We consider two conformal defects close to each other in a free theory, and study what happens as the distance between them goes to zero. This limit is the same as zooming out, and the two defects have fused to another defect. As we zoom in we find a non-conformal effective action for the fused defect. Among other things this means that we cannot in general decompose the two-point correlator of two defects in terms of other conformal defects. We prove the fusion using the path integral formalism by treating the defects as sources for a scalar in the bulk.
In this paper we outline the application of decomposition to condensation defects and their fusion rules. Briefly, a condensation defect is obtained by gauging a higher-form symmetry along a submanifold, and so there is a natural interplay with notions of decomposition, the statement that d-dimensional quantum field theories with global (d-1)-form symmetries are equivalent to disjoint unions of other quantum field theories. We will also construct new (sometimes non-invertible) defects, and compute their fusion products, again utilizing decomposition. An important role will be played in all the…
We present a detailed study of a single vortex in a holographic symmetry breaking phase. At low energies the system flows to an nontrivial conformal fixed point. Novel vortex physics arises from the interaction of these gapless degrees of freedom with the vortex: at low energies the vortex may be understood as a conformal defect in this low energy theory. Defect conformal symmetry allows the construction of a simple infrared geometry describing a new kind of extremal horizon: a Poincare horizon with a small bubble of magnetic Reissner-Nordstrom horizon inside it that carries a single unit of m…
In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects fro…
We consider conformal defects with spins under the rotation group acting on the transverse directions. They are described in the embedding space formalism in a similar manner to spinning local operators, and their correlation functions with bulk and defect local operators are determined by the conformal symmetry. The operator product expansion (OPE) structure of spinning conformal defects is examined by decomposing it into the spinning defect OPE block that packages all the contribution from a conformal multiplet. The integral representation of the block derived in the shadow formalism is faci…
We develop a group theoretical formalism to study correlation functions in defect conformal field theory, with multiple insertions of bulk and defect fields. This formalism is applied to construct the defect conformal blocks for three-point functions of scalar fields. Starting from a configuration with one bulk and one defect field, for which the correlation function is determined by conformal symmetry, we explore two possibilities, adding either one additional defect or bulk field. In both cases it is possible to express the blocks in terms of classical hypergeometric functions, though the ca…
We study defects in non-relativistic conformal field theories. As in the well-studied case of relativistic conformal defects, we find that a useful tool to organize correlation functions is the defect operator expansion (dOPE). We analyze how the dOPE is implemented in theories with a holographic dual, highlighting some interesting aspects of the operator/state mapping in non-relativistic holography.
We study the operator product expansion (OPE) for scalar conformal defects of any codimension in CFT. The OPE for defects is decomposed into "defect OPE blocks", the irreducible representations of the conformal group, each of which packages the contribution from a primary operator and its descendants. We use the shadow formalism to deduce an integral representation of the defect OPE blocks. They are shown to obey a set of constraint equations that can be regarded as equations of motion for a scalar field propagating on the moduli space of the defects. By employing the Radon transform between t…
We consider a free Maxwell field in four dimensions in the presence of a codimension two defect. Reflection positive, codimension two defects which preserve conformal symmetry in this context are very limited. We show only generalized free fields can appear in the defect operator product expansion of the bulk Maxwell field; in particular correlation functions of these defect operators can be evaluated via Wick's Theorem.
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.
Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme.
Defect lines in conformal field theory can be perturbed by chiral defect fields. If the unperturbed defects satisfy su(2)-type fusion rules, the operators associated to the perturbed defects are shown to obey functional relations known from the study of integrable models as T-systems. The procedure is illustrated for Virasoro minimal models and for Liouville theory.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives…
We consider conformal defect solutions in four dimensional $N=2$ gauged supergravity. These solutions are constructed as a warped product of $AdS_2\times S^1$ over an interval with non-trivial electric and magnetic fields. We show for minimal gauged supergravity and for gauged supergravity with vector multiplets and abelian gauging that supersymmetric defect solutions are only possible when the geometry has a conical defect in either the bulk or the boundary metric.
Conformal defects describe the universal behaviors of a conformal field theory (CFT) in the presence of a boundary or more general impurities. The coupled critical system is characterized by new conformal anomalies which are analogous to, and generalize those of standalone CFTs. Here we study the conformal $a$- and $c$-anomalies of four dimensional defects in CFTs of general spacetime dimensions greater than four. We prove that under unitary defect renormalization group (RG) flows, the defect $a$-anomaly must decrease, thus establishing the defect $a$-theorem. For conformal defects preserving …
We discuss conformal manifolds for conformal field theories with boundaries or defects. Using conformal perturbation theory we derive constraints on coefficients appearing in the boundary operator product expansion and three-point functions that need to be satisfied for the existence of marginal couplings. We present several explicit examples where we confirm that $β$-functions vanish using a position space regularization, differential regularization. Where possible, we confirm that our $β$-function results agree with the existing literature.
Unlike conformal boundary conditions, conformal defects of Virasoro minimal models lack classification. Alternatively to the defect perturbation theory and the truncated conformal space approach, we employ open string field theory (OSFT) techniques to explore the space of conformal defects. We illustrate the method by an analysis of OSFT around the background associated to the $(1,2)$ topological defect in diagonal unitary minimal models. Numerical analysis of OSFT equations of motion leads to an identification of a nice family of solutions, recovering the picture of infrared fixed points due …
We use holography to study codimension-2 surface defects in 5d SCFTs engineered by $(p,q)$ 5-brane webs. The three-dimensional defects are realized by D3-branes ending on the brane web. We identify the holographic representation of the defects in Type IIB $AdS_6$ solutions as probe D3-branes, and study conformal and non-conformal defects which, respectively, preserve one half and one quarter of the supersymmetry. For a sample of 5d SCFTs, including the $T_N$ theories, we provide explicit solutions for conformal and non-conformal defects. For the conformal defects we obtain their contribution t…
We study reflection/transmission process at conformal defects by introducing new transport coefficients for conserved currents. These coefficients are defined by using BCFT techniques thanks to the folding trick, which turns the conformal defect into the boundary. With this definition, exact computations are demonstrated to describe reflection/transmission process for a class of conformal defects. We also compute the boundary entropy based on the boundary state.
The critical 2d classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_ε$ and the Kramers-Wannier duality defect $D_σ$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function $Z_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions $Δ_α$ and conformal spins $s_α$ of a distinct set of primary fields (and their descendants, or conformal towe…
Just as exactly marginal operators allow to deform a conformal field theory along the space of theories known as the conformal manifold, appropriate operators on conformal defects allow for deformations of the defects. When a defect breaks a global symmetry, there is a contact term in the conservation equation with an exactly marginal defect operator. The resulting defect conformal manifold is the symmetry breaking coset and its Zamolodchikov metric is expressed as the 2-point function of the exactly marginal operator. As the Riemann tensor on the conformal manifold can be expressed as an inte…
We construct, generalizing appropriately the method applied by J. Teschner in the case of the Virasoro conformal blocks, the braiding and fusion matrices of the Neveu-Schwarz super-conformal blocks. Their properties allow for an explicit verification of the bootstrap equation in the NS sector of the N=1 supersymmetric Liouville field theory.
Some time ago, conformal data with affine fusion rules were found. Our purpose here is to realize some of these conformal data, using systems of free bosons and parafermions. The so constructed theories have an extended $W$ algebras which are close analogues of affine algebras. Exact character formulae is given, and the realizations are shown to be full fledged unitary conformal field theories.
We find the fusion rules for the c_{p,1} series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c_{3p,3}. This leads to the conjecture that (almost) all minimal models with c = c_{p,q…
Two defect lines separated by a distance delta look from much larger distances like a single defect. In the critical theory, when all scales are large compared to the cutoff scale, this fusion of defect lines is universal. We calculate the universal fusion rule in the critical 2D Ising model and show that it is given by the Verlinde algebra of primary fields, combined with group multiplication in O(1,1)/Z_2. Fusion is in general singular and requires the subtraction of a divergent Casimir energy.
We apply the framework of Rychkov-Tan arXiv:1505.00963 to the codimension two twist defect at the Wilson-Fisher fixed point in $4-ε$ dimensions. We obtain the scaling dimensions of the operators on the defect up to the lowest nontrivial order in the $ε$-expansion without using Feynman diagram computation. Our results agree with the known results.
We consider open strings ending on a D5-brane in the pp-wave background, which is realized in the Penrose limit of $AdS_5 \times S^5$ with an $AdS_4\times S^2$ brane. A complete list of gauge invariant operators in the defect conformal field theory is constructed which is dual to the open string states.
Topological defects have long been known to encode symmetries and dualities between physical systems. In the context of string theory, defects have been intensively studied at the level of the worldsheet. Although marked by a number of pioneering milestones, the target space picture of defects is much less understood. In this paper, we show, at the level of the target space, that Poisson-Lie T-duality can be encoded as a topological defect. With this result at hand, we can postulate the kernel capturing the Fourier-Mukai transform associated to the action of Poisson-Lie T-duality on the RR-sec…
We analyze the defect scaling Lee-Yang model from the perturbed defect conformal field theory (DCFT) point of view. First the defect Lee-Yang model is solved by calculating its structure constants from the sewing relations. Integrable defect perturbations are identified in conformal defect perturbation theory. Then pure defect flows connecting integrable conformal defects are described. We develop a defect truncated conformal space approach (DTCSA) to analyze the one parameter family of integrable massive perturbations in finite volume numerically. Fusing the integrable defect to an integrable…
We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate…
We introduce a general method in order to construct the non chiral fusion rules which determine the operator content of the operator product algebra for rational conformal field theories. We are particularly interested in the models of the complementary D-like solutions of modular invariant partition functions with cyclic center Zn. We find that the non chiral fusion rules have a Zn-grading structure.
We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the reflectivity and transmissivity of the defect. Their properties are investigated and they are computed in a number of examples. We obtain a complete answer for all defects in the Ising model and between certain pairs of minimal models. In the case of two conformal field theories with an enhanced symmetry we restrict ourselves to non-trivial defects that can be obtained…
We discuss consequences of the breaking of conformal symmetry by a flat or spherical extended operator. We adapt the embedding formalism to the study of correlation functions of symmetric traceless tensors in the presence of the defect. Two-point functions of a bulk and a defect primary are fixed by conformal invariance up to a set of OPE coefficients, and we identify the allowed tensor structures. A correlator of two bulk primaries depends on two cross-ratios, and we study its conformal block decomposition in the case of external scalars. The Casimir equation in the defect channel reduces to …
We propose a prescription for describing correlation functions in higher-dimensional defect conformal field theories (DCFTs) by those in ancillary conformal field theories (CFTs) without defects, which is a vast generalization of the image method in two-dimensional boundary CFTs. A correlation function of $n$ operators inserted away from a defect in a DCFT is represented by a correlation function of $2n$ operators in the ancillary CFT, each pair of which is placed symmetrically with respect to the defect. For scalar operators, we establish the correspondence by matching the constraints on corr…
Integrable defects in two-dimensional integrable models are purely transmitting thus topological. By fusing them to integrable boundaries new integrable boundary conditions can be generated, and, from the comparison of the two solved boundary theories, explicit solutions of defect models can be extracted. This idea is used to determine the transmission factors and defect energies of topological defects in sinh-Gordon and Lee-Yang models. The transmission factors are checked in Lagrangian perturbation theory in the sinh-Gordon case, while the defect energies are checked against defect thermodyn…
It was well known that there are $e$-particles and $m$-strings in the 3-dimensional (spatial dimension) toric code model, which realizes the 3-dimensional $\mathbb{Z}_2$ topological order. Recent mathematical result, however, shows that there are additional string-like topological defects in the 3-dimensional $\mathbb{Z}_2$ topological order. In this work, we construct all topological defects of codimension 2 and higher, and show that they form a braided fusion 2-category satisfying a braiding non-degeneracy condition.
We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our first main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimensi…
We demonstrate that the fusion algebra of conformal defects of a two-dimensional conformal field theory contains information about the internal symmetries of the theory and allows one to read off generalisations of Kramers-Wannier duality. We illustrate the general mechanism in the examples of the Ising model and the three-states Potts model.
In previous work, we showed that an anomaly in the one point function of marginal operators is related by the Wess-Zumino condition to the Euler density anomaly on a two dimensional defect or boundary. Here we analyze in detail the two point functions of marginal operators with the stress tensor and with the displacement operator in three dimensions. We show how to get the boundary anomaly from these bulk two point functions and find perfect agreement with our anomaly effective action. For a higher dimensional conformal field theory with a four dimensional defect, we describe for the first tim…
In the twisted M-theory setting, various types of fusion of M2 and M5 branes induce coproducts between the algebra of operators on M2 branes and the algebra of operators on M5 branes. By doing a perturbative computation in the gravity side, which is captured by the 5d topological holomorphic $U(1)$ Chern-Simons theory, we reproduce the non-perturbative coproducts.
