The reflection coefficient for minimal model conformal defects from perturbation theory

We consider a class of conformal defects in Virasoro minimal models that have been defined as fixed points of the renormalisation group and calculate the leading contribution to the reflection coefficient for these defects. This requires several structure constants of the operator algebra of the defect fields, for which we present a derivation in detail. We compare our results with our recent work on conformal defects in the tricritical Ising model.


Introduction
Defects in two-dimensional systems have been studied for a long time, see eg [1,2] and references therein. In conformal field theory, attention has been focused primarily on defects which preserve some or all of the conformal symmetry. If the defect lies along the real axis, this can be expressed in terms of the continuity of various quantities. If the holomorphic and anti-holomorphic components T andT of the stress-energy tensor are each separately continuous across the defect, it is said to be topological; if T −T vanishes on the defect, it is called reflecting or factorised and corresponds to some combination of conformal boundary conditions on the upper and lower half planes. These are both examples of the more general case of a conformal defect for which T −T is continuous across the defect.
The Virasoro minimal models are amongst the simplest and most well-studied conformal field theories. The boundary conditions and topological defects have been completely classified in [3] and further studied in [4]. The situation of more general conformal defects is much less clear. The conformal defects in the Ising model were classified in [5] (and in the much simpler Lee-Yang model in [2]), but in general the only results found are either perturbative or numerical [6]. More recently, we have also found exact expressions for conformal defects in the tricritical Ising model [7] (based on ideas in [8]).
There has also been a great deal of study of defects between different conformal field theories, with exact classifications in a few cases [2], exact proposals [9] for defects related to renormalisation group flows, and perturbative calculations [10].
One characteristic of a conformal defect is its transmission coefficient T , or equivalently its reflection coefficient R = 1 − T , which was defined in [2]. These take the values R = 0 for a topological defect and R = 1 for a factorised defect, and 0 < R < 1 for a general conformal defect in a unitary theory [11].
The aim of this paper is to calculate the reflection coefficient for a class of conformal defects in Virasoro minimal models defined as the fixed points of the perturbative renormalisation group flows considered in [6], and to compare this with the values found in [7] for the tri-critical Ising model. The structure of the paper is as follows. In section 2 we define the perturbed defects that we will consider, state their fixed points and outline the calculation of the reflection coefficient for these fixed points. For this calculation we need several of the structure constants of the operator algebra of the defect fields. These are given in [12,13] in terms of topological field theory data but in section 3 we provide an alternative derivation of these constants, extending the results of [14].
In section 4 we calculate the perturbative integrals and in section 5 we give the value of R at the fixed points. Finally we state our conclusions in section 6.
2 The D (r,2) defect and its perturbations We will concern ourselves only with diagonal M p,q Virasoro minimal models, also known as the (A p−1 , A q−1 ) invariant [15]. These are labelled by two co-prime integers (p, q); we shall take p ≥ 2, q ≥ 5. The model has (p − 1)(q − 1)/2 primary fields corresponding to the Virasoro highest weight representations which are labelled by two integers (r, s) with (r, s) (p − r, q − s). We are going to be especially interested in the representation (1,3), and we will write h = h 13 = 2p/q − 1.
The elementary topological defects for this model were classified in [3], and are labelled by the same representations of the Virasoro algebra as the bulk fields. The space of local fields on the defects is also known. If we label the representations by a, then a primary field on the defect is labelled by two representations (a, b) which give its properties under the holomorphic and anti-holomorphic copies of the Virasoro algebra (but see the comment below on the transformation rules for defect fields). The multiplicity M ab of the primary field with labels (a, b) on the defect with label d (which isṼ ab;d d in the notation of [3]) are given in terms of the Verlinde fusion numbers N abc by From the formula (2.1), a general (r, s) defect has (for s > 2 and q large enough) one chiral field of weights (h, 0), one field of weights (0, h) and three fields of weight (h, h). A defect of type (r, 2) is special in that it has one chiral field φ of conformal weights (h, 0), one chiral fieldφ of weights (0, h), but only a two dimensional space of fields {ϕ α } of weights (h, h).
Furthermore, the (r, 2) topological defect can be constructed as the fusion (r, 1) and (1, 2) topological defects and the operator product algebra of fields of type (a, b) = ((1, s)(1, s )) is unaffected by this fusion, in exactly the same way that the action of topological defects on boundaries leaves operator algebras invariant [16]. This means that when considering the algebra of fields generated by the set {1, φ,φ, ϕ α }, we can restrict attention to just the (1, 2) defect.
The fact that there is a two-dimensional space of fields {ϕ a } on the (r, 2) defects allows one to choose a canonical basis of these fields with special properties so that the analysis of the sewing constraints is correspondingly simpler. These sewing constraints have been solved in [14] for the (1, 2) defect in the non-unitary Lee-Yang model, the (A 1 , A 4 ) theory, in which D (1,2) is the only non-trivial defect and {1, φ,φ, ϕ α } are the only non-trivial primary defect fields. In this paper we extend this analysis to the fields {1, φ,φ, ϕ α } on defects of type D (r,2) in all the (A p , A q ) models.
We are interested in the perturbations of the defect D (r,2) by a combination of the fields φ andφ, where the parameters λ andλ are independent. This is a relevant perturbation if h < 1 which is the case if p < q.
One important question is that of the transformation properties of fields on a defect under a conformal transformation. We will use the conventions of [13] which imply that defect fields always transform with the absolute value of the derivative of the conformal map, even if they are "chiral" defect fields. This is possible because the defect defines a direction through the insertion point of the field (the tangent vector along the defect), and so a defect field can pick up an extra phase under a conformal transformation: this is chosen so that all defect fields transform with the absolute value of the derivative of the conformal map. This has the advantage of making the perturbation well-defined on defects that are closed loops and making the correlation function independent of the orientation of the defect at the location of the defect field (as one would expect if the defect is genuinely topological). The question remains whether this choice for the transformation law of "chiral" defect fields is unique: the corresponding situation for a boundary and boundary fields was considered by Runkel [17], and there seems no way to fix it a priori; we stick to the conventions of [13] here for the good reasons cited above.
The expectation values in the perturbed defect D (r,2) (λ,λ) are formally given by This is only formal since there may be UV divergences in the integrals when the insertion points of two fields φ or two fieldsφ meet and IR divergences from integration along the whole real axis. This means that the general procedure of regularisation and renormalisation may be needed to given meaning to the expression (2.3). This is explained in Affleck and Ludwig [18] and applied by Recknagel et al in [19] to the case of boundary perturbations of the unitary minimal models where q = p + 1.
As explained in [6], when y = 1 − h is small and positive, the results of [19] can immediately be applied to the case of defects with the perturbation (2.2) with the prediction (from third order perturbation theory) of three conformal defects at the fixed points The fixed points (i) and (ii) can be identified as the defect D (2,1) (if r = 2) and (more generally) the superposition D (r−1,1) ⊕ D (r+1,1) ; the fixed point (iii) is a potential new conformal defect, denoted by C in [6] in the case of the perturbation of the defect D (1,2) . The value of λ * is given (to first order in y) by where C φ φφ is the three point coupling of the fields φ. Note that λ * depends on the normalisation of φ, but this will cancel in any physical quantities.

The perturbative calculation of the reflection and transmission coefficients
The transmission and reflection coefficients of a conformal defect along the real axis were defined in [2] as where T 1 and T 1 are inserted at the point iY on the upper half-plane, while T 2 and T 2 are inserted at the point −iY . For the unperturbed topological defect, and so R = 0 and T = 1.
For the defect with perturbation (2.2), the expansion of the perturbed quantities using (2.3) gives and so to find the leading order term in R, we only need to calculate the first term in T 1 T 1 and T 2 T 2 . It turns out there are neither UV nor IR divergences in these integrals, their dependence on Y is simply Y −4 and the reflection coefficient R (to leading order) is indeed independent of Y as expected. We shall take Y = 1 from now on.
The consequence is that the only correlation function we need to evaluate is where the insertion points can be in any order. This is equal to by rotation through π.
The analytic structure is simple, (2.14) but the constant C depends on the order of the insertion points {x, x , y, y } and is determined by the operator algebra structure constants, so we now turn to the calculation of some of the structure constants of the local fields on the defect D (r,2) .

The structure constants
In this section we will calculate some structure constants for the (r, 2) defect in the diagonal Virasoro Minimal models. These structure constants can be found in terms of topological field theory data [12,13] which is a general method allowing one to find all the structure constants in the defect theory, but we will not use it here and instead only use elementary properties of the conformal field theory to find the particular structure constants we need for the perturbative calculation of the reflection coefficient in the minimal models.
We note here that we will use the conventions of [13] so that the structure constant C γ αβ is the coefficient of the field φ γ appearing in the OPE of the fields φ α (x) with φ β (y) on the defect oriented opposite to the real line with x > y, which means that this coefficient appears in the OPE of the fields φ α with φ β as they appear along the defect. Rotating by π, we obtain the picture in figure 1.

The bulk theory
The (A p−1 , A q−1 ) Virasoro minimal model has (p − 1)(q − 1)/2 bulk primary fields, of which we are especially interested in the field ϕ of type (1, 3). If we set t = p/q, then h 1, The fusion rules for this field are where χ is of type (1,5) and has conformal weights The structure constant C ϕ ϕϕ clearly depends on the choice of d ϕϕ (see eg [20,21] for different conventions) but the combination is independent of the normalisation.

The defect theory
The defects of the (A p−1 , A q−1 ) Virasoro models are not intrinsically oriented, but the operator product of fields along the defect depends on the ordering of the fields, we shall assume that we can define an orientation for the defects but that all results will be independent of this orientation.
Since the space of fields {ϕ α } of weights (h, h) is only two-dimensional for a defect of type (r, 2), we can take as a basis the fields ϕ L and ϕ R which are the limits of the bulk field ϕ as it approaches the defect from the left or the right respectively as one looks along the defects -see figure 2.
Note that the operator product algebra of the fields {1, φ,φ, ϕ L , ϕ R } does not close on these fields, other fields can arise as well, namely fields with weights (h, h ), (h , h) and (h , h ) which we denote by ψ,ψ and {χ L , χ R } (which again are the limits of the field χ(z,z) as it approaches the defect from the left and the right). Although we should mention the existence of these fields and their occurrence in the operator products of some of the fields {φ,φ, ϕ α }, we will not need any of the structure constants including these fields as they will not contribute to any of the sewing constraints considered later on. Figure 2: The fields ϕ L and ϕ R defined as limits of the bulk field We now define the structure constants between these fields from their operator product expansions (we show the possibility of fields {ψ,ψ, χ α } appearing in an OPE by placing the fields in square brackets [ ]).
If both fields chiral, there are 8 structure constants {d φφ , dφφ, C φ φφ , Cφ φφ , C α φφ , C ᾱ φφ } appearing in the OPEs (recall here that x and y are ordered along the defect) : With one chiral field on the left, there are 12 structure constants {Cφ φα , C φ φα , C β φα , C β φα } in the OPEs likewise there are 12 structure constants {Cφ αφ , C φ αφ , C β αφ , C β αφ } in the OPEs with one field chiral on the right: Finally there are 20 structure constants {d αβ , C φ αβ , Cφ αβ , C γ αβ } in the OPEs involving no chiral fields: Having defined the fifty-two structure constants we need to calculate, we now set about finding relations. The simplest come from the fact that the orientation of the defect is in fact not physical.

Symmetry relations
Since the defect is not intrinsically oriented, our labelling over-counts the structure constants: sixteen constants are related by changing the orientation of the defect, as follows:

Bulk field relations
We can use the fact that ϕ L and ϕ R are the limits of bulk fields to find d LL , d LR , d RL and d RR , as well as C L LL , C R LL , C L RR and C R RR . In the bulk, we have (3.2). Bringing this OPE towards a defect from the left, we obtain We have also found that C but these four constants are not of interest to us.
Likewise, bringing the bulk OPE (3.2) towards a defect from the right, we obtain Finally, using the expression for the defect in terms of projectors [3] D r,2 = r ,s S (r,2),(r ,s) S (1,1),(r ,s)P r ,s , where S (rs)(r s ) is the modular S-matrix given in the appendix, we have where we define γ = 2 cos(2πt) − 1 . which is independent of r, as expected.

Defect -boundary identification
We next use the fact that the OPE algebra of φ along the real axis is the same as that of the boundary field on the (r, 2) boundary -we obtain this identification by bringing the (r, 2) defect next to the identity boundary as considered in [16]. Likewise, the algebra ofφ is also the same as the boundary algebra.
This means that d φφ = dφφ , C φ φφ = Cφ φφ , (3.25) and these values are are given by Runkel's solution to the boundary algebra [21], Note that the structure constant again does not depend on r.

Three-point function constraints
We can express the three point function Taking a and c chiral, this gives the simple relations which, using (3.25) become Taking only a chiral and the two non-chiral fields equal, this gives the slightly more complicated which using (3.23) become Taking a chiral and the other two fields different, we get (3.37) Using d LR = γd ϕϕ , these become Finally, taking only b chiral, we get Looking at the first of these, it becomes Likewise we get which also imply

Bulk field expectation operator product
To find C R LR we use the inner product matrix d αβ of defect fields ϕ L and ϕ R and cyclicity of the three point constant C αβγ defined by (3.47) Using C γ αβ = d γ C αβ and C αβγ = C γβα and the relations (3.19) and (3.21), we get With the inner-product matrix d αβ = ϕ α |ϕ β , and its inverse Likewise, we find all four of these structure constants are equal, Figure 3: The relation between C b La and C b aL from continuity in the bulk.

Continuity of bulk fields
We can relate the structure constants C b aL and C b L,a by moving the insertion point of the field ϕ L from the right of the field a to the left through the bulk. If the defect is oriented along the x axis in the plane, then the field ϕ L can be moved through the upper half plane, as in figure 3.
Likewise, we can relate C b aR and C b R,a by moving the field ϕ R through the lower half plane. Since the OPEs of the bulk field ϕ and the defect field we get the relations We again list the cases according to the number of chiral fields involved: • No chiral fields: we find identities consistent with equation (3.52) (3.58) • If Φ b is chiral and Φ a is not; with ζ = exp(iπh): where the first four structure constants were already found to be zero in equations (3.19) and (3.21).
• If Φ a is chiral and Φ b is not: • If both Φ a and Φ b are chiral:

Unknown constants
We summarise the results so far, distinguishing the structure constants by the number of chiral fields they involve.

No chiral fields
These are all known in terms of the bulk field data: (3.67)

Three chiral fields
These are also all known in terms of the boundary field theory data [21]:

Two chiral fields
The 24 structure constants involving two chiral fields can be written in terms of just two of these, which we can take to be C L φφ , and C L φφ . (3.70) Listing the remaining 22 structure constants:

73)
It will be convenient to introduce κ and Γ to parametrise C L φφ and C L φφ as It will turn out that Γ is real and non-negative and κ is a pure phase. We note that these two structure constants can be found from the results in [13] -they are related to C s defined in [13]:eqn (2.19).

One chiral field
The twenty-four structure constants involving just one chiral field can, using the previous identities, be written in terms of just four: We list the remaining twenty constants here for convenience:

The four-point function sewing constraints
We will use crossing relations for four point correlation functions to find sewing constraints that will enable us to determined the remaining six structure constants {C L φφ , C L φφ , C R φL , C R φL , C L φR , C L φR }. The four-point function Φ a Φ b Φ c Φ d of fields on a defect can be expressed in terms of conformal blocks in two different ways, as illustrated in figure 4 The conformal blocks are functions which satisfy the crossing relations [21]   where the F-matrices are known constants, again given explicitly in [21]. Substituting (3.90) into the expressions in figure 4 leads to further sewing constraints that the structure constants must satisfy.
The simplest relations arise when there is only a single channel in both diagrams, i.e. the sum is over a single pair of weights (h e ,h e ) and a single pair of weights (h g ,h g ). Note that since the space of fields with weights (h, h) is two-dimensional, this does not mean that the OPE has to include only a single field. In all the cases where there is only a single channel, the F -matrix is just the number 1 and so the sewing constraints become just We now list all the non-zero cases in which the fields a, b, c and d are taken from {φ,φ, ϕ α } and for which there is only a single intermediate channel in both diagrams, and state the corresponding equations. We will in fact only use the first eight of these, where there is at most one field of weights (h, h) but we list them all for completeness. The eight we use are: The remaining three which include two fields of type ϕ α but still only have a single intermediate channel are:

Analysis of the sewing constraints
We need to use only the first eight relations. We consider these in turn: • Equation (3.92) Written out in full, this is This is This leads to two equations: for α = L: and for α = R: The first equation becomes: The second equation implies These two equations imply κ 2 = ζ = exp(iπh) .
(3.115) (We will not need to fix the sign of κ as only κ 2 appears in our final answers) • Equation (3.97) These equations imply (for α = L) and (for α = R) which are consistent with the results so far.
• Equation (3.98) These two equations lead to (α = L): and (with α = R) Together, these imply This completes the derivation of the structure constants. They agree with the specific case in [14] (apart from a typo in [14], where it should ρ = exp(iπ/10)).
The remaining crossing relations (3.101) -(3.105) are not needed for the derivation of the structure constants but we have checked that they hold.

The integrals
We want to calculate the leading term in the expansion (2.10), that is The correlation function has the same functional form whatever the order of the fields, but a different constant depending on the order of the insertions. We can restrict to x < x and y < y to get This correlation function is where the constant ∆ depends on the order of the field insertions as in table 4.1 Integration region Order of fields Value of ∆ x < x < y < y φφφφ The values ∆ i are We only need to evaluate three of these integrations, the other three being given by complex conjugation. Furthermore, we only need the leading order term in y in the correlation function, The results are given in table 4.2. Adding all six together, we get Integration region Order of fields Value of the integral x < x < y < y φφφφ − 3πi 16 ∆ 1 x < y < x < y φφφφ − π 2 +3πi 8 ∆ 2 x < y < y < x φφφφ 5 The value of the reflection coefficient for the defect C We now put the various terms together to find the value of R at the fixed point (λ * , λ * ), . (5.1) The leading term in the numerator is 2I and leading term in the denominator is c/16. We can now calculate this for the tri-critical Ising model. In this case, h = 3/5, y = 2/5 and we are far from the small y regime, but we calculate the leading correction and get

Conclusions
We have calculated the leading term in the perturbative expansion of the reflection coefficient for the defect of type (r, 2) in a minimal model. It is believed that a non-trivial conformal defect can be found as a perturbative fixed point of the renormalisation group equations. We have recently found new non-trivial conformal defects in the tri-critical Ising model [7] and it is possible that these are related to the conformal defects found by perturbation theory. We have checked, and the value of R is close enough not to rule this out. It would of course be good to extend this calculation to next-to-leading order where there are UV divergences to be regulated, but so far we have not yet managed this.
We have also calculated defect structure constants for various fields on defects of type (r, 2) extending the results of [14]. These results are not complete -they do not include all fields, and use special properties of the (r, 2) defect, but it would be good to check that these constants in fact agree with the general results of [12] where the same constants were constructed using topological field theory methods.
We would like to thank I. Runkel, C. Schmidt-Colinet and E. Brehm for discussions on defects and their properties and for comments on the manuscript.

A The Virasoro Minimal Models
The Virasoro minimal models occur for c ≡ c(p, q) where p, q are coprime positive integers greater than 1. It is useful to define t = p/q. c is given by c(p, q) = 13 − 6t − 6/t .