UUUITP-07/21
Fusion of conformal defects in four dimensions
Alexander Söderberg
Department of Physics and Astronomy, Uppsala University, Sweden
E-mail: [email protected]
Abstract:
We consider two conformal defects close to each other in a free theory, andstudy what happens as the distance between them goes to zero. This limit is the same aszooming out, and the two defects have fused to another defect. As we zoom in we find anon-conformal effective action for the fused defect. Among other things this means thatwe cannot in general decompose the two-point correlator of two defects in terms of otherconformal defects. We prove the fusion using the path integral formalism by treating thedefects as sources for a scalar in the bulk. a r X i v : . [ h e p - t h ] F e b ontents Recently there have been a lot of research focused on higher dimensional (in dimensionsgreater than two) conformal field theories (CFT’s) in the presence of a boundary (see e.g.[1–4]) or a defect (see e.g. [5–8]). However, the literature regarding higher dimensionalCFT’s in the presence of several defects is scarce [9–12]. We add to this literature this workwhere we consider two scalar Wilson lines in a free theory in four dimensions, and studythe limit in which they intersect. This corresponds to a fusion of the two defects.Fusion of defects have previously been studied in two dimensions [13, 14], in supersymmetrictheories [15–17], and in topological field theories as well as in conformal nets using fusingcategories [18–20]. In this paper, we will provide three different examples of fusing twodefects in four dimensional free theories using the path integral formalism. To our knowledgethis has never been done before. The fusion in these three examples is done in the sameway using the following method:1. Find the two-point function (cid:104) D D (cid:105) for the two defects. This should describe a non-perturbative Casimir effect between the defects, and will be given by the exponentialof an integral over a two-point function of fields on each defect.2. Probe this correlator with a bulk field, (cid:104) D D φ (cid:105) , and study the fusing limit wherethe distance between the defects go to zero. This will be given by integrals over atwo-point function of φ and a field from one of the defects.– 1 –. Find an effective action for the fused defect. The difficulty here lies in identifyingwhat kind of operators appear on the fused defect. In the examples we study, theyare directional derivatives w.r.t. the distance vector between the two defects. Thefused defect should satisfy (cid:104) D D φ (cid:105) = (cid:104) D D (cid:105)(cid:104) D f φ (cid:105) . (1.1)4. Prove the fusing using the path integral formalism by treating the defects as sourcesfor an operator in the bulk. Correlators with two conformal defects have previously been decomposed in conformalblocks corresponding to local operators in the bulk [9, 12]. Only this block decompositionis known. It is therefore interesting to study whether two defects can be decomposed inanother way, which would yield a bootstrap equation for the defect correlators. One mightbelieve that another block decomposition would be in terms of other conformal defects. Aswe study explicit examples of fusion of conformal defects, we find that the fused defect isnot conformal and thus in general there will not be a second decomposition in terms ofconformal defects.A higher dimensional CFT enjoy a SO ( d + 1 , -symmetry in Euclidean space. We saythat a flat (or spherical) p -dimensional defect is conformal if a field localized on it satisfy a SO ( p + 1 , × SO ( d − p ) -symmetry, where SO ( p + 1 , is the conformal symmetry alongthe defect, and SO ( d − p ) is the group of rotations around the defect. The defect itself, asa p -dimensional operator, satisfy SO ( d − p ) -symmetry.The first example we consider are two parallel Wilson lines D ± separated by a distance R in a free theory in four dimensions. These are one-dimensional defects that does not carryany SO (3) -spin D ± = exp (cid:18) λ ± (cid:90) R dxφ ( x ˆ x (cid:107) ± R ˆ x ⊥ ) (cid:19) , λ ± ∈ C . (1.2)Here ˆ x (cid:107) is the unit vector parallel to the defect, and ˆ x ⊥ is one of the three unit vectororthogonal to the defect. In section 2 we study the three-point correlator (cid:104) D + D − φ (cid:105) betweenthe two defects probed with a bulk field φ , and note that it is the same as a two-pointcorrelator (cid:104) D f φ (cid:105) between another defect and the same bulk field (times a non-perturbativeCasimir effect from (cid:104) D + D − (cid:105) between the two defects). From this we deduce that the newdefect D f is a fusion of the other two other defects. We find it to be given by D f = exp (cid:88) n ≥ λ n − + ( − n λ n + n ! R n (cid:90) R dx∂ nx ⊥ φ ( x ) . (1.3)As the distance R between the two defects D ± goes to zero, the defect D f is conformal.However, as we zoom in and pick up perturbations in R , we find dimensionfull coupling This might be more complicated for composite operators. For such operators we also have to be carefulsince they will generate a renormalization group (RG) flow on the defects. – 2 –onstants which break the conformal symmetry. It is thus not possible to decompose thetwo-point correlator (cid:104) D + D − (cid:105) in terms of conformal defects.In section 3 we treat D ± and D f as sources for φ . We are then able to show that the pathintegral is the same regardless of whether we use D + and D − as a source, or D f . Thisproves the fusion.In section 4 and 5 we study two other examples of fusion in free theories. Here we considertwo scalar Wilson loops (non-concentric and concentric) as opposed to Wilson lines. Thedefects are fused in a similar manner, and we can again prove it using the path integralformalism. Our main results in the paper are the fusions at equations (2.20), (4.4) and (5.4).In appendix A we study the block decomposition of the two-point correlators of the circulardefects considered in section 4 and 5. This is done using the method of [9]. We find thatall of the cross-ratios can be expressed in terms of each other, which means that this blockdecomposition does not need to be unique.We conclude in section 6 with some future aspects. Knowing the exponential form of twodefects, we could possibly fuse them using the method presented in this paper. We can write defects as exponentials, and one of the simplest examples is a scalar Wilsonline in d = 4 dimensions [5, 21] D = exp (cid:18) λ (cid:90) R dxφ ( x ) (cid:19) , ∆ φ = 1 . (2.1)Here λ ∈ C is a dimensionless constant. Using Wick’s theorem we find (cid:104) D (cid:105) = 1 + (cid:88) n ≥ λ n n ! (cid:90) R dx ... (cid:90) R dx n (cid:104) φ ( x ) ...φ ( x n ) (cid:105) = (cid:88) n ≥ λ n n n ! (cid:18)(cid:90) R dx (cid:90) R dx (cid:104) φ ( x ) φ ( x ) (cid:105) (cid:19) n = exp (cid:18) λ (cid:90) R dx (cid:90) R dx (cid:104) φ ( x ) φ ( x ) (cid:105) (cid:19) . (2.2)Correlators between local fields behave in the same way as in a homogenous CFT (withoutthe defects). In the free theory it is given by (cid:104) φ ( x ) φ ( x ) (cid:105) = A d | x − x | φ , A d = 1( d − S d = 14 π . (2.3)Here S d is the area of a ( d − -dimensional sphere, and Γ x ≡ Γ( x ) is the Gamma-function.Integrals over fields on the same defects are divergent, so it is convenient to normalize the– 3 – R - R D + D - x ∥ x ⊥ Figure 1 . The two line defects are separated by a distance of R . propagators by dividing with the one-point functions of the defects (cid:104) D (cid:105) N ≡ (cid:104) D (cid:105)(cid:104) D (cid:105) = 1 . (2.4)Now place two defects with a distance R > from eachother (see figure 1) D ± = exp (cid:18) λ ± (cid:90) R dxφ ( x ˆ x (cid:107) ± R ˆ x ⊥ ) (cid:19) ≡ exp (cid:18) λ ± (cid:90) R dxφ ± ( x ) (cid:19) , λ ± ∈ C . (2.5)Here ˆ x (cid:107) is the unit vector along the defect, and ˆ x ⊥ is one of the three unit vectors orthogonalto the defects. It will be convenient for us to use different notations depending on whetherwe integrate over fields on the same defect, or fields from different defects I ± ≡ (cid:90) R dx (cid:90) R dy (cid:104) φ ± ( x ) φ ± ( y ) (cid:105) , J ≡ (cid:90) R dx (cid:90) R dy (cid:104) φ + ( x ) φ − ( y ) (cid:105) . (2.6)A diagrammatic representation of these integrals are in figure 2. The two-point function isgiven by (cid:104) D + D − (cid:105) = exp (cid:18)(cid:90) R dx [ λ + φ + ( x ) + λ − φ − ( x )] (cid:19) = (cid:88) n ≥ n ! (cid:104) (cid:18)(cid:90) R dx [ λ + φ + ( x ) + λ − φ − ( x )] (cid:19) n (cid:105) = (cid:88) n ≥ (cid:0) λ I + + λ − I − + 2 λ + λ − J (cid:1) n n n ! = exp (cid:18) λ I + + λ − I − λ + λ − J (cid:19) . (2.7)This yields the normalized correlator (cid:104) D + D − (cid:105) N ≡ (cid:104) D + D − (cid:105)(cid:104) D + (cid:105)(cid:104) D − (cid:105) = e λ + λ − J . (2.8)The J -integral is found using a Julian-Schwinger parametrization and regularizing one ofthe defects such that it is of finite length L (cid:29) J = lim L →∞ (cid:90) + L − L dy (cid:90) R dx (cid:90) ∞ du e − u ( x − y ) − uR π = lim L →∞ (cid:90) + L − L dy (cid:90) ∞ du e − uR π / √ u = lim L →∞ L πR . (2.9)– 4 – , Figure 2 . A diagrammatic depiction of the integrals I ± , J and K . This yields (cid:104) D + D − (cid:105) N = lim L →∞ e λ + λ − L/ (4 πR ) . (2.10)The non-perturbative dependence in R describes a Casimir effect between the two defects.Let us now probe the two-point correlator with a bulk scalar placed at z ≡ z (cid:107) ˆ x (cid:107) + z ⊥ ˆ x ⊥ , | z ⊥ | > R . (2.11)Note that in the fusion limit R → + this scalar is not squeezed in between the two defects. (cid:104) D + D − φ ( z ) (cid:105) = (cid:88) n ≥ n ! (cid:104) (cid:18)(cid:90) R dx [ λ + φ + ( x ) + λ − φ − ( x )] (cid:19) n φ ( z ) (cid:105) = (cid:88) n ≥ n + 1(2 n + 1)! (cid:104) (cid:18)(cid:90) R dx [ λ + φ + ( x ) + λ − φ − ( x )] (cid:19) n (cid:105)(cid:104) (cid:90) R dx [ λ + φ + ( x ) + λ − φ − ( x )] φ ( z ) (cid:105) = (cid:104) D + D − (cid:105) [ λ + K + ( z ) + λ − K − ( z )] ,K ± ( z ) ≡ (cid:90) R dx (cid:104) φ ± ( x ) φ ( z ) (cid:105) . (2.12)A diagrammatic representation of this integral is in figure 2. This integral can again besolved with a Julian-Schwinger parametrization K ± ( z ) = (cid:90) R dx (cid:90) ∞ du e − u ( z (cid:107) − x ) − u ( z ⊥ ∓ R ) π = 14 π | z ⊥ ∓ R | = (cid:88) n ≥ ( ± R ) n πz n +1 ⊥ . (2.13)It yields the normalized correlator (cid:104) D + D − φ ( z ) (cid:105) N ≡ (cid:104) D + D − φ ( z ) (cid:105)(cid:104) D + (cid:105)(cid:104) D − (cid:105) = lim L →∞ e λ + λ − L/ (4 πR ) (cid:88) n ≥ λ + + ( − n λ − πz n +1 ⊥ R n . (2.14)Now we wish to study whether this correlator can be written in terms of a fused defect (cid:104) D + D − φ ( z ) (cid:105) N ? = (cid:104) D + D − (cid:105) N (cid:104) D f φ ( z ) (cid:105) N . (2.15) Please note that we could have chosen a normalization where we divide with (cid:104) D + D − (cid:105) rather than (cid:104) D + (cid:105)(cid:104) D − (cid:105) . Such normalization would remove the non-perturbative Casimir effect. – 5 –n order to understand this we need to find some kind of operators that the series in (2.14)would correspond to. For this purpose, let us consider the following defect D n = exp (cid:18)(cid:90) R dx∂ n ⊥ φ ( x ˆ x (cid:107) ) (cid:19) , ∂ ⊥ ≡ ∂ x ⊥ = ∂ R . (2.16)If we probe its correlator with a bulk scalar (cid:104) D n φ ( z ) (cid:105) = (cid:104) D n (cid:105) (cid:90) R dx (cid:104) ∂ n ⊥ φ ( x ˆ x (cid:107) ) φ ( z ) (cid:105) = (cid:104) D n (cid:105) lim R → + ∂ n ⊥ K − ( z ) = (cid:104) D n (cid:105) ( − n n !4 πz n +1 ⊥ . (2.17)The normalized correlator is thus (cid:104) D n φ ( z ) (cid:105) N ≡ (cid:104) D n φ ( z ) (cid:105)(cid:104) D n (cid:105) = ( − n n !4 πz n +1 ⊥ ⇒ πz n +1 ⊥ = ( − n n ! (cid:104) D n φ ( z ) (cid:105) N . (2.18)Compare with (2.14) to find (cid:104) D + D − φ ( z ) (cid:105) N = lim L →∞ e λ + λ − L/ (4 πR ) (cid:88) n ≥ λ − + ( − n λ + n ! R n (cid:104) D n φ ( z ) (cid:105) N . (2.19)From this we can deduce that the defects have fused into a single defect D + D − = lim L →∞ e λ + λ − L/ (4 πR ) D f ,D f = exp (cid:88) n ≥ λ − + ( − n λ + n ! R n (cid:90) R dx∂ n ⊥ φ ( (cid:126)x ) . (2.20)It is possible to check that this fusion holds for other probed bulk fields, e.g. (cid:104) D + D − φ ( z ) (cid:105) N = (cid:104) D + D − (cid:105) N (cid:104) D f φ ( z ) (cid:105) N . (2.21)Now let us discuss the fused defect at (2.20). In the strict fusing limit it is conformal, butas we zoom in and pick up perturbations in R , it has dimensionfull coupling constants, R n , and is thus no longer conformal. Among other things this means that it is not possiblefor us to use this fusion for bootstrap purposes. In order to prove that the fusion (2.20) is correct we need to show that it does not matterfor the path integral (which generates all of the correlators) whether we consider the twoline defects, or the fused defect. The path integral is Z [ J ] = (cid:90) D φ exp (cid:34)(cid:90) R d d d x (cid:32) ( ∂φ ) J φ (cid:33)(cid:35) . (3.1) In principle we could instead consider lim R → + ∂ n ⊥ K + . However, as we will see in the next section, thisyields the wrong fusion. – 6 –et us perform a partial integration on the source term and add a zero on the form (cid:0) ∂ − J (cid:1) − (cid:0) ∂ − J (cid:1) . Then we complete the square for φ and perform a partial integrationon the residual (cid:0) ∂ − J (cid:1) -term Z [ J ] = (cid:90) D φ exp (cid:34)(cid:90) R d d d x (cid:32) ( ∂φ ) − ∂ − J ∂φ + (cid:0) ∂ − J (cid:1) − (cid:0) ∂ − J (cid:1) (cid:33)(cid:35) = (cid:90) D φ exp (cid:34)(cid:90) R d d d x (cid:32) (cid:2) ∂ (cid:0) φ − ∂ − J (cid:1)(cid:3) + J ∂ − J (cid:33)(cid:35) . (3.2)Here ∂ − is the Green’s function G . Perform the field redefinition φ ( x ) → φ ( x ) + (cid:0) ∂ − J (cid:1) ( x ) ≡ φ ( x ) + (cid:90) R d d d yG ( x − y ) J ( y ) . (3.3)This gives us the normalized path integral Z [ J ] Z [0] = e ζ [ J ] , ζ [ J ] = (cid:90) R d d d x (cid:90) R d d d y J ( x ) G ( x − y ) J ( y )2 . (3.4)To prove that the fusion (2.20) is correct we need to show that the above path integral forthe two line defects is the same as that for the fused defect. We will do this by writing thedefects as sources. For the two defects we write J b ( x ) = λ + δ ( (cid:126)x ⊥ − (cid:126)R ) + λ − δ ( (cid:126)x ⊥ + (cid:126)R ) + ˜ J ( x ) , (cid:126)R = R ˆ x ⊥ . (3.5)Here (cid:126)x ⊥ is the vector orthogonal to the defects, and ˜ J ( x ) is the actual source for φ . Forthe fused defect we write J f ( x ) = δ ( (cid:126)x ⊥ ) (cid:88) n ≥ λ − + ( − n λ + n ! R n ∂ nx ⊥ + ˜ J ( x ) . (3.6)We want to show that ζ [ J b ] = ζ [ J f ] . (3.7)If we insert (3.5) into (3.4) ζ [ J b ] = (cid:90) R dx (cid:107) (cid:90) R dy (cid:107) (cid:18) λ + λ − G ( (cid:126)s (cid:107) ) + λ + λ − G ( (cid:126)s (cid:107) + 2 (cid:126)R ) (cid:19) ++ (cid:90) R dx (cid:107) (cid:90) R d d d y ˜ J ( y ) (cid:104) λ + G ( (cid:126)s (cid:107) + (cid:126)y ⊥ − (cid:126)R ) + λ − G ( (cid:126)s (cid:107) + (cid:126)y ⊥ + (cid:126)R ) (cid:105) ++ (cid:90) R d d d x (cid:90) R d d d y ˜ J ( x ) G ( x − y ) ˜ J ( y )2 . (3.8)Here (cid:126)x ⊥ , (cid:126)y ⊥ are vectors orthogonal to the defects , and (cid:126)s (cid:107) is the difference between theparallel coordinates along the defects (cid:126)s (cid:107) ≡ ( x (cid:107) − y (cid:107) )ˆ x (cid:107) , (cid:126)y ⊥ = y i ˆ x i ⊥ , (cid:126)x ⊥ = x i ˆ x i ⊥ , i ∈ { , , } . (3.9)– 7 – igure 3 . Two circular defects that are separated by a distance of R . We want to show that the functional (3.8) is the same as (3.6) inserted in (3.4) ζ [ J f ] = (cid:90) R dx (cid:107) (cid:90) R dy (cid:107) (cid:88) m,n ≥ λ R m + n + λ − ( − R ) m + n + 2 λ + λ − ( − R ) m R n m ! n ! ∂ mx ⊥ ∂ ny ⊥ G ( (cid:126)s (cid:107) + (cid:126)x ⊥ − (cid:126)y ⊥ ) (cid:12)(cid:12)(cid:12) (cid:126)x ⊥ = (cid:126)y ⊥ =0 ++ (cid:90) R dx (cid:90) R d d d y ˜ J ( y ) (cid:88) n ≥ λ − R n + λ + ( − R ) n n ! ∂ nx ⊥ G ( (cid:126)s (cid:107) + (cid:126)x ⊥ − (cid:126)y ⊥ ) (cid:12)(cid:12) (cid:126)x ⊥ =0 ++ (cid:90) R d d d x (cid:90) R d d d y ˜ J ( x ) G ( x − y ) ˜ J ( y )2 . This is the Taylor expansion of (3.8) around R = 0 . Here we used that the Green’s functionis symmetric w.r.t. x and yG ( x − y ) = G ( y − x ) ⇒ G ( (cid:126)s (cid:107) , (cid:126)x ⊥ − (cid:126)y ⊥ ) = G ( (cid:126)s (cid:107) , (cid:126)y ⊥ − (cid:126)x ⊥ ) . (3.10)This proves that the defect in (2.20) is indeed the fusion of the line defects in (2.5). In this section we will provide another example of fusion in a free theory. We will considertwo scalar Wilson loops of radius r in d = 4 dimensions at a distance R ≡ | (cid:126)R | > fromeach other (see figure 3) D ± = exp (cid:18) λ ± (cid:90) π dθφ ( r ( c θ ˆ x (cid:107) + s θ ˆ x (cid:107) ) ± (cid:126)R ) (cid:19) ≡ exp (cid:18) λ ± (cid:90) π dθφ ± ( θ ) (cid:19) . (4.1)Here c θ ≡ cos( θ ) , s θ ≡ sin( θ ) and ˆ x j (cid:107) , j ∈ { , } are two of the four unit vectors. (cid:126)R is avector orthogonal to both ˆ x j (cid:107) . We can proceed in the same way as in section 2, encountering– 8 –lightly different integrals corresponding to (2.6) and (2.12) (we probe the defect correlatorwith a bulk field at origo) J = (cid:90) π dθ (cid:90) π dθ (cid:104) φ + ( θ ) φ − ( θ ) (cid:105) = A d ∆ φ (cid:90) π dθ (cid:90) π dθ (cid:90) ∞ du u ∆ φ − Γ ∆ φ e − u ( r (1 − c θ − θ )+2 R ) = π A d ∆ φ − ( r + 2 R ) ∆ φ F (cid:18) ∆ φ , ∆ φ + 12 ; 1; r ( r + 2 R ) (cid:19) = 14 R √ r + R , (4.2) K ± = (cid:90) π dθ (cid:104) φ ± ( θ ) φ (0) (cid:105) = (cid:90) π dθ A d ( r + R ) ∆ φ = 12 π ( r + R )= (cid:88) n ≥ ( − n R n πr n − . (4.3)We find the fusing to be D + D − = e λ + λ − / (4 R √ r + R ) D f ,D f = exp (cid:88) n ≥ λ + + λ − (2 n )! R n (cid:90) π dθ ∇ nR φ ( θ ) . (4.4)Here ∇ R ≡ (cid:126)R · (cid:126) ∇ is the directional derivative w.r.t. the vector (cid:126)R . The fusion is proven inthe same way as in section 3 using the path integral formalism and by treating the defectsas sources for the fundamental scalar φ in the bulk. The last example of fusion that we will study is between two concentric circles. We willagain consider two scalar Wilson loops in d = 4 dimensions in a free theory, separated bya distance R . Unlike the previous section the defects will be concentric (see figure 4), i.e.the Wilson loops will have different radii r ± RD ± = exp (cid:18) λ ± (cid:90) π dθφ (cid:16) ( r ± R )( c θ ˆ x (cid:107) + s θ ˆ x (cid:107) ) (cid:17)(cid:19) ≡ exp (cid:18) λ ± (cid:90) π dθφ ± ( θ ) (cid:19) . (5.1)The integrals we encounter in (cid:104) D + D − (cid:105) and (cid:104) D + D − φ (0) (cid:105) (corresponding to (2.6) and (2.12))are given by J = (cid:90) π dθ (cid:90) π dθ (cid:104) φ + ( θ ) φ − ( θ ) (cid:105) = A d ∆ φ (cid:90) π dθ (cid:90) π dθ (cid:90) ∞ du u ∆ φ − Γ ∆ φ e − u ( r + R − ( r − R ) c θ − θ ) = π A d ∆ φ − ( r + 2 R ) ∆ φ F (cid:18) ∆ φ , ∆ φ + 12 ; 1; ( r − R ) ( r + R ) (cid:19) = 14 πR , (5.2)– 9 – - R r + R D + D - x ∥ x ∥ Figure 4 . The concentric circular defects are separated by a distance of R . K ± = (cid:90) π dθ (cid:104) φ ± ( θ ) φ (0) (cid:105) = (cid:90) π dθ A d ( r ± R ) φ = 12 π ( r ± R ) = (cid:88) n ≥ ( ∓ ) n ( n + 1) R n πr n +2 . (5.3)The fusion is very similar to the line defects in section 2 D + D − = e λ + λ − / (4 πR ) D f ,D f = exp (cid:88) n ≥ λ − + ( − n λ + n ! R n (cid:90) π dθ∂ nR φ ( θ ) . (5.4)Here ∂ nR are derivatives in the Radial direction. This fusion is proven in the same way asin section 3. We have shown how several different defects in Gaussian models can be fused using oneand the same method. This procedure should in principle work for other defects as well.Although we need to be able to express the defects as exponentials. It would be interestingto study whether this method works for fusion of defects with composite operators or higherdimensional defects. One has to be careful though, as such operator will generate an RGflow on the defect.In this work we only considered free theories and it is thus worthwhile to study how theproposed method in this paper can be modified to work in an interacting theory (say with– 10 – quartic interaction in the bulk). This will affect the Wick contractions in equations (2.2),(2.7) and the equation above (2.12).Let us also comment a bit on the conformal block decomposition in [9]. The defect two-point correlators in section 4 and 5 can in principle be decomposed in these blocks, andthere should be three number of independent cross-ratios. However, we find in appendixA that these cross-ratios are not independent from each other for two parallel circles ofcodiemension three in four spacetime dimensions. This means that we cannot guaranteethat this conformal block decomposition is unique. In the case of concentric Wilson loops studied in section 5, we can also (in principle) de-compose the defect two-point correlator using the method in [12]. However, in order to usethis method we first need to find the
Harish-Chandra wave-functions for codimension threedefects ( N = 3 ).It would also be interesting to study whether fusion of defects can be applied to some moreconcrete physical examples. One such example may be the twist defect that appear in thecontext of Rényi entropy [22–24], where the so-called c - function can be found through thefusion limit of two such defects [25]. It is possible that this could be understood as a fuseddefect using the methods of this paper. Acknowledgement
The author is grateful to Marco Meineri and Emilio Trevisani for discussions regarding thisproject and for commenting on the manuscript. He is thankful to the organizers of theBOOTSTRAP 2019 workshop at Perimeter institute, where a majority of this project tookplace. AS is supported by Knut and Alice Wallenberg Foundation KAW 2016.0129.
A Conformal block decomposition
In this appendix we study the conformal block decomposition in [9] of the defect two-pointcorrelators in section 4 and 5. For this purpose we will assume the more general scalarWilson loop D ± = exp (cid:18) λ ± (cid:90) π dθφ ( r ± ( c θ ˆ x (cid:107) + s θ ˆ x (cid:107) ) ± ρ ˆ x ⊥ ) (cid:19) . (A.1)Here the two defects are of different radii r + and r − , and ρ is the distance between thedefects. We get the configuration in section 4 in the limit r + → r − ≡ r with ρ ˆ x ⊥ ≡ (cid:126)R ,and that in section 5 in the limit ρ → with r ± = r ± R . We need the following lightcone As in figure 3, but with different radii on the two circles. I.e. we should treat appendix A as a step closer to an example of this block decomposition. Somethingthat has not been done before to our knowledge. – 11 –ectors to find the conformal cross-ratios P ± = (0 , , , , , ,P ± = (0 , , , , , ,P ± = (cid:18) r ± , ρ r ± − r ± , ± ρr ± , , , (cid:19) . (A.2)We are interested in the matrix M αβ = ( (cid:126)P α + · (cid:126)P γ − )( (cid:126)P γ + · (cid:126)P β − ) = diag (cid:0) , , ξ (cid:1) αβ , ξ = r − + r − ρ r − r + . (A.3)Here α, β, γ ∈ { , , } , where summation over γ is implicit. The three cross-ratios, η a , with a ∈ { , , } , are given by η a = tr ( M a ) = 2 + ξ a . (A.4)As we can see, they can all be expressed in terms of another cross-ratio ξ , and thus they arenot independent from each other. This means that we cannot guarantee that the conformalblock decomposition is unique. We can proceed to follow the procedure in [9] to find anODE (w.r.t. ξ ) for the conformal blocks that we can possibly solve as a series expansion in ξ . However, since we cannot determine whether this decomposition is unique, we will notwrite out any details on this. The interested reader may study the attached Mathematicafile on arXiv. References [1] C. P. Herzog and N. Kobayashi, “The O ( N ) model with φ potential in R × R + ,” JHEP (2020) 126, arXiv:2005.07863 [hep-th] .[2] C. Behan, L. Di Pietro, E. Lauria, and B. C. Van Rees, “Bootstrapping boundary-localizedinteractions,” arXiv:2009.03336 [hep-th] .[3] V. Procházka and A. Söderberg, “Spontaneous symmetry breaking in free theories withboundary potentials,” arXiv:2012.00701 [hep-th] .[4] P. Dey and A. Söderberg, “On Analytic Bootstrap for Interface and Boundary CFT,” arXiv:2012.11344 [hep-th] .[5] M. Billò, V. Gonçalves, E. Lauria, and M. Meineri, “Defects in conformal field theory,” JHEP (2016) 091, arXiv:1601.02883 [hep-th] .[6] A. Söderberg, “Anomalous Dimensions in the WF O( N ) Model with a Monodromy LineDefect,” JHEP (2018) 058, arXiv:1706.02414 [hep-th] .[7] M. Lemos, P. Liendo, M. Meineri, and S. Sarkar, “Universality at large transverse spin indefect CFT,” JHEP (2018) 091, arXiv:1712.08185 [hep-th] .[8] E. Lauria, M. Meineri, and E. Trevisani, “Spinning operators and defects in conformal fieldtheory,” JHEP (2019) 066, arXiv:1807.02522 [hep-th] .[9] A. Gadde, “Conformal constraints on defects,” arXiv:1602.06354 [hep-th] .[10] M. Fukuda, N. Kobayashi, and T. Nishioka, “Operator product expansion for conformaldefects,” JHEP (2018) 013, arXiv:1710.11165 [hep-th] . – 12 –
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