [3] R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990)

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