Open topological defects and boundary RG flows

In the context of two-dimensional rational conformal field theories we consider topological junctions of topological defect lines with boundary conditions. We refer to such junctions as open topological defects. For a relevant boundary operator on a conformal boundary condition we consider a commutation relation with an open defect obtained by passing the junction point through the boundary operator. We show that when there is an open defect that commutes or anti-commutes with the boundary operator there are interesting implications for the boundary RG flows triggered by this operator. The end points of the flow must satisfy certain constraints which, in essence, require the end points to admit junctions with the same open defects. Furthermore, the open defects in the infrared must generate a subring under fusion that is isomorphic to the analogous subring of the original boundary condition. We illustrate these constraints by a number of explicit examples in Virasoro minimal models.


Introduction
In this paper we consider Euclidean two-dimensional quantum field theories on a halfplane which are described by a unitary conformal field theory (CFT) in the bulk, and on the boundary by a perturbed conformal boundary condition. We will assume that the CFT at hand is rational and possesses some non-trivial topological defect lines. Such defects were first considered in [3] and then more extensively in the context of general rational CFTs in [6]. They describe symmetries and dualities of the critical system described by the given CFT [5], [6]. Topological defect lines can be moved around and, if they do not pass though any other observables any correlation function is independent of their position. When they pass through a local bulk operator, generically we obtain a collection of defect segments attached to the main defect and ending on a disorder field located at the insertion. In certain special situations the additional defect segments may be absent and passing the defect through results in an operator with the same Virasoro representation labels but multiplied by some factor. In the simplest situation the original insertion remains intact and we can say that the defect commutes with this bulk insertion. As shown in [11], if we have some defects that commute with a bulk relevant operator then there are interesting consequences for the bulk renormalisation group (RG) flows triggered by this operator. The fusion algebra of such commuting defects between themselves must be robust under the fusion and this places constraints on the end points of the flows (triggered by the same operator with positive or negative coupling) particularly when the flows are massive and the end points may be described by non-trivial topological theories.
For a CFT on a half plane with a conformal boundary condition, if there is a nontrivial boundary relevant operator, we can perturb the boundary condition by this operator triggering a boundary RG flow. Unlike the bulk flows boundary RG flows always end up in a non-trivial conformal boundary condition that at least has the Virasoro identity tower in the boundary spectrum. In the presence of topological defect lines in the bulk CFT we can fuse them with any conformal boundary condition to obtain a new conformal boundary condition which may in general be a superposition of elementary boundary conditions. Based on this construction, an interesting interplay between boundary RG flows and topological defect lines was discussed in [7]. The following general theorem was proved in that paper: given a boundary RG flow from a maximally symmetric conformal boundary condition with label a that ends in a maximally symmetric 1 conformal boundary condition with label b, for any topological defect d there is an RG flow from d × a to d × b where the cross stands for fusion. We will refer to this result as Graham-Watts theorem in the rest of the paper. The perturbing field for the new flow must have the same Virasoro representation properties (and scaling dimension in particular) as the perturbing field in the original flow. As the new starting point may be a direct sum of elimentary boundary conditions there may be many such fields. The precise form of the perturbing field of the new flow has been worked out in [7] for the case of a being elementary and for the general situation it was worked out in [10]. For diagonal modular invariants the action of defect on boundary fields can be also obtained from the action on chiral defect fields which was worked out in [8].
If we know the end-point for a particular boundary flow, using Graham-Watts theorem we find the end-points for other flows obtained via fusion. Thus, in [7], using the results of [19], an extension of perturbative flows triggered by boundary ψ 1,3 fields in minimal models [18] to all Cardy boundary conditions was obtained. It is interesting to note that the g-factors change under fusion according to where g 1 is the g-factor associated with the Cardy boundary condition that has only the identity tower in its spectrum. It is not hard to show that in unitary rational CFTs g 1 is the smallest possible g-factor (see e.g. [14]). Thus, (1.1) implies that fusion with non-trivial topological defect always increases the g-factor. A useful strategy in applying the Graham-Watts theorem may then be to start with a UV boundary condition with a small value of the g-factor, use the g-theorem [12], [13] and symmetries to constrain the end point as much as possible then use fusion to obtain possible end points for flows that start with larger values of g.
In this paper we look at a different usage of topological defects for constraining boundary flows that not merely relates two different flows but directly constraints the possible end points for a given flow. We consider topological junctions of topological defects with a conformal boundary condition. This means that not only the part of the defect line that extends into the bulk but the junction point as well can be moved along the boundary, not changing any correlation functions as long as no boundary insertions are encountered. Such junctions and their properties were considered at length in [10] and we will use the results of that paper extensively. Following [10] we call a topological defect attached to a conformal boundary via a topological junction an open topological defect. When we move such a defect along the boundary with an insertion of a boundary operator present, passing the open defect through the insertion typically results in a configuration with the original insertion replaced by several boundary condition changing fields and new boundary conditions between the insertions and the open defects. But sometimes, for certain defects and boundary fields, no additional fields or boundary conditions arise, the open defect just passes through. In the operator language the defect and the boundary operator commute. In such cases, which are similar to the bulk case considered in [11], we can argue that the end point of the boundary flow must admit a topological junction with the same defect. Moreover, the ring obtained by fusion of such open defects between themselves must be isomorphic to some subring in the infrared boundary condition. This potentially can lead to additional constraints on the end points of RG flows as in the boundary case the fusion ring for open defects in general depends on the boundary condition [10]. Even if the bulk labels are the same the fusion rules may be different.
Another interesting case is when an open defect just multiplies the operator by minus one when passing through it, or in other words when the open defect anti-commutes with the boundary operator. This situation demands that there must be a topological junction with the same defect and the two boundary conditions describing the infrared end point for the two signs of the perturbation. The fusion rules again must be robust (up to isomorphism) and persist into the infrared fixed points. If both commuting and anti-commuting open defects are present they form a Z 2 -graded subring under fusion.
The main goal of this paper is to point to the existence of such constraints on boundary RG flows, to explain how to look for commuting and anti-commuting open defects and to illustrate the resulting constraints on concrete examples. To this end we choose to restrict our constructions to Virasoro minimal models with diagonal modular invariant. Moreover, our main examples of boundary flows will be the flows triggered by boundary ψ 1,3 operators. These flows are integrable and the end points of the flows are known. This allows us to check that the constraints we derive from open defects are satisfied. In addition we consider a flow triggered by a boundary ψ 1,2 operator in the pentacritical Ising model. This flow is believed to be integrable but the end points are not known. We derive some constraints on the possible end points.
The main body of the paper is organised as follows. In section 2 we discuss generalities about topological defects and their junctions with boundary conditions. We fix our normalisation conventions and derive a commutation relation for an open defect and a boundary operator. In section 3 we discuss the constraints on RG flows arising from open defects commuting or anti-commuting with the perturbing operator. In section 4 we work out explicit examples in the tetracritical and the pentacritical Ising models. In section 5 we discuss some specifics for flows triggered by boundary condition changing operators. For such flows there may be special linear combinations of different open defects (with the same Virasoro labels) that commute with the perturbing field. We give some explicit examples of this. The appendix contains some useful relations between the diagonal minimal model fusion matrices.

Open topological defects
Throughout the paper for simplicity we restrict ourselves to the case of unitary Virasoro minimal models with diagonal modular invariant. All constructions can be generalised to a general rational CFT. For the minimal models both topological defects [3] and the elementary conformal boundary conditions [1] are labeled by the same pairs of integers from the Kac table as the chiral operators. In this section we will just use the letters: a, b, c, . . . for such labels. Boundary fields linking a boundary condition a on the left with a boundary condition b on the right are built on Virasoro representations i ∈ a × b. We denote such fields as ψ . On the diagrams below we will omit the upper indices of boundary operators as those can be read off from the boundaries.
Three elementary topological defects labelled by a, b, c can be joined together if that is permitted by the fusion, that is if a ∈ b × c. Defect networks can be simplified via a sequence of elementary moves. The latter equates two networks as depicted on Figure  1. This was shown in [9] using the topological field theory approach of [6]. As emphasised in [10] one does not have to choose the F -matrices appearing on Figure 1 to be the same as the conformal block F -matrices. The latter are fixed if we canonically normalise the conformal blocks. To do concrete calculations we are going to use the conformal block F -matrices calculated in [15], [16], so we are going to assume that the defect junctions are normalised in such a way that the defect F -marices are those of the conformal blocks. We also assume that the identity defect can be attached at any point and can be moved freely without changing anything.
We are further interested in topological defects that can end topologically on a given conformal boundary condition. This means that the ending should behave as a local operator of dimension zero. It is not hard to see that for an elementary boundary condition with a label a and an elementary defect with label d the junction is topological if the fusion d × a contains a. To see this we can deform the defect keeping the junction pinned down so that the defect fuses with the boundary on one side of the junction. The junction then looks like a boundary condition-changing operator between a and d × a. There is a dimension zero such operator if d × a contains a. Equivalently a × a should contain d and therefore the set of admissible defect labels d is the same as the set labelling boundary operators. This observation generalises to junctions which have two different elementary boundary conditions on either side of the junction: a and b. The junction is topological if there is a fusion vertex linking a, b and d. Such a junction is depicted on Figure 2. Each junction of an elementary defect and an elementary boundary comes with a choice of coupling that can be thought of as a choice of normalisation of the corresponding junction field. We are going to choose the normalisation for the open defects and the boundary fields as described in [10]. The conventions of [10] include additional factors for the junctions of defects with boundaries which arise from taking a defect stretched parallel to the boundary and partially fusing its right or left half with the boundary. To distinguish between the two types of fusion it will be convenient to orient our defects assuming that the defect outgoing from the boundary was fused on the left and the defect coming into the boundary was fused on the right. The orientation will be marked by arrows on the diagrams. Furthermore, to signify the presence of these additional factors we will add a bullet on the junction when depicting it. The factors themselves are presented on Figure 3.  d the corresponding operator acting on the radial quantisation states on a half plane with the boundary condition a.
With the factors given on Figure 3 two simple relations hold. Firstly, a defect arc attached to the boundary with no insertions can be shrunk leaving no additional factors. This is illustrated on Figure 4. Manipulations with junctions of defects with a boundary can be lifted to junctions between topological defects by representing the boundary conditions with label a as fusions between D a and the identity boundary condition. A boundary operator with Virasoro label i can be traded for the defect labelled by i ending with a defect ending field located on the identity boundary condition. This is shown on Figure 6. The general expression for coefficients α ab i has been calculated in [10] (see equation (B.7) of that paper). Once the F -matrices appearing in defect junctions have been fixed, these coefficients can be explicitly calculated. In this paper we use the conformal block F -matrices so in principle α ab i are fixed but at no point in our calculations we need to use their explicit form. Using Figure 6 we calculate, following [10], the action of an open defect on boundary operators. The latter is obtained by surrounding a boundary operator by the defect arch and shrinking the arch onto the operator. This can be calculated by a sequence of moves shown on Figure 7 where we consider the most general boundary condition changing operator. The final factors X a ′ a ′′ i,aã that appear on Figure 7 are An alternative derivation of this result can be done using the three-dimensional topological quantum field theory representation developed in [2] and [4]. Using Figure 7 we can derive a commutation relation between an open defect and an insertion of a boundary operator ψ i . To that end we need to pass the defect junction through ψ i from left to right. This can be done by creating an arc around the insertion of ψ i , partially fusing a portion of the defect to the right of the insertion and finally shrinking the arc onto the boundary field. This is depicted on Figure 8.  We can also conclude from the orthogonality relation (A.6) that these are the only interesting situations for RG flows originating from an elementary boundary condition, there cannot be a commutation up to a non-trivial rescaling of ψ i . The latter however are possible when boundary condition changing fields are involved (see section 5). The open defects ending on a given elementary conformal boundary condition a are closed under fusion. Curiously, as found in [10], the fusion rule is not the usual bulk fusion rule but depends on the boundary a.   [10]. Among other general properties of (2.5), (2.6) we note the following identities where S i j is the modular S-matrix.

Constraints on boundary RG flows
Suppose now that we take an elementary conformal boundary condition labelled by a and perturb it by a relevant operator ψ(t) with a coupling λ. The perturbed boundary correlators are where counter terms may be needed to cancel divergences.
is independent of s as long as it does not cross any of x 1 , . . . x k . This means that D d . The same fusion rules will be valid also in the deformed theory, so, in addition to admitting topological junctions with defects labelled by the same d's, these open defects must form the same fusion algebra up to possibly a change in normalisation. Given that the fusion algebra in general depends on the boundary condition this may place some additional constraints on the IR BCFT.
Consider now an open defect D [a] d that anti-commutes with ψ. Let us place the corresponding junction at a point s on the boundary and consider a perturbation with a coupling λ to the left of s and with a coupling −λ to the right of s. A deformed correlation function in such a configuration can be written as d and ψ that this expression is independent of s as long as s does not cross any of x i . Taking λ to the infrared fixed point we get a topological junction of the defect labeled by d and the two conformal boundary conditions that describe the IR endpoints of the flow in the positive and negative λ directions. Thus, for each anti-commuting defect there must exist a topological junction between the two end-points of the flows in the positive and negative direction and the same defect. If we take all open defects ending on a that either commute or anti-commute with ψ i they form a Z 2 -graded algebra with respect to fusion. The corresponding algebra between the infrared fixed points must be isomorphic (up to possibly a change in normalisations).
It should be noted that both of these constraints generalise in a straightforward manner to the case when a is a direct sum of elementary boundary conditions. We can also consider both of the above situations in the Hamiltonian language. To that end consider an infinite strip of width L with the boundary condition a put on both ends. Let 0 ≤ σ ≤ L be the coordinate across the strip and −∞ < τ < ∞ be the coordinate along the strip. For τ being Euclidean time the Hilbert space can be decomposed into Virasoro irreducible representations V i as .
The primary fields φ r,s are labelled by two integers 1 ≤ r ≤ m − 1, 1 ≤ s ≤ m from the Kac table with the identification The same set of integers label the bulk defects D r,s as well as the elementary conformal boundary conditions which we will denote as (r, s).
The fusion rules are summarised in the following equation The fusion ring contains two subrings generated by fields of the form φ 1,s and φ r,1 respectively. The two subrings intersect over a subring generated by the identity field and the operator φ if m is odd. For such boundary conditions we can introduce the S-charge for the boundary fields that according to (2.2) is given by where the boundary label a is ( m 2 , s) or (r, m+1 2 ) depending on the parity of m and i ∈ a × a. This charge is equal to ±1 due to the orthogonality relation (A.6) and the fusion rule (m − 1, 1) × a = a.
If we are perturbing an S-invariant boundary condition by a charge 1 boundary field then, by virtue of the Graham-Watts theorem, we expect each end point of the flow to be S-invariant. If we perturb by a charge -1 field then the end points of the flow are interchanged by the action of S. For example in the tricritical Ising model, that corresponds to m = 4, we have two spin reversal invariant Cardy boundary conditions: (2, 2) and (2, 1). The latter boundary condition is stable while the (2, 2) boundary condition, also known as the disordered boundary condition, admits two relevant boundary fields: ψ 1,2 and ψ 1,3 . The first field has the S-charge -1 while the second one has charge 1. The boundary RG flows in the tricritical Ising model that start from the elementary boundary conditions were worked out in [17]. Both ψ 1,2 and ψ 1,3 perturbations of the disordered boundary condition are integrable and their end points are given on the following diagrams.
(2, 1) (1, 1) It is straightforward to check that the endpoints satisfy the requirements for the action of S. Below we will be particularly interested in boundary flows triggered by perturbing the boundary condition (r, s) by the boundary field ψ . For large values of m these flows were studied in [18] where the end points were identified using g-theorem. The end points in the non-perturbative regime were found in [7] with the help of Graham-Watts theorem, which was put forward in that paper, and using the results of [19]. The general rule for the end points of the ψ 1,3 flows that start from elementary boundary conditions can be summarised in the following two expressions where one expression corresponds to a positive choice of the coupling and the other to the negative choice. To the best of our knowledge it has not been fixed in general which answer corresponds to which sign. The expressions (4.12) and (4.13) are interchanged under the action of the field identification (4.2). Commutators of boundary fields with open defects can be computed using the general expression (2.1). The fusion matrices for the diagonal minimal models can be calculated recursively following [15] (see also [16] for a closed expression).

Tetracritical Ising model
The first example of a non-trivial open defect that commutes with a relevant operator on an elementary boundary condition appears in the tetracritical Ising model that is the unitary minimal model with m = 5. This model has 10 primary fields and thus the same number of topological defects and elementary conformal boundary conditions. We focus on the ψ 1,3 boundary field where we know the end points of the flows. All elementary boundary conditions have a ψ 1,3 boundary field except for the 4 boundary conditions of the form (r, 1), 1 ≤ r ≤ 4. Table 1 shows the open defects that have a topological junction with a given boundary condition and that commute or anti-commute with ψ 1,3 .

Pentacritical Ising model
Pentacritical  (3,2), the end points must be also spin-reversal invariant. Noting that the g-factors satisfy g 3,3 > g 3,2 > g 3,1 we see that an end point of the ψ 1,3 flows from (3, 2) is either degenerate or is given by the (3, 1) boundary condition that is spin-reversal invariant.
We also have a boundary field ψ 1,2 present on the boundary condition (3,3). The corresponding boundary flows are believed to be integrable but the end points are not known. We find that D [3,3] 3,1 anti commutes and S [3,3] commutes with ψ 1,2 . The two end points of these flows must be spin reversal symmetric and admit a topological junction between them and D 3,1 . One of the three situations then must be realised: either one or both end-points are degenerate (with further constraints) or they are both (3,2) or they are both (3,1). (We also checked that the fusion rule for D [3,1] 3,1 with itself is of the type (4.27) -same as in the UV boundary condition (3,3).)

Flows from direct sums of boundary conditions
So far we have discussed the implication of open defects commuting with the perturbation for the flows originating from an elementary boundary condition. This can be generalised to flows from direct sums of elementary boundary conditions triggered by boundary condition changing operators. Examples of such flows, including those triggered by ψ 1,3 operators, were studied in [20]. The commutator with an open defect has been calculated on Figure 8.
A new feature of direct sum boundary conditions is that the perturbing field can be a linear combination of several components and a commuting (or anti-commuting) open defect can be a particular linear combination of defects with the same Virasoro label but linking different sets of elementary boundary conditions. One way to generate such linear combinations is by starting with a commuting open defect on an elementary boundary condition and fuse it with a closed defect. Suppose D We can fuse the junction with a closed string defect D s on either side of the junction. Such a fusion done on the left is illustrated on Figure 9. where the coefficients Y L; i,j a,s,d are easily computed using the results of [10] (see Figure  8 of [10] in particular). As the final configuration on Figure 9 is only an intermediate result, we omit the explicit expression for Y L; i,j a,s,d . At this stage it is important for us to note that, as a consequence of the commutation of D of the fusion of D s with the boundary a. Note that (5.2) is true for any fixed label j. Now, picking a configuration given by (5.1) with a fixed label j we can further fuse it with the closed defect D s on the right side of the junctions. Using steps similar to those on Figure 9  is an overall normalisation factor. Similarly, we can do the above fusion in the reversed order, that is first fusing with a closed defect on the right, singling out . Fusing the boundary with a closed defect D 1,2 we obtain the direct sum (2, 1) ⊕ (2, 3). Up to an overall factor the boundary field ψ  are normalised as in [10]. Since (1, 2) × (3, 1) = (3, 2) we have only one value j = (3, 2) in (5.4), (5.7). Using (5.4)  . (5.14) The coefficients of these linear combinations are quite ugly so we do not present them here, but we checked that they span the linear subspace generated by the elementary open defects listed in (5.14). Indeed, a separate calculation shows that each of the defects in (5.14) commutes with Ψ ′ .

Acknowledgements
The author thanks M. Buican, I. Runkel and C. Schmidt-Colinet for useful discussions and I. Runkel for comments on the draft.
A Some identities for the minimal model fusion matrices where S ab stand for the elements of the modular S-matrix.