c-Theorem and the topology of 2D QFTs

Volume 212, number 1

PHYSICS LETTERS B

15 September 1988

c - T H E O R E M A N D T H E T O P O L O G Y O F 2D Q F T s Cumrun VAFA Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 20 June 1988

In this note we briefly discuss some topological structures of the space of 2D QFTs. Our main tool is the consideration of Zamolodchikov's c function as a Morse function on this space.

One of the main obstacles in the development o f string theory is the lack of understanding of what corresponds to a "configuration space" for strings. This is clearly a very important issue if we wish to study non-perturbative phenomena in string theories. In particular if we wish to talk about tunneling phenomena in string theory the zeroth order step is to figure out what are the allowed intermediate configurations. So far most attempts have centered around choosing a particular space as the background for string propagation, and considering functionals on the loop space as the configuration space. Such an approach is unnatural from the string point of view which is expected to predict its own background. Also there are some string vacua which do not correspond to strings propagating on any background. For all these reasons we have to consider other alternatives. An attractive suggestion in this connection is to consider the space of all 2D QFTs which will be denoted by Q as the configuration space [ 1-3 ] (unless otherwise stated, we take Q to include only the unitary theories). In such a proposal, the conformally invariant QFTs play a special role, as they provide us with acceptable perturbative vacua for strings (if we impose the further condition that c, the central charge of the Virasoro algebra, be equal to 26). The "c-theorem" of Zamolodchikov [ 4 ] is a very interesting result for the above approach and has been considered in a number of works [ 5 ]. This theorem tells us that there is a function c which is defined for an arbitrary point of Q, which is critical at points pe Q, 28

Oec(p) =0, if and only if p is a conformally invariant theory in which case c ( p ) is the same as the central change of the Virasoro algebra for the conformal theory p. Moreover, Zamolodchikov shows that c decreases along the renormalization group trajectories (capturing the irreversibility of the renormalization group flow similar to the irreversibility of thermodynamics captured by the H-theorem). O f course the fixed points o f the renormalization group flows are nothing but the conformal theories. If p is a conformal theory, we can consider nearby theories, which are obtained from p by addition of a term gi~b~ to the action. Then it is shown in ref. [4] that c(gi) = c ( p ) - 3 ( 2 - d i ) g 2 + O ( g 3) ,

where di denotes the dimension o f the operator q~i at the conformal theory p. The operators whose expectation values decrease c are the relevant operators. These are the operators whose dimension is less than 2 (except for the trivial identity operator which does not create a flow). Operators whose dimension is 2 are called the marginal operators. The rest o f the operators are called irrelevant and correspond to the directions for which c increases. The c-theorem has been applied in particular to renormalization group flows for minimal unitary models [ 6-8 ]. What we wish to do in this paper is to use this function to discuss some topological aspects of Q.

0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Volume 212, n u m b e r 1

PHYSICS LETTERS B

As is well known in Morse theory ~l the knowledge of the critical points of a function on a manifold, and the behaviour of the function near the critical point carries a lot of the global topological properties of the space. Even though Morse theory has been best understood for the finite dimensional spaces, it has also been used in some infinite dimensional cases. Our space Q is infinite dimensional ~2, and strictly speaking one is not guaranteed that the Morse theory will be applicable. To make any progress in the description of the topology of Q using c we will assume that Morse theory could be used in this context. There are, in addition, some further technical difficulties which we will discuss later in the paper. We shall first briefly remind the reader of the Morse theory in the finite dimensional context. If we consider a surface shown in fig. 1, and consider the "height" function f which is the distance of a point on that surface from the ground (located relative to the ground as shown in fig. 1 ), there are 6 extremal points (which we denote by Pi). We consider the formal variable t and make a polynomial out of it as follows: we consider the number of directions in which f i s decreasing at each of the critical points. If ni denotes this number at Pi (which is called the index of f a t P~), we form the polynomial M ( f ) = Z t n' , P,

~t A nice introduction to Morse theory is in ref. [9]. ~2 The dimension of Q at a Q F T is equal to the n u m b e r of spinless fields of that theory.

7/////Z///////

Fig. 1. The f u n c t i o n f which is defined as the height of each point of the surface from the ground, has six critical points.

15 September 1988

which is called the Morse function associated t o f For the example under consideration it is given by M ( f ) = l + 4 t + t 2. To each manifold M one also associates a polynomial called the Poincar6 polynomial defined by

P(M) = ~ d i m H k ( M ) tk . k

In the example under consideration P ( M ) = 1 + 4 t + t 2. So we see that in this case M ( f ) = P ( M ) . In general this is not true for an arbitrary function f For instance the height function defined for fig. 2 gives M ( f ) = l + 5 t + 2 t 2. Morse theory tells us that in general

M ( f ) - P ( M ) = (1 + t ) Q ( t ) ,

(1)

where Q(t) is a polynomial with positive coefficients. A function for which M ( f ) = P ( M ) is called a perfect Morse function. For every compact finite dimensional manifold M one can construct perfect Morse functions. As a simple application of ( 1 ) we see that i f M ( f ) is an even function o f t then it must be a perfect Morse function (lacunary principle), which implies that we can read off the dimension of the homologies of M from the coefficients of M ( f ) . In the case M is infinite dimensional, the Poincar6 polynomial will in general be a power series in t. Let us return to the case of 2D QFTs, namely the space Q. We see that if we view the c-function as a Morse function on Q, then by Zamolodchikov's theorem the

X/H/T///////////// Fig. 2. The f u n c t i o n f ( a g a i n viewed as a height function) has two extra critical points,

29

Volume 212, number 1

PHYSICS LETTERS B

critical points o f it are nothing but the conformal theories. Moreover, the index o f the Morse function at each o f the critical points is the n u m b e r o f the relevant operators o f the corresponding conformal theory. F o r conformal theories which have marginal directions we have a slight difficulty in assigning a Morse index. W h a t we will have to do for such theories is to perturb the c-function slightly, to resolve the degeneracy, and then consider the p e r t u r b e d function as the Morse function. We will not go into this technicality in this paper. The question arises as to what extent can we compute the Morse polynomial associated to c? I f we knew all the conformal theories a n d the n u m b e r o f relevant operators for each one we would be able to c o m p u t e the Morse p o l y n o m i a l associated to c. But we lack a full classification o f c o n f o r m a l theories, and so we do not know all the critical points o f c. However, for c < 1 we do have a classification o f unitary conformal theories [ 10 ], and therefore we can calculate the Morse function associated with that subspace o f Q. F o r c < 1 the space o f conformal theories comes in three series, the A - D - E series. The r e n o r m a l i z a t i o n group flows seem to take the A series to itself [ 8 ]. If so this suggests that we can restrict our attention to the flow o f the A series. We therefore obtain a space with one critical point for each integer m > 1 with the value o f c given by c= 1

6 r e ( m + 1) '

m> 1 .

The spectrum o f the operators for these theories are known. In particular one can easily read off the number o f relevant operators. This is given by 2 ( m - 2 ). So if we form the Morse polynomial associated to this subspace we obtain 1 1q-te+t4+ .... l_t2,

(2)

where 1 corresponds to the trivial conformal theory with c = 0 , t 2 comes from the Ising m o d e l which has two relevant operators o f d i m e n s i o n 1/8 a n d 1, the next term l 4 c o m e s from the tri-critical Ising m o d e l which has four relevant operators, a n d so on. Using the lacunary principle one easily sees that for this subspace o f Q (i.e., for c < 1 and the flow o f the A series), c is a perfect Morse function. It turns out that ( 2 ) is the Poincar6 p o l y n o m i a l o f 30

15 September 1988

two familiar spaces: CP ~ and loop space o f SU (2). These two spaces are topologically different even though they share the same Poincar6 polynomial. They have the same homology but not the same topology. By looking at the global topology o f the ren o r m a l i z a t i o n group flows the relevant directions o f the tricritical Ising model, which is four dimensional, we can decide which ( i f either) o f the two spaces [ CP °° or the loop space o f SU ( 2 ) ] is relevant for the corresponding subspace o f Q. O f course, we have to b e a r in m i n d that there is no obvious reason why the one associated to conformal theories under consideration should be related to either o f these two famous examples. In fact there are infinitely m a n y topologically distinct spaces with the same Poincar6 p o l y n o m i a l given by (2). At any rate, this suggests that the topology o f the renormalization group flows may be quite interesting. The way the topology makes its appearance in the renormalization group flows is through the appearance o f identifications o f points when one considers the naive p a r a m e t r i z a t i o n s o f Q in terms o f the expectation values o f fields. It would be worthwhile to try to u n d e r s t a n d such identifications ~3 Before going further, we wish to discuss some implications o f the Morse idea in the study o f renormalization group flows. So far in the literature the emphasis has been to consider a single flow line from one conformal theory to another. F o r the c < 1 unitary models, since each critical point has two extra relevant operators c o m p a r e d to the next critical point, it means that there must be a one p a r a m e t e r family o f flows from one conformal theory to the adjacent one (i.e., we can turn on two o f the fields with arbitrary relative strengths and flow from one conformal theory to another. The relative strength labels the one p a r a m e t e r family). This should correspond to turning on the o d d p o l y n o m i a l s (such as the magnetic field) in the L a n d a u - G i n z b u r g classification o f critical p h e n o m e n a . It would be interesting to investigate this flow. It is not clear a priori why there should be a sensible truncation o f the space for c < 1 theories. In particular, non-trivial topological information at the b o u n d a r y at c = 1, m a y have to be taken into account ~3 Such identifications have begun to appear in the study of renormalization group flows for c< 1 theories [8 ].

Volume 212, n u m b e r I

P H Y S I C S LETTERS B

tO correctly apply the Morse theoretic approach ~4. Another technical point that we have avoided to discuss is the question o f symmetries o f conformal theories. Conformal theories could have some symmetries which force similar symmetries on the space of flows (for example all the A series include a Z2 symmetry, which in the L a n d a u - G i n z b u r g classification corresponds to the parity o f the power of the field). In such cases the space Q will have orbifold singularities. We have avoided a discussion of this point, by leaving somewhat vague what we mean exactly by Q. Our assumption is that we are dealing with a covering of Q which resolves these singularities, but does not introduce any additional critical points. So far we have been discussing the flow o f the A series. Even though there are indications that the renormalization group flows may keep one within this class, it may also be that the renormalization group flow takes one from the A series to the other series (it is known [8 ] that one can flow from the D - E series to the A series). At any rate we can write the Morse polynomial for all c < 1 theories. In other words we can include the additional critical points corresponding to the D - E series. The polynomial we find is not the Poincar6 polynomial of any space known to us. It would also be interesting to ask questions about other series of conformal theories. For example we can form the Morse polynomials corresponding to various W Z W models. In such cases, however, we have no guarantee that the renormalization group flows keep us within the class we started with. At any rate, it would be interesting to see whether we can attach any significance to the corresponding Morse polynomial. We would like to move to a different issue, and ask whether Q is connected. This is a very important issue, for if Q is not connected and is to be identified as the configuration space of strings, then strings may lose some of their predictive power, in that if we start a universe in one component o f Q it has no way of getting to some other components. We will now show that ifc is a perfect Morse function on Q, then Q cannot be connected! Consider the conformal theory o f **4 In fact, similarities that appear between the classification of c < 1 conformal theories, and c = 1 theories may be an indication that there is interesting topological information at c = 1

theories, which may have to be taken into account.

15 September 1988

the Monster [ 11 ]. This is an asymmetric orbifold with c = 24 obtained by modding the toroidal Leech lattice compactification by a reflection x--, - x on the left-moving coordinates, without touching the rightmovers (this theory has been studied recently in ref. [ 12 ] ). This theory is interesting from the point of view of conformal theories in that it does not contain any relevant operators! If we consider the theory with c = 0 in connection with this theory one o f two things could happen. If we draw the space Q (thinking of c as a height function) we have one o f the two cases shown in fig. 3. In the case (a) the space Q is disconnected, which means there is no way to get from the Monster theory to the trivial theory through intermediate 2D QFTs. In other words, the Monster theory and the trivial theory live on different components o f Q. In case (b) there is a third conformal theory with c > 24 which flows both to the Monster theory and to the trivial theory. In this case we see that the c-function is not a perfect Morse function, as it has critical points which could be removed (see fig. 3). We do not know which o f these two possibilities occur, but we find it an important question to settle one way or the other. Note that if the c-function turns out to be a perfect Morse function, we can attach a topological significance to each conformal theory. Namely the conformal theories would be in one to one correspondence with cohomology classes o f the space of 2D QFTs. So far our considerations have been classical. We may wish to think of renormalization group flows as q u a n t u m tunnelings between string vacua. In view o f the fact that Morse theory finds a natural physical interpretation in the work of Witten [ 13 ] one could try doing the same here. Namely we can view the cC

~ i II

C=24

C=O

M0nsler Trivial (a)

C-24 C=24 C:C ~ / / / / ~ ' l o n s t e r Trivial (b)

Fig. 3. In the case (a) the space Q is disconnected, in the case (b) c is not a perfect Morse function, and can be deformed to a function with fewer critical points (dashed curve). 31

Volume 212, number 1

PHYSICS LETTERS B

function playing the role o f the " m a g n e t i c field" on the space Q, and, considering the non-linear supersymmetric sigma model on Q, such an interpretation may be quite attractive in terms o f tunneling phen o m e n a between different vacua o f strings. But even before making such connections with string theory, we have to address some aspects o f strings which we have not properly taken into account yet. We have mainly focused on unitary conformal theories with arbitrary values ofc. If we wish to deal with string theory we have to change two things: First we should allow for non-unitary theories. This would allow minkowskian solutions to string theory. Secondly we should fix the value o f c = 26, and a d d ghosts to the system, so the net c = 0 . In such a context Zam o l o d c h i k o v ' s c-theorem would not be applicable, a n d in particular along the r e n o r m a l i z a t i o n group trajectories c could increase or decrease. This is so much the better if we wish to describe tunneling processes between string vacua, since they have the same c ( n a m e l y c = 0 ) . But there are a n u m b e r o f issues which have to be settled first: H o w much "non-unirarity" is allowed in the space we are considering? A related question is whether we can define an abstract notion o f what is the signature o f a general conformal theory (so that for the bosonic strings the solution to string theory corresponds to theories with signature 24)? To what extent are the ghosts allowed to mix in with the rest o f the coupling? W h a t is the meaning o f " t i m e " in connection with the r e n o r m a l i z a t i o n group flow parameter (this has been raised by Friedan [ 1 ] ) ? Clearly a lot o f issues will have to be straightened out before we come up with an acceptable q u a n t u m theory o f strings. However, we hope to have convinced the reader o f the i m p o r t a n c e o f the study o f the topology o f the renormalization group flows a n d its potential relevance for t h e q u a n t u m theory o f strings.

32

15 September 1988

We have greatly benefited from discussions with R. Bott, D. Friedan, D. Kazhdan, E. Martinec, S. Shenker, C. Taubes a n d E. Witten. This research was s u p p o r t e d in part by N S F contract PHY-82-15249, a n d by a fellowship from the H a r v a r d Society o f Fellows.

Note added. After completing this work we received a p a p e r [ 14 ] which deals with similar ideas.

References [ 1] D. Friedan, unpublished. [2] A.M. Polyakov, Phys. Scr. T15 (1987) 191. [3] T. Banks and E. Martinec, Nucl. Phys. B 293 (1987) 733. [4] A.B. Zamolodchikov, JETP Lett. 43 (1986) 731. [5] A.A. Tseytlin, Phys. Len. B 194 (1987) 63; V. Periwal, Princeton preprint PUPT- 1079; N.E. Mavromatos and J.L. Miramontes, Phys. Lett. B 212 (1988) 33; N.E. Mavromatos, Oxford preprint 11/88. [6] A.B. Zamolodchikov, Sov. J. Nucl. Phys. 44 (1987) 529. [ 7 ] A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 ( 1987 ) 687. [ 8 ] D. Kastor, E. Martinec and S. Shenker, unpublished. [9] R. Bott, Lecture at Carg~se Summer Institute, 1979, on Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980). [10] D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; J.L. Cardy, Nucl. Phys. B 270 (1986) 186; A. Capelli, C. Itzykson and J.-B. Zuber, Nucl. Phys. B 280 (1987) 445; D. Gepner, Nucl. Phys. B 287 (1987) 111. [ 11 ] I.B. Frenkel, J. Lepowskiand A. Meurman, Proc. Natl. Acad. Sci. USA 81 (1984) 32566. [ 12 ] L. Dixon, P. Ginsparg and J. Harvey, preprint PUPT- 1088, HUTP-88/A013. [13] E. Witten, Nucl. Phys. B 202 (1982) 253. [ 14] S.R. Das, G. Mandal and S.R. Wadia, preprint TIFR-TH88/33.