Closed/open string diagrammatics
Abstract.
We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bioperad on the level of homology.
Introduction
There has been considerable activity towards a satisfactory diagrammatics of open/closed string interaction and the underlying topological field theories. For closed strings on the topological level, there are the fundamental results of Atiyah and Dijkgraaf [1, 2], which are nicely summarized in [3]. The topological open/closed theory has proved to be trickier since there have been additional unexpected axioms, notably the Cardy condition [4, 5, 6, 7, 8]; this algebraic background is again nicely summarized in [9].
In closed string field theory [10, 11, 12], there are many new algebraic features [13, 14, 15], in particular, coupling to gravity [16, 17] and a BatalinVilkovisky structure [18, 19]. This BV structure has the same origin as that underlying string topology [20, 21, 22, 23, 24] and the decorated moduli spaces [25, 26, 27].
In terms of open/closed theories beyond the topological level, many interesting results have been established for branes [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] and Gepner models in particular [46, 47, 48]. Mathematically, there has also been work towards generalizing known results to the open/closed setting [49, 50, 51, 52, 53].
We present a model which accurately reflects the standard phenomenology of interacting open/closed strings and which satisfies and indeed rederives the “expected” equations of open/closed topological field theory and the BVstructure of the closed sector. Furthermore, the model allows the calculation of many new equations, and there is an infinite algorithm for generating all of the equations of this theory on the topological level. A finite set of equations, four of them new, are shown to generate open/closed duality.
The rough idea is that as the strings move and interact, they form the leaves of a foliation, the “string foliation”, on their worldsheets. Dual to this foliation is another foliation of the worldsheets, which comes equipped with the additional structure of a “transverse measure”; as we shall see, varying the transverse measure on the dual “measured foliation” changes the combinatorial type of the string foliation.
The algebra of these string interactions is then given by gluing together the string foliations along the strings, and this corresponds to an appropriate gluing operation on the dual measured foliations. The algebraic structure discovered is new, and we axiomatize it (in Appendix A) as a “closed/open” or “c/o structure”. This structure is present on the topological level of string interactions as well as on the chain level. On the homology level, it induces the structure of a modular bioperad, which governs c/o string algebras (see Appendix A and Theorem 4.4).
Roughly, a measured foliation in a surface is a collection of rectangles of some fixed widths and unspecified lengths foliated by horizontal lines (see Appendix B for the precise definition). One glues such a collection of rectangles together along their widths in the natural measurepreserving way (cf. Figure 5), so as to produce a measured foliation of a closed subsurface of . In the transverse direction, there is a natural foliation of each rectangle also by its vertical string foliation, but this foliation has no associated transverse measure. In effect, the physical length of the string is the width of the corresponding rectangle. A measured foliation does not determine a metric on the surface, rather, one impressionistically thinks of a measured foliation as describing half of a metric since the widths of the rectangles are determined but not their lengths (see also §B.1 for more details).
Nevertheless, there is a condition that we may impose on measured foliations by rectangles, namely, a measured foliation of by rectangles is said to quasi fill if every component of complementary to the rectangles is either a polygon or an exactly oncepunctured polygon. The cell decomposition of decorated Riemann’s moduli space for punctured surfaces [54, 55, 56, 57, 58] has been extended to surfaces with boundary in [26], and the space of quasi filling measured foliations by rectangles again turns out to be naturally homotopy equivalent to Riemann’s moduli space of (i.e., classes of structures on surfaces with one distinguished point in each hyperbolic geodesic boundary component; see the next section for further details). Thus, in contrast to a measured foliation impressionistically representing half a metric, a quasi filling measured foliation actually does determine a conformal class of metrics on . See the closing remarks for a further discussion of this “passage from topological to conformal field theory”.
Figure 1 Foliations for several string interactions, where the strings are represented by dashed lines and the dual measured foliation by solid lines. The white regions in parts ab are for illustration purposes only
More explicitly in Figure 1, each boundary component comes equipped with a nonempty collection of distinguished points that may represent the branes, and the labeling will be explained presently. That part of the boundary that is disjoint from the foliation and from the distinguished points has no physical significance: the physically meaningful picture arises by replacing each distinguished point in the boundary by a small distinguished arc (representing that part of the interaction that occurs within the corresponding brane) and collapsing to a point each component of the boundary disjoint from the foliation and from the distinguished arcs.
Since the details for general measured foliations may obfuscate the relevant combinatorics and phenomenology of strings, we shall restrict attention for the most part to the special measured foliations where each nonsingular leaf is an arc properly embedded in the surface. The more general case is not without interest (see Appendix B).
The natural equivalence classes of such measured foliations are in onetoone correspondence with “weighted arc families”, which are appropriate homotopy classes of properly and disjointly embedded arcs together with the assignment of a positive real number to each component (see the next section for the precise definition). Furthermore, the quotient of this subspace of foliations by the mapping class group is closely related to Riemann’s moduli space of the surface (again see the next section for the precise statement).
A windowed surface is a smooth oriented surface of genus with punctures and boundary components together with the specification of a nonempty finite subset of each boundary component, for , and we let denote the set of all distinguished points in the boundary of and let denote the set of all punctures. The set of components of is called the set of windows.
In the physical context of interacting closed and open strings, the open string endpoints are labeled by a set of branes in the physical target, and we let denote this set of brane labels, where we assume . In order to account for all possible interactions, it is necessary to label elements of by the power set (comprised of all subsets of ). In effect, the label denotes closed strings, and the label denotes the formal intersection of the corresponding branes. This intersection in the target may be empty in a given physical circumstance.
A branelabeling on a windowed surface is a function
where denotes the disjoint union, so that if for some , then is the unique point of in its component of . A branelabeling may take the value at a puncture. (In effect, revisiting windowed surfaces from [61] now with the additional structure of a branelabeling leads to the new combinatorial topology of the next sections.)
A window on a windowed surface branelabeled by is called closed if the endpoints of coincide at the point and ; otherwise, the window is called open.
To finally explain the string phenomenology, consider a weighted arc family in a windowed surface with branelabeling . To each arc in the arc family, associate a rectangle of width given by the weight on , where is foliated by horizontal lines as before. We shall typically dissolve the distinction between a weighted arc and the foliated rectangle , thinking of as a “band” of arcs parallel to whose width is the weight. Disjointly embed each in with its vertical sides in so that each leaf of its foliation is homotopic to rel . Taken together, these rectangles produce a measured foliation of a closed subsurface of as before, and the leaves of the corresponding unmeasured vertical foliation represent the strings.
Thus, a weighted arc family in a branelabeled windowed surface represents a string interaction. Given such surfaces with weighted arc families and a choice of window of , for , suppose that the sum of the weights of the arcs in meeting agrees with the sum of the weights of the arcs in meeting . In this case as in open/closed cobordism (see e.g. [9]), we may glue the surfaces along their windows respecting the orientations so as to produce another oriented surface , and because of the condition on the weights, we can furthermore combine and to produce a weighted arc family in (cf. Figure 5). This describes the basic gluing operations, namely, the operations of a c/o structure on the space of all weighted arc families in branelabeled windowed surfaces (cf. Section 2 for full details). Furthermore, these operations descend to the chain and homology levels as well (cf. Section 2 and Appendix A).
As we shall explain (in Section 3), the degree zero indecomposables of the c/o structure are illustrated in Figures 3 and 4, and further useful degree one indecomposables are illustrated in Figure 6 (whose respective parts ae correspond to those of Figure 1.)
Relations in the c/o structure of weighted arc families or measured foliations are derived from decomposable elements, i.e., from the fact that a given surface admits many different decompositions into “generalized pairs of pants” (see the next section), so the weighted arc families or measured foliations in it can be described by different compositions of indecomposables in the c/o structure.
We shall see that all of the known equations of open/closed string theory, including the “commutative and symmetric Frobenius algebras, GerstenhaberBatalinVilkovisky, Cardy, and center (or knowledge)” equations, hold for the c/o structure on chains on weighted arc families (cf. Figures 711).
Furthermore, we shall derive several new such equations (cf. Figure 12) and in particular a set of four new equations which together with known relations generate closed/open string duality (see Theorem 3.1).
Indeed, it is relatively easy to generate many new equations of string interactions in this way, and we shall furthermore (in Section 3.2) describe an algorithm for generating all equations of all degrees on the topological level, and in a sense also on the chain level.
We turn in Section 4 to the algebraic analysis of Section 3 and derive independent sets of generators and relations in degree zero on the topological, chain, or homology levels. In particular, this gives a new nonMorse theoretic calculation of the open/closed cobordism group in dimension two [5, 9]. Several results on higher degree generators and relations are also presented, and there is furthermore a description of algebras over our c/o structure on arc families.
Having completed this “tour” of the figures and this general physical discussion of the discoveries and results contained in this paper, let us next state an “omnibus” theorem likewise intended to summarize the results mathematically:
Theorem For every branelabeled windowed surface , there is a space of mapping class group orbits of suitable measured foliations in together with geometrically natural operations of gluing surfaces and measured foliations along windows. These operations descend to the level of piecewiselinear or cubical chains for example.
These operations furthermore descend to the level of integral homology and induce the structure of a modular bioperad, cf. [9]. Algebras over this bioperad satisfy the expected equations as articulated in Theorem 4.4.
Furthermore, new equations can also be derived in the language of combinatorial topology: pairs of “generalized pants decompositions” of a common branelabeled windowed surface give rise to families of relations.
In degree zero on the homology level, we rederive the known presentation of the open/closed cobordism groups [5, 9], and further partial algebraic results are given in higher degrees. In particular, several new relations (which have known transformation laws) are shown to act transitively on the set of all generalized pants decompositions of a fixed branelabeled windowed surface.
This paper is organized as follows. 1 covers the basic combinatorial topology of measured foliations in branelabeled windowed surfaces and their generalized pants decompositions leading up to a description of the indecomposables of our theory, which in a sense go back to the 1930’s. 2 continues in a similar spirit to combinatorially define the spaces underlying our algebraic structure on the topological level as well as the basic gluing operations on the topological level. The operations on the chain level then follow tautologically. The operations on the homology level require the analysis of certain fairly elaborate flows, which are defined and studied in Appendix C and also discussed in 2. In 3 continuing with combinatorial topology, we present generators, relations, and finally prove the result that appropriate moves act transitively on generalized pants decompositions. 4 finally turns to the algebraic discussion of the material described in 3 and explains the precise sense in which the figures actually represent traditional algebraic equations; 4 furthermore presents our new algebraic results about string theory. Closing remarks in particular include a discussion of how one might imagine our results extending from topological to conformal field theory.
Appendix A gives the formal algebraic definition and basic properties of a c/o structure, and Appendix B briefly surveys Thurston’s theory of measured foliations from the 197080’s and describes the extension of the current paper to the setting of general measured foliations on windowed surfaces. It is fair to say that Appendix A could be more appealing to a mathematician than a physicist (for whom we have tried to make Appendix A optional by emphasizing the combinatorial topology in the body of the paper), and that the physically speculative Appendix B should probably be omitted on a first reading in any case.
Appendix C defines and studies certain flows which are fundamental to the descent to homology as described in Appendix A. Nevertheless, the discussion of the flows and their salient properties in Appendix C is independent of the technical aspects of Appendix A (since chains are interpreted simply as parameterized families); in a real sense, Appendix C is the substance of this paper beyond the combinatorial topology, algebraic structure, and phenomenology, so we have strived to keep it generally accessible.
1. Weighted arc families, branelabeling, and generalized pants decompositions
1.1. Weighted arc families in branelabeled windowed surfaces.
In the notation of the introduction, consider a windowed surface , with punctures , boundary distinguished points and windows , together with a branelabeling . Define the sets
Fix some brane label , and define the branelabeling to be the constant function on with value ; corresponds to the “purely open sector with a spacefilling branelabel”. On the other hand, the constant function with value corresponds to the “purely closed sector”.
It is also useful to have the notation , where is the cardinality of a set . For instance, a pair of pants with one distinguished point on each boundary component is a surface of type , while the data of the windowed surface includes the specification of one point in each boundary component as well. One further point of convenient notation is that we shall let simply denote a surface of genus with punctures and boundary components when there is a unique distinguished point on each boundary component.
Define a arc in to be an arc properly embedded in with its endpoints in so that is not homotopic fixing its endpoints into . For example, given a distinguished point , consider the arc lying in a small neighborhood that simply connects one side of to another in ; is a arc if and only if .
Two arcs are parallel if they are homotopic rel , and a arc family is the homotopy class rel of a collection of arcs, no two of which are parallel. Notice that we take homotopies rel rather than rel .
A weighting on an arc family is the assignment of a positive real number to each of its components.
Let denote the geometric realization of the partially ordered set of all arc families in . is described as the set of all projective positively weighted arc families in with the natural topology. (See for instance [25] or [59] for further details and Figure 2 below for an illustrative example.)
The (pure) mapping class group of is the group of orientationpreserving homeomorphisms of pointwise fixing modulo homotopies pointwise fixing . acts naturally on by definition with quotient the arc complex
We shall also require the corresponding deprojectivized versions: is the space of all positively weighted arc families in with the natural topology, and
Figure 2 The arc complex is homeomorphic to a circle . We omit the common label at each point of to avoid cluttering the figure. There are exactly the two orbits of arcs on the right and left. These can be disjointly embedded in the two distinct ways at the top and bottom. As the parameter on the bottom varies in the range , there is described a projectively weighted arc family, that is, the two disjoint arcs determine a onedimensional simplex in , and likewise for the parameter on the top. The two onesimplices are incident at their endpoints as illustrated to form a circle . Furthermore, , , and with the primitive mapping classes acting by translation by one on .
It will be useful in the sequel to employ a notation similar to that in Figure 2, where parameterized collections of arc families are described by pictures of arc families together with functions next to the components, where the functions represent the parameterized evolution of weights. We shall also typically let the icon denote either a puncture or a distinguished point on the boundary as in Figure 2.
In contrast to Figure 2, if we instead consider the purely open sector with spacefilling branelabel , for , then there is yet another orbit of arc encircling the boundary distinguished point. In this case, is homeomorphic to the join of the circle in Figure 2 with the point representing this arc, namely, is homeomorphic to a twodimensional disk.
For another example of an arc complex, take the branelabeling on , for which again . There is a unique orbit of singleton arc, and there are two possible orbits of arc families with two component arcs illustrated in Figure 3. Again, is homeomorphic to a circle. If , then is homeomorphic to a threedimensional disk.
To explain the connection with earlier work, consider the purely closed sector on . Let denote the subspace of corresponding to all projective positively weighted arc families so that each component of is either a polygon or an exactly oncepunctured polygon, i.e., quasi fills ; was shown in [26] to be proper homotopy equivalent to a natural bundle over Riemann’s moduli space of the bordered surface as defined in the Introduction provided is not an annulus (, ).
Let denote the subspace of corresponding to all projective positively weighted arc families so that each window of (i.e., each boundary component) has at least one arc in incident upon it. The spaces comprise the objects of the basic topological operad studied in [25].
1.2. Generalized pants decompositions.
A generalized pair of pants is a surface of genus zero with boundary components and punctures, where , with exactly one distinguished point on each boundary component, that is, a surface of type , , or .
A (standard) pants decomposition of a windowed surface is (the homotopy class of) a collection of disjointly embedded essential curves in the interior , no two of which are homotopic, together with a condition on the complementary regions to in .
To articulate this condition, let us enumerate the curves in , choose disjoint annular neighborhoods of in , for , and set . Just for the purposes of articulation, let us also choose on each boundary component of a distinguished point. We require that each component of is a generalized pair of pants or a boundaryparallel annulus of type , for some .
Simple Euler characteristic considerations give the following lemma.
Lemma 1.1.
For a windowed surface , there are many windows. The real dimension of is . Furthermore, there are curves in a pants decomposition of and generalized pairs of pants complementary to an annular neighborhood of the pants curves.
If is a branelabeling on the windowed surface , then a generalized pants decomposition of is (the homotopy class of) a family of disjointly embedded closed curves in the interior of and arcs with endpoints in , no two of which are parallel, so that each complementary region is one of the following indecomposable branelabeled surfaces:

a triangle with no vertex branelabeled by ;

a generalized pair of pants , , or with all points in the boundary branelabeled by ;

a oncepunctured monogon with puncture branelabeled by and boundary distinguished point by ;

an annulus with at least point of labeled by .
For instance, if every brane label is empty, then a generalized pants decomposition is a standard pants decomposition. At the other extreme, if every brane label is nonempty, then admits a decomposition into triangles and oncepunctured monogons, a socalled quasi triangulation of , cf. [58]; see Figures 12b and 13 for examples. Provided there is at least one nonempty brane label, we may collapse each boundary component with empty brane label to a puncture to produce another windowed surface from . A quasi triangulation of can be completed with branelabeled annuli to finally produce a generalized pants decomposition of itself.
Thus, any branelabeled windowed surface admits a generalized pants decomposition. Furthermore, any collection of disjointly embedded essential curves and arcs connecting nonempty brane labels so that no two components are parallel can be completed to a generalized pants decomposition.
1.3. Indecomposables.
We shall introduce standard foliations on indecomposable surfaces which are the basic building blocks of the theory, and we begin with the annulus in Figure 3.
Figure 3 The twist flow on .
In the notation of Figure 3, consider the purely closed sector with branelabeling on a fixed annulus of type . Define a oneparameter “Dehn twist flow” , for , on , as illustrated in the figure, where denotes the sum of the weights of the arcs in . Letting denote the right Dehn twist along the core of the annulus, one extends to all positive real values of by setting , where denotes the integral part of , and likewise for negative real values of .
Figure 4 illustrates the remaining building blocks of the theory. Notice that branelabeled with some taking value on the boundary is absent from Figure 4 and implicitly from the theory since is empty.
A fact going back to Max Dehn in the 1930’s is that “free” homotopy classes rel in a fixed pair of pants of type are determined by the three “intersection numbers” , namely, the number of endpoints of component arcs in each respective boundary component, subject to the unique constraint that is even. Two representative cases are illustrated in Figure 4e, and the full partially ordered set is illustrated in Figure 4d. There are conventions in the pair of pants that have been suppressed here insofar as the “arc connecting a boundary component to itself goes around the right leg of the pants”; see Figure 4f and see [60] for details.
Figure 4 Indecomposables. We depict the geometric realization of in part a for some branelabeling whose image does not contain , and which is simply omitted from the figure. There is the unique element of depicted in part b when the branelabel on the boundary is nonempty. For the branelabeling on indicated in part c, we consider instead the homotopy classes of arc families rel , rather than rel as before. Likewise, for the branelabeling on , which we omit from the figure, we consider again the homotopy classes of arc families rel , where by definition, and depict the geometric realization in part d.
One further remark is that arc families in all generalized pairs of pants are also implicitly described by Figures 4df, where punctures correspond to boundary components with no incident arcs.
1.4. Standard models of arc families.
Suppose that is a generalized pants decomposition of a branelabeled windowed surface , where has curve components and arc components . Let denote a fixed annular neighborhood of for , and set .
In order to parameterize weighted arc families, we must make several further choices, as follows. Choose a framing to the normal bundle to each curve , which thus determines an identification of the unit normal bundle to in with the standard annulus . In turn, the unit normal bundle is also identified with the neighborhood , and there is thus an identification of with determined by the framing on . Furthermore, choose homeomorphisms of each generalized pair of pants component of with some standard generalized pair of pants . Choose an embedded essential arc once and for all in , and likewise choose standard models for arc families in (say, with the conventions for twisting as in Figure 4f). Let us call a generalized pants decomposition together with this specification of further data a basis for arc families.
Given , choose a representative weighted arc family that meets each component of transversely a minimal number of times, let denote the sum of the weights of the arcs in that meet counted with multiplicity (and without a sign), and let denote the analogous sum for the arcs .
Theorem 1.2.
Fix a basis for arc families with underlying generalized pants decomposition of a branelabeled windowed surface , and adopt the notation above given some . Then under the identifications with the standard annulus and standard pants , is represented by a weighted arc family that meets complementary regions to in in exactly one of the configurations shown in Figure 4 and meets each in for some welldefined , where is weighted by . Furthermore, a point of is uniquely determined by its coordinates for and , for .
Proof.
Since the arcs in a generalized pants decomposition connect points of and the components of an arc family avoid a neighborhood of , intersections with triangles and oncepunctured monogons are established. We may homotope an arc family to a standard model in each pair of pants; the twisting numbers are then the weighted algebraic intersection numbers (with a sign) with in each annulus (all arcs oriented from toptobottom or bottomtotop of the annulus); see the “DehnThurston” coordinates from [61], [60] for further details. ∎
Corollary 1.3.
In the notation of Theorem 1.2, any parameterized family in is represented by one that meets complementary regions to in in parameterized families of the configurations shown in Figure 4 and meets each in , where depends upon the parameters, for . Furthermore, a parameterized family is uniquely determined by its parameterized coordinates for and , for .
Notice that in either case of the theorem or the corollary, the intersection numbers on any triangle satisfy all three possible weak triangle inequalities.
2. C/O string operations on weighted arc families
Recall that a window in a branelabeled windowed surface is closed if its closure is an entire boundary component of and the distinguished point complementary to is branelabeled by , and otherwise the window is open.
Given a positively weighted arc family in , let us furthermore say that a window is active if there is an arc in the family with an endpoint in , and otherwise the window is inactive.
In order to most directly connect with the usual phenomenology of strings, we shall require all windows to be active, but the more general case of operations on inactive windows is not uninteresting, specializes to the treatment here, and will be discussed in Appendix B.
Given a positively weighted arc family in , we may simply collapse each inactive window, or consecutive sequence of inactive windows in a boundary component, to a new distinguished point on the boundary, where the branelabeling of the resulting distinguished point is the union of all the brane labels on the endpoints of the windows collapsed to it. In case a boundary component consists entirely of inactive windows, then it is collapsed to a new puncture, which is again branelabeled by the union of all the brane labels on the collapsed boundary component. Thus, given any positively weighted arc family in , there is a corresponding positively weighted arc family in a corresponding surface so that each window is active. (This is one explanation for why we branelabel by the powerset of branes, namely, in order to effectively take every window to be active.)
For any windowed surface , define
where the disjoint union is over all branelabelings on . The basic objects of our topological c/o structure are
where the disjoint union is over all orientationpreserving homeomorphism classes of windowed surfaces.
Figure 5 The c/o operations on measured foliations.
If , then define the weighting of an active window to be the sum of the weights of arcs in with endpoints in , where we count with multiplicity (so if an arc in has both endpoints in , then the weight of this arc contributes twice to the weight of ).
Suppose we have a pair of arc families in respective windowed surfaces and a pair of active windows in and in , so that the weight of agrees with the weight of . Since are oriented surfaces, so too are the windows oriented. In each operation, we identify windows reversing orientation, and we identify certain distinguished points.
To define the open and closed gluing () and selfgluing () of along the windows , we identify windows and distinguished points in the natural way and combine foliations. In closed string operations, we “replace the distinguished point, so there is no puncture” whereas with open string operations, “distinguished points always beget either other distinguished points or perhaps punctures”. In any case whenever distinguished points are identified, one takes the union of brane labels (the intersection of branes) at the new resulting distinguished point or puncture.
More explicitly, the general procedure of gluing defined above specializes to the following specific operations on the :
Closed gluing and selfgluing See Figure 5a. Identify the two corresponding boundary components of and , identifying also the distinguished points on them and then including this point in the resulting surface . inherits a branelabeling from those on in the natural way. We furthermore glue and together in the natural way, where the two collections of foliated rectangles in and which meet and have the same total width by hypothesis and therefore glue together naturally to provide a measured foliation of a closed subsurface of . (The projectivization of this gluing operation is precisely the composition in the cyclic operad studied in [25]; we have deprojectivized and included the weighting condition in the current paper in order to allow selfgluing of closed strings as well.)
Open gluing The surfaces and are distinct, and we identify to to produce . There are cases depending upon whether the closure of and is an interval or a circle. The salient cases are illustrated in Figure 5bd. In each case, distinguished points on the boundary in and are identified to produce a new distinguished boundary point in , and the brane labels are combined, as is also illustrated. As before, since the weight on agrees with the weight on , the foliated rectangles again combine to provide a measured foliation of a closed subsurface of .
Open selfgluing There are again cases depending upon whether the closure of or is a circle or an interval, but there is a further case as well when the two intervals lie in a common boundary component and are consecutive. Other than this last case, the construction is identical to those illustrated in Figure 5bd. In case the two windows are consecutive along a common boundary component, again they are identified so as to produce surface with a puncture resulting from their common endpoint as in Figure 5ef, where the puncture is branelabeled by the label of this point, and the foliated rectangles combine to provide a measured foliation of a closed subsurface of .
At this stage, we have only constructed a measured foliation of a closed subsurface of , and indeed, will typically not be a weighted arc family. By Poincaré recurrence, the subfoliation comprised of leaves that meet corresponds to a weighted arc family in . Notice that the weight of any window uninvolved in the operation agrees with its  or weight, so in particular, every window of is active for .
Let us already observe here that the part of that we discard to get can naturally be included (as we shall discuss in Appendix B). Furthermore, notice that a gluing operation never produces a “new” puncture branelabeled by .
The assignment of in to in , for completes the definition of the various operations. Associativity and equivariance for bijections are immediate, and so we have our first nontrivial example of a c/o structure (see Appendix A for the precise definition):
Theorem 2.1.
Together with open and closed gluing and self gluing operations, the spaces form a topological c/o structure. Furthermore, this c/o structure is branelabeled by and is a c/o structure, where is the genus and is the Eulercharacteristic.
Proof.
See Appendix A for the definitions and the proof. ∎
Corollary 2.2.
The open and closed gluing operations descend to operations on the PL chain complexes of giving them a chain level c/o structure.
Proof.
We define a “chain level c/o structure” in such a manner that this follows immediately from the previous theorem; see Appendix A for details. ∎
Theorem 2.3.
The integral homology groups comprise a modular bioperad when graded by genus for closed gluings and selfgluings and by Eulercharacteristicminusone for open gluings and selfgluings.
Proof.
In contrast to the previous corollary, this requires more than just a convenient definition since we must first show that the gluing operations descend to the level of homology; specifically, given homology classes in and , we must find representative chains that assign a common weight on the windows to be glued.
This is accomplished by introducing two continuous flows on for each window , namely, , for for non selfgluing and , for for selfgluing, where is a fixed branelabeling on the windowed surface . In effect for non selfgluing, simply scales in the action on so that the weight of window is unity at time one. To describe the key attributes of the more complicated flow for selfgluing, suppose that is any other window of and where the weight of is less than the weight of .
There is a welldefined “critical” value of so that the weight of first agrees with the weight of ; furthermore, the function is continuous in .
These flows are defined and studied in Appendix C, and the theorem then follows directly from Proposition A.2. ∎
3. Operations, Relations, Duality
3.1. Operations
Operations may be conveniently described by weighted arc families, or by parameterized families of weighted arc families. If a parameterized family of arc families depends upon real parameters, then we shall say that it is an operation of degree . In order to establish notation, the standard operations in degrees zero and one are illustrated in Figure 6.
In this figure, the distinguished points on the boundary come with an enumeration that we have typically suppressed. Only for clarity for the bracket in Figure 6k do we indicate the enumeration of the distinguished points with the numerals “1” and “2”; we shall omit such enumerations in subsequent figures since they can be inferred from the incidence and labeling of arcs in the figure.
It is worth remarking that the BV operator is none other than the projection to orbits of the Dehn twist operator discussed in Section 1.3.
Figure 6 Standard operations of degrees zero and one. If there is no branelabel indicated, then the label is tacitly taken to be . See Section 4.1 for the traditional algebraic interpretations.
3.2. Relations
Relations in the c/o structure on or its chain complexes can be described and derived by fixing some decomposable windowed surface , choosing two generalized pants decompositions of and specifying an arc family or a parameterized family of arc families in . Each of and decompose into indecomposable surfaces and annular neighborhoods of the pants curves.
According to Theorems 1.2 and 2.1, thus admits two different descriptions as iterated compositions of operations in the c/o structure, and these are equated to derive the corresponding algebraic relations. We shall abuse notation slightly and simply write an equality of two pictures of , one side of the equation illustrating and in and the other illustrating and ; we shall explain the algebraic interpretations in the next section. As with operations, a relation on a parameter family of weighted arc families is said to have degree .
Accordingly, Figures 7 and 8 illustrate all of the standard relations of twodimensional open/closed cobordism (cf. [5, 7, 8, 9]). In particular, notice that the “Whitehead move” in Figure 7a corresponds to associativity of the open string operation. The Cardy equation in Figure 7e depends upon the two generalized pants decompositions of the surface with no empty brane labels, where consists of a single simple closed curve, and is an ideal triangulation.
Figure 7 Open/closed cobordism relations.
The Frobenius equation is more interesting since it consists of two pairs , for , where is comprised of a weighted arc family with each window active and an ideal triangulation of a quadrilateral; see Figure 7c at the far left and right. Perform the unique possible Whitehead move on to get , for . In fact, the pairs and are not identical, rather they are homotopic in , as is also illustrated in Figure 7c.
Figure 8 Further open/closed cobordism relations: associativity and the Frobenius equation in the closed sector.
Figure 9 Closed sector relations: compatibility of bracket and composition.
Figure 10 Homotopy for onethird of the BV equation.
The other closed sector relations were already confirmed in [25] and are rendered in Figures 811 in the current formalism, where all branelabelings are tacitly taken to be ; furthermore all boundaryparallel pants curves are omitted from the figures (except in Figure 11 for clarity). The Frobenius equation is again degree one, and the BV equation itself is degree two.
Figure 11 The homotopy BV equation. There are three summands in Figure 11, each of which is parameterized by an interval, which together combine to give the sides of a triangle. For each side of this triangle, there is the homotopy depicted in Figure 10 with the appropriate labeling. Glue the three rectangles from Figure 10 to the three sides of the triangle in the manner indicated. The BV equation is then the fact that one boundary component of this figure (9 terms) is homotopic to the other boundary component (3 terms); Figure 11 of [25] renders this entire homotopy.
Figure 12 New relations.
It is thus straightforward to discover new relations, and several such relations of some significance are indicated in Figure 12. Figure 12a illustrates a degree one equation on called the “BV sandwich”, which can be succinctly described by “close an open string, perform a BV twist, and then open the closed string”. Oneparameter families of weighted arc families on two triangles are combined by parameterized open string gluing to produce a closed string BV twist sandwiched between closing/opening the string. The significance of the relations in Figure 12b and the justification for the choice of terminology will be explained in the next section.
Here is an algorithm for deriving all of the relations in degree zero on the topological level: Induct over the topological type of the surface and over the orbits of all pairs of generalized pants decompositions of it. (Though there are only finitely many orbits of singleton generalized pants decompositions, there are infinitely many orbits of pairs.) In each indecomposable piece, consider each of the possible building blocks illustrated in Figures 34. Among these countably many equations are all of the degree zero equations of the topological c/o structure.
To derive all higher degree relations on the topological level, notice that each indecomposable surface has (the geometric realization of) its arc complexes of some fixed rather modest dimension. Thus, parameterized families may be described as specific parameterized families in each building block, for instance, in the coordinates of Corollary 1.3. Such parameterized families can be manipulated using known transformations (see the next section) to explicitly relate coordinates for different generalized pants decompositions and derive all topological relations.
A fortiori, topological relations hold on the chain level (and likewise for the chain and homology levels as well). For parameterized families, there is again the analogous exhaustively enumerative algorithm, but one must recognize when two parameterized families are homotopic, which is another level of complexity.
It is thus not such a great challenge to discover new relations in this manner. The remaining difficulties involve systematically understanding not only higher degree equations like the BV sandwich but also in determining a minimal set of relations, and especially in understanding the descent to homology.
3.3. Open/closed duality
We seek a collection of combinatorially defined transformations or “moves” on generalized pants decompositions of a fixed branelabeled windowed surface, so that finite compositions of these moves act transitively. In particular, then any closed string interaction (a standard pants decomposition of a windowed surface branelabeled by the emptyset) can be opened with the “opening operator” illustrated in Figure 6d, say with a single branelabel ; this surface can be quasi triangulated, giving thereby an equivalent description as an open string interaction.
In particular, the two moves In Figure 13ab were shown in [58] to act transitively on the quasi triangulations of a fixed surface, and likewise the two “elementary moves” on and of Figure 13cd were shown in [62] to act transitively on standard pants decompositions, where we include also the generalized versions of Figure 13d on with and Figure 13c on as well (though this includes some nonwindowed surfaces strictly speaking).
Figure 13 Four combinatorial moves, where absent brane labels are arbitrary.
For another example, the Cardy equation can be thought of as a move between the two generalized pants decompositions depicted in Figure 7e, and likewise for the four new relations in Figure 12b.
Theorem 3.1.
Consider the following set of combinatorial moves: those illustrated in Figure 13 together with the Cardy equation Figure 7e, and the four closed/open duality relations Figure 12b. Finite compositions of these moves act transitively on the set of all generalized pants decompositions of any surface.
Proof.
In light of the transitivity results mentioned above by topological induction, it remains only to show that the indicated moves allow one to pass between some standard pants decomposition and some quasi triangulation of a fixed surface of type . This follows from the fact that on any surface other than those in Figure 7e and 12b, one can find in a separating curve separating off one of these surfaces. Furthermore, one can complete to a standard pants decomposition so that there is at least one window in the same component of as . Choose an arc in connecting a window to ; the boundary of a regular neighborhood of corresponds to one of the enumerated moves, and the theorem follows by induction. ∎
It is an exercise to calculate the effect of these moves on the natural coordinates in Theorem 1.2 in the case of the quadrilateral and the oncepunctured monogon , and the calculation of the new duality relations on generalized pairs of pants , for , and on is implicit in Figure 4. The calculation of the first elementary move on is also not so hard, but the formulas are unfortunately incorrectely rendered in [67]; see [61] or [71]. The calculation of the second elementary move on , a problem going back to Dehn, was solved in [61].
4. Algebraic properties on the chain and homology levels
4.1. Operations on the chain level
The moves discussed in the last section give rise to relations on the chain level as well. As explained in Appendix A upon fixing a chain functor , a chain may be thought of as a parameterized family of arc families, i.e., as a suitable continuous function , where represents a tuple of parameters. The gluing operations on the chain level can furthermore be thought of as gluing in families, where the gluing is possible when the weights of the appropriate windows agree in the two families. As mentioned previously, any relation on the topological level gives rise to a relation of degree zero on the chain level. Some of the relations we discuss will be only up to homotopy, i.e., of higher degree.
Given any , i.e., given any suitable parameterized family on a component of , say with underlying surface , we may fix a window on and regard as an operation in many different ways:
()ary operation: given chains , each on a surface with distinguished window, glue them to all windows of except ; the inputs yield the output ;
dual unary operation: given a chain on a surface with distinguished window, glue it to along to produce the chain we shall denote ; the input yields the output .
More generally, we may partition the windows of the underlying surfaces into inputs and outputs to obtain more exotic operations associated with chains. (The mathematical structure of PROPs were invented to formalize this structure; for a review see [64, 65].)
For instance, let us explain the sense in which the constant chain in Figure 6b describes the binary operation of multiplication. Taking the base of the triangle as the distinguished window , consider families with distinguished window and with distinguished window , where the brane labels at the endpoints of are and of are . These chains can be glued to the constant family if and only if the weight of on its distinguished window is constant equal to and the weight of on its distinguished window is constant equal to . Let the base of the triangle in Figure 6b be the window 0, the side the window 1, and the side the window 2. The chain operation is defined as . Notice that the resulting chain will have constant weight on its window 0. It is in this sense that we shall regard the constant chain as a binary multiplication.
On the other hand acts as a comultiplication as well: given with brane labels on its distinguished window, we have .
4.1.1. Degree 0 indecomposables and relations
Degree zero chains are generated by zerodimensional families, that is, by points of the spaces .
For the indecomposable branelabeled surfaces of §1.2, the relevant degree 0 chains are enumerated in Figure 6ae and 6ln. They become explicit operators by fixing the distinguished window to be the lower side in 6a, the base in 6b and the outside boundary in 6ce and 6ln: this is the algebraic meaning of the illustrations in Figure 6. For example, 6b and 6e give the respective open and closed binary multiplications and , while 6a and 6c give the respective identities and on their domains of definition, namely, families whose weight on the distinguished window is constant equal to . The subscripts indicate compatibility for chain gluing and selfgluing in the chain level c/o structure, and the superscripts denote branelabels in the open sector.
It follows from Figures 7a and 8a, that the multiplications and are associative:
(4.1)  
(4.2) 
where means the usual composition of operations. The dual unary operations to these multiplications satisfy the Frobenius equations up to homotopy as shown in Figure 7c for the open sector:
(4.3) 
and in Figure 8b for the closed sector:
(4.4) 
The “closing” operation of Figure 7d acts as a unary operation which changes one window from open to closed. Its dual “opening” operation changes one closed to one open window. It follows from Figure 7b that is an algebra homomorphism:
(4.5) 
The image of lies in the center, as in Figure 7d:
(4.6) 
where interchanges the tensor factors, namely, interchanges the two nonbase sides of the triangle, and it satisfies the Cardy equation in Figure 7e:
(4.7) 
The operators in Figures 6ln are puncture operators, which are “shift operators for the puncture grading”.
Figure 14 Homotopy of Equation (4.8) to Equation (4.6), where the dotted lines indicate generalized pants decompositions.
We finally express the center equation in a less symmetric but more familiar form:
(4.8) 
As indicated in Figure 14, the equation (4.8), which is represented by the far left and right figures, is equivalent on the chain level to two copies of our equation (4.6), represented by the two equalities in the figure, which holds on the nose.
4.1.2. Degree one indecomposables and relations
The known degree one operations are given by the binary operation of Figure 7g and the operation of Figure 7h. These operations are related, indeed, they satisfy the GBV equations up to homotopy: is a homotopy preLie operation whose induced homotopy Gerstenhaber structure coincides with the induced homotopy Gerstenhaber structure of the homotopy BV operator (see Figures 911). This completes the discussion of know relations.
There is a degree one chain which is of interest, namely, the family which is generated by the BV sandwich setting and in Figure 12a. This is supported on the annulus with brane labels A and B. We shall call this operator . The BV sandwich equation then gives the equality of chain level operators
(4.9) 
In the same spirit, there is another degree one chain which is associated to for . Although this chain is not closed, it appears naturally as follows. Using the BV sandwich relation for the chain , we see that it also decomposes as:
(4.10) 
Thus, admits the two expressions (4.9) and (4.10), so the chain , which is a kind of “BVsquared in the open sector”, likewise admits the two expressions corresponding to the two different generalized pants decompositions of :
See the closing remarks for a further discussion of this operator .
Lemma 4.1.
Suppose that is an indecomposable branelabeled windowed surface. If is a triangle or a oncepunctured monogon, then is contractible. For an annulus, is homotopy equivalent to a circle, and for a generalized pair of pants with boundary components, is homotopy equivalent to the Cartesian product of circles.
Proof.
The claims for triangles and oncepunctured monogons are clear from Figure 4ab. For the degree one indecomposables, we first have the annuli branelabeled by or by ; the free generator of the first homology of the former is precisely the BV operator , while the free generator of the latter is , where is the window labeled by . In each case, we have that is homotopy equivalent to a circle. For , we again have homotopy equivalent to a circle as in Figure 2, with the free generator .
For the generalized pairs of pants, first notice that the set of all homotopy classes of families of projective weighted arcs in a generalized pair of pants with boundary components (where the arc family need not meet each boundary component) is homeomorphic to the join of circles. (In effect, a point in the circle determines a projective foliation of the annulus as in Figure 3, and one deprojectivizes and combines as in Figure 4d to produce a foliation of the pair of pants.). The complement of two spaces in their join is homeomorphic to the Cartesian product of the two spaces with an open interval, and the lemma follows. In fact, the first homology of is freely generated by , and , and the first homology of is freely generated by , , and . ∎
For a final chain calculation, consider the degree two chain defined by which is another type of “BVsquared operator in the open sector” arising on the surface with branelabeling given by on the boundary and by at the puncture. In fact, generates , where by Lemma 4.1.
Tautologically, can be written as the sum of two nonclosed chains given by
(4.11) 
Furthermore, we may homotope each of the operators and into “traces over multiplications” in the following sense, where we concentrate on with the parallel discussion for omitted. Consider the homotopy of arc families in depicted in Figure 15, which begins with and ends with the indicated family. Cutting on the dotted lines in Figure 15 decomposes each surface into a hexagon, and these hexagons may be triangulated into four triangles corresponding to four multiplications. Thus, each of the operations and is given as the double trace over a quadruple multiplication. Again, see the closing remarks for a further discussion of these operators.
Figure 15 The operators and the homotopy of .
4.2. Algebraic properties on the homology level
Since is connected for any windowed surface with branelabeling , we conclude
the sum over all homeomorphism classes of branelabeled surfaces with closed and open windows. It follows that the degree zero relations on the homology level are precisely those holding on the chain level up to homotopy.
This observation together with Theorem 3.1 implies the result of [5, 9] that the open/closed cobordism group admits the standard generators with the complete set of relations depicted in Figures 7 and 8: associativity, algebra homomorphism, the Frobenius equations, center and Cardy together with duality.
Lemma 4.2.
Each component of is contractible, hence the homology is concentrated in degree .
Proof.
Consider the foliation which has a little arc around each of the points of with constant weight one. We can define a flow on , by including these arcs with any element and then increasing their weights to one while decreasing the weights of all the original arcs to zero. ∎
In particular, the first open BV operator itself thus vanishes on the homology level, while the second open BV operator and even its “square” do not. (The situation may be different in conformal field theory as discussed in the closing remarks.)
Let be the respective images in homology of the chains , define to be the images of the puncture operators in homology, and the image in homology of .
Just as chains can be regarded as operators on the chain level, so too homology classes can be regarded as operators on the homology level.
Proposition 4.3.
The degree zero operators on homology are precisely generated by the degree zero indecomposables and provided , where denotes the set of punctures. If , then one must furthermore include the operators . The degree zero relations on homology are precisely those given by the moves of Theorem 3.1. All operations of all degrees supported on indecomposable surfaces are generated by the degree zero operators and .
Proof.
Theorem 4.4.
Suppose . Then an algebra over the modular bioperad is a pair of vector spaces which have the following properties: is a commutative Frobenius BV algebra , and is a colored Frobenius algebra (see e.g., [9] for the full list of axioms). In particular, there are multiplications and a nondegenerate metric on which makes each into a Frobenius algebras.
Furthermore, there are morphisms which satisfy the following equations: letting denote the dual of , the morphism permuting two tensor factors, and letting be arbitrary nonempty branelabels, we have