Enrico M. Brehm
We investigate perturbatively tractable deformations of topological defects in two-dimensional conformal field theories. We perturbatively compute the change in the $g$-factor, the reflectivity, and the entanglement entropy of the conformal defect at the end of these short RG flows. We also give instances of such flows in the diagonal Virasoro and Super-Virasoro Minimal Models.
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The behaviour of boundary conditions under relevant bulk perturbations is studied for the Virasoro minimal models. In particular, we consider the bulk deformation by the least relevant bulk field which interpolates between the mth and (m-1)st unitary minimal model. In the presence of a boundary this bulk flow induces an RG flow on the boundary, which ensures that the resulting boundary condition is conformal in the (m-1)st model. By combining perturbative RG techniques with insights from defects and results about non-perturbative boundary flows, we determine the endpoint of the flow, i.e. the …
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 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.
We investigate a class of models in 1+1 dimensions with four fermion interaction term. At each order of the perturbation expansion, the models are ultraviolet finite and Lorentz non-invariant. We show that for certain privileged values of the coupling constants, Lorentz symmetry is restored, and indeed the model turns out to be conformally invariant. This phenomenon is both quantum mechanical and non-perturbative.
It is shown that the master equation of the affine-Virasoro construction on the unitary affine algebra naturally emerges in the fusion algebra of the nonunitary level $k$ WZNW model. Operators corresponding to solutions of the master equation are suitable for performing one-parametrical renormalizable perturbation around the given conformal nonunitary WZNW model. In the large $|k|$ limit, the infrared fixed point of the renormalization group beta function is found. There are as many infrared conformal points as $\frac{1}{2}N(N+1)$, where $N$ is the total number of solutions of the master equat…
Perturbing a Virasoro minimal model by the (1,3) primary bulk field results in an integrable field theory. In this paper, an infinite set of commuting conserved charges is obtained by considering defects: a one-parameter family of perturbed defect operators is given, and it is shown that these operators mutually commute, that they commute with the Hamiltonian of the perturbed CFT, and that they satisfy a T-system functional relation. The formulation in terms of perturbed defects is a modification of the original prescription for such charges by Bazhanov, Lukyanov and Zamolodchikov.
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…
Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be descr…
In this paper we study a simple example of a two-parameter space of renormalisation group flows of defects in Virasoro minimal models. We use a combination of exact results, perturbation theory and the truncated conformal space approach to search for fixed points and investigate their nature. For the Ising model, we confirm the recent results of Fendley et al. In the case of central charge close to one, we find six fixed points, five of which we can identify in terms of known defects and one of which we conjecture is a new non-trivial conformal defect. We also include several new results on ex…
It has been known for some time that the (1,3) perturbations of the (2k+1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of the (3k+-1,3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W3 and (bosonic) WBC2 = W(2,4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relatin…
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 …
Perturbations of B-type defects in Landau-Ginzburg models are considered. In particular, the effect of perturbations of defects on their fusion is analyzed in the framework of matrix factorizations. As an application, it is discussed how fusion with perturbed defects induces perturbations on boundary conditions. It is shown that in some classes of models all boundary perturbations can be obtained in this way. Moreover, a universal class of perturbed defects is constructed, whose fusion under certain conditions obey braid relations. The functors obtained by fusing these defects with boundary co…
We propose defect lines as a useful tool to study bulk perturbations of conformal field theories, in particular to analyse the induced renormalisation group flows of boundary conditions. As a concrete example we investigate bulk perturbations of N=2 supersymmetric minimal models. To these perturbations we associate a special class of defects between the respective UV and IR theories, whose fusion with boundary conditions indeed reproduces the behaviour of the latter under the corresponding RG flows. v2: Some explanations added in section 4, minor changes.
We examine codimension--1 topological defects whose associated worldline is geodesically embedded in $\AdS_{2}$. This discussion extends a previous study of exact analytical solutions to the equations of motion of topological defects in $\AdS_{n}$ in a particular limit where the masses of the scalar and gauge field vanish. We study the linear perturbations about the zeroth order kink-like solution and verify that they are stable. We also discuss general features of the perturbation expansion to all orders.
In the context of two-dimensional rational conformal field theories we consider topological junctions of topological defect lines with boundary conditions. We refer to such junctions as open topological defects. For a relevant boundary operator on a conformal boundary condition we consider a commutation relation with an open defect obtained by passing the junction point through the boundary operator. We show that when there is an open defect that commutes or anti-commutes with the boundary operator there are interesting implications for the boundary RG flows triggered by this operator. The end…
We explore perturbations to a kink-like (codimension 1) topological defect whose world brane is $AdS_{q}$ embedded into $AdS_{q+1}$. Previously, we found solutions in the limit the mass of the scalar field vanishes. In this article we extend a calculation previously done in $AdS_{2}$ to higher-dimensional embedding spaces and find that all perturbations to the mass of the field are stable to first order as expected in a theory with topological defects. We find that the equation of motion to the correction strongly resembles a problem well-known in quantum mechanics.
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 describe a general way of constructing integrable defect theories as perturbations of conformal field theory by local defect operators. The method relies on folding the system onto a boundary field theory of twice the central charge. The classification of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal field theories. These include local defect perturbations of all $c<1$ minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the for…
With a view to understanding extended-BMS symmetries in the framework of the $AdS_4/CFT_3$ correspondence, asymptotically AdS geometries are constructed with null impulsive shockwaves involving a discontinuity in superrotation parameters. The holographic dual is proposed to be a two-dimensional Euclidean defect conformal field localized on a particular timeslice in a three-dimensional conformal field theory on de Sitter spacetime. The defect conformal field theory generates a natural action of the Virasoro algebra. The large radius of curvature limit $\ell\to\infty$ yields spacetimes with nont…
Non-conformal Line Defect (Shell Operator) in AdS$_3$/CFT$_2$: Spinning and Higher Point Correlators
本文将 AdS$_3$/CFT$_2$ 中薄壳缺陷的混沌匹配推广至旋转缺陷两点函数与多标量缺陷高点函数，并用引力在壳作用量验证精确一致。
We consider reflection and transmission of interfaces which implement renormalisation group flows between conformal fixed points in two dimensions. Such an RG interface is constructed from the identity defect in the ultraviolet CFT by perturbing the theory on one side of the defect line. We compute reflection and transmission coefficients in perturbation theory to third order in the coupling constant and check our calculations against exact constructions of RG interfaces between coset models.
Given a 4d ${\mathcal N}=2$ supersymmetric theory with an ${\mathcal N}=(2,2)$ supersymmetric surface defect, a marginal perturbation of the bulk theory induces a complex structure deformation of the defect moduli space. We describe a concrete way of computing this deformation using the bulk-defect OPE.
We classify and study defects in 2d Jackiw-Teitelboim gravity. We show these are holographically described by a deformation of the Schwarzian theory where the reparametrization mode is integrated over different coadjoint orbits of the Virasoro group. We show that the quantization of each coadjoint orbit is connected to 2d Liouville CFT between branes with insertions of Verlinde loop operators. We also propose an interpretation for the exceptional orbits. We use this perspective to solve these deformations of the Schwarzian theory, computing their partition function and correlators. In the proc…
We consider two level $k$ WZNW models coupled to each other through a generalized Thirring-like current-current interaction. It is shown that in the large $k$ limit, this interacting system can be presented as a two-parameter perturbation around a nonunitary WZNW model. The perturbation operators are the sigma model kinetic terms with metric related to the Thirring coupling constants. The renormalizability of the perturbed model leads to an algebraic equation for couplings. This equation coincides with the master Virasoro equation. We find that the beta functions of the two-parameter perturbat…
We discuss special perturbations of the gauged level $k$ WZNW model inspired by the $σ$-model perturbation of the nonunitary WZNW model. In the large $k$ limit there is a second conformal point in the vicinity of the ultarviolet fixed point. At the second critical point the conformal model has a rational Virasoro central charge which no longer corresponds to a coset construction. In spite of this fact the perturbative conformal model appears to be a unitary system as long as the underlying coset construction at the ultraviolet critical point is unitary.
We study the effect of bulk perturbations of N=(2,2) superconformal minimal models on topological defects. In particular, symmetries and more general topological defects which survive the flow to the IR are identified. Our method is to consider the topological subsector and make use of the Landau-Ginzburg formulation to describe RG flows and topological defects in terms of matrix factorizations.
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.
We consider defect operators in scalar field theories in dimensions $d=4-ε$ and $d=6-ε$ with self-interactions given by a general marginal potential. In a double scaling limit, where the bulk couplings go to zero and the defect couplings go to infinity, the bulk theory becomes classical and the quantum defect theory can be solved order by order in perturbation theory. We compute the defect $β$ functions to two loops and study the Renormalization Group flows. The defect fixed points can move and merge, leading to fixed point annihilation; and they exhibit a remarkable factorization property whe…
Integrable RG Flows on Topological Defect Lines in 2D Conformal Field Theories
本研究提出一族含参数的可积格点模型，通过改变谱参数非均匀性实现二维共形场论中不同拓扑缺陷线，并结合贝特ansatz与数值方法研究其间的重整化群流，同时探讨了与各向异性Kondo模型的关联。
We demonstrate that the one-loop dilatation generator for the scalar sector of a certain perturbation of N=4 Super Yang-Mills with fundamentals is the Hamiltonian of an integrable spin chain with open boundary conditions. The theory is a supersymmetric defect conformal field theory (dCFT) with the fundamentals in hypermultiplets confined to a codimension one defect. We obtain a K-matrix satisfying a suitably generalized form of the boundary Yang-Baxter equation, study the Bethe ansatz equations and demonstrate how Dirichlet and Neumann boundary conditions arise in field theory, and match to ex…
Renormalization Group domain walls are natural conformal interfaces between two CFTs related by an RG flow. The RG domain wall gives an exact relation between the operators in the UV and IR CFTs. We propose an explicit algebraic construction of the RG domain wall between consecutive Virasoro minimal models in two dimensions. Our proposal passes a stringent test: it reproduces in detail the leading order mixing of UV operators computed in the conformal perturbation theory literature. The algebraic construction can be applied to a variety of known RG flows in two dimensions.
Topological defects may play the role of the curvatons. We propose a new mechanism of generating density perturbations from cosmological defects in inflationary models. We show several examples in which defects play crucial role in generating density perturbations.
The integrable perturbation of the degenerate boundary condition (d) by the $ϕ_{1,3}$ boundary field generates a renormalization group flow down to the superposition of Cardy boundary states (+)&(-). Exact Thermodynamic Bethe Ansatz (TBA) equations for all the excited states are derived here extending the results of a previous paper to this case. As an intermediate step, the non-Cardy boundary conformal sector (+)&(-) is also described as the scaling limit of an A_4 lattice model with appropriate integrable boundary conditions and produces the first example of superposition of finitized Viraso…
A manifestly covariant equation is derived to describe the second order perturbations in topological defects and membranes on arbitrary curved background spacetimes. This, on one hand, generalizes work on macroscopic strings in Minkowski spacetime and introduces a framework for studing in a precise manner membranes behavior near the black hole horizon and on the other hand, introduces a more general framework for examining the stability of topological defects in curved spacetimes.
Codimension one defects in free scalar field theory
本文通过匹配条件与微扰重求和两种互补方法，系统研究了自由标量场论中余维一缺陷的关联函数、反常维数、融合、量子化、g-定理、Cardy公式及淬火等性质，为更复杂相互作用理论奠定基础。
We derive the equations of motion for scalar metric perturbations in a particular nonsingular bouncing cosmology, where the big bang singularity is replaced by a spacetime defect with a degenerate metric. The adiabatic perturbation solution is obtained for nonrelativistic hydrodynamic matter. We get the same result by working with conformal coordinates. This last method is also valid for vector and tensor metric perturbations, and selected results are presented. We, finally, discuss several new effects from the linear perturbations of this nonsingular bouncing cosmology, such as across-bounce …
In this thesis we consider four dimensional N=2 superconformal field theories, in presence of line defects such as Wilson loops. In this set up, using supersymmetric localization, we compute many observables, such as the vacuum expectation value of the Wilson loop, correlation functions with chiral operators and the emitted radiation of the charged particle moving along the Wilson loop trajectory. These results are achieved by exploiting the localized matrix model on a four sphere, and checked against a perturbative computation on the flat space in N=1 superspace formalism, up to four loops in…
Recently Gaiotto [1] considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet ones for a given renormalization group flow. In this paper we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the IR conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect field…
Non-local charges from perturbed defects via SymTFT in 2d CFT
该研究从无扰理论的三维SymTFT视角出发，提出对易条件得到兼容的体和缺陷扰动，使受扰线缺陷守恒并构成幺半范畴，为极小模型的标准可积扰动提供非局域守恒荷族，还为其他体扰动找到解并对比局域守恒荷，为无需局域守恒荷的可积性提供新方法。
Using $q$-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p+1) models of conformal field theory perturbed by the operator $phi_{2,1}$. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime $1^+$ of $A_{p-1}$ dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for …
Interaction of asymmetric $ϕ^6$ kinks with a spatially localized $\mathcal{PT}$-symmetric perturbation is investigated numerically. It has been shown that when the kink (antikink) hits the defect from the gain side, a final velocity of the kink decreases (increases), while for the kink and antikink coming from the opposite direction their final velocities remain unchanged. It is also found that when the kink interacts with the defect from the gain side multiple pair of the kink-antikink are formed from small amplitude waves (phonons) in the final states depending on the initial velocity of the…
Defect Conformal Manifolds along RG Domain Walls between $\mathbb Z_N$-Parafermions and Minimal Models
通过追踪非可逆对称性并利用“幻影流”的代数结构，研究了 $\mathbb{Z}_N$ 顺铁磁理论到 Virasoro 极小模型 $\mathcal{M}_{N+1}$ 的重整化群畴壁，揭示了缺陷上连续共形流形的动态生成及其在 $N$ 大极限下的传输率消失。
Influences of topological defect and dislocation on conductivity behavior of charge carries in external electromagnetic fields are studied. Particularly the quantum Hall effect is investigated in detail. It is found that the nontrivial deformations of spacetime due to topological defect and dislocation produce an electric current at the leading order of perturbation theory. This current then induces a deformation on the Hall conductivity. The corrections on the Hall conductivity depend on the external electric fields, the size of the sample and the momentum of the particle.
We investigate the relation between instantons and monopoles in the Laplacian Abelian Gauge using analytical methods in the continuum. Our starting point is the fact that the 't Hooft instanton with its high symmetry leads to a pointlike defect with Hopf invariant one. In order to generalise this result we partly break the symmetry by a local perturbation. We find that for generic configurations near the 't Hooft instanton the defects become loops. The analytical results show explicitly that these defects are magnetic monopoles with unit charge. In addition, the monopoles are twisted to accoun…
Notes on a Surface Defect in the $O(N)$ Model
本文研究$O(N)$模型中由局域二次扰动定义的表面缺陷，通过epsilon和大$N$展开证明其长距离流向非平凡缺陷共形场论，并验证了$b$-定理。
We explore a $C$-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate $C$-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases…
Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment fa…
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