Nonperturbative dynamics for abstract (p,q) string networks Fotini Markopoulou and Lee Smolin 8 9 Center for Gravitational Physics and Geometry 9 1 Department of Physics n The Pennsylvania State University a University Park, PA, USA 16802 J 6 2 v 8 4 1 2 1 7 9 December 2, 1997 / h t - p e h : v i X r a ABSTRACT We describeabstract(p,q)stringnetworks whicharethestringnetworks of Sen without the information about their embedding in a background spacetime. The non-perturbative dynamical formulation invented for spin networks, in terms of causal evolution of dual triangulations, is applied on them. The formal transition amplitudes are sums over discrete causal his- tories that evolve (p,q) string networks. The dynamics depend on two free SL(2,Z) invariant functions which describe the amplitudes for the local evo- lution moves. email addresses: [email protected], [email protected] 1 1 Introduction Inaveryinterestingpaper[1],Senintroducesthenotionofanetworkof(p,q) strings and hypothesizes that they might play a role in a non-perturbative formulation of string theory analogous to the role SU(2) spin networks play in non-perturbative quantum general relativity[2, 3]. These are composed of (p,q) strings that meet at trivalent vertices introduced by Schwarz[4] and shown to be BPS states in [5]. This suggestion has been explored in several recent papers[6, 7]. In this note we take up Sen’s suggestion and show that the (p,q) string networks fit naturally into a class of generalizations of spin networks that have been studied as possible non-perturbative quantum the- ories of gravity[8]. The dynamics of spin networks that one of us suggested in [9] may be applied directly to (p,q) string networks to yield a formal non- perturbative background independent formulation of their dynamics. This formulation is in turn closely related to path integral formulations of non- perturbativequantumgravitybasedondeformationsoftopologicalquantum field theories[10, 11, 12, 13, 14, 15]. Penrose originally invented spin networks to provide a simple realization of the idea that the quantum geometry of spacetime must be discrete and based only on algebra and combinatorics, with all continuum notions aris- ing in the continuum limit[16]. As introduced by Sen in [1], (p,q) string networks depend on an embedding in a flat background 9+1 dimensional spacetime. However if they are to serve as the basis of a non-perturbative formulation of string theory the (p,q) string networks should be removed from this context and postulated to exist as background independent com- binatorial entities from which the continuum description is to be derived as a suitable approximation. The possibility of this is suggested by the strik- ing fact that, just as in Penrose’s original formulation [16], angles for the embedding of the (p,q) string network in flat space are derived purely from the combinatorics of the network[1]. This raises the question of how the dynamics and observable algebra of the theory are to be defined in the absence of a background manifold. In [17, 9, 8] we have studied this problem for non-embedded extensions of spin-networks. This led to the introduction of a general theory with two basic features: (1) The spin networks are extended to 2-dimensional labeled surfaces and the space of states is constructed algebraically. (2) Dynamics is expressed in terms of local moves that generate histories as combinatorial structures which share several of the properties of Lorentzian spacetimes including causal structure and many-fingered time. 2 As shown by Schwarz [4] and elaborated for networks by Krogh and Lee [6], (p,q) strings can be understood as labeled 2-dimensional surfaces. This leads directly to the application of the dynamical formulation introduced in [8, 9] to string networks. We must emphasize that we have not yet shown whether the dynamics we propose has any connection with the dynamics of (p,q) string networks embedded in background manifolds. The dynamics in the form proposed here is specified by two free SL(2,Z) invariant functions. In the future, these may be determined by the condition that they match the dynamics predicted by M theory, but this has not yet been done. Inthenextsectionweintroducetheabstract(p,q)stringnetworks,which obeythecombinatoricsofthe(p,q)networksofSen[1]butarenotembedded in any background manifold. In section 3 we describe non-perturbative dynamics for these abstract (p,q) networks. 2 Abstract (p,q) string networks Spin networks (and their extensions to general quantum groups) can be un- derstood in the general framework first developed for conformal field theory and topological field theory [19, 20, 21] in which states associated to various manifolds are constructed from algebraic operations. It is easy to see that (p,q) string networks can be formulated in the same framework. We consider sets {(p ,q )} consisting of n relatively prime pairs of in- i i tegers. Each pair may be visualized as the first homology classes of a T2 as in [6], but the only role this will play in the abstract formalism is that the torus parameter τ may come into the SL(2,Z) invariant functions that specifythedynamics. Wemay notethattherelatively primepairs(p,q)also labeltherationalnumbersor, alternatively, theprojectiverepresentations of the mapsof the circle to itself1. As a result, thereis a naturalmultiplication defined on them, which is (p1,q1)⊗(p2,q2) = (p1+p2,q1+q2) = r(p3,q3) → (p3,q3) (1) where r, a positive integer, is the greatest common factor of p1 + p2 and q1+q2. Associativity follows from the uniqueness of prime factorization. Thebasicideaofacategorical constructionofaTQFT orconformalfield theory[19, 20, 21] is that sets of labels, which are representations of some 1 Wethank Louis Crane for this observation. 3 (p ,q ) 3 3 (p ,q ) (p ,q ) 1 1 2 2 Figure 1: The representation of the product (1) as a three string vertex. algebra, are mapped to manifolds such that the product is represented by cobordisms. In this case we will associate the sets {(p ,q )} to sets of circles i i sothat thealgebrathey generate ismappedtocobordismsof two manifolds. The product (1) is symbolized by a trinion, or 3-punctured sphere, with an object(p,q)associatedwitheachincomingpunctureand(−p,−q)associated with the one outgoing puncture as in Figure (1). When r 6= 0 we say there exists a morphismfrom (p1,q1)⊗(p2,q2) to (p3,q3). The product(1) is thus the algebraic counterpart of the three string vertex, which when the strings are represented as tubes becomes the trinion. Under reversal of orientation of a puncture we require (p,q) → (−p,−q). Taking all the orientations consistently a trinion is labeled by three relatively prime pairs (p ,q ) such i i that, 3 3 p = q = 0 (2) X i X i i=1 i=1 A trinion with labels (p ,q ) satisfying (2) will be called a good trinion. i i An abstract (p,q) string network is then described by a compact surface S with m punctures constructed from sewing good trinions together along the punctures,as shown in Figure(2). Theresultis a surfaceS with aset of circles c which decompose it into a set of good trinions, with labels (p,q) α α on the punctures c of the trinions. A closed (p,q) string network is one α without free punctures. Anon-degenerate stringnetwork willbeonethathasatrinion decompo- sition in which no two trinions meet on more than one circle. We may note that the string networks of Sen [1] are non-degenerate because the strings are straight lines in flat spacetime. We will thus restrict ourselves to the consideration of non-degenerate abstract string networks. To the set of non-degenerate abstract (p,q) string networks we associate a Hilbert space HSN constructed as follows. Given each surface S with its S circles c there is a space V spanned by a basis of states |S,c ;(p,q) i. α cα α α 4 Figure 2: A genus 4 surface cut into six trinions B3 by circles c . I α We choose the inner product h|i so that these are orthonormal [8]. We then define HSN = VS (3) M cα where the sum is over all labeled surfaces (S,c ). 2 α What distinguishes theories of the type we are about to propose from 3-d topological quantum field theory is that there are operators which act on HSN to change the topology of the surface S. For example, consider Υ1 and Υ2 to be two open (p,q) string nets, each with n,freepunctures,withlabels (p,q) , k = 1,...,p. ThusΥ1 andΥ2 have k the same boundary, but are otherwise different. There is then an operator CΥ1,Υ2 which acts on each state in the basis |S,cα;(p,q)αi by looking for componentsof(S,(p,q) )whichareisomorphicto(Υ1,(p,q) ), andreplaces α k it by a different 2-surface (Υ2,(p,q) ), which has the same boundary, but k otherwise differs. Now,theremaybemorethanoneplacesin(S,(p,q) )where(Υ1,(p,q) ) α k is recognized as a subset. There are then several maps r : (Υ1,(p,q) ) → (S,(p,q) ). (4) I k α For each I the map r picks out a set of n non-intersecting circles cI, k = I k 1,...,n in S. Cutting S on these circles decomposes it into the two pieces r (Υ1) and S −r (Υ1) . We may then sew Υ2 onto S −r (Υ1) along the I I I (cid:0) (cid:1) (cid:0) (cid:1) circles cI as they are also identical with the set of punctures of Υ2. We may k call this state | S −r (Υ1) ∪Υ2i with the dependence on the circles and I (cid:0) (cid:1) 2 All of this can be expressed compactly by saying that there is a functor from the cobordism category Cob whose objects are sets of circles and whose morphisms are two manifolds with boundary to the category of relatively prime pairs of integers with the product (1). The general formulation of this kind of correspondence is described in [20, 19, 21]. 5 Figure 3: A local substitution move. labelings understood. The operator CΥ1,Υ2 is then defined by CΥ1,Υ2|Si = X|(cid:16)S −rI(Υ1)(cid:17)∪Υ2i (5) I A slightly more complicated definition can be made when S is not compact. AnexampleofsuchasubstitutionmoveisgiveninFig. (3). Theseoperators are local as they only change local pieces of a string network which are isomorphic to a given piece. We now show that a particular set of such operators may be used to formulate the dynamics of the abstract (p,q) string networks. 3 Dynamics Following the proposal made for causal evolution of spin networks in [9] we may give a general form for dynamics of (p,q) string networks that is locally causal. We note that in a background independent formalism the notion of locality must be based on the states themselves. For the 2 + 1 case it is actually more convenient to describe the dy- namics in the dual picture [9]. An abstract trivalent string network |S,j i α may be equivalently described by a labeled triangulation T, (See Fig. (4)) in which the sides of the triangles carry the (p,q) labels and the condition (2) is satisfied by the three (p,q) pairs around each triangle. If the string network is closed, the corresponding triangulated surface is compact. We may also note that non-degeneracy of the string network implies that the triangulation isnon-degenerateinthesensethatnotwotriangles sharemore than one edge. The issue of the evolution of a network is now translated to the evolution of a 2-dimensional triangulation. Now, the generating local 6 Figure 4: Triangulations dual to trivalent networks evolution moves from a non-degenerate 2-dimensional triangulation to an- other one, whichleave invariantthetopology of thetriangulated surface, are the Pachner moves ([22], which were defined originally for PL manifolds). These have already been used to describe causal spacetime histories of spin networks [9] (and in the euclidean case they can be found in [11, 14]. Let us use them here to write down a formal path-integral evolution for abstract (p,q) networks. The Pachner moves are transitions from an initial (p,q) triangulation T i to a final one T . They are labelled tetrahedra having the triangles of T f i and T as their boundaries. The different moves are the different ways of f gluing a tetrahedron on n connected triangles of T and get 4−n connected i triangles in T , n ≤ 3. The new edges which are created by the move must f be labeled consistently with (1) given the labelings of the remaining edges. See Fig.(5). For (p,q) networks, these moves are particularly simple. The 1 → 3 move is always possible. The inverse, 3 → 1 requires that there are three triangles in T around a single vertex, as in Fig. (5). A 2 → 2 move is i only possible when labels of the four external sides add up to (0,0). By iterating the moves one constructs a sequence of string networks Γ0,Γ1,... = {Γi}. As in general relativity in which local evolution of initial data generates a spacetime, any such sequence generates a history. Also as in general relativity, a given history may be generated many ways by local evolution moves. For this reason it is best to define it abstractly, as follows. A history M between an initial string network Γ0 and a final string network Γ is then a three dimensional simplicial complex consisting of N f tetrahedraτI suchthat∂M= Γ0∪Γf (whereweidentifythestringnetwith 7 C C E G F A B A B A B A B C D C D Figure 5: The 2+1 dimensional Pachner moves. its dual triangulation). Each edge of M is labeled by a pair (p,q) such that the conditions (2) are satisfied on every face. Thefaces of each history make up a causal set3 that is defined as follows. The faces of each tetrahedra are divided into a past set P and a future set, F so that for two faces f1 ∈ P and f2 ∈ F of the same tetrahedron we have f1 < f2 (where < indicates the partial ordering). We then extend this to all faces in M so that f1 < fr if thereisasequenceofr faces fi suchthatfi < fi+1 withinsometetrahedron. Thus each history has a discrete causal structure. Given the causal structure of a history M there is a natural definition of a spacelike surface. A discrete spacelike surface, ∆, is a collection of triangles inMwithoutboundaryandwithnotwo triangles causally related. Becausethecausalstructureislocal, eachhistoryMcanbefoliatedinmany different ways into sequences of spacelike surfaces (abstract (p,q) networks), Ti,∆1,...,∆I,...,Tf. If a history was generated by a sequence Γi of string networks each of these is a spacelike slice, but the history M they construct will have many spacelike slices not in the list {Γ }. Thus, the theory has a i discrete analogue of the multi-fingered time of general relativity. To each tetrahedron T in M we can associate an amplitude, which is the amplitude for a local transition between two string networks. The am- plitude A [τ,(p ,q ) ] will then depend on the labels of the edges of that TI i i I tetrahedron. To beconsistent with stringtheory it should besome SL(2,Z) 3A causal set is a partially ordered set, where the order relation stands for causal- ity, which has no closed causal loops. The formulation of discrete quantum theories of gravity with such causal structureshas been studied by Sorkin and collaborators[23] and ’tHooft[24]. 8 (p ,q ) (p n ,q n ) (p ,q ) 1 1 2 2 (p ,q ) (p ,q ) 1 1 2 2 (p1 -pn ,q 1 -q n ) (p2 +pn ,q 2 +q n ) (p3 ,q3 ) (p3 ,q 3) (p1 ,q1 ) (p3 ,q 3) (p1 ,q 1) (p3 ,q3 ) (p ,q ) (p ,q ) 5 5 6 6 (p2 ,q2 ) (p4 ,q4 ) (p2 ,q2 ) (p4 ,q4 ) Figure 6: The allowed Pachner moves for (p,q) string networks. invariant function of the the string theory parameter τ and the (p,q) in- tegers on its faces. There will also be in general different amplitudes for different causal structures on the tetrahedron, T , this is indicated by the I dependence on T in the form of the amplitude A [τ,(p ,q ) ]. I TI i i I If the history M contains N tetrahedra T , the amplitude for M is local I when it is the product N A[M] = A [τ,(p ,q ) ]. (6) Y TI i i I I=1 For the 1 → 3 Pachner move, A1→3[τ,pi,qi] must be an SL(2,Z) invariant function of the parameter τ, the three incoming labels (p ,q ), i = 1,2,3, i i symmetricinthei,andaninternalpair(p ,q )thatrunsaroundtheinternal n n loop, subject to (2). (See Fig. (6).) The amplitude for the inverse 3 → 1 moves is by hermiticity the complex conjugate of the 1 → 3 moves. For the 2 → 2 move the amplitude A2→2[τ,pi,qi] is a real function of τ and the four 4 external edges (p ,q ), subject to the constraint that (p ,q ) = (0,0). i i i=1 i i P (See Fig. (6).) An invariant choice of A1→3[τ,pi,qi] and A2→2[τ,pi,qi] defines a non-perturbative (p,q) network theory. Using the operators of the last section, a hermitian operator H on HSN can be defined [8] such that < T2|H|T1 > vanishes unless T2 dbiffers from T1 by the action of one Pachner mobve, in which case it is equal to the corresponding amplitude, A [τ,(p ,q ) ]. TI i i I The result of the action H|T i is then the linear combination of the i states reached by the possiblebPachner moves from |T i each multiplied by i 9 the corresponding amplitude A [τ,(p ,q ) ] 4. Given T and T then, the TI i i I i f transition amplitude between the two is ATi→Tf = hTi|eıtH|Tfi. (7) Expansion of (7) gives the amplitude as a sum over histories M. t is a formal parameter which scales the amplitudes and has no interpretation as a physical time. (One measure of physical time is the number, N, of tetrahedra in each history, as this is related, in the continuum limit, to an integral of a positive function over the spacetime history.) We note that formally the evolution defined by (7) is unitary. We do not know anything more about the choices of the amplitudes. It is possible that consistency with perturbative string theory plus SL(2,Z) invariance will constrain or determine them5. Alternately, one may try to search the space of amplitudes for choices that lead to critical behavior, resulting in a continuum limit. The appropriate critical behavior should be related to directed percolation, for reasons explained in [17]. There is also a result that suggests that perturbative string theories may be derived from the perturbation theory of theories of the kind we have considered here, in the case that they have a good continuum limit[18]. 4 Conclusion What we have described here is a proposal for a formal path integral evo- lution of abstract (p,q) networks based on two steps. The first is the con- struction of a space of states HSN composed of abstract graphs with the same labels and combinatorics as a large class of BPS states which are networks of (p,q) strings embedded in 9+1 dimensional Minkowski space. This fits into the categorical framework behind much recent work in non- perturbative quantum gravity and topological field theory. The second step is a proposal for the dynamics of the theory based on a set of local moves. The exact dynamics of the theory is not prescribed, but its choice reduces to the specification of two SL(2,Z) invariant functions. 4Forthe details of thisconstruction in thegeneral case, see [8]. 5Wethank ShyamoliChaudhuriand Djordje Minic for conversations about this. 10