Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

Quantum open string theory with manifest closed string factorization

Bar ton Z w i e b a c h l

Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 8 November 1990

We present a quantum theory of covariant open and closed strings that satisfies manifestly the full BV master equation. The classical action includes interacting open strings and free closed strings. All open-closed and closed string self-interactions appear at the quantum level. The string diagrams, arising from a new minimal area problem, make all amplitudes manifestly factorizable in every open and closed string channel. As by-products we obtain a global closed string symmetry of classical open string theory, and a gauge invariant action for open strings propagating in closed string backgrounds.

1. I n t r o d u c t i o n

The covariant open string field theory built by Witten [ 1 ] in the BRST approach [2 ], has been an extremely successful model of a field theory of strings. It is based on a three string vertex that cou- ples symmetrically three open strings. Its off-shell loop amplitudes, however, are not manifestly factorizable in closed string channels and show spurious closed string poles [ 3 ]. This made it impossible to extract an off-shell closed string field theory from the open string theory. In addition, it is not clear yet whether or not the classical open string field theory satisfies the full master equation of Batalin and Vilkovisky (BV). If it did, consistent BRST quantization to all orders in the loop expansion would be guaranteed. The problem is a delicate issue ofregularization, and, as studied by Thorn [4], the quantum action may require the addition of an infinite series of interactions.

The main result in the present paper is a covariant quantum theory o f open and closed strings whose off- shell amplitudes, showing closed string poles as well as the open string poles, factorize manifestly in every channel. By construction, this theory satisfies the BV

1 Supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069.

master equation so its quantization is not proble- matic. This open-closed theory is naturally related to the closed string field theory [ 5-9 ]. The double of an open-closed string diagram is always a closed string diagram. The open-closed string diagrams arise as solutions o f a new minimal area problem involving open and closed curves.

The classical part of the open-closed theory is a nonpolynomial rearrangement of ref. [ 1 ], where ex- tra strips on the open string vertices prevent the ap- pearance of closed string poles at the loop level. It includes, in addition, free closed string fields. This classical action violates the BV master equation ex- plicitly and quantum interactions are needed to re- store the invariance. This is the origin of the open- closed and pure closed string vertices. The loop string diagrams, as expected, are different from those of ref. [ 1], since the latter when doubled are not always consistent closed string diagrams [ 9,10 ]. The open- closed theory is in clear correspondence with the light- cone theory of open strings. The work of ref. [ 1 1 ] can be shown to imply that the closed string kinetic term belongs to the classical light-cone action, and the open-closed and closed string self-interactions ap- pear in the quantum part. Thus gauge fixing the open- closed field theory to the light-cone is free of concep- tual difficulties. It is intriguing that the gauge sym- metry of open-closed string theory does not include

22 Elsevier Science Publishers B.V. (North-Holland)

Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

nonlinear coordinate transformations but only its li- nearized part, due to the presence of a free spin two particle. At the quantum level the theory has BRST transformations corresponding to fully nonlinear co- ordinate transformations. The gauge symmetry does not become nonlinear since in the BV formalism only BRST transformations are well defined for the quan- tum action.

Two interesting applications can be developed. Both use vertices coupling open and closed strings via a disk. The explicit form of these vertices is simple to find. In the first application, following Hata and Nojiri [ 12 ], we give a symmetry transformation for open string fields parametrized by a "global" closed string parameter. This symmetry appears to be the closed string extension of Poincar6 transformations (for superstrings it would include the string exten- sion of global supersymmetry). In the second appli- cation, we give a classical open string field theory de- scribing the propagation of open strings on a closed string background ~Vo. It is interesting that open string gauge invariance requires that T0 satisfies the closed string field theory equation of motion.

As in our previous work, the explicit form of some of the higher genus vertices is not yet known. The minimal area problem is expected to determine them [7]. The results announced in this letter are estab- lished in complete detail in refs. [ 13,14 ].

2. A minimal area problem and open-closed string vertices

Our objective is to define open-closed string dia- grams whose double are closed string diagrams. Re- call hat closed string diagrams use the metric of min- imal area under the condition that all nontrivial closed curves are longer or equal to 2n [ 8,7 ]. We pro- pose the following problem: Given a genus g Rie- mann surface R with b boundaries, n punctures in the interior and m punctures on the boundary compo- nents (g, b, n, m >1 O) the open-closed string diagram uses the metric p, o f minimal (reduced) area under the condition that the length o f any nontrivial Jordan open curve in R, be greater or equal to g and that the length o f any nontrivial Jordan closed curve be greater or equal to 2g. Here, a nontrivial open curve is an open curve that cannot be contracted away keeping its

endpoints at the boundaries. In contrast with the string diagrams of ref. [ 1 ], there cannot be short closed curves, we insist that closed curves in the open surface be longer than 2n. Thus closed string poles will arise from long tubes, as desired for manifest closed string factorization. Conditions are also im- posed on the nontrivial open curves to obtain open string factorization and because upon doubling these curves become nontrivial closed curves. We can show that a diagram on an open Riemann surface R whose double is a closed string diagram, is an open-closed string diagram on R [ 14 ].

Let us give some examples. The first concerns the open-closed vertex, shown in fig. 1 a. An open string of length n, suddenly becomes a closed string of length 2n, by creating an extra piece of length n. As strange as this may appear at first sight, this is the natural open-closed covariant vertex since its double, shown to the right, is the covariant three closed string ver- tex. As a second example we consider a vertex cou- pling two open strings to a closed string in fig. 2a. The pattern of string overlaps for this diagram is shown to the right. The length a~2 of the segment AB must be shorter than ½n, otherwise the nontrivial open curve CAD would be shorter than n. The vertex has one modular parameter a~2~ [0, In] and its double is a closed string polyhedron. Our final example cor- responds to a disk with a closed string puncture and represents a closed string going into the vacuum (fig. 3). A closed string suddenly stops leaving a bound- ary. It is an open-closed string diagram because it is the metric of minimal area for a punctured disk. This unusual interaction is necessary for complete facto- rization. It has no analog in the light-cone field the- ory, but it is familiar from Fock space states describ- ing boundary conditions in open string theory [ 15 ].

E

D [

B D_~ C P

(a)

E

Fig. 1. (a) The open-closed transition vertex in covariant open- closed string theory. An open string of length n suddenly becomes a closed string of length 2n. (b) The double of the open-closed vertex is the three closed string vertex.

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Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

(a)

B

Ib) Fig. 2. (a) The interaction vertex for two open strings and one closed string. The two open strings, have some mutual overlap (the segment AB), the rest of the open strings create part of the emerging closed string. This interaction has one modular param- eter, which can be taken to be the length of the segment AB. (b) The pattern of string overlaps in this interaction.

o,~-- 7r -.~

Fig. 3. The closed-boundary vertex. It represents an incoming closed string that just stops, leaving a boundary. This interaction is needed for complete factorization of amplitudes in covariant open-closed quan tum theory.

3. Quantum open-closed string theory

Our objective is a quantum field theory whose Feynman graphs construct the string diagrams aris- ing from the minimal area problem. The construc- tion will be done in three steps. The first step consists in showing that the string diagrams can be built with vertices and propagators. In this step one isolates the vertices, in fact, a vertex for each moduli space. The second step will be writing down a geometrical con-

sistency condition relating all the string vertices of the open-closed theory. Finally, as a third step, the con- sistency condition, is used to construct the string ac- tion and show that it satisfies the full BV master equation.

3. I. Extracting vertices from string diagrams

We proceed order by order in the dimensionality of the corresponding moduli space. At dimension zero we have four relevant surfaces: (i) the disk with three open string punctures, for which the string diagram is the Witten vertex but with external strips of length n attached to the external legs, (ii) the disk with one open and one closed string puncture, corresponding to the open-closed vertex shown in fig. 1, the vertex is defined to include a strip of length n for the exter- nal open string, and a stub of length n for the external closed string, (iii) the disk with a single closed string puncture, representing the closed string going into the vacuum, whose string diagram in fig. 3 implies that the vertex is a cylindrical stub of length n, and finally (iv) the three punctured sphere, which uses the sym- metric closed string vertex with stubs of length n on each leg. These are the basic elementary vertices of the open-closed theory.

As in ref. [ 7 ], we assume that near enough to the closed string punctures and near enough to the open string punctures, the minimal area diagrams have cylinders of circumference 2n and strips of length n, respectively. The place where the metric stops being that of a flat cylinder or a flat strip, fixes the normal- ization of the local coordinates at the punctures. We also assume that if the external strips or cylinders are truncated, leaving stubs or strips of at least length n, the resulting surfaces are sill surfaces of minimal area. Both assumptions can be proven when the minimal area metrics arise from quadratic differentials (which is the case for all punctured disks and spheres, and for a fraction of every moduli space). Further evi- dence for this assumption was given in ref. [ 7 ]. Now we proceed with sewing. The crucial fact is that sew- ing of minimal area surfaces gives surfaces of mini- mal area [ 7 ]. The stubs and strips in the vertices pre- vent the appearance of closed curves shorter than 2n or nontrivial open curves shorter than n in the sewn surface. Thus, when we look at a particular moduli space, and construct all inequivalent sewing config-

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Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

Fig. 4. An open string one loop tadpole with one external open string. The parameter T is the length of the internal propagator. As T~0 we obtain a singular limit.

urations, there cannot be overcounting O f surfaces, since all metrics are different and metrics of minimal area are unique. The surfaces missing to complete the modular region, with their minimal area metrics, and external legs amputated down to stubs and strips of length n, are taken to define the corresponding ver- tex. This procedure is carried out order by order in the dimensionality o f moduli space, and allows us to extract vertices for every moduli space.

It should be noted that the three open string vertex was modified to include strips to prevent the appear- ance of a collapsed tadpole at the one loop level (fig. 4). It is not clear whether the contribution to the BV equation from this degenerate surface vanishes. The strips on the external legs prevent now the full col- lapse and further non-singular interactions are needed. Since we have changed the three string ver- tex, in order to restore the correct covering of open string amplitudes, the above step by step procedure introduces N>~4 open string vertices (via a disk) consisting of all the Feynman graphs built with the original three string vertex, with all internal propa- gators shorter or equal to 2n, and with strips on the external legs. These define the rearranged theory.

3.2. Consistency condition on string vertices

The general vertex isolated above corresponds to a set o f surfaces (with metrics) in some moduli space. The vertex is specified by the genus G, the number N of closed string punctures, the number B of boundary components, and the number mi>~ 0 o f open string punctures at each boundary component . We denote

by M the total number of open string punctures: M = Y f=~mi. The total number of punctures in the surface, ( N + M ) , must be at least one. This interac- tion vertex is represented by a blob. The curly lines are closed strings, each heavy dot is a boundary, and the straight lines are open strings.

This general vertex fills a region of the correspond- ing rnoduli space. The fact that the vertices generate Feynman graphs that cover correctly moduli space implies that they satisfy a consistency condition [ 9 ]: the set of string diagrams in the boundary of the re- gion of moduli space covered by the vertex must match with the string diagrams obtained with Feyn- man graphs using lower point vertices and one col- lapsed propagator. This condition for the case o f open-closed string theory is shown in fig. 5. For sim- plicity we are only considering the case of orientable surfaces. On the left-hand side we have the boundary of the general vertex. On the right-hand side we have five type of configurations. The first configuration

1 m B

2 "" I

N t

~ m n 2

n l n

2 -- t --

N

Fig. 5. The geometrical consistency condition of vertices for an open-closed string theory. The boundary of the region of moduli space covered by a vertex must coincide with all possible config- urations built with a single collapsed propagator.

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Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

corresponds to a collapsed open string propagator. In this process we lose one boundary and two open strings, thus we sum over all possible blobs such that: B~ +B2 = B + 1, and Ml +342 = M + 2 . The second configuration has a collapsed closed string propaga- tor. The next three configurations involve loop like sewing. The last one, involving a closed string, is straightforward. For the open ones there are two cases to be distinguished; the two open strings to be sewn might be in the same boundary or in different bound- aries. Suppose the two open strings are in the same boundary. If they are sewn the number of open strings decreases by two, and the number of boundaries in- creases by one. Therefore, the relevant blob in this case has (G, N, B - 1, M + 2 ). If the two strings are in different boundaries again we lose a boundary and two open string punctures, this time, however, the ge- nus increases by one. Thus the relevant blob for this case is one with ( G - 1, N, B+ 1, M + 2 ) .

3.3. Open-closed string action and the master equation

Now we can set up a quantum action S for open and closed string fields • and ~u, respectively. The geometrical consistency conditions will fix the order of h, and the power of g (the open string coupling constant) for each interaction. Moreover, it guaran- tees that the action S satisfies the BV master equa- tion; (OS/Ofb)(OS/O(~)=ih02S/O~O~, where ~ de- notes all fields in the theory. Expanded in open and closed string fields and letting S = Y~phPS p, we find

(OSPl OS p2 OS pl OSP2~

w , + 0*W :

0 2 S p - 1 0 2 S p - 1 = i ~ + i - - (1)

O~)iO~_)i '

where the repeated index i denotes integration over string coordinates. This equation is recognized to be a representation, using some conformal field theory, of the geometrical consistency condition of fig. 5. The terms on the right-hand side correspond to collapsed propagators between punctures in the same vertex, and the terms on the left-hand side correspond to col- lapsed propagators between punctures in two differ- ent vertices. The geometrical equation contains 0~t: as one term in the equation. We know that the 0 op-

erator is realized in the field theory by the sum of BRST operators acting on the vertex [16,6]: < ~Pl ~ Q = < 8~°1. Thus the 8~terms arises from the left-hand side ofeq. ( 1 ), when one of the S' refers to the kinetic terms in the string action.

We take the full (rearranged) classical open string theory to belong to S O . It follows that the closed string kinetic term must also belong to the classical action. Consider any interaction in S p with a fixed value of p, corresponding to some particular vertex ~o. All terms giving rise to 8~/r must appear in the same equation. Therefore if S" contains both open and closed fields, the action of both open and closed BRST operators must appear in the same equation. Since the open BRST operator Q appears in S °, its contri- bution to 0 f r arises from a term of the form (8S"/ 0~)(0S°/8qb). The contribution from the closed BRST operator () must arise from a term of the form (8SP/8~) (8SP'/O~). The homogeneity of eq. (1) implies that p ' = 0, and therefore () belongs to S °.

Let us now derive the value of p for any interac- tion. Since all surfaces without closed string punc- tures can be built with open string Feynman rules alone, one can show that in this case p = 2 G + B - 1. Consider the following case of the geometrical equation

- 0 - = - ~ - ~( • (2) p=i p=O

The interaction on the LHS has p = l. Because of eq. ( 1 ) the tree like configuration must have equal total p and the loop like configuration must have a value smaller by one unit. We deduce that the open-closed string vertex has p = 1. Each time we add a closed string we increase the total p by half a unit, therefore the complete expression for p reads p=2G+B+ ½N- 1. The power of the open string coupling con- stant g in front of each elementary interaction can be derived similarly. Each interaction is accompanied by a factor (g)q where q = 4 G + 2 N + 2 B + M - 4 = 2 p + ( N + M - 2 ) .

We can now give the complete action S= ~p=o.l/2.L..hPS ~', satisfying explicitly order by order the BV master equation:

~5 ~' B,M , G,N,B r n l . . . m o = 0 . . . o ~

2G+B--I+N/2=p N+M>~I

(3)

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Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

G,N j G,N ml SB, M = ~[~B,M ( t~ ) ... ( ~b ) m. ~tN

///dim .At' x ""O bff t[JN ~ ~- f ~ i~-_l dmi~i) Oq~l""O~MObr-~l / c~

(4)

where M = Y~ ~=~m,. In eq. (4) we have spelled out the various elements entering the definition of the vertices. We have an integral over the region ~ of the corresponding moduli space of dimension dim J¢'( = 6 G - 6 + 2N+ 3 B + M ) filled by the vertex. The modular parameters are the m~ and the associ- ated antighost insertions are the 4 . These bring the net ghost number in the correlator to ( 6 - 6 G - 3B), as it should be. The vertex operators O representing the fields are inserted into the surface using the com- plex coordinates induced at the punctures by the minimal area metric. For closed strings they have ghost number two, and for open strings ghost number one. In the above sum there are some special cases. We do not include the one punctured sphere o,1 So.o, nor the disk with one open string puncture o.o $1,,. The ki- netic terms are also not defined by eq. (4). They are given by S°: ° = fq~Q~, and S°:o 2 =fTO.bff ~ Trespec- tively. The last exception is the closed/boundary ver- tex of fig. 3 corresponding to S°:o ~ . It requires an extra ghost insertion because of the presence of a confor- mal Killing vector (see ref. [ 13 ] ).

The BRST transformations are found from the field equations 8a(bs- ~)=8S /8~ , 88q~=fS/Sq~. The string fields are decomposed into field and antifield bffT~~/+c+q/* and qb=~+CoO*, and the gauge fixed action in the Siegel gauge is given by S = SI ¢.=~.=o (following the notation ofref. [ 17 ] ). The (on-shell) nilpotent BRST transformations are given by ~B~/=~B~ffI~,*=0*=0 and ~B~b=SB~9[¢*=0.=O. By construction, they leave invariant the gauge fixed ac- tion plus the log of the measure.

As announced, the classical part contains the rear- ranged open string theory plus free closed strings. The quantum part of the action begins at h 1/2 where we find the classical three closed string vertex and all couplings of a single closed string via a disk to m open strings (for m = 0 , this is the closed boundary ver- tex). To order h, we find a five punctured sphere, two closed strings coupling to m >10 open strings via a disk, and all numbers of open strings coupling through an-

nuli. It is interesting that at p = 3 we get both a five punctured sphere, and a one punctured torus. Thus the classical and quantum interactions of pure closed string theory get mixed here. To extract a classical in- teracting closed string theory from the above quan- tum action, one must define the gravitational self- coupling x = h 1/2g2.

4. Applications for open-closed interactions

The simplest kind of open-closed interactions, namely the interactions via a disk, have interesting physical applications. These vertices satisfy proper- ties that arise as particular cases of the general geo- metrical consistency conditions in fig. 5. Two appli- cations will be sketched below.

4. I. Global closed string symmetry of classical open strings

Particle field theories on flat Minkowski space have global spacetime symmetries. These can be under- stood as the remnant of general coordinate transfor- mations, when gravity is turned off. It should not be surprising that the set of global "spacetime" symme- tries of classical open string theory is parametrized by closed strings. Following the interesting work of Hata and Nojiri [ 12], who found such symmetry for the covariantized light-cone theory, we will use the open-closed vertices to find the symmetry transfor- mation for Witten's open string theory. The vertex coupling N open strings to a closed string via a disk is

(1 2 ...N; l~[ N~/.

(5)

where to the right we show the patterns of overlap (see ref. [ 13 ] for details). The open-closed vertex satisfies (1; lc ] ( Q + Q ) = 0 . The BRST property of the higher vertices is summarized by the following subcase of fig. 5 (since our considerations are classi- cal, we take vertices with no stubs):

(6)

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Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

The symmetry transformation of the open string field is given by

8n IAt ) = ( ( 1'; lclA~ )

+n=z ~ g(n-t)(l '2""n; lclAlA2""A")) l l ' l ) '

(7)

where A is a closed string field. Under this transfor- mation the action LP°=AQA+ ]A 3 varies by 8ALP ° = -- 2 ( QA I ~ ( A ) ) where I ~ ( A ) ) is a closed string field nonlinear on the open string field

I,~(A) )

1 g~,,_~) = ~ n (12.. .n;l 'clA,. . .A,)llclc). (8)

r l = l

Invariance requires the "global" condition (~A = 0. This symmetry transformation is expected to contain Poincar6 transformations, as was the case in ref. [ 12 ]. It would be interesting to elucidate the significance of the additional symmetry transformations con- tained in this string symmetry.

the full closed string equation of motion Q~Uo+ ~u~ . . . . 0.

The above situation is not unfamiliar in particle field theory. A gravitino can be coupled to back- ground gravity if the background is Ricci fiat. In string theory consistent string propagation on external backgrounds forces the backgrounds to satisfy field equations [ 18 ]. The above action may be useful to understand better the essential question of back- ground independence in string field theory [ 19 ]. It is clear that the open-closed couplings allow us to de- fine a family of" tensors" such as the I ~ ( A ) ) , with interesting transformation properties. In particular I ~ ( A ) 7, since it couples linearly to the closed string field, and thus to the graviton, is some kind of en- ergy-momentum tensor for open string fields.

Acknowledgement

I wish to acknowledge and thank D. Freedman, S.J. Gates, T. Kugo, C. Schubert, A.S. Schwarz, C.B. Thorn, S. Yost and M. Wolf for useful conversations.

4.2. Open string theory on closed string backgrounds References

The above results suggest that we can couple an open string to a closed string background ~o by add- ing to the lagrangian LP ° the term LP ~ = ( ~o I ~ ( A ) ) . Under an open string gauge transformation I ~ ( A ) ) goes into Q acting on something plus a term propor- tional to the equation of motion. The first term van- ishes if we require that Q~Uo= 0 and the second term can be taken care of by modifying the gauge transfor- mation of the open string field. This does not stop here since 5° ~ is not invariant under the modified gauge transformation. The full action includes all possible couplings of open to closed strings via a disk

5 ° = ~ ° + ~ ( ~ u " l ~ ( A ) ) , (9) n = l

where i ~ ( A ) ) is a ket in the direct product of n closed strings, and is completely nonlinear in A. It is built out of vertices of the form ( 1... m; lc ... ncl, for all values of m. The action is invariant under the modified gauge transformation when the linearized condition Q~Uo= 0, found above, gets generalized to

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