Topological defects in open string field theory
Toshiko Kojita, Carlo Maccaferri, Toru Masuda, Martin Schnabl
aa r X i v : . [ h e p - t h ] D ec MISC-2016-09
Topological defects in open string field theory
Toshiko Kojita ( a,b )1 , Carlo Maccaferri ( c )2 , Toru Masuda ( a,d )3 , Martin Schnabl ( a )4 ( a ) Institute of Physics of the ASCR, v.v.i.Na Slovance 2, 182 21 Prague 8, Czech Republic ( b ) Maskawa Institute for Science and Culture, Kyoto Sangyo Univ.,Motoyama, Kamigamo, Kita-Ku, Kyoto-City, Kyoto, Japan ( c ) Dipartimento di Fisica, Universit´a di Torino and INFN Sezione di TorinoVia Pietro Giuria 1, I-10125 Torino, Italy ( d ) Department of Physics, Nara Women’s UniversityKita-Uoya-Nishimachi, Nara, Nara, Japan
Abstract
We show how conformal field theory topological defects can relate solutions of openstring field theory for different boundary conditions. To this end we generalize the resultsof Graham and Watts to include the action of defects on boundary condition changingfields. Special care is devoted to the general case when nontrivial multiplicities arise upondefect action. Surprisingly the fusion algebra of defects is realized on open string fieldsonly up to a (star algebra) isomorphism. Email: kojita at cc.kyoto-su.ac.jp Email: maccafer at gmail.com Email: masudatoru at cc.nara-wu.ac.jp Email: schnabl.martin at gmail.com ontents S [ D Ψ] . . . . . . . . . . . . . . . . . . . . . . . . 405.1.2 Computation of the Ellwood invariant . . . . . . . . . . . . . . . . 425.1.3 KMS and KOZ boundary state . . . . . . . . . . . . . . . . . . . . 44 Introduction and summary
In the past 16 years there has been quite a lot of progress in charting out the space ofpossible solutions of the classical equations of motion of open string field theory (OSFT)[1] by both numerical [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] as well as analytic tools [13, 14,15, 16, 17, 18, 19, 20, 21, 10, 22, 23] by which new exact solutions have been found oranalyzed [15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. See[42, 43, 44, 45, 46, 47] for reviews.The OSFT action S OSF T = − g o (cid:20) h Ψ ∗ Q B Ψ i + 13 h Ψ ∗ Ψ ∗ Ψ i (cid:21) , (1.1)can be formulated for an arbitrary system of “D-branes”, coincident or not, and describedby a generic Boundary Conformal Field Theory (BCFT, see [48, 49, 50, 51, 52] for reviews)for composite or fundamental boundary conditions.Obviously, to describe all solutions for the bewildering space of theories based on arbi-trary BCFT is a difficult task. But beside the intrinsic importance of the classification ofOSFT solutions, this program can potentially help in the discovery of new D-brane sys-tems, by encoding new world-sheet boundary conditions into the gauge invariant contentof OSFT solutions [10, 20, 21]. Numerical approaches are useful on a case-by-case basis,especially when one does not know what to expect, i.e. when the problem of classifyingall conformal boundary conditions for a given bulk CFT is unsolved. Analytic solutionsare scarce and until recently they essentially described only the universal tachyon vacuumor marginal deformations. A notable progress has been achieved with the solution [38], byErler and one of the authors, which can be written down explicitly for any given pair ofreference and target BCFT’s. The existence of this solution gives evidence that OSFT candescribe the whole landscape of D-branes that are consistent with a given closed stringbackground. However since the solution requires the knowledge of the OPE between theboundary condition changing operators between the two BCFT’s, it does not directly helpin the problem of discovering new BCFT’s.It would be nice to have an organizing principle by which we could simply relatesolutions in the same or possibly different theories. Solution generating techniques arescarce and problematic [53, 54, 55]. It is well known however that symmetries can be usedto generate new solutions. Given a star algebra automorphism SS ( ψ ∗ χ ) = S ( ψ ) ∗ S ( χ ) , (1.2)commuting with the BRST operator Q B one can see that if Ψ is a solution of the equationof motion, then so is S Ψ. The operator S can correspond to a discrete symmetry, or a3ontinuous symmetry. In the latter case one has a family of such operators S α which ariseby exponentiation of the infinitesimal generator, a star algebra derivative PP ( ψ ∗ χ ) = ( P ψ ) ∗ χ + ψ ∗ ( P χ ) . (1.3)Indeed, assuming [ Q B , P ] = 0 and setting S α = e αP , one finds that S α maps solutions tosolutions. The symmetry generator P is often given by a contour integral of a spin onecurrent and upon exponentiation it can be interpreted as a topological defect operator.Even in the case of discrete symmetries, the operator S can be viewed as a so calledgroup-like topological defect operator [56, 57]. The main goal of this paper is to extend this analogy further. For every topologicaldefect in a given BCFT we construct an operator D which maps the state space of oneBCFT into another, in such a way that D ( ψ ∗ χ ) = D ( ψ ) ∗ D ( χ ) . (1.4)What makes a defect topological is that the defect operator commutes with the energymomentum tensor, and hence also commutes with the BRST charge Q B . Then it imme-diately follows that if Ψ is a classical solution of OSFT for a given BCFT, then D Ψ is asolution of OSFT built upon another BCFT.The explicit action of the defect operator D on the open string fields turns out to bequite tricky. In general the string field algebra is not given by a single BCFT Hilbert spacewith a single boundary condition but is given rather by a direct sum of A - B bimodules L a,b H ( ab ) , where a and b label the boundary conditions for the endpoints of open stringstretched by two D-branes. The algebras A and B represent a set of boundary fields ona D-brane for the a and b boundary condition respectively, and are themselves bimoduleswith the left and right multiplication provided by the operator product. As the defectoperator must commute with the Virasoro generators (single surviving copy on the upperhalf-plane) it must act as D d φ abi = X a ′ ,b ′ X dabia ′ b ′ φ a ′ b ′ i . (1.5)An important feature is that this maps boundary operators intertwining between two givenboundary conditions into a sum of operators intertwining between different boundaryconditions allowed by the defect fusion rules of the theory. It thus maps, in general, theoriginal star algebra into a bigger star algebra. This contrasts with the action of thedefect operators on the closed string Hilbert space where it maps the whole space intoitself [58]. Topological defects have played a prominent role in the recent development of two-dimensional CFT,see also [58, 59, 60, 61, 62]. D d = a ∈ d × ab ∈ d × b a φ i b a i Figure 1: Action of the topological defect on a boundary field φ abi can be described byenclosing it with an open defect attached to the boundary. The result is a collection(direct sum) of boundary fields φ a ′ b ′ i in all possible new boundary conditions allowed byfusion. The dots at the junctions represent simple normalization factors determined insubsection 4.2.For the sake of simplicity and concreteness we limit our discussion in this paper totopological defects of minimal models with diagonal partition function. These have beenfully classified [58] and they are in one-to-one correspondence with primary operators.For these defects we find from the distributivity requirement (1.4), in a canonical nor-malization for boundary fields (3.23), that the X coefficients are given in terms of thenormalized g -functions g ′ (2.22) and the normalized 6J-symbols (2.36) X dabia ′ b ′ = ( g ′ a g ′ b g ′ a ′ g ′ b ′ ) (cid:20) a, a ′ , db ′ , b, i (cid:21) . (1.6)For the special case of a = b this reduces, up to a normalization, to the result of Grahamand Watts [60].The action of the defect on boundary fields can be conveniently understood in termsof defects attached onto the boundary as in Figure 1. The defect endpoints can be freelymoved along the boundary without changing correlators as long as the defect does notcross any operator insertion. The junction point can be viewed as an insertion of theidentity operator (and not a traditional boundary condition changing operator) up to anormalization factor which we determine in subsection 4.2.Quite surprisingly however, the fusion rules of the open defect operators are twisted byan orthogonal similarity transformation when multiple boundary condition are generatedby the defect. So instead of D d D c = X e N edc D e (1.7)which holds for the defect action on bulk states, the action on boundary operators obeys D d D c = U dc M e N edc D e ! U − dc . (1.8)5ere U is a matrix, which for fixed c and d describes a discrete transformation in the spaceof multiplicity labels, which for fixed initial and final boundary conditions has its rowslabeled by the intermediate boundary condition created by D c , and columns labeled by e the summation parameter of the direct sum on the right hand side. When we include alsothe initial and the final boundary conditions as part of the multiindices, i.e. { a, a ′ , a ′′ } and [ e ; a, a ′′ ] respectively, the U matrix becomes orthogonal and it is surprisingly givenby the Racah symbol (2.40)( U dc ) { a a ′ a ′′ } [ e ; b, b ′′ ] = δ ab δ a ′′ b ′′ (cid:26) c, a, a ′ a ′′ , d, e (cid:27) . (1.9)From the explicit formula [10] for the boundary state in terms of the OSFT classicalsolution it follows that applying the defect operator to the open string field results in aboundary state encircled by the defect operator which gives a new consistent boundarystate of the kind considered by Graham and Watts [60]. In formulas || B D Ψ ii = D || B Ψ ii . (1.10)Therefore, assuming that a given solution Ψ X → Y describes BCFT Y in terms of BCFT X ,upon action of the defect operator D , it will describe BCFT D Y in terms of BCFT D X , i.e. D Ψ X → Y = Ψ D X →D Y . (1.11)As a byproduct of our analysis we discover the interesting relation D dj ( a ) B ij g a = X a ′ ∈ d × a X daaia ′ a ′ ( a ′ ) B ij g a ′ , (1.12)relating bulk to boundary structure constants for different boundary conditions, where D is the coefficient of the defect operator (2.13) acting on a spinless bulk field (labeled by j ). In order to be as self-contained as possible the paper includes some review materialand it is organized as follows. In section 2 we review the basic definitions and propertiesof defects and defect networks in two dimensional CFT. We also introduce the dualitymatrices as generic solutions to the pentagon identity (which is a consequence of thetopological structure of the networks and the basic fusion rules of the defects) and therelated 6J and Racah symbols. In section 3 we review the basic construction of boundarystates in diagonal minimal models and how topological defects act on them. In additionwe review Runkel’s derivation of the boundary OPE coefficients and identify a particu-larly useful normalization for boundary fields. Section 4 is devoted to the main resultsof our work, namely the construction of open topological defect operators as maps be-tween two boundary operator algebras which is compatible with the OPE. We present6wo independent derivations of our results, one which is algebraic and which builds onthe initial analysis by Graham and Watts [60] and one which is geometric and uses theproperties of defect networks in presence of boundaries. Both our constructions clearlyshow that the composition of open topological defect operators follows the fusion rulesonly up to a similarity transformation whose precise structure is encoded in the Racahsymbols. In section 5 we consider open topological defects as new operators in OSFTwhich map solutions to solutions. We show that the way OSFT observables are affectedby the action of defects is consistent with the BCFT description and the interpretationof OSFT solutions as describing a new BCFT using the degrees of freedom of a referenceBCFT. In section 6 we concretely present our constructions in the explicit example of theIsing model BCFT. Few appendices contain further results which are used in the maintext. Consider two, generally distinct, 2d CFT’s glued along a one-dimensional interface. Weassume that energy is conserved across this interface. Let ( x , x ) be the coordinates of thesystem and the interface is placed at x = 0. From the conservation law ∂ T + ∂ T = 0,we see that if T is continuous then the total energy is conserved ∂∂x Z dx T = Z dx ∂∂x T = 0 . (2.1)Then we require that the momentum density T = T is continuous across the interface T ( x , x ) (cid:12)(cid:12) x → = T ( x , x ) (cid:12)(cid:12) x → − . (2.2)Introducing the complex coordinates z = x + ix and ¯ z = x − ix , the above conditionis written as lim x → (cid:0) T ( z ) − ¯ T (¯ z ) (cid:1) (cid:12)(cid:12) z = x + ix = lim x → (cid:0) T ( z ) − ¯ T (¯ z ) (cid:1) (cid:12)(cid:12) z = x − ix . (2.3)This condition also means that the system has invariance under conformal transformationswhich leave the shape of the defect line untouched, and the interface enjoying (2.3) iscalled a conformal interface or a conformal defect. The gluing condition (2.3) is usuallyimplemented by giving a rule for how fields of these two CFTs are related at the interface,and conformal defects give a mapping from a field configuration of one theory to that of theother theory. The concept of conformal defects comes from the study of one dimensionalimpurity system [63, 64]. For a recent discussion of conformal defects see [65, 66, 67, 68].7here are two special classes of conformal defects: the factorized defects and thetopological defects. The factorized defects are purely reflective with respect to the energyflow, and the two CFTs do not communicate at all. This condition is given by requiringthe energy current T to be zero at the defect, or equivalentlylim x → T ( z ) (cid:12)(cid:12) z = x + ix = lim x → ¯ T (¯ z ) (cid:12)(cid:12) z = x + ix , lim x → T ( z ) (cid:12)(cid:12) ¯ z = x − ix = lim x → ¯ T (¯ z ) (cid:12)(cid:12) ¯ z = x − ix . (2.4)From (2.4) we see that the system is reduced to two separated BCFTs sharing the defectline as their common boundary. Also note that a conformal boundary can be viewed asan example of a factorized defect between a given bulk CFT and an empty c = 0 theory.On the other hand, topological defects are purely transmissive with respect to theenergy, and this condition is expressed by the momentum conservation across the defect.From the conservation law ∂ T + ∂ T = 0, we see that if T is continuous across thedefect, the momentum is conserved. In complex coordinates, this condition is given bylim x → T ( z ) (cid:12)(cid:12) z = x + ix = lim x → T ( z ) (cid:12)(cid:12) z = x − ix , lim x → ¯ T (¯ z ) (cid:12)(cid:12) ¯ z = x + ix = lim x → ¯ T (¯ z ) (cid:12)(cid:12) ¯ z = x − ix . (2.5)The energy momentum tensor does not see the defect, since its components are continuousacross the defect line. Therefore, continuous deformations of topological defects do notchange the value of correlation functions.A familiar example of a topological defect appears in the Ising model CFT, which isequivalent to the minimal model M (3 ,
4) with three primary fields { , ε, σ } . In addition,there exists the disorder field µ which however is not mutually local with the spin field σ . The correlation functions containing both spin fields and disorder fields have branchcuts on their Riemann surface, and they are represented by disorder lines which connecta pair of disorder fields. Clearly the value of such correlation functions does not changeby continuous deformations of disorder lines.In the above example, topological defects are curve segments with both end-points atdisorder fields but we can consider more generic configurations in which defects join orend on a boundary, as we will review and discuss later. A particularly class of defect operators are closed topological defects which are associatedto homotopy classes of cycles on a punctured surface. The closed topological defectsgive rise naturally to closed string operators D which act on bulk fields by encirclingthem with the defect. Since both the holomorphic and antiholomorphic component ofthe energy momentum are continuous across the defect, the encircling defect loop can8e arbitrarily smoothly deformed and the defect action should also commute with theVirasoro generators [ L n , D ] = [ ˜ L n , D ] = 0 , ∀ n. (2.6)By Schur’s lemma the action of the defect operator on the bulk states must be constant onevery Verma module. Then we can concentrate on the action of D on the bulk primaryoperator φ ( i, ¯ i,α, ¯ α ) ( z, ¯ z ), where i and ¯ i are labels for the Virasoro representation of theholomorphic and antiholomorphic part and α and ¯ α are corresponding multiplicity labels.Let a be a label classifying the topological defects then, following Petkova and Zuber [58],we can write D a φ ( i, ¯ i,α, ¯ α ) ( z, ¯ z ) = D a ( i, ¯ i,α, ¯ α ) φ ( i, ¯ i,α, ¯ α ) ( z, ¯ z ) , (2.7)with constant coefficients D a ( i, ¯ i,α, ¯ α ) . This can be written by D a = X ( i, ¯ i,α, ¯ α ) D a ( i, ¯ i,α, ¯ α ) P ( i, ¯ i,α, ¯ α ) , (2.8)where P ( i, ¯ i,α, ¯ α ) is the projector on the Verma module labeled by ( i, ¯ i, α, ¯ α ).To determine the coefficients D a ( i, ¯ i,α, ¯ α ) , let us consider the modular transformation ofthe torus partition function with a pair of closed topological defect lines. There are twoways for evaluating this. One way is to consider time slices parallel to the defect line,obtaining the following expression Z a | b =Tr (cid:16) ( D a ) † D b ˜ q L − c ¯˜ q ¯ L − c (cid:17) = X ( j, ¯ j,α, ¯ α ) ( D a ( j, ¯ j,α, ¯ α ) ) ∗ D b ( j, ¯ j,α, ¯ α ) χ j (˜ q ) χ ¯ j (¯˜ q ) . (2.9)The other way is to consider time evolution along the defect line. Let V i ¯ i ; xy denotes themultiplicity of the Virasoro representation ( i ¯ i ) appearing in the spectrum in this timeslicing H a | b = V i ¯ i ; ab R i ⊗ ¯ R ¯ i , (2.10)then the partition function is written as Z a | b = X ( j, ¯ j,α, ¯ α ) V j ¯ j ; ab χ j ( q ) χ ¯ j (¯ q ) . (2.11)From modular invariance of the Virasoro characters, we can connect (2.9) and (2.11), andobtain a bootstrap equation V i ¯ i ; ab = X j ¯ j S ji S ¯ j ¯ i D a ( i, ¯ i,α, ¯ α ) (cid:16) D b ( i, ¯ i,α, ¯ α ) (cid:17) ∗ , (2.12)9hich in paticular states that the rhs should be a positive integer.For the diagonal minimal models there is a simple solution to (2.12) D ai = S ai S i , V i ¯ i ; ab = X k N aik N k ¯ ib . (2.13)We can check this result with the help of the Verlinde formula [69] N kij = X l S il S jl S ∗ kl S l . (2.14)On the right hand side of the first equation of (2.13), index a runs over the irreduciblerepresentations of the Virasoro algebra, and from this expression, we see that there are asmany distinct topological defects as primary operators. Plugging back to (2.8), we obtainthat D a = X i S ai S i P i . (2.15)It then follows that these operators obey the fusion algebra D a D b = X N cab D c , (2.16)just like the conformal families of the primary fields[ φ a ] × [ φ b ] = X c N cab [ φ c ] . (2.17)This follows by a simple computation X i S ai S i P i ! X j S bj S j P j ! = X i S ai S i S bi S i P i = X i X c N cab S ci S i P i , (2.18)where in the last equality we used Verlinde formula (2.14). To get a more conceptual understanding of topological defects, it is useful to considernetworks of topological defects, where the defects are allowed to join in trivalent vertices.Let us state now some minimalistic assumptions: defects can be decomposed into elemen-tary ones. The labels of the elementary defects define naturally an associative algebra.For two elementary defects a and b we define the product of labels a × b as a free sumof labels of the elementary defects which arise upon fusing defect a with defect b . Theassociativity follows from the topological nature of defects.10 bc dp = a bc d = a bc d non-elementarytopologicaldefect Figure 2: The topological nature of defects implies that an“s-channel” configuration withan elementary defect should be equivalent to a “t-channel” configuration with a composite,but still topological defect. a bc dp q = q F pq a bc d a bc d Figure 3: Elementary defect network move.Another important assumption is that there is always an identity defect, labeled as 1or , which can be freely drawn or attached anywhere without changing anything. The most powerful property of topological defects, is that they can be freely deformed,without changing the value of any correlator, as long as they do not cross the positionof any operator insertion or another defect. Therefore, a small piece of defect networkin the“s-channel”-like configuration, shown on the left hand side of Figure 2, composedof four defects joined by an intermediate one, can be deformed into an alternate “t-channel”-like configuration where the new defect will in general no longer be elementaryone. Decomposing it into a linear combination of elementary defects, defines a set ofa priori unconstrained coefficients F pq (cid:20) a bc d (cid:21) , see Figure 3. In many circumstances thesecoefficients are known, but in order to be self-contained and perhaps more general, let usignore this knowledge and proceed by following the consequences of consistency. At this point our conventions differ from some of the literature on the subject e.g. [70]. a b c da b c d a b c da b c d a b c de e ee e sp pq qr rsF ps b ca q ttF sr c db t F qr c dp e F pt b ra e F qt s da e Figure 4: Pentagon identity as a consistency condition for fusion of defects.final configuration along two different paths depicted by arrows in Figure 4, one finds thecelebrated pentagon identity X s F ps (cid:20) b ca q (cid:21) F qt (cid:20) s da e (cid:21) F sr (cid:20) c db t (cid:21) = F qr (cid:20) c dp e (cid:21) F pt (cid:20) b ra e (cid:21) . (2.19)By the MacLane coherence theorem, this equation is enough to guarantee the consistencyof the fusion rule in Figure 3 for any possible defect network. This is not to say that otheridentities are not of interest, but that those required for consistency are implied by thepentagon identity. This identity differs from the one given in [72] by transposition of columns in every F . For unorienteddefects this difference is immaterial as F pq (cid:20) a bc d (cid:21) = F pq (cid:20) b ad c (cid:21) . For oriented defects one has to specifycarefully an orientation. Our implicit orientation (always downwards in Figure 4) is the same as in [73]and others [74], but differs from the one required to match the formulas of Moore and Seiberg [72, 71].
12 simple property of the F following from our definition and the natural normalizationfor the trivial identity defect is that F pq (cid:20) a bc d (cid:21) = 1 whenever 1 ∈ { a, b, c, d } . (2.20)Using this in the pentagon identity by setting e = 1 (which requires also q = d , a = t and p = r ) one finds the orthogonality relation X s F ps (cid:20) b ca d (cid:21) F sr (cid:20) c db a (cid:21) = δ pr . (2.21)For p = r the left hand side of (2.19) still makes sense, but the right hand side wouldcontain fusion matrix elements with non-admissible label, so the only consistent value forthe left hand side is zero. Alternatively, one can of course apply the elementary moveagain to the right hand side of Figure 3 viewed sideways.Let us now consider defect networks containing closed loops. An arbitrary such net-work can be reduced via the elementary move in Figure 3 to a network without any closedloops. The simplest defect network with a loop is of course a single loop, but let us startwith a slightly more general configuration shown in Figure 5 of a bubble with two exter-nal defect lines attached. As long as there are no operators put inside the bubble, it can i jab = i jab i jab abi jab i i = δ ij F i a ba b = k F k a ba b ⇓ k Figure 5: Defect bubble network. When no operators are present inside, the topologi-cal bubble can be shrank to zero size, yielding a numerical factor symmetric under theexchange of a and b labels.shrink leaving behind a pure number. This already tells, that the two external lines mustcarry the same label otherwise, upon shrinking, one would expect a defect changing oper-ator insertion. These operators carry nontrivial conformal weight and thus the correlatorscaling properties would contradict those expected for a network of topological defects.13ttaching an auxiliary line of identity defect (see Figure 5) allows us to find a simpleexpression for the bubble in terms of the F -matrix. In rotation invariant theories forunoriented defects the numerical factor F i (cid:20) a ba b (cid:21) must be symmetric in a and b , as followsby considering the 180 ◦ degree rotation.As a corollary we find that a bubble with no defects attached, which is the same thingas if two identity defects were attached, is equivalent to an overall factor g ′ a = 1 F (cid:20) a aa a (cid:21) , (2.22)which we identify with the normalized g -function of the defect. Another corollary is an expression for the sunset diagram in Figure 6. Since it can be abc = 1 F a aa a F a b cb c = θ ( a , b , c ) Figure 6: Defect sunset network gives rise to an S symmetric factor θ ( a, b, c ).viewed as a symmetric bubble on a loop in two possible ways, it follows that θ ( a, b, c ) = θ ( a, c, b ) = θ ( c, b, a ) , (2.23)and hence it enjoys the full S permutation symmetry. Analogously, we can introduce˜ θ ( a, b, c ) = 1 F (cid:20) a aa a (cid:21) F a (cid:20) b bc c (cid:21) , (2.24)which, as a consequence of pentagon identity (setting t = q = 1 together with s = e = d = a , r = b and p = c in (2.19)) satisfies g ′ a g ′ b g ′ c = θ ( a, b, c )˜ θ ( a, b, c ) , (2.25)and hence ˜ θ ( a, b, c ) also possess the S permutation symmetry.Another peculiar identity which can be obtained by the sequence of moves in Figure 7is In the context of the minimal models this is equal to S a S , where S is the modular S -matrix. It isthus the value of the g -function of the boundary condition associated to the defect via the folding trick,normalized by the g -function of the trivial defect. In non-unitary theories (e.g. for the Lee-Yang model)or theories with oriented defects this quantity may coincide with the usual normalized g -function onlyup to a sign. = c ∈ a × b F c a ba b a ab bc = c ∈ a × b c F a aa a × F b bb b = c N cab F c cc c Figure 7: Derivation of the Verlinde-like formula. In the second step we use the S symmetry of the bubble factor θ ( a, b, c ).1 F (cid:20) a aa a (cid:21) F (cid:20) b bb b (cid:21) = X c N cab F (cid:20) c cc c (cid:21) . (2.26)It might come as a surprise that this equation is a consequence of the polynomial pentagonidentity (2.19). To see that it is indeed the case, one may proceed in two steps. Startingwith the orthogonality relation (2.21), setting p = r = 1, adjusting accordingly the otherindices, and using the relation (2.25) one derives easily (2.26).For the minimal models, the relation (2.26) is in fact nothing but the Verlinde formulafor the first row (or column) of the S -matrix S a S b S = X c N cab S c . (2.27)thanks to the relation between the modular S -matrix and the F -matrices (which followsfrom the formulas in [71], see also e.g. (E.9) in [83]) S ij S = X k ∈ i × j e πi ( h i + h j − h k ) F (cid:20) k kk k (cid:21) , (2.28)which for i = 1 simplifies to g ′ j = S j S = (cid:16) F (cid:20) j jj j (cid:21) (cid:17) − .The next natural step is to consider a defect loop with three external defect linesattached, as in Figure 8. Applying the elementary defect network move to any pair ofvertices connected by an internal defect line, we find the elementary vertex with a bubbleon one of the external lines, and as before we can replace the bubble by the correspondingfactor. In general this would yield a Z symmetric expression which follows directly fromthe pentagon identity (2.19) by setting t = 1 which requires also s = d , r = b and e = a with the help of (2.25). For unoriented parity-invariant defects in parity invariant CFT’sthe symmetry is enhanced to S . 15 bi jkc jki = F ck j ia b × F k a ba b − Figure 8: Triangular defect network. When no operators are present inside, the topologicaltriangle can be shrank to zero size, yielding a numerical factor with S permutationsymmetry under the exchange of ( i, a ), ( j, b ) and ( k, c ) labels.The symmetries of the F matrices are actually much larger and can be nicely mani-fested by considering a defect network in the shape of tetrahedron, see Figure 9. It canbe drawn as such on a Riemann sphere, but the resulting identities should have universalvalidity. Let us now choose any triangular face, e.g. ( abc ), and shrink it to a point picking a bc ij k Figure 9: Tetrahedral defect network. When no operators are present inside the faces,the topological tetrahedron can be shrank to zero size, yielding a numerical factor with S permutation symmetry under the exchange of the faces with ( a, b, c ), ( a, j, k ), ( b, k, i )and ( c, i, j ) labels.up the triangle factor as in Figure 8. This results in the sunset diagram, see Figure 6,which we have already evaluated. Combining the factors, one quickly arrives at (cid:20) i, j, ka, b, c (cid:21) TET ≡ F ck (cid:20) j ia b (cid:21) F k (cid:20) a ba b (cid:21) F k (cid:20) i ji j (cid:21) F (cid:20) k kk k (cid:21) = 1 g ′ k θ ( a, b, k ) θ ( i, j, k ) F ck (cid:20) j ia b (cid:21) . (2.29)The notation is such that the labels in the upper row always form an admissible triplet,i.e. i ∈ j × k .We could have chosen an arbitrary face of the tetrahedron for reducing the triangleand due to the Z cyclicity of the defect triangle in Figure 8, we would have obtained oneout of three possible expressions. Altogether we get 12 different expressions which must16e equal to each other, and which correspond to the orientation preserving subgroup A of the tetrahedral group S . The Z subgroup cyclically permutes the columns (cid:20) i, j, ka, b, c (cid:21) TET = (cid:20) j, k, ib, c, a (cid:21) TET = (cid:20) k, i, jc, a, b (cid:21) TET , (2.30)while the Z × Z subgroup is switching upper and lower labels simultaneously in twodifferent columns (cid:20) i, j, ka, b, c (cid:21) TET = (cid:20) i, b, ca, j, k (cid:21) TET = (cid:20) a, b, ki, j, c (cid:21) TET = (cid:20) a, j, ci, b, k (cid:21) TET . (2.31)The equality of the three expressions (2.30) follows already from the pentagon identity,but (2.31) does not. The reason is that in deriving (2.31) we have assumed that all defectswere unoriented. For oriented defects some labels in the identity (2.31) must be replacedby the conjugate labels to account for the change of orientation.In the special case of parity invariant defects in parity invariant theories the tetrahe-dral defect network is invariant under the full tetrahedral group S which in addition tothe generators of A contains also 12 transformations combining rotations with a singlereflection. The additional identities can be generated with the help of (cid:20) i, j, ka, b, c (cid:21) TET = (cid:20) j, i, kb, a, c (cid:21) TET . (2.32)These are the symmetries of the classical or quantum Wigner’s 6J symbol. This objecthowever differs from the 6J symbol by a tetrahedral invariant normalization factor (seediscussion in section 2.2.2), and resembles thus an object often called T ET in the liter-ature, see e.g. [76] or [75]. Another difference would arise in the case of oriented defect,where the tetrahedral invariance of the defect network would be broken. As follows from the definition of the F -matrix, see Figure 3, invariance under 180 ◦ rotationfor unoriented defects implies F pq (cid:20) a bc d (cid:21) = F pq (cid:20) d cb a (cid:21) . (2.33)Similarly parity invariance of both the theory and the defect (with respect to any axis)implies a stronger condition F pq (cid:20) a bc d (cid:21) = F pq (cid:20) c da b (cid:21) = F pq (cid:20) b ad c (cid:21) . (2.34) Also because of the fact, that up to the normalization factor, this object obeys the pentagon identitywithout signs, it is more reminiscent of the classical Racah W-coefficient.
17f course, the two identities (2.34) together imply (2.33). All these identities are true inthe Virasoro minimal models.Mathematically, the symmetry of the F matrix (2.33) is equivalent to the condition(2.31) under the assumption of the pentagon identity, or in particular (2.30). Analogously,under the same assumption, the symmetries (2.34) are equivalent to (cid:20) i, j, ka, b, c (cid:21) TET = (cid:20) b, a, kj, i, c (cid:21) TET and (cid:20) i, j, ka, b, c (cid:21)
TET = (cid:20) j, i, kb, a, c (cid:21) TET respectively.
From the defect network manipulations we have seen that (cid:20) i, j, ka, b, c (cid:21)
TET obeys the sametetrahedral symmetries as classical or quantum Wigner symbol. Such an object is byno means unique, since a product over the four tetrahedron vertices of an S invariantfunction of the three corresponding edges will always have the tetrahedral symmetry. Aparticularly useful combination is what we call the normalized 6J symbol (cid:20) i, j, ka, b, c (cid:21) = 1 p θ ( i, j, k ) θ ( i, b, c ) θ ( a, j, c ) θ ( a, b, k ) (cid:20) i, j, ka, b, c (cid:21) TET (2.35)= 1 g ′ k s θ ( a, b, k ) θ ( i, j, k ) θ ( i, b, c ) θ ( a, j, c ) F ck (cid:20) j ia b (cid:21) , (2.36)which enjoys also the full tetrahedral symmetry for unoriented defects. If it were not forthe prefactor 1 /g ′ k it would have obeyed the pentagon identity, since F pq (cid:20) a bc d (cid:21) → Λ( a, b, q )Λ( c, q, d )Λ( c, a, p )Λ( p, b, d ) F pq (cid:20) a bc d (cid:21) , (2.37)is an exact symmetry of the pentagon identity (2.19) for an arbitrary function Λ( i, j, k ) ofan admissible triplet . Due to cyclicity of θ , one can arrange their arguments, so that thefull prefactor under the square root in (2.36) can be viewed as a gauge transformation.Hence the 6J symbol obeys the following pentagon-like identity X s g ′ s (cid:20) c, b, sa, q, p (cid:21)(cid:20) d, s, ta, e, q (cid:21)(cid:20) d, c, rb, t, s (cid:21) = (cid:20) d, c, rp, e, q (cid:21)(cid:20) r, b, ta, e, p (cid:21) . (2.38)When one of the entries equals 1, the 6J symbol simplifies (cid:20) i, c, b , b, c (cid:21) = 1 p g ′ b g ′ c . (2.39) Further discussion of this “gauge symmetry” observed by Moore and Seiberg [71] is relegated toappendix A. g ′ a g ′ b = P c N cab g ′ c .This system of equations in rational CFT’s has as many solutions as we have labels. Everysolution corresponds to a single column of the modular S -matrix normalized by its firstelement. Then we can solve the pentagon identities (2.38) by imposing the symmetrieson the 6J symbol. We have checked that for Lee-Yang model, Ising model and tricriticalIsing model these over-determined polynomial systems have a unique solution.A fundamental property of the 6J symbol is gauge invariance under the symmetry(2.37) which holds for any Λ which is S -invariant and subject to the additional conditionΛ(1 , a, a ) = 1.Another very useful object which will play an important role in the following sectionsof this paper is what we call Racah symbol following the terminology of Coquearaux (cid:26) i, j, ka, b, c (cid:27) ≡ p g ′ k g ′ c (cid:20) i, j, ka, b, c (cid:21) . (2.40)It obeys the full pentagon identity, X s (cid:26) c, b, sa, q, p (cid:27)(cid:26) d, s, ta, e, q (cid:27)(cid:26) d, c, rb, t, s (cid:27) = (cid:26) d, c, rp, e, q (cid:27)(cid:26) r, b, ta, e, p (cid:27) , (2.41)and can be therefore identified with the F matrix in a given special gauge (cid:26) c, b, sa, q, p (cid:27) = F Rac ps (cid:20) b ca q (cid:21) . (2.42)A peculiarity of this gauge is that F Rac1 i (cid:20) a ba b (cid:21) = F Rac i (cid:20) a ab b (cid:21) = s g ′ i g ′ a g ′ b . (2.43)When one of the entries of the first two columns equal 1, the Racah symbol simplifies (cid:26) i, c, b , b, c (cid:27) = 1 . (2.44)This object, like the 6J symbol, is also invariant under the gauge transformation(2.37). Just like a generic solution to the pentagon identity, the Racah symbols obey theorthogonality condition X q (cid:26) b, a, qc, d, p (cid:27)(cid:26) c, a, sb, d, q (cid:27) = δ ps . (2.45) What Coquearaux [76] calls geometrical Racah symbols, or Carter et al [75] call the 6j symbol areour F matrices. F matrices. This has to be contrasted with the F matrices arising from thetransformations of the conformal blocks which are uniquely determined once the confor-mal blocks are normalized, by giving the coefficient of their leading term. It would beinteresting to know whether there is a specific normalization choice which also fixes thegauge for the defect networks. This may involve a careful study of defect changing fields,which goes beyond the scope of this paper. Conformal boundary conditions in 2D CFT’s have to satisfy a number of consistencyconditions spelled out explicitly in [77, 78]. This section is a review of some of theseconsistency conditions in diagonal minimal models and of the action of topological defectson the fundamental boundary states.
From the bulk perspective, conformal boundary conditions in 2D CFT’s are encoded in theconformal boundary states which are required to obey a number of necessary conditions.The most elementary requirement of preserving the conformal symmetry forbids the two-dimensional energy and momentum to flow through the boundary, which leads to thegluing condition (cid:0) L n − ¯ L − n (cid:1) || B ii = 0 . (3.1)The set of linearly independent solutions was written down by Ishibashi [79]. The Ishibashistates are in one-to-one correspondence with spinless bulk primaries V α | V α ii = X IJ M IJ ( h α ) L − I ¯ L − J | V α i (3.2)= X n | n, α i ⊗ | n, α i (3.3)= h h α L − ¯ L − + · · · i | V α i . (3.4)The multi-indices I, J , with I = { i , ..., i n } appearing in the first line label the non-degenerate descendants in the conformal family of V α , and M IJ ( h α ) is the inverse of theGram matrix h V α | L I L − J | V α i . We denoted L I = L i L i . . . L i n and L − I = L − i n L − i n − . . . L − i .The Ishibashi state (see the second line) is as a sum over a basis of states (which are or-thonormal wrt the Gram matrix) in the Verma module over the chiral part of the primary V α . 20 highly non-trivial consistency requirement is given by Cardy’s condition. Considerthe partition function on a finite cylinder with two boundary conditions a and b . Viewed inthe “closed string” channel, the diagram can be interpreted as a matrix element betweentwo boundary states || a ii and || b ii . In the “open string” channel it becomes a trace overthe Hilbert space of the CFT with the two boundary conditions a and b hh a || ˜ q ( L +¯ L − c ) || b ii = Tr H open ab (cid:0) q L − c (cid:1) , (3.5)where q = e πiτ , ˜ q = e − πi/τ , (3.6)and τ = R/L is given by the radius and length of the cylinder.It is well known that for minimal models with diagonal partition function Cardy’scondition is solved by a set of fundamental boundary states, explicitly given by [80] || B i ii = X j S ji q S j | j ii , (3.7)where S ji are the entries of the modular matrix and i and j denote the Virasoro represen-tation which are present in the minimal model. Therefore in this case there is a one-to-onecorrespondence between chiral primaries and fundamental boundary conditions. The mostgeneral boundary condition consistent with Cardy’s condition is obtained by taking posi-tive integer linear combinations of the above fundamental boundary states, in the case ofdiagonal minimal models. Boundary operators generally change the boundary conditions. The multiplicity of aboundary operator in the Virasoro representation k , changing the boundary conditionsfrom i to j is the integer coefficient N kij appearing in the fusion rules of the theory φ i × φ j = X k N kij φ k . (3.8)While one could extract the spectrum of boundary operators from the cylinder am-plitude between two boundary states, to compute their OPE structure constants one hasto resort to the 4-pt conformal bootstrap. To this end, let us consider a 4pt boundaryfunction G ( abcd ) ijkl ( ξ ) ≡ (cid:10) I ◦ φ abi (0) φ bcj (1) φ cdk ( ξ ) φ dal (0) (cid:11) UHP , (3.9)21here I ( z ) = − z . We can compute it in two ways using different OPE channels G ( abcd ) ijkl ( ξ ) = (cid:10) I ◦ φ abi (0) φ bcj (1) φ cdk ( ξ ) φ dal (0) (cid:11) = (cid:10) h ◦ φ dal (0) h ◦ I ◦ φ abi (0) h ◦ φ bcj (1) h ◦ φ cdk ( ξ ) (cid:11) = ξ h i + h j − h l − h k (cid:10) I ◦ φ dal (0) φ abi (1) φ bcj (1 − ξ ) φ cdk (0) (cid:11) = ξ h i + h j − h l − h k G ( dabc ) lijk (1 − ξ ) , (3.10)where h ( z ) = z − ξz . We now express the four point functions in terms of the structureconstants and the four point conformal blocks G ( abcd ) ijkl ( ξ ) = X p C ( abc ) pij C ( cda ) pkl G ( aca ) pp F ( i, j, k, l ; p )( ξ ) , (3.11)where the conformal blocks are given by the formula F ( i, j, k, l ; p )( ξ ) = X I,J β I ( h i , h j , h p ) β J ( h k , h l , h p ) G IJ ( h p ) ξ h p + | J |− h k − h l , (3.12)where I is a Virasoro multiindex, G IJ is the matrix of inner products in a highest weightrepresentation of Virasoro algebra, and the β coefficients are defined via φ abi ( x ) φ bcj ( y ) = X p,I C ( abc ) pij β I ( h i , h j , h k )( x − y ) h i + h j − h k −| I | L − I φ acp ( y ) . (3.13)Notice that F do not depend on any normalization of boundary operators.The conformal blocks in ξ can be linearly related to the the conformal blocks in 1 − ξ via precisely chosen F matrices F ( k, l, i, j ; p )( ξ ) = X q F blocks pq (cid:20) l ik j (cid:21) F ( i, l, k, j ; q )(1 − ξ ) . (3.14)Then, using (3.11) and (3.14), we find from (3.10) C ( dab ) pli C ( bcd ) pjk G ( dbd ) pp = X q F blocks qp (cid:20) l ik j (cid:21) C ( abc ) qij C ( cda ) qkl G ( aca ) qq , (3.15)where the two point functions are given in terms of the three point functions as G ( aca ) pp = C ( aca ) 1 pp g a . (3.16)This can be further simplified to C ( abd ) lip C ( bcd ) pjk = X q F blocks qp (cid:20) l ik j (cid:21) C ( abc ) qij C ( acd ) lqk . (3.17) Use C ( cda ) qkl C ( aca ) 1 qq = C ( acd ) lqk C ( ada ) 1 ll and C ( aba ) 1 ii g a = C ( bab ) 1 ii g b . that this equation can be exactly solved by setting C ( abc ) kij = F blocks bk (cid:20) a ci j (cid:21) , (3.18)thanks to the pentagon identity.We can find a more convenient expression for the structure constants by changing thenormalization of the boundary operators. To this end, let us express F blocks pq (cid:20) a bc d (cid:21) in termsof the normalized 6J symbols F blocks pq (cid:20) a bc d (cid:21) = g ′ q s θ ( b, d, p ) θ ( c, a, p ) θ ( c, d, q ) θ ( b, a, q ) (cid:20) b, a, qc, d, p (cid:21) . (3.19)Runkel’s solution then becomes C ( abc ) kij = F blocks bk (cid:20) a ci j (cid:21) = g ′ k s θ ( c, j, b ) θ ( i, a, b ) θ ( i, j, k ) θ ( c, a, k ) (cid:20) c, a, ki, j, b (cid:21) (3.20)= s g ′ k g ′ b s θ ( c, j, b ) θ ( i, a, b ) θ ( i, j, k ) θ ( c, a, k ) (cid:26) c, a, ki, j, b (cid:27) (3.21)= vuuut θ ( c,j,b ) √ g ′ c g ′ j g ′ b θ ( i,a,b ) √ g ′ i g ′ a g ′ b θ ( i,j,k ) √ g ′ i g ′ j g ′ k θ ( c,a,k ) √ g ′ c g ′ a g ′ k (cid:26) c, a, ki, j, b (cid:27) . (3.22)From here we see that there is a special choice of normalization of the boundary fields, inwhich ˆ C ( abc ) kij = s p g ′ i g ′ j g ′ k θ ( i, j, k ) (cid:26) c, a, ki, j, b (cid:27) . (3.23)The Racah symbol is gauge invariant, but the object θ ( i, j, k ) has to be computed from F blocks pq (cid:20) a bc d (cid:21) , i.e. θ ( i, j, k ) = g ′ i F blocks1 i (cid:20) j kj k (cid:21) . (3.24) Generalizations have been studied in [83, 84, 85]. This expression for the boundary structure constants is particularly interesting since the square ofthe prefactor coincides with the bulk structure constants, C bulk ijk = √ g ′ i g ′ j g ′ k θ ( i,j,k ) in a canonical normalizationfor bulk operators where the coefficient of the two point functions are set universally to one times thesphere partition function. .3 Defect action on boundary states As we have seen in section 2.1, defects act naturally on bulk operators by encircling them.Cardy boundary states are (non-normalizable) states in the space of bulk operators || B a ii = X i S ai √ S i | i ii , (3.25)so, analogously, the action of the defect operator is given by [60] D a || B b ii = X c N cab || B c ii , (3.26)as follows by a computation almost identical to section 2.1, using again Verlinde formulaand the fact that the projectors obey P i | j ii = δ ij | j ii .Now we can easily offer an alternative proof for the observation of [11] that OSFTmakes predictions for the coefficients of the boundary states || B X ii = P β B βX | β ii B βX B βY B βR = X Z N ZXY B βZ , (3.27)under the assumption that the reference D-brane || R ii allowed an OSFT solution de-scribing || X ii and that such a solution could have been re-interpreted on a D-brane || Y ii sharing the relevant Verma modules as || R ii .Assuming that the defect operator acts as a multiple of the identity on each Vermamodule and that as an operator it is selfconjugate (or possibly antiselfconjugate) underBPZ conjugation then the coefficients of the boundary states satisfy h V β | D || R iih V β || R ii = h V β | D || X iih V β || X ii , (3.28)and hence B βDR B βR = B βDX B βX , (3.29)where DR and DX stands for D-branes obtained by fusing defect D onto R or X branes.The new DX brane itself is either fundamental or should be an integer linear combinationof such and therefore B βX B βDR B βR = B βDX = X Z N ZDX B βZ , (3.30)which matches the formula (3.27) derived from OSFT by reinterpreting a solution Ψ R → X on the DR brane. It is one of the goals of this paper to explain this coincidence.24 Attaching defects to boundaries
In this section we define and study the action of topological defects on boundary fields.This was partially done by Graham and Watts [60] for boundary operators which donot change the boundary conditions. We will generalize their algebraic approach to thefull open string spectrum, including the important case of boundary condition changingoperators. In addition we will provide an independent geometric derivation using defectnetworks.We determine how an open string defect acts as an operator mapping the boundaryoperator algebra of a system of boundary conditions to a closed subset of the operatoralgebra of a new system of boundary conditions. Then we study the composition of suchoperators and we show how is this related to the fusion rules of the theory. We first proceedin a completely algebraic way, imposing the condition that the OPE must commute withthe action of an open topological defect. This is a non-trivial constraint that can besolved for the coefficients defining the open string defect (see later) and, together withan appropriate twist-invariance condition, allows to uniquely determine such coefficients,thanks to the pentagon identity. Then we show that the composition of open topologicaldefects is governed by the fusion rules of the theory but, differently from the closedstring case, there is a non trivial rotation in the Chan-Paton’s labels corresponding tocoincident final boundary conditions. This rotation is in fact a similarity transformation.In the second subsection we show that our algebraic results can be independently obtainedin a purely geometric way by manipulating the involved defect networks with boundary.Our goal is to define an action of defects on the open string Hilbert space D : H open → H open . (4.1)This general action is further specified by decomposing H open into fundamental boundaryconditions H open = M a,b H ( ab ) , (4.2)where, when a = b , the corresponding states are boundary condition changing fields. Let d be a label for a topological defect, then the open string topological defect is a linearmap D d : H ( ab ) → M a ′ ∈ d × ab ′ ∈ d × b H ( a ′ b ′ ) . (4.3)This map is injective but in general is not surjective (the defect maps from a given openstring Hilbert space, onto a “bigger” one). This is how open strings feel the fact that a25opological defect, in general, maps a single D-brane into a system of multiple D-branes,according to the fusion rules of the underlining bulk CFT. As in the bulk case, Schur’slemma implies that the operator D restricted to H ( a,b ) → H ( a ′ ,b ′ ) should be a multipleof the identity on every Verma module Vir i in H ( a,b ) . Assuming that all states can beobtained by acting Virasoro operators on primary states, which is true in unitary CFT’s,the open string defect action is fully specified by D d φ abi = X a ′ ∈ d × ab ′ ∈ d × b X dabia ′ b ′ φ a ′ b ′ i , (4.4)where the φ ’s are the boundary primary fields, allowed by the involved boundary condi-tions. In this subsection we will determine the above-defined X -coefficient in an algebraic way.Then we will inspect how the composition of open topological defects is related to thefusion of defects in the bulk and to the fusion rules of the theory. A simple consistency condition for the action of open-string defects has been introducedby Graham and Watts [60] D d (cid:0) φ abi ( x ) φ bcj ( y ) (cid:1) = (cid:0) D d φ abi ( x ) (cid:1) (cid:0) D d φ bcj ( y ) (cid:1) . (4.5)Using the general ansatz (4.4) as well as the operator product expansion, the constraint(4.5) takes the explicit form X dacka ′ c ′ C ( abc ) kij = X b ′ ∈ d × b C ( a ′ b ′ c ′ ) kij X dabka ′ b ′ X dbckb ′ c ′ , (4.6)where C ( abc ) kij are the boundary structure constants. Restricting ourselves to the A n seriesof the minimal models, see section 3.2, the boundary structure constants can be writtenas C ( abc ) kij = n abi n bcj n ack s p g ′ i g ′ j g ′ k θ ( i, j, k ) blocks (cid:26) c, a, ki, j, b (cid:27) , (4.7)where n abi are generic normalizations of boundary fields with the convention that n abi = 1for the canonical normalization (3.23). With this explicit form of the structure constants26t immediately follows that (4.6) admits a general solution X dabia ′ b ′ = n abi n a ′ b ′ i (cid:26) b ′ , i, ad, a ′ , b (cid:27) , N ( d, a, a ′ ) N ( d, b, b ′ ) (4.8)thanks to the pentagon identity (2.41). The pentagon identity doesn’t fix the constants N ( d, c, c ′ ), but their are in fact fixed by imposing the parity condition X dabia ′ b ′ = X dbaib ′ a ′ , (4.9)which, using the properties of the Racah symbol in section 2.2.2, gives (cid:18) N ( d, a, a ′ ) N ( d, b, b ′ ) (cid:19) = s g ′ a ′ g ′ b g ′ a g ′ b ′ , (4.10)so that we can take N ( x, y, z ) = q (cid:26) x, x, z, z, y (cid:27) = (cid:18) g ′ y g ′ x g ′ z (cid:19) . (4.11)The X coefficients take the explicit form X dabia ′ b ′ = n abi n a ′ b ′ i F Rac di (cid:20) a ba ′ b ′ (cid:21) q F Rac1 a ′ (cid:20) a da d (cid:21) F Rac1 b ′ (cid:20) b db d (cid:21) F Rac1 i (cid:20) a ba b (cid:21) (4.12)= n abi n a ′ b ′ i ( g ′ a g ′ b g ′ a ′ g ′ b ′ ) (cid:20) a, a ′ , db ′ , b, i (cid:21) . (4.13)Notice that, differently from the boundary structure constants, the defect coefficients X don’t depend on the crossing symmetry properties of the conformal blocks, since only theRacah symbols are involved in their definitions. Let us now consider the fusion of topological defects on general boundary fields. To thisend, we need to calculate the subsequent action of D c and D d on φ abi . As we saw in (4.4),after the first action of D c there are multiple boundary conditions in general, and it isnatural to arrange the r.h.s. of (4.4) into a matrix regarding a ′ and b ′ as matrix indices. In this work we consider only 2D CFT’s which are separately invariant under C , P and T discretesymmetries. The parity symmetry P is related to the twist symmetry in the corresponding SFT [86]. D c φ abi = b ′ . . . b ′ n a ′ X cabia ′ b ′ φ a ′ b ′ i . . . X cabia ′ b ′ n φ a ′ b ′ n i ... ... . . . ... a ′ m X cabia ′ m b ′ φ a ′ m b ′ i . . . X cabia ′ m b ′ n φ a ′ m b ′ n i , (4.14)where a ′ i ∈ c × a and b ′ j ∈ c × b . The number of labels a ′ i is given by the number of nonzero N cai s, and the number of labels b ′ j is that of N cbj s. The right hand side is a m × n matrix,and m and n are given by m = X i N caa ′ i , n = X j N cbb ′ j , (4.15)respectively. We also write this equivalently as ( D c φ abi ) a ′ b ′ = X cabia ′ b ′ φ a ′ b ′ i . Similarly, afterthe subsequent action of D d , we have D d D c φ abi = b ′ . . . b ′ n a ′ M a ′ b ′ . . . M a ′ b ′ n ... ... . . . ... a ′ m M a ′ m b ′ . . . M a ′ m b ′ n , (4.16)where the submatrix M a ′ p b ′ q is given by M a ′ p b ′ q ≡ D d (cid:16) X cabia ′ p b ′ q φ a ′ p b ′ q i (cid:17) = b ′′ . . . b ′′ t a ′′ X cabia ′ p b ′ q X da ′ p b ′ q ia ′′ b ′′ φ a ′′ b ′′ i . . . X cabia ′ p b ′ q X da ′ p b ′ q ia ′′ b ′′ t φ a ′′ b ′′ t i ... ... . . . ... a ′′ s X cabia ′ p b ′ q X da ′ p b ′ q ia ′′ s b ′′ φ a ′′ s b ′′ i . . . X cabia ′ p b ′ q X da ′ p b ′ q ia ′′ s b ′′ t φ a ′′ s b ′′ t i . (4.17)This is a s × t matrix, and s and t are given by s = X i N da p ′ a ′′ i , t = X j N db ′ q b ′′ j , (4.18)and the size of the matrix D d D c φ abi is given by X a ′ p ∈ c × a s × X b ′ q ∈ c × b t = X i, p N da ′ p a ′′ i N caa ′ p ! × X j, q N db ′ q b ′′ j N cbb ′ q ! . (4.19)28rom (4.16) and (4.17) we see that to identify the position of components in ( D d D c φ abi ),we need to refer to both the intermediate boundary condition ( a ′ b ′ ) and the final boundarycondition ( a ′′ b ′′ ). We then introduce a composite label { a a ′ a ′′ } and express the aboveresult as (cid:0) D d D c φ i (cid:1) { a a ′ a ′′ }{ b b ′ b ′′ } ≡ (cid:16) D d (cid:0) D c φ abi (cid:1) a ′ b ′ (cid:17) a ′′ b ′′ = X cabia ′ b ′ X da ′ b ′ ia ′′ b ′′ φ a ′′ b ′′ i . (4.20)For the defect action on the bulk space we have D d D c = P e N edc D e . For the actionon the boundary operators we have to replace the ordinary sum by a direct sum, sincedifferent defects map to different Hilbert spaces M e ∈ d × c (cid:0) D e φ abi (cid:1) a ′′ b ′′ = f M e . . . f M e k , (4.21)where f M e j ≡ D e j φ abi = X e j abia ′′ b ′′ φ a ′′ b ′′ i . . . X e j abia ′′ b ′′ h φ a ′′ b ′′ h i ... . . . ... X e j abia ′′ g b ′′ φ a ′′ g b ′′ i . . . X e j abia ′′ g b ′′ h φ a ′′ g b ′′ h i . (4.22)Now we introduce the labels [ e ; a, a ′′ ] to represent M e ∈ d × c D e φ i ! [ e ; a, a ′′ ][ f ; b, b ′′ ] ≡ (cid:0) D e φ abi (cid:1) a ′′ b ′′ δ ef = X eabia ′′ b ′′ φ a ′′ b ′′ i δ ef . (4.23)Clearly the two expressions (4.20) and (4.23) are different, but notice that they have thesame dimensions thanks to the identity X k N ij k N klm = X k N imk N klj . (4.24)That is, as explained in (4.19), the number of the labels { a a ′ a ′′ } is given by P a ′′ (cid:16)P a ′ N aca ′ N a ′ da ′′ (cid:17) ,while the number of the labels [ e ; a, a ′′ ] is given by P a ′′ (cid:16)P e N cde N aea ′′ (cid:17) . These two num-bers are equal, as explained e.g. in [71]. Similarly, we also conclude that the number ofthe labels { b b ′ b ′′ } and that of the labels [ e ; b, b ′′ ] are the same.This suggests that there might be a similarity transformation linking the two matrices, (cid:0) D d D c φ i (cid:1) { a a ′ a ′′ }{ b b ′ b ′′ } = " U dc M e ∈ d × c D e φ i ! U − dc { a a ′ a ′′ }{ b b ′ , b ′′ } , (4.25)29here U dc is a real invertible matrix with matrix indices { a a ′ a ′′ } and [ e ; ˜ a, ˜ a ′′ ]. Sub-stituting (4.20) for ( D d D c φ i ) and (4.23) for (cid:0)L e ∈ d × c D e φ i (cid:1) , this equation is expressedas X da ′ b ′ ia ′′ b ′′ X cabia ′ b ′ = X [ e ; ˜ a ˜ a ′′ ] X [ f ; ˜ b, ˜ b ′′ ] U { a a ′ a ′′ } [ e ; ˜ a ˜ a ′′ ] dc X e ˜ a ˜ bi ˜ a ′′ ˜ b ′′ δ ef ( U − dc ) [ f ; ˜ b, ˜ b ′′ ] { b b ′ b ′′ } . (4.26)Since X dabia ′ b ′ is the Racah symbol with some extra factors (4.8), the equation (4.26) is againreminiscent of the pentagon identity (2.41). In fact, we find that the following U dc is asolution( U dc ) { a a ′ a ′′ } [ e ; ˜ a, ˜ a ′′ ] = N ( c, a, a ′ ) N ( d, a ′ , a ′′ ) M ( d, e, c ) N ( e, a, a ′′ ) ( c, a, a ′ a ′′ , d, e ) ( a = ˜ a ) and ( a ′′ = ˜ a ′′ ) , a = ˜ a ) or ( a ′′ = ˜ a ′′ ) , (4.27)where the factor N ( x, y, z ) is the same as that appearing in (4.8), and M ( x, y, z ) is anonzero arbitrary real number. To check (4.26), notice that the inverse matrix U − dc isgiven by ( U − dc ) [ e ; a, a ′′ ] { a a ′ a ′′ } = M ( d, e, c ) N ( e, a, a ′′ ) N ( c, a, a ′ ) N ( d, a ′ , a ′′ ) (cid:26) a ′′ , a, ec, d, a ′ (cid:27) , (4.28)as can be checked from the orthogonality relation (2.45). A natural choice for M ( x, y, z )is M ( x, y, z ) = N ( x, y, z ) = q F Rac y (cid:20) z zx x (cid:21) = (cid:18) g ′ y g ′ z g ′ x (cid:19) , (4.29)which makes U dc an orthogonal matrix( U dc ) { a a ′ a ′′ } [ e ; ˜ a, ˜ a ′′ ] = ( U − dc ) [ e ; ˜ a, ˜ a ′′ ] { a a ′ a ′′ } . (4.30)This can be easily checked by substituting (4.29) into (4.27) and (4.28), obtaining( U dc ) { a a ′ a ′′ } [ e ; a, a ′′ ] = (cid:26) c, a, a ′ a ′′ , d, e (cid:27) , (4.31)and, using the tetrahedral symmetry of the Racah symbol( U − dc ) [ e ; a, a ′′ ] { a a ′ a ′′ } = (cid:26) a ′′ , a, ec, d, a ′ (cid:27) = (cid:26) c, a, a ′ a ′′ , d, e (cid:27) = ( U dc ) { a a ′ a ′′ } [ e ; a, a ′′ ] . (4.32)A comment on the appearance of the matrix structures in the above discussion. Wehave arranged the elements of the matrices in a particular way as in (4.16), (4.17) and(4.21) for illustrative purpose, but we don’t have to necessarily adhere to this orderingof rows and columns. Indeed, from (4.27) we see that the mixing only occurs when30 a, b ) = (˜ a, ˜ b ) and ( a ′′ , b ′′ ) = (˜ a ′′ , ˜ b ′′ ), and with a suitable ordering of the columns and therows we can bring U dc into a block-diagonal form. The net mixing is therefore given by X da ′ b ′ ia ′′ b ′′ X cabia ′ b ′ = X e, f U a ′ e (cid:20) d ca ′′ a (cid:21) X eabia ′′ b ′′ δ ef U b ′ f (cid:20) d cb ′′ b (cid:21) , (4.33)where U a ′ e (cid:20) d ca ′′ a (cid:21) = U { a a ′ a ′′ } [ e ; a a ′′ ] dc = (cid:26) c, d, ea ′′ , a, a ′ (cid:27) . (4.34)In section 4, we will explicitly work out this block-diagonalization in the example of theIsing model CFT. Imagine a disk correlator with a number of bulk operator insertions. Placing a topologicaldefect parallel to the boundary and sufficiently close to it, so that there are no bulk oper-ators between the defect and the boundary, one can smoothly deform the defect so that itfuses onto the boundary without affecting any correlator. From the bulk perspective, aswe reviewed in Section 2, these correlators can be viewed as overlaps of the new boundarystate D || B ii with the vacuum excited by the vertex operators. Already by consideringdisk amplitudes without operator insertions, we find a number of interesting relations,illustrated in Figure 10 g b = X b ′ ∈ d × b F (cid:20) d dd d (cid:21) g b ′ = X b ′ ∈ d × b g b ′ g ′ d , (4.35)from which it follows (assuming the existence of the identity boundary condition) that thenormalized g function of the defect is in fact the g function of the corresponding boundarycondition, normalized by the g -function of the identity boundary condition g ′ d ≡ F (cid:20) d dd d (cid:21) = g d g , (4.36)or, considering the “sunset” disk diagrams g b F b (cid:20) d b ′ d b ′ (cid:21) = g b ′ F b ′ (cid:20) d bd b (cid:21) , (4.37)whose actual numerical value depends on the chosen gauge for the F matrices. We remindthat consistency of defect network manipulations only imply that the involved F matricesobey the pentagon identity and therefore the gauge for the F matrices used for defectmanipulations is not fixed. This has to be constrasted with the F blocks matrices entering31 b d b d bb b b = S S d == 1 F b d bd b = 1 F b d bd b b ∈ d × b S S d Figure 10: Defects can be used to derive a relation between the g -functions of differentboundary conditions. Defect attached to the boundary can be shrank in two differentdirections, producing consistent answer thanks to the identities for the fusion matrix F .the boundary structure constants which, as we have reviewed in section 3.2, imply a veryspecific gauge choice once the conformal blocks are canonically normalized.Now, what happens by attaching a defect to a boundary, when there are boundaryoperators present? The conformal weight of such operators cannot change, so they mustbecome new boundary operators in the same Virasoro representation, but interpolatingbetween the new boundary conditions, and possibly modified by a new normalizationconstant.To understand what happens to the boundary operators it is convenient to proceed insteps. First, imagine to partially fuse a defect d on a boundary segment b . Such a fusionbrings in a nontrivial factor F b ′ (cid:20) d bd b (cid:21) , see Figure 11. d b = b ∈ d × b F b d bd b d bb b d Figure 11: Partial fusing of a defect onto a boundary.This factor can be assigned to the left and right junctions between the defect, theoriginal boundary and the new boundary and it is natural to distribute it evenly betweenthe two junctions. This implies that when a d defect fuses on a boundary a to give asuperposition of boundary conditions a ′ = d × a , the involved junction must be accom-panied by a factor of q F a ′ (cid:20) d ad a (cid:21) . This factor will be represented by a boldface dot at the32 a a aa d d d = X a ′ ∈ d × a r F a ′ h d ad a i = X a ′ ∈ d × a Figure 12: A defect fusing onto a boundary. Every produced fundamental boundarycondition is accompanied by a junction factor which is graphically represented as a thickdot. D d φ abi = = a ∈ d × ab ∈ d × b F b d bd b = a ∈ d × ab ∈ d × b F a d ad a d dd a bφ i a a bφ i ba a bφ i b Figure 13: The geometric description of a defect action on a boundary field. Notice the √ F factors at the junctions.junction, see Figure 12. The action of a defect on an open string state is explicitly definedin Figure 13Once the above geometrical defintion is given, defect distributivity D d (cid:0) φ abi ( x ) φ bcj ( y ) (cid:1) = (cid:0) D d φ abi ( x ) (cid:1) (cid:0) D d φ bcj ( y ) (cid:1) , (4.38)follows rather naturally. Since the defect can be partially fused onto the boundary asin Figure 14 (using the rule in Figure 11), the only issue one has to take care of is thenormalization factor F b ′ (cid:20) d bd b (cid:21) which is accounted by the non-trivial normalization of thejunctions. It remains to compute the explicit X coefficients of the defect action D d φ abi = X a ′ ∈ d × a X b ′ ∈ d × b X dabia ′ b ′ φ a ′ b ′ i . (4.39)33 a a bφ i dd ba φ i b c cb cφ j cφ j = b ∈ d × b F b d bd b a d a a bφ i d b cb cφ j = b ∈ d × b Figure 14: Defect distributivity. In the second line the factor F b ′ (cid:20) d bd b (cid:21) has been absorbedinto the two c -number insertions at the junction points denoted by thick dots.In order to make use of the needed defect network manipulations, we uplift the boundaryconditions and the boundary insertions into a defect network with a line carrying the i representation and ending on a chiral defect-ending field, placed at the boundary withidentity boundary conditions. This is explained in detail in appendix B. In our settingthis move is essentially equivalent to a corresponding three-dimensional manipulation inthe topological field theory description of defects in RCFT [57], see in particular [70], butit has its own two-dimensional description given in B. After this topological move, thedefect coefficient X can be easily computed as in Figure 15. The α coefficients explicitlydepend on the chosen normalization for boundary fields (with the convention that n abi =1for the canonical choice (3.23) )and on the chosen gauge for defect networks, see (B.7),giving in total X dabia ′ b ′ = n abi n a ′ b ′ i s γ ( i, a ′ , b ′ ) γ ( i, a, b ) q F a ′ (cid:20) d ad a (cid:21) F di (cid:20) a ba ′ b ′ (cid:21) F i (cid:20) a ba b (cid:21) q F b ′ (cid:20) d bd b (cid:21) = n abi n a ′ b ′ i q F Rac1 a ′ (cid:20) d ad a (cid:21) F Rac di (cid:20) a ba ′ b ′ (cid:21) F Rac1 i (cid:20) a ba b (cid:21) q F Rac1 b ′ (cid:20) d bd b (cid:21) = n abi n a ′ b ′ i ( g ′ a g ′ b g ′ a ′ g ′ b ′ ) (cid:20) b ′ , a ′ , ia, b, d (cid:21) . (4.40)Notice in particular that the gauge dependent factors in the α ’s (B.7) conspire togetherwith the gauge dependence of the defect manipulation in Figure 15 to give an overallgauge invariant result which only depends on the normalization choice for the boundaryfields, as it should: acting a defect on a boundary field doesn’t depend on the defect’sgauge. 34 a bφ i d b D d φ abi a b = = α abi a a bφ i d b = α abi F a d ad a a a bφ i d b F b d bd b = α abi F a d ad a F di a ba b F i a ba b F b d bd b a φ i b ii = α abi α a bi F a d ad a F di a ba b F i a ba b F b d bd b a φ i bX dabia b φ a bi Figure 15: Defect network manipulations determining the defect coefficients X dabia ′ b ′ . Noticethat the boundary insertion i is traded for a defect ending on the boundary. Along sucha defect an ab -bubble is collapsed, after F -crossing on the original defect line d . Thetwo junctions at which the d defect joins the boundary corresponds to square roots of F matrix elements. 35 .2.2 Defect fusion from network manipulations Let us now see how we can use similar manipulations to reduce the composition of twodefects c and, subsequently, d to a direct sum of defects e , in the fusion of c and d . Westart with a general boundary field Φ and act the defects on it D d D c Φ = X a,b X i X a ′ ,b ′ X a ′′ ,b ′′ (cid:16) D d (cid:0) D c φ abi (cid:1) a ′ b ′ (cid:17) a ′′ b ′′ = X { a,a ′ ,a ′′ } X { b,b ′ ,b ′′ } (cid:0) D d D c Φ (cid:1) { a,a ′ ,a ′′ } , { b,b ′ ,b ′′ } . (4.41)Then we can perform the manipulations shown in Figure 16 D d D c φ abi a , a , a b , b , b = { }{} a a bφ i d ba bca bφ i d a bca ba be ( U dc ) aa a [ e ; aa ] U Tdc [ e ; bb ] bb bf ∈ d × c D f φ ( ab ) i [ e ; aa ][ e ; bb ] }{{ } φ i F b e c db b F b c bc b F b d bd b = e ∈ c × d F a d ad a F a c ac a F e d cd c F a e ae a F a e d ca a F b e c db b F b c bc b F b d bd b F e d cd c F b e be b = e ∈ c × d F a d ad a F a c ac a F a e d ca a e Figure 16: Defect network manipulations determining the fusion rules of open stringdefects. ( U dc ) { aa ′ a ′′ } [ e ; a,a ′′ ] = vuuut F a ′′ (cid:20) d a ′ d a ′ (cid:21) F a ′ (cid:20) c ac a (cid:21) F a ′′ (cid:20) e ae a (cid:21) F e (cid:20) d cd c (cid:21) F a ′ e (cid:20) d ca ′′ a (cid:21) = (cid:26) c, d, ea ′′ , a, a ′ (cid:27) , (4.42)36hich coincides with (4.31) Notice that in the geometric approach there naturally appearsthe transpose of the U matrix on the right D d D c Φ = U dc M e ∈ d × c D e Φ ! U Tdc . (4.43)Before closing this section, let us comment on one particularly surprising aspect of therelation (4.43). It is a bit reminiscent of some sort of generalized non-abelian projectiverepresentation of the closed string defect algebra D d D c = P e ∈ d × c D e and one mayask whether the corresponding 2-cocycle condition is satisfied. This is equivalent to thecondition of associativity of the defect algebra on the open string fields (cid:0) D e D d (cid:1) D c = D e (cid:0) D d D c (cid:1) . (4.44)To prove associativity, following the steps in Figure 17, one has to show that X f U a ′′ f (cid:20) e da ′′′ a ′ (cid:21) U b ′′ f (cid:20) d eb ′ b ′′′ (cid:21) U a ′ h (cid:20) f ca ′′′ a (cid:21) U b ′ h (cid:20) c fb b ′′′ (cid:21) = X g U a ′ g (cid:20) d ca ′′ a (cid:21) U b ′ g (cid:20) c db b ′′ (cid:21) U a ′′ h (cid:20) e ga ′′′ a (cid:21) U b ′′ h (cid:20) g eb b ′′′ (cid:21) . (4.45)To see that, let us start by writing the pentagon identity for the Racah symbols (2.41) ina more convenient form (cid:26) p, e, rd, c, q (cid:27)(cid:26) r, e, pa, b, t (cid:27) = X s (cid:26) c, b, sa, q, p (cid:27)(cid:26) d, s, ta, e, q (cid:27)(cid:26) d, t, sb, c, r (cid:27) . (4.46)Using this identity we can express the product of the two factors on the left hand sidedepending on the a -type labels (any of the labels a, a ′ , a ′′ and a ′′′ ) U a ′′ f (cid:20) e da ′′′ a ′ (cid:21) U a ′ h (cid:20) f ca ′′′ a (cid:21) = (cid:26) d, e, fa ′′′ , a ′ , a ′′ (cid:27)(cid:26) c, f, ha ′′′ , a, a ′ (cid:27) = (cid:26) a ′ , a ′′′ , fe, d, a ′′ (cid:27)(cid:26) f, a ′′′ , a ′ a, c, h (cid:27) = X g (cid:26) d, c, ga, a ′′ , a ′ (cid:27)(cid:26) e, g, ha, a ′′′ , a ′′ (cid:27)(cid:26) e, h, gc, d, f (cid:27) . (4.47)This last expression upon multiplication by the remaining b -type terms from the left handside of (4.45) can now easily by summed over label f using the relation (4.46) and one endsup precisely with the right hand side of (4.45). This concludes the proof of associativity. In this section we would like to study how topological defects act on OSFT solutions.As we have already stated, OSFT provides a new way to explore the possible conformal Instead of the usual representation on vectors up to a phase, this behaves as a representation onmatrices up to a similarity transformation. a bφ i d c D e D d D c φ abi a , a , a b , b , b = ea ba bba a bφ i ca bb a bφ i ea ba ba bφ i a bU a f e da a U b f d eb b U a g d ca a U b g c db b f ghU a h f ca a U b h c fb b { {} } U a h e ga a U b h g eb b Figure 17: Proof of associativity. Fusing together three defects attached to a boundaryin two different ways results in a consistency condition (4.45) for the U matrix. Theresulting condition follows from the pentagon identity for the Racah symbol (2.41).boundary conditions of a bulk CFT, by solving the equations of motion. Let us thenbriefly review how can we use OSFT to analyze BCFT’s with central charge c differentfrom 26 [10, 11]. The open string star algebra is factorized in the Hilbert space of the c = −
26 ghosts’ BCFT, with standard boundary conditions, and in the matter c = 2638CFT, whose boundary conditions can be generic. In our application of OSFT to BCFTwe will further assume that the matter BCFT is the tensor product of a BCFT c (whoseproperties we wish to study) times a compensating “spectator” BCFT − c BCFT total = BCFT c ⊗ BCFT − c ⊗ BCFT ghost . (5.1)In the total corresponding star algebra we restrict to the subalgebra where only descen-dants of the identity in the spectator sector are excited. Then we can search for classicalsolutions with the most general ansatz at ghost number oneΨ = X a,b X i X I,J,K a i ( ab ) IJK L c − I | φ abi i ⊗ L R − J | i ⊗ L gh − K c | i , (5.2)where L c − I = L − i n L − i n − · · · L − i for 0 ≤ i ≤ i ≤ · · · i n , and J, K are defined in the sameway. The label i runs over all Virasoro representations of BCFT c that are allowed by thepair of boundary conditions a , b (with, possibly, non-trivial multiplicities). In the case ofdiagonal minimal models we have that i ∈ a × b . If some fundamental boundary conditionappears multiple times (trivial multiplicities), then the coefficients a i ( ab ) IJK are matrices inthe degeneracy labels (Chan-Paton factors).A possible approach to search for new boundary conditions in the CFT c factor (5.1) isto solve the OSFT equation of motion with the ansatz (5.2). In particular the boundarystate corresponding to these new boundary conditions can be computed from OSFT gaugeinvariant observables [10].In the previous section we have defined the action of topological defects on genericboundary operators including those which change the boundary conditions. This thennaturally defines an action of a topological defect on open string fields D d Ψ = X a,b X i X I,J,K a i ( ab ) IJK L c − I (cid:0) D d | φ abi i (cid:1) ⊗ L R − J | i ⊗ L gh − K c | i , (5.3)because [ L matter n , D ] = 0 . (5.4)It follows that, noticing that [ b n , D ] = 0 and [ c n , D ] = 0,[ Q, D ] = 0 . (5.5)It is also not difficult to establish that D d ( φ ∗ χ ) = ( D d φ ) ∗ ( D d χ ) ∀ φ, χ. (5.6)Indeed, assuming the BFCT of interest to us is unitary and therefore its total Hilbert spaceis spanned by the direct sum of the Verma modules over the primaries, this condition just39ollows from the compatibility with the OPE (4.5) and the conservation laws of the starproduct [13, 14, 15]. Concretely, given two descendants string fields L − I φ abi (0) | i and L − J φ bcj (0) | i , we can express their star product schematically as L − I φ abi (0) | i ∗ L − J φ bcj (0) | i = X K V KIJ ( h i , h j ) L − K e P v k L − k φ abi ( x ) φ bcj ( y ) | i , (5.7)where the coefficients V KIJ , v k as well as the insertion points x and y are explicitly knownor calculable. Acting with D d , one can bring it through all the Virasoro generators, useformula (4.5) and use again formula (5.7) to reassemble the left hand side.Therefore open topological defects map solutions to solutions in OSFT Q Ψ + Ψ ∗ Ψ = 0 → Q ( D Ψ) + ( D Ψ) ∗ ( D Ψ) = 0 . (5.8)The main issue is now the physical interpretation of these new solutions. In this subsection we will derive the following key result: given a solution Ψ X → Y whichshifts the open string background from BCFT X to BCFT Y , we will show that the solution D Ψ X → Y shifts from BCFT DX to BCFT DY , where the subscript denotes the boundary con-ditions obtained by fusing the defect D on the X and Y boundary conditions respectively.In formulas D Ψ X → Y = Ψ DX → DY . (5.9)To do so we will evaluate the OSFT observables of D Ψ X → Y and show that they fully agreewith the observables of the r.h.s of (5.9). In particular we will show that the boundarystate of D Ψ X → Y is just the result of the defect action on the boundary state of Ψ X → Y || B D Ψ ii = D || B Ψ ii . (5.10)Notice that in OSFT there appears a natural interplay between the open string defectoperator D and its closed string counterpart D . S [ D Ψ]Before studying the full boundary state, it is instructive to look at the simpler case ofthe OSFT action. Carrying out the summation on
I, J and K in (5.2) we can write thestring field as Ψ = X a,b X i Ψ abi , (5.11)40here the label i and ab denote the Virasoro representation and boundary conditions ofBCFT c . In order to treat both the quadratic and cubic terms in the action (1.1) at thesame time, it is useful to prove the more general statementTr[( D d Ψ ) ∗ · · · ∗ ( D d Ψ n )] = g d g Tr[Ψ ∗ · · · ∗ Ψ n ] , (5.12)where Tr is the usual Witten integral, combined with trace over Chan-Paton factors. Bydefect distributivity ( D d Ψ ) ∗ · · · ∗ ( D d Ψ n ) = D d (Ψ ∗ · · · ∗ Ψ n ) , (5.13)it is enough to show that Tr[ D d Ψ] = g d g Tr[Ψ] . (5.14)Using the results of previous sub-sections the l.h.s. can be easily evaluated asTr[ D d Ψ] = X a X i X a ′ ∈ d × a X daaia ′ a ′ Tr[Ψ { a,a ′ } , { a,a ′ } i ]= X a X i δ i X a ′ ∈ d × a Tr[Ψ { a,a ′ } , { a,a ′ } ] , (5.15)since only the boundary Virasoro representation i = 1 can contribute to the Wittenintegral. Moreover, on general grounds, X daa a ′ a ′ = 1 since a defect always maps the identityto the identity, as can be easily checked in the explicit example (4.8). To continue wesimply notice that Tr[Ψ { a,a ′ } , { a,a ′ } ] = g a ′ g a Tr[Ψ aa ] , (5.16)since the two traces involve the same operator algebra and only differ in the normalizationof the vacuum amplitudes g a ≡ h i ( a )disk , and similarly for a ′ . We therefore haveTr[ D d Ψ] = X a Tr[Ψ aa ] X a ′ ∈ d × a g a ′ g a ! . (5.17)Using the Pasquier algebra for the g -functions, see equation (2.26) and (4.36) X a ′ ∈ d × a g a ′ = g d g a g , (5.18)concludes the proof of (5.12).In the next section we will see an alternative geometric approach making use of defect-network manipulations, Figure 19. 41hat we derived holds at the level of the Witten integral therefore, remembering thatthe defect trivially commutes with the BRST charge, it is immediate to see that for anyopen string field Ψ we have S OSFT [ D d Ψ] = g d g S OSFT [Ψ] . (5.19)Suppose now we have a solution Ψ X → Y which describes BCFT Y as a state in BCFT X .This means in particular we have a solution whose action is given by S OSFT [Ψ X → Y ] = 12 π ( g X − g Y ) . (5.20)It follows that the defect-acted solution will have an action given by S OSFT [ D d Ψ X → Y ] = g d g S OSFT [Ψ X → Y ] = 12 π (cid:18) g d g X g − g d g Y g (cid:19) . (5.21)The difference entering into the above equation is nothing but the difference in theidentity-coefficients ( g -functions) of the two boundary states obtained by acting the closedstring defect D d on the source and target boundary states || B X ii , || B Y ii , connected bythe solution Ψ X → Y D d || B X ii = D d ( g X | ii + · · · ) = g d g X g | ii + · · · (5.22) D d || B Y ii = D d ( g Y | ii + · · · ) = g d g Y g | ii + · · · . (5.23)This is the first non-trivial check that (5.9) indeed holds. Let us now see how the Ellwood invariant is affected by the defect action. To start with,it is useful to derive an identity involving the bulk-boundary structure constants ( a ) B ij , the g -functions and the open and closed defect coefficients. The required identity is obtainedby computing a disk amplitude with a spinless bulk field V j ( z, ¯ z ), a boundary field φ aai ( x )and a defect loop d encircling the bulk field. See Figure 18.The correlator can be computed by acting the closed string defect on the closed stringfield V j , or by partially attaching the defect to the boundary, producing an open stringdefect acting on the boundary field φ i ( x ). Calling, in generality, D di the defect coefficient,i.e. D = P i D di P i , where P i projects in the i -th Verma module in the bulk, we get theidentity D dj ( a ) B ij g a = X a ′ ∈ d × a X daaia ′ a ′ ( a ′ ) B ij g a ′ , (5.24)42 = φ i V j D dj b d φ i V j bφ i φ i V j V j = b ∈ d × b = b ∈ d × b X dbbib b b b d = b ∈ d × b V j d b φ i b Figure 18: Two equivalent ways of computing a bulk-boundary correlator in presence ofa closed string defect d .which in the case of diagonal minimal models reads S dj S j ( a ) B ij g a = X a ′ ∈ d × a F Rac aa ′ (cid:20) a di a ′ (cid:21) ( a ′ ) B ij g a ′ . (5.25)Thanks to this relation it is easy to algebraically show thatTr V j [ D d Ψ] = X a X i X a ′ ∈ d × a X daaia ′ a ′ Tr V j [Ψ { a,a ′ } , { a,a ′ } i ]= X a X i X a ′ ∈ d × a F Rac aa ′ (cid:20) a di a ′ (cid:21) ( a ′ ) B ij g a ′ ( a ) B ij g a Tr V j [Ψ { a,a ′ } , { a,a ′ } i ]= X a X i S dj S j Tr V j [Ψ aai ] = X a,α X i ∈ a × a Tr D d V j [Ψ aai ]= Tr D d V j [Ψ] , (5.26)where in the second line we have used that the two involved open/closed couplings (carry-ing the same Virasoro labels but different boundary conditions) only differ by the overall Boundary fields are here canonically normalized (3.23). In the special case where the boundary fieldcarries the identity representation it coincides with (4.42) of [83]. In this case we have ( a ) B j g a = S aj √ S j and analogously for the new boundary conditions a ′ , and so the relation reduces to the Verlinde formula S dj S j S aj = X a ′ ∈ d × a S a ′ j . r V D d Ψ = a a ∈ d × a f (1) ◦ Ψ aa V d aa = a a ∈ d × a F a d ad a f (1) ◦ Ψ aa V d af (1) ◦ Ψ aa V d ap = a f (1) ◦ Ψ aa V d a = Tr D d V [Ψ]= a , p a ∈ d × ap ∈ d × d F a d ad a F a p a ad d Figure 19: Defect networks manipulations for the Ellwood invariant of a defect-acted openstring-field. The dashed line corresponds to the identification of the left and right partof the open string, via the identity conformal map f (1) ( z ) = (cid:0) iz − iz (cid:1) . Notice that thanksto the junctions normalizations and the consequent F -matrix orthogonality relation, onlythe identity defect p = 1 stretches between the boundary and the defect d , making d agenuine closed string defect. When V = 1 this also gives a geometric proof of (5.14).bulk-boundary structure-constant B and the g -functions, due to the identical operatoralgebra involved in their calculation. Again, this can be shown geometrically manipulat-ing defect networks, as shown in Figure 19, and the result is correctly independent on thegauge used for defect manipulations. Having obtained in generality how an Ellwood invariant is affected by the action of anopen string defect, let us see how the OSFT boundary state constructed in [10] (KMSfrom now on) behaves under the defect action. The KMS approach gives a simple recipeto directly compute the coefficients of the matter Ishibashi states of the boundary stateassociated to a given solution Ψ in terms of a minimal generalization of the Ellwoodinvariant. In the setting of this paper, where we are assuming that the matter BCFTis the tensor product of a diagonal rational BCFT of central charge c and a “spectator”sector of central charge (26 − c ), the required generalization of the Ellwood invariantis simply achieved by assuming that the spectator sector contains a free boson (call it Y ) with Dirichlet boundary conditions, see [11, 10] for further details. Then the KMS44onstruction is usefully summarized as || B Ψ ii KMS = X j n j Ψ | V j ii ! ( c ) ⊗ || B ii (26 − c ) ⊗ || B ii ghost (5.27) n j Ψ = (2 πi ) Tr ˜ V j [Ψ − Ψ tv ] (5.28)˜ V j ( z, ¯ z ) = c ¯ c V j e i √ − h j Y ( z, ¯ z ) . (5.29)Since we have shown that Tr V [ D Ψ] = Tr DV [Ψ] for any string field, the KMS constructionapplied to the solution D Ψ will obviously give || B D Ψ ii KMS = D || B Ψ ii KMS . (5.30)At last let’s also consider the other available construction of the boundary state in OSFT,given by Kiermaier, Okawa and Zwiebach (KOZ) [21]. KOZ geometrically constructa BRST-invariant ghost number three closed string state obeying the level matching b − = L − = 0. This closed string state is conjectured to be BRST equivalent to the BCFTboundary state and in fact the two coincide for many known analytic solutions. Themain ingredient of this construction is a choice of half-propagator strip in the backgroundof a classical solution Ψ, whose left edge and right edge are glued together, to form anannulus-like surface, which is used to build the closed string state || B Ψ ii KOZ = e π s ( L +¯ L ) I s Pexp (cid:20) − Z s dt [ L R ( t ) + {B R ( t ) , Ψ } ] (cid:21) . (5.31)The various objects entering the above definition are defined in [21], but for us it issufficient to recall that the quantity Pexp[ ... ] represents a half-propagator strip of lenght s , in the background of the classical solution Ψ, and that the symbol H s identifies the initialleft edge of the strip with the final right edge. The internal boundary of this annulus-likesurface (corresponding to the propagation of the open string midpoint) defines a closedstring state, the KOZ boundary state.If, instead of the original classical solution Ψ on BCFT X we replace the defect-actedsolution D Ψ on BCFT DX , then by defect distributivity on the star product (and theobvious commutativity with the b -ghost insertions) we will recover a closed string defectextending along the midpoint line, which is nothing but a closed-string defect operatoracting on the original KOZ boundary state, see Figure 20 || B D Ψ ii KOZ = e π s ( L +¯ L ) I s Pexp (cid:20) − Z s dt [ L R ( t ) + {B R ( t ) , D Ψ } ] (cid:21) = D || B Ψ ii KOZ . (5.32) A simple consequence of this is that acting with a topological defect on a tachyon vacuum solution,the new solution solution will still be the tachyon vacuum (the boundary state will still vanish), althoughexpressed with the degrees of freedom of the new BCFT D ( X ). ∞ n =0 B D Ψ KOZ ∼ ∞ n =0 B R , Ψ (1) B R , Ψ (2) B R , Ψ ( n ) B R , Ψ ( n ) closedstringpatch closedstringpatch B R , Ψ (2) B R , Ψ (1) d × X d × X d × X dd d d = D B
Ψ KOZ
X XX
Figure 20: Pictorial representation of the KOZ boundary state for a defect-acted solution(the boundary integrals in the “Pexp” operation, as well as other details, are not shown, asthey are not important for our argument). Thanks to the defect distributivity, the variousopen string defects result in a single defect-loop encircling the closed string coordinatepatch.
In this section we would like to illustrate our general findings on the concrete exampleof open string field theory for the Ising model [11]. This is the simplest unitary diagonalminimal model and has c = . It has three irreducible Virasoro representations, denotedas , ε and σ with well-known fusion rules ε × ε = ε × σ = σ (6.1) σ × σ = + ε. The bulk fields are all spinless and also labeled by , ε and σ . There are three possiblefundamental boundary conditions also denoted as , ε and σ which describe the fixed ( ± )and free boundary conditions for the spins in the underlying lattice model. We will referto the conformal boundary conditions of the Ising model as the Ising “D-branes”.46 .1 Defect action on Ising boundary fields In the Ising model there are three (fundamental) topological defects. They can act onbulk field via closed-string defect operators, with the following composition rules D ε = D (6.2) D σ D ε = D ε D σ = D σ (6.3) D σ = D + D ε , (6.4)which realize the fusion rules (6.1).Now we would like to construct the open string defect operators and study theircomposition rules according to section 4.The general boundary field on a system of N -branes, N ε ε -branes, and N σ σ -braneshas the form Ψ = ε σ L ( ) P ( ε ) ε Q ( σ ) σε ¯ P ( ε ) ε M ( εε ) R ( εσ ) σσ ¯ Q ( σ ) σ ¯ R ( σε ) σ N ( σσ ) + N ( σσ ) ε , (6.5)where the ( a, b ) component is a N a × N b matrix with respect to the Chan-Paton indices. Wehave chosen this presentation since topological defect operators are blind to Chan-Patonfactors. The upper indices inside the parenthesis represent the left and the right boundaryconditions, which are also indicated outside the matrix for later convenience,and the lowerindex is the Virasoro label. Each entry of (6.5) is a generic matrix-valued state in theVerma module indicated by the corresponding subscript, and allowed by the boundaryconditions. This expression can also represent an open string field of the form (5.2, 5.11).As a useful example, let us study the fusion of two σ defect operators, D σ , on theboundary field (or open string field) (6.5). From (4.4) and the concrete value of the defectcoefficients X for the Ising model BCFT (see appendix C), applying the D σ on the openstring field (6.5) results in a matrix with a larger size D σ Ψ = { σ } { εσ } { σ } { σε }{ σ } L ( σσ ) √ P ( σσ ) ε / Q ( σ ) σ / Q ( σε ) σ { εσ } √ ¯ P ( σσ ) ε M ( σσ ) / R ( σ ) σ − / R ( σε ) σ { σ } / ¯ Q ( σ ) σ / ¯ R ( σ ) σ N ( ) √ N ( ε ) ε { σε } / ¯ Q ( εσ ) σ − / ¯ R ( εσ ) σ √ N ( ε ) ε N ( εε ) . (6.6)47ere indices { ab } ( a, b = , ε, σ ) outside the matrix keep track of changes of the leftand the right boundary conditions; for example, the (1,1) component of the matrix hasthe indices { σ }{ σ } , and the boundary condition of the corresponding entry has beenchanged as D σ : L ( ) L ( σσ ) . (6.7)the (1,2) component of the matrix has the indices { σ }{ εσ } , and correspondingly, D σ : P ( ε ) ε X σ εεσσ P ( σσ ) ε , (6.8)where X σ εεσσ = √ , and so on. Note that the action of D σ increases the number of branesin general, because it changes the σ -brane into a -brane and an ε -brane. Now the systemhas N σ -branes, N σ ε -branes and ( N + N ε ) σ -branes.Applying D σ again we obtain D σ Ψ = { σ } { σε } { εσ } { εσε } { σ σ } { σεσ }{ σ } L ( ) P ( ε ) ε √ Q ( σ ) σ √ Q ( σ ) σ { σε } L ( εε ) P ( ε ) ε Q ( εσ ) σ − Q ( εσ ) σ { εσ } P ( ε ) ε M ( ) R ( σ ) σ − R ( σ ) σ { εσε } ¯ P ( ε ) ε M ( εε ) √ R ( εσ ) σ √ R ( εσ ) σ { σ σ } √ ¯ Q ( σ ) σ ¯ Q ( σε ) σ ¯ R ( σ ) σ √ ¯ R ( σε ) σ N ( σσ ) N ( σσ ) ε { σεσ } √ ¯ Q ( σ ) σ − ¯ Q ( σε ) σ − ¯ R ( σ ) σ √ ¯ R ( σε ) σ N ( σσ ) ε N ( σσ ) , (6.9)which has an equal number of rows and columns given by 2( N + N ε + N σ ).On the other hand, ( D ⊕ D ε ) Ψ is given by [ ; ] [ ; εε ] [ ; σσ ] [ ε ; ε ] [ ε ; ε ] [ ε ; σσ ][ ; ] L ( ) P ( ε ) ε Q ( σ ) σ [ ; εε ] ¯ P ( ε ) ε M ( εε ) R ( εσ ) σ [ ; σσ ] ¯ Q ( σ ) σ ¯ R ( σε ) σ N ( σσ ) + N ( σσ ) ε [ ε ; ε ] L ( εε ) P ( ε ) ε √ Q ( εσ ) σ [ ε ; ε ] P ( ε ) ε M ( ) √ R ( σ ) σ [ ε ; σσ ] √ ¯ Q ( σε ) σ √ R ( σ ) σ N ( σσ ) − N ( σσ ) ε . (6.10)This is clearly not equal to (6.9). While the closed defect operators D d obey the defectalgebra (6.4) which is strictly isomorphic to the Verlinde fusion algebra, this is not thecase for the open string defect operators D d . As discussed in section 4, we need to take48nto account the similarity transformation (4.31) and (4.32) in order to connect (6.10)with (6.9). In this case, the similarity transformation is given by U σσ = [ ; , ] [ ; ε,ε ] [ ; σ,σ ] [ ε ; ,ε ] [ ε ; ε, ] [ ε ; σ,σ ] { σ } { σε } { εσ } { εσε } { σ σ } √ √ { σεσ } √ − √ (6.11)= [ ; ] [ ε ; ε ] [ ε ; ε ] [ ; εε ] [ ; σσ ] [ ε ; σσ ] { σ } { σε } { εσ } { εσε } { σ σ } √ √ { σεσ } √ − √ = (cid:0) U − σσ (cid:1) T . (6.12)The reader can easily check that D σ Ψ = U σσ [( D ⊕ D ε ) Ψ] U − σσ . (6.13)Similarly, to investigate the relation (6.3), we calculate D σ D ε Ψ D σ D ε Ψ = { εσ } { ε σ } { σσ } { σσε }{ εσ } L ( σσ ) √ P ( σσ ) ε / Q ( σ ) σ − / Q ( σε ) σ { ε σ } √ ¯ P ( σσ ) ε M ( σσ ) / R ( σ ) σ / R ( σε ) σ { σσ } / ¯ Q ( σ ) σ / ¯ R ( σ ) σ N ( ) −√ N ( ε ) ε { σσε } − / ¯ Q ( εσ ) σ / ¯ R ( εσ ) σ −√ N ( ε ) ε N ( εε ) (6.14)and D ε D σ Ψ D ε D σ Ψ = { σσ } { εσσ } { σε } { σ ε }{ σσ } L ( σσ ) − √ P ( σσ ) ε / Q ( σ ) σ / Q ( σε ) σ { εσσ } − √ ¯ P ( σσ ) ε M ( σσ ) − / R ( σ ) σ / R ( σε ) σ { σε } / ¯ Q ( σ ) σ − / ¯ R ( σ ) σ N ( ) √ N ( ε ) ε { σ ε } / ¯ Q ( εσ ) σ / ¯ R ( εσ ) σ √ N ( ε ) ε N ( εε ) . (6.15)49omparing these expressions with (6.6), we find U σε = [ σ ; σ ] [ σ ; εσ ] [ σ ; σ ] [ σ ; σε ] { εσ } { ε σ } { σσ } { σσε } − = ( U σε ) − , (6.16) U εσ = [ σ ; σ ] [ σ ; εσ ] [ σ ; σ ] [ σ ; σε ] { εσ } { ε σ } − { σε } { σ ε } = ( U εσ ) − . (6.17) Now let us discuss the action of the defect operators on the classical solutions of OSFT.For illustration, let us focus on the σ -brane of the Ising model and let us consider thecorresponding OSFT. At lowest nontrivial level the string field takes the form Ψ = tc | i + ac | ε i for which the potential is [11] V ( t, a ) = − t − a + 27 √ t + 2716 ta . (6.18)From here one can already see the four critical points given by the perturbative as wellas tachyon vacuum, and further two solutions related by a Z -symmetry describing theother two fundamental boundary conditions. Going to higher levels, the string field will keep the formΨ σ → = ψ σσ + ψ σσε , (6.19)where ψ σσ and ψ σσε will stand for the two components in the identity and ε Verma modulesrespectively. Correspondingly, the equations of motion split into two independent sets: Qψ σσ + ψ σσ ∗ ψ σσ + ψ σσε ∗ ψ σσε = 0 (6.20) Qψ σσε + ψ σσ ∗ ψ σσε + ψ σσε ∗ ψ σσ = 0 . (6.21) At higher levels one may find other solutions, see the discussion in [11]. ψ σσ and ψ σσε obey the equations of motion as well: ψ σσ − ψ σσε , (6.22)and (cid:18) ψ σσ ± ψ σσε ± ψ σσε ψ σσ (cid:19) . (6.23)Intuitively, following our discussion in section 4, it is clear that first of these solutionsshould be the result of applying the D ε defect to (6.19). Actually, we obtain X εσσ σσ = − X εσσεσσ = 1 from (4.12), then D ε Ψ σ → results in (6.22).The remaining two solutions in (6.23) also fit in with our discussion: from the analysisin section 5, we see that the solution D σ Ψ σ → describes a σ -brane in the theory around asystem with a -brane and an ε -brane. We then denote it by Ψ + ε → σ ,Ψ + ε → σ ≡ D σ Ψ σ → . (6.24)From (4.12), we see that D σ ψ σσ = (cid:18) X σσσ ψ X σσσ εε ψ εε (cid:19) = (cid:18) ψ ψ ǫǫ (cid:19) , (6.25) D σ ψ σσε = (cid:18) X σσσε ε ψ ǫε X σσσεε ψ ε ε (cid:19) = (cid:18) ψ εε ψ ε ε (cid:19) . (6.26)and Ψ + ε → σ is given by Ψ + ε → σ = (cid:18) ψ ψ εε ψ ε ε ψ εε (cid:19) . (6.27)Further acting D σ or D ε D σ , we obtain the two solutions in (6.23) D σ Ψ + ε → σ = (cid:18) ψ σσ + ψ σσε + ψ σσε ψ σσ (cid:19) , (6.28) D ε D σ Ψ + ε → σ = (cid:18) ψ σσ − ψ σσε − ψ σσε ψ σσ (cid:19) . (6.29)Note that these two solutions are related by a similarity transformation, D ǫ D σ Ψ + ε → σ = U εσ ( D σ Ψ + ε → σ ) U − εσ , (6.30)where U εσ = U { σσ } [ σ ; σ ] εσ U { εσσ } [ σ ; εσ ] εσ ! = (cid:18) − (cid:19) . (6.31)51he matrix which is used here to connect these two solutions is nothing but the U matrixdiscussed in section 4.1.2 and which is a part of (6.17) we derived in the first half ofthis section. Furthermore,we can consider the components of U as proportional to theidentity string field, and then we can regard (6.30) as a gauge transformation in OSFTΛ( Q B + Ψ)Λ − .The discussion here can be generalized and we can prove the following statement:consider a Verlinde fusion algebra a × b = P e N abe e and let Ψ be a general classicalsolution of OSFT. Then two classical solutions Ψ ′ = D a D b Ψ and Ψ ′′ = ⊕ e N abe D e Ψ arerelated by a gauge transformation with a constant gauge parameter, which is given bythe U matrix, Ψ ′ = U ab ( Q B + Ψ ′′ ) U − ab . (6.32)This is true in general because the components of U are proportional to the identity stringfield and thus vanishes upon action of the BRST charge. Note that there is no problemin considering the components of U to be proportional to the identity string field, for iftwo final boundary conditions are not the same ( a ′′ = ˜ a ′′ ), the corresponding componentsbecome zero, as showed in (4.27). This is consistent with the fact that the identity Vermamodule cannot connect different boundary conditions.Similarly, we can produce different classical solutions by acting defect operators onΨ σ → . We here summarize solutions obtained by acting combination of D σ ’s and D ε ’swith the number of D σ ’s less than three:Ψ σ → D ε (cid:15) (cid:15) D σ / / Ψ + ε → σ D σ / / D ε (cid:8) (cid:8) Ψ σ → + ε D σ / / D ε (cid:15) (cid:15) . . . Ψ σ → ε D ε O O D σ / / Ψ ′ + ε → σ D ε T T D σ / / Ψ σ → ε + D ε O O D σ / / . . . (6.33)where Ψ σ → ε = ψ σσ − ψ σσε , (6.34)Ψ ′ + ε → σ = (cid:18) ψ − ψ εε − ψ ε ε ψ εε (cid:19) , (6.35)besides Ψ σ → + ε ≡ D σ Ψ + ε → σ and Ψ σ → ε + ≡ D ε D σ Ψ + ε → σ in (6.28) and (6.29), respec-tively. The relation between Ψ + ε → σ and Ψ ′ + ε → σ is also given by a U matrix, D σ D ε Ψ σ → ε = U σε ( D σ Ψ σ → ε ) U − σε , (6.36)52hat is, Ψ + ε → σ = (cid:18) − (cid:19) Ψ ′ + ε → σ (cid:18) − (cid:19) . (6.37)Starting from Ψ → σ , another series of classical solutions can be obtained. Since thestate Ψ → σ satisfy the boundary condition , the whole solution will be solely composedby the identity Verma module Ψ → σ = e ψ . (6.38)Applying open string defect operators D ε and D σ again and again, we obtain the followingsequence : Ψ → σ D ε (cid:15) (cid:15) D σ / / Ψ σ → + ε σ / / D ε (cid:8) (cid:8) Ψ + ε → σ D σ / / D ε (cid:8) (cid:8) Ψ σ → · +2 ε D ε (cid:8) (cid:8) D σ / / . . . Ψ ε → σ D ε O O D σ / / Ψ ′ σ → + ε σ / / D ε (cid:8) (cid:8) Ψ ′ + ε → σ D σ / / D ε (cid:8) (cid:8) Ψ ′ σ → · +2 ε D ε (cid:8) (cid:8) D σ / / . . . (6.39)where Ψ ε → σ = e ψ εε , (6.40)Ψ σ → + ε = Ψ ′ σ → + ε = e ψ σσ , (6.41)Ψ + ε → σ = Ψ ′ + ε → σ = e ψ e ψ εε ! , (6.42)Ψ σ → + ε = Ψ ′ σ → + ε = e ψ σσ e ψ σσ ! . (6.43)It is interesting that we constructed these solutions without using ε Verma module, whichis a possible excitation on a σ -brane, or on a ( + ε )-brane system.Following the same line of reasoning, we can generally prove that for a × b = c with a , b and c general conformal boundary conditions, Ψ a → c can be constructed strictly withinthe identity Verma module, for Ψ a → c = D a Ψ → b . This is a rather nontrivial fact derivedby considering defect action on the classical solutions. Here we have distinguished Ψ σ → + ǫ and Ψ ′ σ → + ǫ because D σ (Ψ → σ − Ψ ε → σ ) = 0 as topologicaldefects do not kill boundary fields [60]. Conclusions
In this paper, starting from CFT topological defects, we have build new operators actingon the open string star algebra, and we have used them to generate new solutions in OSFT.To this end we have carefully studied the action of topological defects on boundaries andboundary fields, extending to the full boundary operator algebra the results of Grahamand Watts [60]. We have also provided a clear geometric construction of open topologicaldefects using defect networks. Our geometric picture is two-dimensional and it doesn’trequire the 3D topological description of [57]. The action of defects on boundary fieldsturns out to be much more involved than the action on bulk fields and the compositionof open defect operators follows the fusion rules only up to a similarity transformationin Chan-Paton space. The pentagon identity is crucial for the consistency of open defectoperators and their fusion. The concrete results we presented are valid for diagonalminimal models but the idea of using topological defects to generate new solutions isclearly very general and applies to any string background.We have payed special attention to the issue of gauge freedom in the definition ofthe F -matrices, which arise from the symmetries of the pentagon identity. We do notuse the auxiliary concept of chiral vertex operators and we take conformal blocks to becanonically normalized as in [81] which fixes the “gauge” of the F -matrices. However, inthe computation we have performed, we did not have to specify any gauge choice for the F -matrices involved in defect network manipulations: our results concerning the actionof topological defects on boundary fields and their composition are explicitly independentof the gauge chosen for defect networks. Combining defects and disorder operators maygive further constraints relating the defect F -matrices with the F -matrices coming fromthe transformation of the conformal blocks.A simple generalization of our results should be given by the study of open topologicaldefects in RCFT’s with charge conjugation modular invariant, the original setting of [60].In this case the representations won’t be self-conjugate and one will have to pay attentionto the orientation inside defect networks. More work would be needed to address RCFTwith non charge conjugation partition function (the simplest example being the Pott’smodel).It would be interesting to explore open topological defects in non rational CFT’s(although a complete classification of them is not available). For example, a typicalproperties of defects in non rational CFT’s is that they can posses a moduli space as ithappens for boundary conditions. It would be also useful to extend our work to conformaldefects [67] to get more general solution generating techniques in OSFT.A more interesting (and difficult) question, and actually one of the motivation for the54resent research, is whether defects can be used as part of the building blocks for con-structing OSFT solutions. This is certainly a new arena where interesting new structuresmay appear and which could overcome some limitations of the known solutions such as[38].Open topological defects are very natural objects inside the open string star-algebrawhich are relevant to the description of the open string landscape. We hope that ourresearch is a useful step towards a better understanding of the space of solutions of OSFT. Acknowledgments
We would like to thank C. Bachas, Z. Bajnok, I. Brunner, T. Erler, S. Fredenhagen, J.Fuchs, D. Gaiotto, M. Kudrna, M. Rapˇc´ak, I. Runkel, Y. Satoh, C. Schweigert, A. Senand R. Tateo for useful discussions at various stages of this work. TK, CM and TM thankthe Academy of Science of Czech Republic for kind hospitality and support during vari-ous stages of this work, while MS thanks Torino University for the same. Both CM andMS thank the organizers of the 2015 workshop “Gauge theories, supergravity and super-strings” in Benasque, for providing a stimulating environment during part of this research.The research of CM is funded by a
Rita Levi Montalcini grant from the Italian MIUR.The research of MS is supported by the Czech funding agency grant P201/12/G028.
A Comments on Moore and Seiberg “gauge symme-try”
An important property of the pentagon equation is its huge symmetry called a bit mis-leadingly a gauge symmetry F pq (cid:20) a bc d (cid:21) → Λ( a, b, q )Λ( c, q, d )Λ( c, a, p )Λ( p, b, d ) F pq (cid:20) a bc d (cid:21) , (A.1)where Λ( i, j, k ) is an arbitrary function of an admissible triplet. One can redefine Λ( i, j, k )by multiplying it with φ ( i ) φ ( j ) φ ( k ) to get F pq (cid:20) a bc d (cid:21) → φ ( p ) φ ( a ) φ ( a ) φ ( q ) φ ( p ) φ ( q ) Λ( a, b, q )Λ( c, q, d )Λ( c, a, p )Λ( p, b, d ) F pq (cid:20) a bc d (cid:21) , (A.2)so that the symmetry looks even bigger. Imposing that our normalization condition F cb (cid:20) bc d (cid:21) = 1 is preserved by the gauge transformation, implies Λ(1 , b, b ) = Λ( c, , c ) , ∀ b, c ,55o that both expressions are label-independent, and hence we can freely normalize themto 1, i.e. Λ(1 , b, b ) = Λ( c, , c ) = Λ(1 , ,
1) = 1 . (A.3)Under this condition g ′ a = (cid:16) F (cid:20) a aa a (cid:21) (cid:17) − are gauge invariant.Imposing that gauge transformation preserve the 180 ◦ invariance condition F pq (cid:20) a bc d (cid:21) = F pq (cid:20) d cb a (cid:21) implies that Λ must be cyclically invariant. The specular conditions F pq (cid:20) a bc d (cid:21) = F pq (cid:20) c da b (cid:21) and F pq (cid:20) a bc d (cid:21) = F pq (cid:20) b ad c (cid:21) would similarly imply that Λ is permutation invariant.An important question however, is to what extent is this freedom physical. Here wewish to stress, that while different physical quantities in BCFT might be related by sucha transformation, in general there is no gauge freedom when a concretely defined quantityis concerned.In particular the structure constants of boundary CFT are given by one concretegauge choice of F , which cannot be changed by a choice of normalization of the boundaryoperators. Changing their normalization as φ abi → ˜ φ abi = n abi φ abi , (A.4)the structure constants transform as well C ( abc ) pij → ˜ C ( abc ) pij = n abi n bcj n acp C ( abc ) pij , (A.5) C ( aca ) 1 pp → ˜ C ( aca ) 1 pp = n acp n cap n aa C ( aca ) 1 pp . (A.6)It is easy to check that the equation (3.15) is obeyed also for the transformed structureconstants, and that it is not required to transform Fn abi n bcj n acp n cdk n dal n cap n acp n cap = n bcj n cdk n bdq n dal n abi n dbq n bdq n dbq . (A.7)So the freedom in normalization of the boundary operators has no relation to the choiceof gauge for F blocks pq (cid:20) j ki l (cid:21) , which in fact is fixed in conformal field theory as we saw above.In general there are two special gauges for the solutions of the pentagon identity (plusthe identity condition and specular conditions): The blocks gauge and the Racah gauge .The blocks gauge has been discussed in section 3.2. The Racah gauge is defined as F Rac pq (cid:20) a bc d (cid:21) = (cid:26) b, a, qc, d, p (cid:27) . (A.8)56t has the special property F Rac11 (cid:20) a aa a (cid:21) F Rac1 a (cid:20) b cb c (cid:21) = 1 p g ′ a g ′ b g ′ c . (A.9)The Racah-gauge F -matrix is among other things important for expressing the boundarystructure constants in terms of the bulk structure constants, see formula (4.7). B Un-fusing defects from boundaries
In this appendix we will derive a useful manipulation which we have used in section 4.2.1to determine the defect coefficients X dabia ′ b ′ in the geometric approach. We will assume thatall fundamental boundary conditions in the game can be obtained by fusing a topologicaldefect on a particular reference boundary condition, denoted as . Consider a boundaryfield φ abi changing an a boundary to a b boundary. By our assumption this is topologicallyequivalent to a network of defects with a leg ending on a defect ending field, carrying the i Virasoro representation and sitting at the boundary, as in Figure 21. a bφ i a bφ i a bφ i a bφ i = α abi == i Figure 21: A boundary field in the i Virasoro representation can be traded for a defectnetwork ending on a defect ending field carrying the same Virasoro representation butsitting on a boundary with the boundary condition.The equality holds up to an unknown three-label coefficient α abi = α bai , symmetricin the boundary condition labels for parity reasons. By the triviality of the identityrepresentation we must have α aa = 1 . (B.1)This is consistent with the fact that the normalized g function of the defect is related tothe disk partition function as h i ( a ) = h i (1) F (cid:20) a aa a (cid:21) = h i (1) g ′ a = g a . (B.2)We can determine the coefficients α abi by computing a boundary three point function intwo ways, either by using the (given) boundary structure constants, or by manipulatingdefects after having performed the move in Figure 21, as represented in Figure 22.57 i jk ca = α abi α bcj α cak bi jkca Figure 22: Two equivalent ways of computing a boundary three-point function. The α coefficients are sensitive to both the normalization choice for boundary fields and the F -matrix gauge used in defect manipulations.After factoring out the universal dependence on the insertion points, we are left withthe equality C ( abc ) kij C ( aca ) kk g ′ a = G ijk α abi α bcj α cak F bk (cid:20) a ci j (cid:21) F k (cid:20) a ca c (cid:21) . (B.3)The new quantity G ijk is the non-trivial coefficient of the three point function involvingthe elementary defect network with boundary shown in Figure 23. i j k = G ijk f ( x i , x j , x k ) i j k Figure 23: A simple three-point function which depends on the gauge chosen for defectmanipulation. The function f ( x i , x j , x k ) is the part of the three-point-function which iscompletely fixed by conformal invariance.We will now see that consistency will relate G and the α ’s to the choice of normalizationof the boundary fields and the chosen gauge for the defect networks.Suppose we made a generic rescaling of the canonically normalized boundary fieldswith structure constants given by (3.23), φ abi → n abi φ abi so that58 ( abc ) kij = n abi n bcj n ack s p g ′ i g ′ j g ′ k θ ( i, j, k ) blocks (cid:26) c, a, ki, j, b (cid:27) , (B.4)Then rewriting the rhs of (B.3) by expressing the generic F -matrices in terms of theRacah symbols, using (2.36) and (2.40), and isolating the unknowns α and G to the rightwe get n abi n bcj n ack p γ ( i, j, k ) blocks s γ ( i, j, k ) γ ( i, a, b ) γ ( j, b, c ) γ ( k, a, c ) = G ijk α abi α bcj α cak , (B.5)where we have defined for convenience γ ( i, j, k ) ≡ θ ( i, j, k ) p g ′ i g ′ j g ′ k . (B.6)Notice that specifying this quantity corresponds to pick a gauge for the F -matrices usedfor defect manipulation. Racah gauge corresponds to γ ( i, j, k ) = 1, for an admissibletriplet. The generic solution to equation (B.5), enforcing α abi = α bai , is then given by α abi = β i n abi p γ ( i, a, b ) , (B.7) G ijk = 1 β i β j β k s γ ( i, j, k ) γ ( i, j, k ) blocks . (B.8)Notice the undetermined parameters β i . Their presence is easily explained: they are just anormalization choice for the defect ending field appearing in Figure 21. This undeterminednormalization always cancels in the defect manipulations we consider in this paper and itis consistent to set it to 1. C Ising data
In this appendix, we present explicit data for the Ising model to facilitate concrete calcu-lations.The standard F matrices (the one entering in the transformation properties of thecanonically normalized conformal blocks, and called F blocks in the main text) are given bythe expressions below for ξ = . The Racah coefficients are given by the same expressionsfor ξ = 1. Note that the parameter ξ corresponds to the gauge-freedom of Moore andSeiberg, but we stress again that there is no such freedom in the transformation property59f the 4-pt conformal blocks, once the coefficient of their leading term is canonically setto 1. The general ξ -dependent F -matrices read F (cid:20) ε εε ε (cid:21) = 1 , F (cid:20) σ σσ σ (cid:21) = − F εε (cid:20) σ σσ σ (cid:21) = 1 √ F ε (cid:20) σ σσ σ (cid:21) = ξ √ , F ε (cid:20) σ σσ σ (cid:21) = 1 √ ξ (C.2) F σ (cid:20) ε σε σ (cid:21) = F σ (cid:20) σ εσ ε (cid:21) = ξ, F σ (cid:20) ε εσ σ (cid:21) = F σ (cid:20) σ σε ε (cid:21) = 1 ξ (C.3) F σσ (cid:20) ε σσ ε (cid:21) = F σσ (cid:20) σ εε σ (cid:21) = − . (C.4)The modular S-matrix is (the order of rows and columns is 1, ε , σ ) S =
12 12 q − q q − q . (C.5)The normalized g -functions are therefore g ′ = 1 , g ′ ε = 1 , g ′ σ = √ . (C.6) g ′ a = 1 F (cid:20) a aa a (cid:21) (C.7)We can obtain the symmetric defect coefficients ( X dabia ′ b ′ = X dbaib ′ a ′ ) (4.12) by using theabove F -matrices in the Racah gauge. Defects action on boundary fields are given by X ε εε = X εεε = X εσσ σσ = X σ σσ = X σεε σσ = X σσσ = X σσσ εε = 1 , on ε boundary fields X ε εεε = − X εσσεσσ = 1 , X σ εεσσ = 1 √ , X σσσε ε = √ , and on σ boundary fields are obtained by X ε σσεσ = 1 √ , X εεσσ σ = √ , X σ σσσ = − X σεσσσε = 12 / X σ σσσε = 12 / , X σεσσσ = 2 / . eferences [1] E. Witten, “Noncommutative Geometry And String Field Theory,” Nucl. Phys. B , 253 (1986).[2] A. Sen and B. Zwiebach, “Tachyon condensation in string field theory,” JHEP (2000) 002 doi:10.1088/1126-6708/2000/03/002 [hep-th/9912249].[3] N. Berkovits, A. Sen and B. Zwiebach, “Tachyon condensation in superstringfield theory,” Nucl. Phys. B (2000) 147 doi:10.1016/S0550-3213(00)00501-0[hep-th/0002211].[4] N. Moeller, A. Sen, B. Zwiebach, “D-branes as tachyon lumps in string field theory,”JHEP (2000) 039. [hep-th/0005036].[5] A. Sen and B. Zwiebach, “Large marginal deformations in string field theory,” JHEP (2000) 009 [hep-th/0007153].[6] Y. Michishita, “Tachyon lump solutions of bosonic D-branes on SU(2) group mani-folds in cubic string field theory,” Nucl. Phys. B (2001) 26 [hep-th/0105246].[7] A. Bagchi, A. Sen, “Tachyon Condensation on Separated Brane-Antibrane System,”JHEP (2008) 010. [arXiv:0801.3498 [hep-th]].[8] J. L. Karczmarek and M. Longton, “SFT on separated D-branes and D-brane trans-lation,” JHEP (2012) 057 [arXiv:1203.3805 [hep-th]].[9] M. Kudrna, T. Masuda, Y. Okawa, M. Schnabl and K. Yoshida, “Gauge-invariantobservables and marginal deformations in open string field theory,” JHEP (2013) 103 [arXiv:1207.3335 [hep-th]].[10] M. Kudrna, C. Maccaferri and M. Schnabl, “Boundary State from Ellwood Invari-ants,” JHEP (2013) 033 [arXiv:1207.4785 [hep-th]].[11] M. Kudrna, M. Rapcak and M. Schnabl, “Ising model conformal boundary conditionsfrom open string field theory,” arXiv:1401.7980 [hep-th].[12] M. Kudrna and C. Maccaferri, “BCFT moduli space in level truncation,” JHEP (2016) 057 doi:10.1007/JHEP04(2016)057 [arXiv:1601.04046 [hep-th]].[13] L. Rastelli and B. Zwiebach, “Tachyon potentials, star products and universality,”JHEP (2001) 038 doi:10.1088/1126-6708/2001/09/038 [hep-th/0006240].6114] M. Schnabl, “Wedge states in string field theory,” JHEP (2003) 004[hep-th/0201095].[15] M. Schnabl, “Analytic solution for tachyon condensation in open string field theory,”Adv. Theor. Math. Phys. (2006) 433 [arXiv:hep-th/0511286].[16] Y. Okawa, “Comments on Schnabl’s analytic solution for tachyon condensation inWitten’s open string field theory,” JHEP (2006) 055 [arXiv:hep-th/0603159].[17] I. Ellwood and M. Schnabl, “Proof of vanishing cohomology at the tachyon vacuum,”JHEP (2007) 096 [hep-th/0606142].[18] T. Erler, “Split String Formalism and the Closed String Vacuum, II,” JHEP (2007) 084 doi:10.1088/1126-6708/2007/05/084 [hep-th/0612050].[19] T. Kawano, I. Kishimoto and T. Takahashi, “Gauge Invariant Overlaps for Clas-sical Solutions in Open String Field Theory,” Nucl. Phys. B (2008) 135doi:10.1016/j.nuclphysb.2008.05.025 [arXiv:0804.1541 [hep-th]].[20] I. Ellwood, “The Closed string tadpole in open string field theory,” JHEP (2008) 063 [arXiv:0804.1131 [hep-th]].[21] M. Kiermaier, Y. Okawa and B. Zwiebach, “The boundary state from open stringfields,” arXiv:0810.1737 [hep-th].[22] M. Murata and M. Schnabl, “On Multibrane Solutions in Open String Field Theory,”Prog. Theor. Phys. Suppl. (2011) 50 [arXiv:1103.1382 [hep-th]].[23] T. Baba and N. Ishibashi, “Energy from the gauge invariant observables,” JHEP (2013) 050 doi:10.1007/JHEP04(2013)050 [arXiv:1208.6206 [hep-th]].[24] T. Erler and M. Schnabl, “A Simple Analytic Solution for Tachyon Condensation,”JHEP (2009) 066 [arXiv:0906.0979 [hep-th]].[25] M. Schnabl, “Comments on marginal deformations in open string field theory,” Phys.Lett. B (2007) 194 [hep-th/0701248 [HEP-TH]].[26] M. Kiermaier, Y. Okawa, L. Rastelli and B. Zwiebach, “Analytic solutionsfor marginal deformations in open string field theory,” JHEP (2008) 028[hep-th/0701249 [HEP-TH]]. 6227] E. Fuchs, M. Kroyter and R. Potting, “Marginal deformations in string field theory,”JHEP (2007) 101 doi:10.1088/1126-6708/2007/09/101 [arXiv:0704.2222 [hep-th]].[28] M. Kiermaier and Y. Okawa, “Exact marginality in open string field theory: AGeneral framework,” JHEP (2009) 041 [arXiv:0707.4472 [hep-th]].[29] L. Bonora, C. Maccaferri and D. D. Tolla, “Relevant Deformations in OpenString Field Theory: a Simple Solution for Lumps,” JHEP (2011) 107[arXiv:1009.4158 [hep-th]].[30] M. Kiermaier, Y. Okawa and P. Soler, “Solutions from boundary condition changingoperators in open string field theory,” JHEP (2011) 122 [arXiv:1009.6185 [hep-th]].[31] M. Murata and M. Schnabl, “Multibrane Solutions in Open String Field Theory,”JHEP (2012) 063 [arXiv:1112.0591 [hep-th]].[32] H. Hata and T. Kojita, “Winding Number in String Field Theory,” JHEP (2012) 088 doi:10.1007/JHEP01(2012)088 [arXiv:1111.2389 [hep-th]].[33] H. Hata and T. Kojita, “Singularities in K-space and Multi-brane Solutions in Cu-bic String Field Theory,” JHEP (2013) 065 doi:10.1007/JHEP02(2013)065[arXiv:1209.4406 [hep-th]].[34] T. Masuda, T. Noumi and D. Takahashi, “Constraints on a class of classical solutionsin open string field theory,” JHEP (2012) 113 doi:10.1007/JHEP10(2012)113[arXiv:1207.6220 [hep-th]].[35] T. Erler, “Analytic solution for tachyon condensation in Berkovits‘ open superstringfield theory,” JHEP (2013) 007 [arXiv:1308.4400 [hep-th]].[36] T. Takahashi and S. Tanimoto, “Marginal and scalar solutions in cubic openstring field theory,” JHEP (2002) 033 doi:10.1088/1126-6708/2002/03/033[hep-th/0202133].[37] C. Maccaferri, “A simple solution for marginal deformations in open string field the-ory,” JHEP (2014) 004 doi:10.1007/JHEP05(2014)004 [arXiv:1402.3546 [hep-th]].[38] T. Erler and C. Maccaferri, “String Field Theory Solution for Any Open StringBackground,” JHEP (2014) 029 [arXiv:1406.3021 [hep-th]].6339] C. Maccaferri and M. Schnabl, “Large BCFT moduli in open string field theory,”JHEP (2015) 149 doi:10.1007/JHEP08(2015)149 [arXiv:1506.03723 [hep-th]].[40] I. Kishimoto, T. Masuda, T. Takahashi and S. Takemoto, “Open String Fields as Ma-trices,” PTEP (2015) no.3, 033B05 doi:10.1093/ptep/ptv023 [arXiv:1412.4855[hep-th]].[41] N. Ishibashi, I. Kishimoto and T. Takahashi, “String field theory solution correspond-ing to constant background magnetic field,” arXiv:1610.05911 [hep-th].[42] C. B. Thorn, “String Field Theory,” Phys. Rept. , 1 (1989).[43] W. Taylor and B. Zwiebach, “D-branes, tachyons, and string field theory,”hep-th/0311017.[44] A. Sen, “Tachyon dynamics in open string theory,” Int. J. Mod. Phys. A (2005)5513 doi:10.1142/S0217751X0502519X [hep-th/0410103].[45] E. Fuchs and M. Kroyter, “Analytical Solutions of Open String Field Theory,” Phys.Rept. (2011) 89 doi:10.1016/j.physrep.2011.01.003 [arXiv:0807.4722 [hep-th]].[46] M. Schnabl, “Algebraic solutions in Open String Field Theory - A Lightning Review,”Acta Polytechnica 50, no. 3 (2010) 102 [arXiv:1004.4858 [hep-th]].[47] Y. Okawa, “Analytic methods in open string field theory,” Prog. Theor. Phys. (2012) 1001.[48] J. L. Cardy, “Boundary conformal field theory,” hep-th/0411189.[49] V. B. Petkova and J. -B. Zuber, “Conformal boundary conditions and what theyteach us,” hep-th/0103007.[50] I. Runkel, “Boundary problems in conformal field theory,” PhD thesis, [51] M. Gaberdiel, “Boundary conformal field theory and D-branes”, Lecturesgiven at the TMR network school on UTFFFFD Nonperturbative methodsin low dimensional integrable models UTFFFFD, Budapest, 15-21 July 2003. [52] A. Recknagel, V. Schomerus, “Boundary Conformal Field Theory and the WorldsheetApproach to D-Branes,” Cambridge University Press (2013)6453] M. Schnabl, “String field theory at large B field and noncommutative geometry,”JHEP (2000) 031 doi:10.1088/1126-6708/2000/11/031 [hep-th/0010034].[54] I. Ellwood, “Singular gauge transformations in string field theory,” JHEP (2009) 037 doi:10.1088/1126-6708/2009/05/037 [arXiv:0903.0390 [hep-th]].[55] T. Erler and C. Maccaferri, “Connecting Solutions in Open String FieldTheory with Singular Gauge Transformations,” JHEP (2012) 107doi:10.1007/JHEP04(2012)107 [arXiv:1201.5119 [hep-th]].[56] J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, “Kramers-Wannierduality from conformal defects,” Phys. Rev. Lett. (2004) 070601doi:10.1103/PhysRevLett.93.070601 [cond-mat/0404051].[57] J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, “Duality and de-fects in rational conformal field theory,” Nucl. Phys. B (2007) 354doi:10.1016/j.nuclphysb.2006.11.017 [hep-th/0607247].[58] V. B. Petkova and J. B. Zuber, “Generalized twisted partition functions,” Phys. Lett.B (2001) 157 doi:10.1016/S0370-2693(01)00276-3 [hep-th/0011021].[59] C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, “Permeable conformalwalls and holography,” JHEP (2002) 027 doi:10.1088/1126-6708/2002/06/027[hep-th/0111210].[60] K. Graham and G. M. T. Watts, “Defect lines and boundary flows,” JHEP (2004) 019 [hep-th/0306167].[61] J. Fuchs, M. R. Gaberdiel, I. Runkel and C. Schweigert, “Topological defects forthe free boson CFT,” J. Phys. A (2007) 11403 doi:10.1088/1751-8113/40/37/016[arXiv:0705.3129 [hep-th]].[62] N. Drukker, D. Gaiotto and J. Gomis, “The Virtue of Defects in 4D GaugeTheories and 2D CFTs,” JHEP (2011) 025 doi:10.1007/JHEP06(2011)025[arXiv:1003.1112 [hep-th]].[63] E. Wong and I. Affleck, “Tunneling in quantum wires: A Boundary conformal fieldtheory approach,” Nucl. Phys. B (1994) 403. doi:10.1016/0550-3213(94)90479-0[64] M. Oshikawa and I. Affleck, “Boundary conformal field theory approach to the crit-ical two-dimensional Ising model with a defect line,” Nucl. Phys. B (1997) 533doi:10.1016/S0550-3213(97)00219-8 [cond-mat/9612187].6565] C. Bachas and M. Gaberdiel, “Loop operators and the Kondo problem,” JHEP (2004) 065 doi:10.1088/1126-6708/2004/11/065 [hep-th/0411067].[66] T. Quella, I. Runkel and G. M. T. Watts, “Reflection and transmission forconformal defects,” JHEP (2007) 095 doi:10.1088/1126-6708/2007/04/095[hep-th/0611296].[67] C. Bachas and I. Brunner, “Fusion of conformal interfaces,” JHEP (2008) 085doi:10.1088/1126-6708/2008/02/085 [arXiv:0712.0076 [hep-th]].[68] Z. Bajnok, L. Holl and G. Watts, “Defect scaling LeeYang model fromthe perturbed DCFT point of view,” Nucl. Phys. B (2014) 93doi:10.1016/j.nuclphysb.2014.06.019 [arXiv:1307.4536 [hep-th]].[69] E. P. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal FieldTheory,” Nucl. Phys. B (1988) 360.[70] I. Runkel, “Perturbed Defects and T-Systems in Conformal Field Theory,” J. Phys.A (2008) 105401 [arXiv:0711.0102 [hep-th]].[71] G. Moore and N. Seiberg, “Lectures on RCFT,” in: Physics, geometry, and topology(Banff, AB, 1989), ed. H.C. Lee, NATO Adv. Sci. Inst. Ser. B Phys., 238, Plenum,New York, 1990, 263-361.[72] G. W. Moore and N. Seiberg, “Classical and Quantum Conformal Field Theory,”Commun. Math. Phys. (1989) 177.[73] A. N. Kirillov and N. Y. Reshetikhin, “Representations of the algebra U(q)(sl(2, qorthogonal polynomials and invariants of links,” In *Kohno, T. (ed.): New develop-ments in the theory of knots* 202-256.[74] L. Alvarez-Gaume, C. Gomez and G. Sierra, “Quantum Group Interpretation of SomeConformal Field Theories,” Phys. Lett. B (1989) 142.[75] J. S. Carter, D. E. Flath, M. Saito, “The classical and quantum 6j symbols,” Prince-ton University Press, 1995.[76] R. Coquereaux, “Racah-Wigner quantum 6j symbols, Ocneanu cells forA(N) diagrams, and quantum groupoids,” J. Geom. Phys. (2007) 387doi:10.1016/j.geomphys.2006.03.010 [hep-th/0511293].6677] D. C. Lewellen, “Sewing constraints for conformal field theories on surfaces withboundaries,” Nucl. Phys. B (1992) 654. doi:10.1016/0550-3213(92)90370-Q[78] G. Pradisi, A. Sagnotti and Y. S. Stanev, “Completeness conditions for boundary op-erators in 2-D conformal field theory,” Phys. Lett. B (1996) 97 doi:10.1016/0370-2693(96)00578-3 [hep-th/9603097].[79] N. Ishibashi, “The Boundary and Crosscap States in Conformal Field Theories,”Mod. Phys. Lett. A (1989) 251.[80] J. L. Cardy, “Boundary Conditions, Fusion Rules and the Verlinde Formula,” Nucl.Phys. B (1989) 581.[81] P. di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” Springer-Verlag (1996)[82] I. Runkel, “Boundary structure constants for the A series Virasoro minimal models,”Nucl. Phys. B (1999) 563 doi:10.1016/S0550-3213(99)00125-X [hep-th/9811178].[83] R. E. Behrend, P. A. Pearce, V. B. Petkova and J. B. Zuber, “Boundary conditionsin rational conformal field theories,” Nucl. Phys. B (2000) 525 [Nucl. Phys. B (2000) 707] doi:10.1016/S0550-3213(99)00592-1, 10.1016/S0550-3213(00)00225-X [hep-th/9908036].[84] G. Felder, J. Frohlich, J. Fuchs and C. Schweigert, “The Geometry of WZW branes,”J. Geom. Phys. (2000) 162 doi:10.1016/S0393-0440(99)00061-3 [hep-th/9909030].[85] G. Felder, J. Frohlich, J. Fuchs and C. Schweigert, “Correlation functions and bound-ary conditions in RCFT and three-dimensional topology,” Compos. Math. (2002)189 doi:10.1023/A:1014903315415 [hep-th/9912239].[86] H. Yang and B. Zwiebach, “A Closed string tachyon vacuum?,” JHEP0509