String integrability of defect CFT and dynamical reflection matrices
NNORDITA 2021-014
String integrability of defect CFTand dynamical reflection matrices
Georgios Linardopoulos ∗ and Konstantin Zarembo , † Institute of Nuclear and Particle Physics, National Centre for ScientificResearch, ”Demokritos”, 153 10, Agia Paraskevi, Greece Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100Copenhagen, Denmark [email protected], [email protected]
Abstract
The D3-D5 probe-brane system is holographically dual to a defect CFTwhich is known to be integrable. The evidence comes mainly from thestudy of correlation functions at weak coupling. In the present work weshed light on the emergence of integrability on the string theory side. Wedo so by constructing the double row transfer matrix which is conservedwhen the appropriate boundary conditions are imposed. The correspond-ing reflection matrix turns out to be dynamical and depends both on thespectral parameter and the string embedding coordinates. ∗ Also at the Department of Nuclear and Particle Physics, NKU of Athens, Greece. † Also at ITEP, Moscow, Russia. a r X i v : . [ h e p - t h ] F e b Introduction
The string sigma model on AdS × S [1] is an integrable two-dimensionalfield theory [2]. Integrability has important implications for the AdS/CFTcorrespondence extending, via holography, to four dimensions [3] and givingrise to powerful tools to explore the duality at the non-perturbative level.The sigma model ordinarily describes closed strings which are automat-ically integrable. Integrability of a string ending on a D-brane is not soobvious because it can be broken by the boundary conditions at the string’sendpoint. The question of which D-branes in AdS × S are integrable isnon-trivial and was addressed in detail in [4].D-branes arise in a variety of holographic setups. An example we will beconcerned with is field theory in the presence of a domain wall. The setupis realized by imposing Nahm-pole boundary conditions [5], which break thegauge symmetry from SU ( N + k ) to SU ( N ) on one side of the domain wall.The system preserves scale invariance and gives rise to a defect CFT (dCFT)[6, 7]. In string theory, the domain wall is a footprint of the D3-D5 inter-section. Its holographic dual is a probe D5-brane embedded in AdS as a4-dimensional hyperplane [6]: x = κz, κ ≡ πk √ λ ≡ tan α, (1.1)where λ is the ’t Hooft coupling and the angle α specifies the inclination ofthe D5-brane relative to the hyperplane x =
0. In the standard Poincar´ecoordinates of AdS , ds = dx µ dx µ + dz z , (1.2)the hyperplane x = κz has the AdS geometry. The S embedding of theD5-brane is an equatorial S with k units of internal gauge field flux: F = k ε ijk x i dx j ∧ dx k , i, j, k = , , , (1.3)where x i are the 6d Cartesian coordinates describing the embedding S ⊂ S .The non-magnetic D5-brane with k = α and nonzeromagnetic flux does not fall into the classification scheme of [4]. Yet thereis overwhelming evidence that integrability persists for any value of k . The2vidence comes mainly from the field-theory side where efficient integrability-based techniques have been developed to compute one-point functions of localoperators, in perturbation theory [9] and beyond [10] (these developments aresummarized in [11]). Moreover, integrability bootstrap solves for dCFT cor-relation functions at any coupling [12, 13]. Lack of basic understanding whythe D3-D5 system is integrable makes this picture incomplete, we believe.Our goal is to fill this gap.The conserved charges of an integrable system with a boundary are en-coded in the double row transfer matrix [14] which is built from the Laxconnection and the reflection matrix. The latter typically has constant nu-merical entries. Under this assumption, an elegant classification scheme ofintegrable boundary conditions has been put forward [4] establishing a one-to-one link between integrable D-branes and Z automorphisms of the un-derlying symmetry algebra.Constancy and independence from the spectral parameter are very nat-ural assumptions. However, there are examples of integrability-preservingboundary conditions, going back to the work of Corrigan and Sheng [15],that are not described by constant reflection matrices. In principle, thereflection matrix can depend on the spectral parameter, or the dynamicalvariables, or both. The dynamical reflection matrices arise, for example, in O ( N ) models with Robin boundary conditions [16, 17]. Robin (i.e. mixedNeumann-Dirichlet) boundary conditions are precisely the ones that describea string attached to a D-brane with internal magnetic flux. This explains,perhaps, why the classification of [4] missed the magnetized D3-D5 system,and points towards the dynamical character of reflection in this case.Since both AdS and S are symmetric spaces, we start by briefly review-ing integrability in symmetric cosets with boundaries. We then constructthe reflection matrix of a string ending on the D5-brane (1.1), (1.3). This isdone separately in § and in § , because the correspondingequations of motion decouple in the conformal gauge. In section 5 we discusssymmetries and in section 6 we comment on how to include the fermions. A symmetric coset space G / H is defined by a Z decomposition of its sym-metry algebra, g = h ⊕ h . The sigma model current gets decomposed as J = g − dg = J + J . (2.1)3auge transformations act on g ∈ G from the right g → gh , under which J transforms as a gauge field and J as a matter field in the adjoint. Theequations of motion are equivalent to the flatness of the Lax connection [18]: L ( x ) = J + x + x − J − xx − ⋆ J ≡ J + A ( x ) , (2.2)where x is the spectral parameter. Defining the fixed frame current, j = gJ g − , (2.3)the spectral parameter-dependent part A ( x ) of the Lax connection (2.2) takesthe following form, in the fixed frame: a = gAg − = x − ( j − x ∗ j ) . (2.4)The connection a ( x ) depends only on the matter current j . It is also flat: da + a ∧ a = . (2.5) Integrable boundary conditions
String dynamics takes place for σ > σ = An infinite tower ofconserved charges can be constructed by expanding the monodromy matrix M( x ) = g ( )—→ P exp ⎛⎝ ∞ ∫ ds L σ ( s ; x )⎞⎠ (2.6)around appropriate values of the spectral parameter x , see § M( x ) = − a τ ( x )M( x ) . (2.7) More details can be found in many relevant reviews, e.g. [19]. We are considering a semi-infinite string. This is not a real restriction because inte-grability is broken (or preserved) locally. For example, if the two ends of the string areattached to the D-brane, integrability imposes two independent conditions at each of theendpoints. σ = T ( x ) = M t (− x ) U ( x )M( x ) . (2.8)Apart from its explicit dependence on the spectral parameter x , the reflectionmatrix U can also be dynamical. In other words, U may depend on theembedding coordinates at the string’s endpoint and through them implicitlyon time. The time derivative of the double row monodromy matrix (2.8)follows from (2.7):˙ T ( x ) = M t (− x ) ( ˙ U − a tτ (− x ) U ( x ) − U ( x ) a τ ( x )) M( x ) . (2.9)If the time derivative (2.9) vanishes, the double row transfer matrix will gen-erate infinitely many conserved charges. Therefore the boundary conditionsat σ = U ( x ) ! = a tτ (− x ) U ( x ) + U ( x ) a τ ( x ) , (2.10)where the symbol ! = denotes restriction to the boundary at the string’s end-point σ =
0, i.e. A ! = B iff A ( τ, ) = B ( τ, ) . (2.11)Plugging the connection (2.4) into the integrability condition (2.10) we obtain˙ U ! = x − [ { j tτ U + U j τ } + x { j tσ U − U j σ } ] . (2.12)In the simplest case, the reflection matrix U is a constant matrix that de-pends neither on the spectral parameter x nor on time τ . In this case theintegrability condition (2.12) reduces to j tτ U + U j τ ! = j tσ U − U j σ ! = . (2.13) We assume that the currents are taken in some representation of the symmetry algebraso that the monodromy matrix is literally a matrix in this representation. M t then denotesmatrix transposition. More abstractly, M t can be a Z transformation on G C induced byan anti-unitary involution of the Lie algebra g . D-brane in AdS The coset AdS = SO ( , )/ SO ( , ) can be realized by means of the 5-dimensional Dirac matrices γ a in the (− + + + +) signature. The denominatoralgebra so ( , ) is spanned by γ ab , while so ( , ) contains both γ ab and γ a ,i.e. so ( , ) = ⟨ γ a , γ ab ⟩ . The coset decomposition is thus h = h ⊕ h , h = ⟨ γ ab ⟩ , h = ⟨ γ a ⟩ , (3.1)where a, b = , . . . ,
4. The Dirac matrices satisfy γ ta = K − γ a K , γ tab = −K − γ ab K . (3.2)In the chiral representation K = γ , but we will never need this explicitform. Below, we will also split the Dirac matrices into the SO ( , ) compo-nents γ µ , µ = . . . γ , as well as use the chiral projectorsΠ ± = ± γ . (3.3)The integrability condition (2.12) can be concisely formulated in terms ofthe transposition brackets, which are defined as ⟨ A, B ⟩ ± = K A t K − B ± BA, (3.4)for any two matrices A , B . The integrability condition (2.12) is then equiv-alent to d ̂ U dτ ! = x − (⟨ j τ , ̂ U ⟩ + + x ⟨ j σ , ̂ U ⟩ − ) , (3.5)where ̂ U = K U . (3.6)The bracket of the Dirac matrices follows readily from their transpositionproperties (3.2): ⟨ γ a , Γ ⟩ ± = [ γ a , Γ ] ± , ⟨ γ ab , Γ ⟩ ± = − [ γ ab , Γ ] ∓ , (3.7)where Γ is any matrix. Some useful identities of the transposition bracketsare listed in appendix A. The standard generators of the conformal algebraare D = γ , P µ = Π + γ µ , K µ = Π − γ µ , L µν = γ µν . (3.8)6 oset representative and current Choosing the coset representative g = e P µ x µ z D , (3.9)the Z decomposition of the moving frame current (2.1) takes the form J = z γ µ dx µ , J = z ( γ dz + γ µ dx µ ) . (3.10)To verify that (3.9) correctly describes AdS , we note that the quadraticform of the Lie algebra tr [ J ] = dx µ dx µ + dz z , (3.11)reproduces the Poincar´e metric (1.2). The fixed frame current (2.3) followsby elementary Dirac algebra: j = z [ ( zdz + xdx ) ( D − xP ) + ( z + x ) P dx + Kdx + L µν x µ dx ν ] , (3.12)where the single summations over covariant indices have been omitted (e.g. xdx ≡ x µ dx µ ) and x ≡ x µ x µ . Boundary conditions
The longitudinal and transverse coordinates of thetilted AdS brane (1.1) are:longitudinal ∶ x , , , x ∥ ≡ x sin α + z cos α (3.13)transverse ∶ x ⊥ ≡ x cos α − z sin α. (3.14)The string boundary conditions on the D5-brane are Neumann for the lon-gitudinal coordinates ( x , , , x ∥ ) and Dirichlet for the transverse coordinate( x ⊥ ), i.e. ´ x , , = ´ x ∥ ! = ( Neumann ) (3.15)˙ x ⊥ ! = x ⊥ ! = ( Dirichlet ) . (3.16)Below we determine a reflection matrix ̂ U that satisfies the integrabilitycondition (3.5) upon imposing (3.15)–(3.16), thus showing that the D5-braneis integrable in AdS . 7 .1 Vertical brane Let us first consider the non-magnetic D5-brane for which k =
0. (1.1) thenimplies that the inclination of the brane relative to the hyperplane x = α =
0. The brane is perpendicular to the x axis. At the string’sendpoint, j τ ! = z [ ( z ˙ z + x i ˙ x i ) ( D − x j P j ) + ( z + x i ) ˙ x j P j + ˙ x i K i + x i ˙ x j L ij ] (3.17) j σ ! = ´ x z [( z + x i ) P + K + x i L i ] , (3.18)where i, j = , ,
2. Note that the conformal generators (3.8) making up thefixed frame current (3.12) split into two disjoint groups, those that appearin j τ and those that appear in j σ . The two equations in (2.13) thereforedecouple and can be solved separately.In terms of the transposition brackets (3.4), the integrability condition(2.13) is equivalent to a set of seven equations: ⟨ D, ̂ U ⟩ + = , ⟨ P i , ̂ U ⟩ + = , ⟨ K i , ̂ U ⟩ + = , ⟨ L ij , ̂ U ⟩ + = ⟨ P , ̂ U ⟩ − = , ⟨ K , ̂ U ⟩ − = , ⟨ L i , ̂ U ⟩ − = . (3.20)Given that the brane is perpendicular to the x axis, its normal four-vectorcan be written as n µ = ( , , , ) , so that the most natural solution of (3.19)–(3.20) is ̂ U = n µ γ µ = γ . (3.21)Indeed, by using (3.7), (A.2) one can readily check that (3.21) satisfies (3.19)–(3.20) and thus (3.5) holds for the constant reflection matrix, in accordancewith the findings of [4]. We now consider an arbitrary inclination angle α . Introducing the longitu-dinal coordinates, y µ = ( x i , x ∥ ) , (3.22)the string boundary conditions (3.15)–(3.16) can be written as´ y µ ! = ˙ x ⊥ ! = x ⊥ ! = . (3.23)8e will use the longitudinal coordinates y µ alongside the spacetime four-vector x µ = ( x i , x ) . Noting the identity x + z = y + x ⊥ , (3.24)the values of the fixed frame currents (3.12) on the boundary (at x ⊥ = j τ ! = z [ y µ ˙ y µ ( D − x ν P ν ) + y ˙ x µ P µ + ˙ x µ K µ + x µ ˙ x ν L µν ] (3.25) j σ ! = ´ x ⊥ cos α z ( y P + K + x µ L µ ) . (3.26)Because the equation ⟨ j τ , γ ⟩ + ! = ⟨ j τ , γ ⟩ + is a total derivative: ⟨ j τ , γ ⟩ + ! = κ dSdτ , S ≡ x µ γ µ − Π + − y Π − z . (3.27)The matrix S satisfies two remarkable identities, which hold once x ⊥ is setto zero: ⟨ j τ , S ⟩ + ! = − ˙ S, ⟨ j σ , S ⟩ − ! = . (3.28)These can be checked by using (3.7) and the formulae in appendix A. Theseidentities suggest the following ansatz for the reflection matrix ̂ U : ̂ U = γ + CS, (3.29)where C is a constant that may depend on the spectral parameter. Theansatz goes through the integrability condition (3.5) by virtue of (3.27),leaving behind an algebraic equation for C : C = x − ( κ − C ) ⇒ C = κ x + . (3.30)This leads to the following solution for the reflection matrix: ̂ U = γ + κ x + x µ γ µ − Π + − ( x + z ) Π − z . (3.31)The reflection matrix (3.31) is dynamical (i.e. it depends on the embeddingcoordinates of the string) and carries a non-trivial dependence on the spec-tral parameter x . According to the general analysis of coset models withboundaries [17, 20], the reflection matrix can be polynomial in the spectralparameter of at most degree two. This is also true for our solution, becausemultiplication with x + x .9 D-brane in S The 5-dimensional Dirac matrices γ ` a in the (+ + + + +) signature, representthe coset S = SO ( )/ SO ( ) . The numerator algebra so ( ) is formed by γ ` a and their commutators γ ` a ` b , while the denominator algebra so ( ) is spannedby γ ` a ` b . The coset decomposition reads h = h ⊕ h , h = ⟨ γ ` a ` b ⟩ , h = ⟨ γ ` a ⟩ , (4.1)where ` a, ` b = , . . . ,
5. As before, the matrices γ ` a , γ ` a ` b satisfy (3.2), (3.7). Coset representative and current
The coset parametrization of S is g = n + iγ ` a n ` a , (4.2)where n = cos θ , n ` a = m ` a sin θ , m ` a m ` a = . (4.3)The coset variables n ` a , n are quite distinct from the S coordinates x a , x ( a = , . . . , parametrization, x a = m ( a − ) sin θ, x = cos θ, (4.4)we obtain the map x a = n n ( a − ) , x = n − . (4.5)The Z components of the moving frame current (2.1) that follow from thecoset representative (4.2) are J = γ ` a ` b n ` a dn ` b , J = iγ ` a ( n dn ` a − n ` a dn ) . (4.6)As a crosscheck, the quadratic form − tr [ J ] = dθ + sin θ ( dm ` a dm ` a ) = ∑ a = dx a , (4.7)correctly reproduces the S metric. The overall minus sign is related to thefact that the currents (4.6) of the 5-sphere occupy a block in the matrices ofthe AdS × S supercurrents, the supertrace of which reproduces the metricof the full space. The fixed frame current (2.3) is given, in the case of S , by j = i ( n − ) n dn ` a γ ` a − i ( n + ) dn n ` a γ ` a − n n ` a dn ` b γ ` a ` b . (4.8)10 oundary conditions The longitudinal and transverse coordinates of theS ⊂ S component of the D5-brane arelongitudinal ∶ x ∥ = ( x , x , x ) (4.9)transverse ∶ x ⊥ = ( x , x , x ) . (4.10)The string boundary conditions on the 2-sphere are Dirichlet for the trans-verse coordinates x ⊥ and, due to the internal flux (1.3), Neumann-Dirichletfor the longitudinal coordinates x ∥ :´ x = κ ( x ˙ x − x ˙ x ) (4.11)´ x = κ ( x ˙ x − x ˙ x ) , ˙ x = ˙ x = ˙ x = x = κ ( x ˙ x − x ˙ x ) . (4.13)In compact form these boundary conditions read´ x ∥ − κ ( x ∥ × ˙ x ∥ ) ! = x ⊥ ! = . (4.15)The 5-sphere coordinates x a , x also obey, ∑ a = x a ! = , x = x = x = . (4.16)In terms of the coset variables n ` a , n , the string boundary conditions (4.11)–(4.13) become n ´ n + n ´ n = κn ( n ˙ n − n ˙ n ) (4.17) n ´ n + n ´ n = κn ( n ˙ n − n ˙ n ) ˙ n = ˙ n = ˙ n = n ´ n + n ´ n = κn ( n ˙ n − n ˙ n ) , (4.19)while also n + n + n
23 ! = , n = n = , n = √ . (4.20)11 ntegrable boundary conditions To show that the D5-brane is inte-grable in S , we need to specify a reflection matrix ̂ U which satisfies theintegrability condition (3.5) upon imposing the boundary conditions (4.11)–(4.16), or equivalently (4.17)–(4.20). The values of the fixed frame currents(4.8) on the x ⊥ = σ = j τ ! = − n ` i ˙ n ` j γ ` i ` j (4.21) j σ ! = − i ´ n n ` i γ ` i − n ` i ( ´ n γ ` i + ´ n γ ` i ) − √ κ n ` i n ` j ˙ n ` k (cid:15) ` j ` k ` (cid:96) γ ` i ` (cid:96) . (4.22)Plugging (4.21)–(4.22) into the integrability condition (3.5), one can provethat the reflection matrix ̂ U = γ + κ xx + n ` i γ ` i n , (4.23)where ` i = , ,
3, satisfies it. Therefore the string boundary conditions (4.11)–(4.20) on the 5-sphere are integrable.
As we have already mentioned, the Taylor expansions of the monodromy ma-trices (2.6), (2.8) lead to infinite sets of conserved charges. The expansion at x = ∞ in particular, generates the conserved charges of the global symmetry.The aim of the present section is to determine the set of global symmetriesof the string sigma model on AdS × S that is preserved by the D5-brane.We first note that the monodromy matrix (2.6) can be written as: M( x ) = g ( ) —→ P exp ⎛⎝ ∞ ∫ ds L σ ( s ; x )⎞⎠ = —→ P exp ⎛⎝ ∞ ∫ ds a σ ( s ; x )⎞⎠ , (5.1)where a σ is the σ -component of the fixed frame Lax connection (2.4) a σ ( x ) = x − ( j σ − x j τ ) . (5.2)Taylor-expanding the path-ordered exponential (5.1) around x = ∞ leads to —→ P exp ⎛⎝ ∞ ∫ ds a σ ⎞⎠ = − x ∞ ∫ dsj τ + x ⎡⎢⎢⎢⎢⎣ ∞ ∫ dsj σ + ∞ ∫ s ∫ dsds ′ j ′ τ j τ ⎤⎥⎥⎥⎥⎦− . . . (5.3) Assuming appropriate boundary conditions at σ = ∞ . —→ P exp ⎛⎝ ∞ ∫ ds a σ ⎞⎠ = exp ( ∞ ∑ r = (− x ) r + Q r ) = − x Q + x ( Q + Q )− . . . (5.4)The first charge in the above hierarchy is just the Noether charge of the globalbosonic symmetry SO ( , ) × SO ( ) of the string sigma model on AdS × S : Q = ∞ ∫ dsj τ . (5.5) Double row monodromy matrix
It may seem that the double row mon-odromy matrix (2.8) generates the same number of conserved charges as themonodromy matrix (5.1), leading to the wrong conclusion that boundaries donot break any symmetries. In practice however, some charges get cancelledby folding (i.e. through the construction (2.8)) and are simply not there insystems with boundaries. In more detail, we need to expand the monodromymatrix T ( x ) = ←— P exp ⎛⎝ ∞ ∫ ds a tσ ( s ; − x )⎞⎠ U ( x )—→ P exp ⎛⎝ ∞ ∫ ds a σ ( s ; x )⎞⎠ (5.6)in 1 / x . Taking into account the general form of the AdS × S reflectionmatrices (3.31), (4.23), ̂ U ( x ) = ̂ U + x + ( x ̂ U + ̂ U ) , (5.7)as well as the expansion (5.3), we get the expansion of the double row mon-odromy matrix ˆ T ≡ K T around x = ∞ :ˆ T ( x ) = ̂ U + x ⎛⎝ ̂ U + ∞ ∫ ds ⟨ j τ , ̂ U ⟩ − ⎞⎠ + . . . (5.8)In order to identify the conserved charges we setˆ T ( x ) = ̂ U + x ˜ Q + x ( ˜ Q + ˜ Q ) + . . . , (5.9)13nding in particular, for the first conserved charge˜ Q = ̂ U + ∞ ∫ ds ⟨ j τ , ̂ U ⟩ − . (5.10)By comparing the bulk Noether charge (5.5) with the first conservedcharge (5.10) on the boundary, we can determine the fraction of the globalbosonic symmetry SO ( , ) × SO ( ) of the AdS × S string sigma modelthat is preserved by the D5-brane. The preserved symmetries correspond tothe set of generators for which the transposition bracket ⟨ j τ , ̂ U ⟩ − is nonzero.The respective charges are not eliminated by folding.Interestingly, the symmetries that are preserved by the D5-brane are de-termined by ̂ U and are thus independent of k . The conserved charges on S receive an extra contribution from ̂ U that is localized on the brane, implyingthat the endpoint of the string carries an R-charge.For AdS , (3.31) gives ̂ U = γ , ̂ U =
0, while the transposition identities(3.20) imply ⟨ P , ̂ U ⟩ − = ⟨ K , ̂ U ⟩ − = ⟨ L i , ̂ U ⟩ − = , (5.11)for i, j = , ,
2. These are the broken conformal generators. The preservedAdS symmetries are generated by { D, P i , K i , L ij } , (5.12)which spans the SO ( , ) subgroup of SO ( , ) . The domain wall preservesthe group of 3-dimensional conformal transformations, in agreement with theAdS geometry of the D5-brane in AdS.For S , the reflection matrix (4.23) gives ̂ U = γ , ̂ U = κn ` i γ ` i / n ,whereas ⟨ γ ` i , ̂ U ⟩ − = ⟨ γ ` i , ̂ U ⟩ − = ⟨ γ ` i , ̂ U ⟩ − = , (5.13)for ` i = , ,
3. Therefore the preserved S symmetry consists of the generators { γ ` i ` j , γ , γ , γ } . (5.14)The Dirac matrices { γ , γ , γ } satisfy the so ( ) algebra, and so do γ ` i ` j . Theunbroken symmetry group is thus SO ( )× SO ( ) which again agrees with theD5-brane geometry in S . The boundary itself carries an R-charge which isproportional to ̂ U and belongs to the broken part of the symmetry algebra.14 Including the fermions
Including the fermions is rather straightforward [4]. The symmetry algebra ofAdS × S is embedded in psu ( , ∣ ) , the Dirac-matrix representation beingbest suited for this purpose, the Z symmetry is replaced by Z [21] andtransposition by supertransposition. The reflection matrix is just the directsum of the AdS and S components (3.31), (4.23) that were computed above: U = [ U U ′ ] , (6.1)where the U block of the matrix corresponds to AdS, and U ′ to the sphere.The Lax connection is built from the Z components of the current J = g − dg = J + J + J + J , (6.2)where g is now the group element of P SU ( , ∣ ) , the currents J and J arebosonic (or even), while J , J are fermionic (or odd). The Lax connectionreads [2] L ( x ) = J + x + x − J − xx − ∗ J + √ x + x − J + √ x − x + J . (6.3)The flatness of L ( x ) is equivalent to the full set of equations of motion thatfollows from the AdS × S superstring action. The monodromy matrix isagain given by (2.6), where the Lax connection (2.2) is replaced by (6.3). Inthe double row construction (2.8), transposition gets replaced by supertrans-position, for consistency with Grassmann grading: T ( x ) = M st (− x ) U ( x )M( x ) . (6.4)In complete analogy with the bosonic case that was treated in the previoussection, the supercharges Q that are broken by the string boundary conditionsare determined from the condition: ⟨ Q, ̂ U ⟩ − = , (6.5)where ⟨⋅ , ⋅⟩ now denotes the supertransposition bracket ⟨ A, B ⟩ ± = KA st K − B ± BA, (6.6)15hat generalizes (3.4). Moreover, we have defined K = [K K] , ̂ U = [ γ γ ] . (6.7)Given that the odd elements of psu ( , ∣ ) are of the form Q = [ Q−Q † γ ] , Q st = [ K − γ KQ ∗ Q t ] , (6.8)it follows from (6.5) that the broken supercharges obey the reality condition, Q ∗ = K − γ Q γ K . (6.9)The reality condition (6.9) singles out exactly half of the supercharges whichare broken by the boundary conditions. The other half remains unbroken.We conclude that the brane (1.1), (1.3) is one-half BPS, in accordance withthe supergravity analysis of [22]. The reflection matrix that defines the hierarchy of conserved charges of astring ending on a D5-brane is maximally complicated since it depends, notonly on the spectral parameter, but also on dynamical variables. In quantumtheory the D5-brane carries internal degrees of freedom [12], since the ele-mentary excitations of the string form bound states upon reflection from theboundary. There are k such bound states [12]. The same parameter k con-trols both the inclination of the brane in AdS and the magnetic flux in S .In the classical regime of string theory the parameter k is very large, scalingnaturally as k ∼ √ λ . We believe that the proliferation of bound states andthe dynamical character of the reflection matrix are not unrelated. Anotherindication that some degrees of freedom localize on the brane is the boundarycontribution to the R-charge that appears in (5.10).There are other classes of integrable boundary conditions of the string inAdS × S that describe a variety of states and operators in the dual gaugetheory. Those associated with constant reflection matrices are completelyclassified [4]. It would be interesting to see which of them admit deforma-tions with dynamical reflection matrices. Extending the analogy with theD5-brane, we expect the deformation parameter to be quantized at finitecoupling and to correspond to the dimension of the boundary Hilbert space.16 cknowledgements We would like to thank M. de Leeuw and C. Kristjansen for interesting discus-sions. G.L. received funding from the Hellenic Foundation for Research andInnovation (HFRI) and the General Secretariat for Research and Technol-ogy (GSRT), in the framework of the first post-doctoral researchers support ,under grant agreement No. 2595. The work of K.Z. was supported by thegrant ”Exact Results in Gauge and String Theories” from the Knut and AliceWallenberg foundation and by RFBR grant 18-01-00460 A.
A Transposition bracket identities
Here we list a number of identities obeyed by the transposition brackets (3.4).The following formula follows directly from (3.7): ⟨ Π s γ µ , Γ ⟩ r = Π − s [ γ µ , Γ ] r + s r [ γ , Γ ] + γ µ . (A.1)Several particular cases of (A.1), used in the main text, are: ⟨ Π ± γ µ , γ ν ⟩ + = η µν Π ∓ ⟨ Π ± γ µ , γ ν ⟩ − = γ µν Π ∓ ⟨ Π ± γ µ , Π ± ⟩ + = γ µ ⟨ Π ± γ µ , Π ∓ ⟩ + = ⟨ Π ± γ µ , Π ± ⟩ − = ∓ γ γ µ ⟨ Π ± γ µ , Π ∓ ⟩ − = . (A.2) References [1] R. R. Metsaev and A. A. Tseytlin, “Type IIB superstring action in
AdS × S background” , Nucl. Phys. B533, 109 (1998) , hep-th/9805028 .[2] I. Bena, J. Polchinski and R. Roiban, “Hidden symmetries of the AdS × S superstring” , Phys. Rev. D69, 046002 (2004) , hep-th/0305116 .[3] J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = , JHEP 0303, 013 (2003) , hep-th/0212208 .[4] A. Dekel and Y. Oz, “Integrability of Green-Schwarz sigma models withboundaries” , JHEP 1108, 004 (2011) , .
5] D. Gaiotto and E. Witten, “Supersymmetric boundary conditions in
N = , J. Stat. Phys. 135, 789 (2009) , .[6] A. Karch and L. Randall, “Open and closed string interpretation of SUSYCFT’s on branes with boundaries” , JHEP 0106, 063 (2001) , hep-th/0105132 .[7] O. DeWolfe, D. Z. Freedman and H. Ooguri, “Holography and defectconformal field theories” , Phys. Rev. D66, 025009 (2002) , hep-th/0111135 . • K. Nagasaki, H. Tanida and S. Yamaguchi, “Holographic interface-particlepotential” , JHEP 1201, 139 (2012) , . • K. Nagasaki andS. Yamaguchi, “Expectation values of chiral primary operators inholographic interface CFT” , Phys. Rev. D86, 086004 (2012) , .[8] O. DeWolfe and N. Mann, “Integrable open spin chains in defect conformalfield theory” , JHEP 0404, 035 (2004) , hep-th/0401041 . • N. Mann andS. E. Vazquez, “Classical open string integrability” , JHEP 0704, 065 (2007) , hep-th/0612038 . • D. H. Correa and C. A. S. Young, “Reflecting magnonsfrom D7 and D5 branes” , J. Phys. A41, 455401 (2008) , . • D. H. Correa, V. Regelskis and C. A. S. Young, “Integrable achiral D5-branereflections and asymptotic Bethe equations” , J. Phys. A44, 325403 (2011) , . • N. MacKay and V. Regelskis, “Achiral boundaries and thetwisted Yangian of the D5-brane” , JHEP 1108, 019 (2011) , .[9] M. de Leeuw, C. Kristjansen and K. Zarembo, “One-point functions indefect CFT and integrability” , JHEP 1508, 098 (2015) , . • I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen and K. Zarembo, “One-point functions in AdS/dCFT from matrix product states” , JHEP 1602, 052 (2016) , . • I. Buhl-Mortensen, M. de Leeuw,A. C. Ipsen, C. Kristjansen and M. Wilhelm, “One-loop one-point functionsin gauge-gravity dualities with defects” , Phys. Rev. Lett. 117, 231603 (2016) , . • M. de Leeuw, C. Kristjansen and S. Mori, “AdS/dCFTone-point functions of the SU(3) sector” , Phys. Lett. B763, 197 (2016) , . • I. Buhl-Mortensen, M. de Leeuw, A. C. Ipsen, C. Kristjansenand M. Wilhelm, “A quantum check of AdS/dCFT” , JHEP 1701, 098 (2017) , . • M. de Leeuw, C. Kristjansen and G. Linardopoulos, “Scalarone-point functions and matrix product states of AdS/dCFT” , Phys. Lett. B781, 238 (2018) , .[10] I. Buhl-Mortensen, M. de Leeuw, A. C. Ipsen, C. Kristjansen andM. Wilhelm, “Asymptotic one-point functions in gauge-string duality withdefects” , Phys. Rev. Lett. 119, 261604 (2017) , .
11] M. de Leeuw, A. C. Ipsen, C. Kristjansen and M. Wilhelm, “Introduction tointegrability and one-point functions in
N = , Les Houches Lect. Notes 106, 352 (2019) , . • M. de Leeuw, “One-point functions in AdS/dCFT” , J. Phys. A53, 283001 (2020) , . • G. Linardopoulos, “Solvingholographic defects” , PoS CORFU2019, 141 (2020) , .[12] S. Komatsu and Y. Wang, “Non-perturbative defect one-point functions inplanar N = super Yang-Mills” , Nucl. Phys. B958, 115120 (2020) , .[13] T. Gombor and Z. Bajnok, “Boundary states, overlaps, nesting andbootstrapping AdS/dCFT” , JHEP 2010, 123 (2020) , . • T. Gombor and Z. Bajnok, “Boundary state bootstrap and asymptoticoverlaps in AdS/dCFT” , .[14] E. K. Sklyanin, “Boundary conditions for integrable equations” , Funct. Anal. Appl. 21, 164 (1987) .[15] E. Corrigan and Z.-M. Sheng, “Classical integrability of the O(N) nonlinearsigma model on a half line” , Int. J. Mod. Phys. A12, 2825 (1997) , hep-th/9612150 .[16] I. Aniceto, Z. Bajnok, T. Gombor, M. Kim and L. Palla, “On integrableboundaries in the 2 dimensional O ( N ) σ -models” , J. Phys. A50, 364002 (2017) , .[17] T. Gombor, “New boundary monodromy matrices for classical sigmamodels” , Nucl. Phys. B953, 114949 (2020) , .[18] H. Eichenherr and M. Forger, “On the dual symmetry of the nonlinearsigma models” , Nucl. Phys. B155, 381 (1979) .[19] K. Zarembo, “Integrability in sigma-models” , Les Houches Lect. Notes 106, 205 (2019) , .[20] T. Gombor, “On the classification of rational K-matrices” , J. Phys. A53, 135203 (2020) , .[21] N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Zwiebach, “Superstring theory on AdS × S as a coset supermanifold” , Nucl. Phys. B567, 61 (2000) , hep-th/9907200 .[22] K. Skenderis and M. Taylor, “Branes in AdS and pp-wave spacetimes” , JHEP 0206, 025 (2002) , hep-th/0204054 ..