E. Corrigan, C. Zambon
A simple, basic, argument is given, based solely on energy-momentum considerations to recover conditions under which a_r affine or conformal Toda field theories can support defects of integrable type. Associated triangle relations are solved to provide expressions for transmission matrices that generalize previously known examples calculated for the sine-Gordon model and the a_2 affine Toda model.
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In this talk some classical and quantum aspects concerning a special kind of integrable defect - called a jump-defect - will be reviewed. In particular, recent results obtained in an attempt to incorporate this defect in the affine Toda field theories, in addition to the sine-Gordon model, will be presented.
Defects and type D relativistic Toda lattice for some 5d gauge theories
本文通过折叠ADHM构造得到SO瞬子的杨图描述，并证明5d折叠规范理论中的正则单值缺陷对应于D型相对论Toda晶格的定态波函数，从而建立了Bethe规范对应。
We study quantum integrability of affine Toda theories with a line of defect. In particular, we focus on the problem of constructing quantum higher-spin conserved currents in models defined by two A_r^{(1)} Toda theories separated by a non-trivial type-I defect. For a suitable choice of the defect potential these theories are known to be classically integrable, that is they possess an infinite set of higher-spin conserved charges in involution. Studying the corresponding conservation laws at quantum level we discover that anomalies arise, which we compute exactly at all orders in the coupling …
Affine Toda field theories with a purely transmitting integrable defect are considered and the model based on a_2 is analysed in detail. After providing a complete characterization of the problem in a classical framework, a suitable quantum transmission matrix, able to describe the interaction between an integrable defect and solitons, is found. Two independent paths are taken to reach the result. One is an investigation of the triangle equations using the S-matrix for the imaginary coupling bulk affine Toda field theories proposed by Hollowood, and the other uses a functional integral approac…
Some remarks on D-branes and defects in Liouville and Toda field theories
本文分析了广义对角模型中的 Cardy-Lewellen 方程，利用共形场论性质证明其在非有理模型中形式简单，并将其应用于 Liouville 和 Toda 场论，验证了缺陷满足 cluster 方程，推导了所有 $sl(n)$ Toda 场论的方程并推广了边界态形式。
We derive non-abelian Toda field theories (NATFTs) from a 4d Chern-Simons (CS) theory with two order defects by employing a certain asymptotic boundary condition. The 4d CS theory is characterized by a meromorphic 1-form $ω$\,. We adopt $ω$ with two simple poles and no zeros, and each of the order defects is located at each pole. As a result, an anisotropy parameter $β^2$ can be identified with the distance between the two defects. As examples, we can derive the (complex) sine-Gordon model and the Liouville theory.
A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on $a_r^{(1)}$ data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is i…
The concept of fusing rules for defects in a^(1)_r affine Toda field theories is introduced in the classical and quantum contexts. This idea is employed in finding a new transmission matrix for the a^(1)_3 theory which obeys the triangle relations and the crossing-unitarity relations.
A folding process is applied to fused a^(1)_r defects to construct defects for the non-simply laced affine Toda field theories of c^(1)_n, d^(2)_n and a^(2)_2n at the classical level. Support for the hypothesis that these defects are integrable in the folded theories is provided by the observation that transmitted solitons retain their form. Further support is given by the demonstration that energy and momentum are conserved.
The Liouville integrability of the generalised type II defects is investigated. Full integrability is not considered, only the existence of an infinite number of conserved quantities associated with a system containing a defect. For defects in affine Toda field theories (ATFTs) it is shown that momentum conservation is very likely to be a necessary condition for integrability. The defect Lax matrices which guarantee zero curvature, and so an infinite number of conserved quantities, are calculated for the momentum conserving Tzitzéica defect and the momentum conserving $D_4$ ATFT defect. Some a…
Type II integrable defects with more than one degree of freedom at the defect are investigated. A condition on the form of the Lagrangian for such defects is found which ensures the existence of a conserved momentum in the presence of the defect. In addition it is shown that for any Lagrangian satisfying this condition, the defect equations of motion, when taken to hold everywhere, can be extended to give a Bäcklund transformation between the bulk theories on either side of the defect. This strongly suggests that such systems are integrable. Momentum conserving defects and Bäcklund transformat…
Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang-Baxter equations, and second by solving a linear intertwining relation between a finite dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite dimensional representation expressed in terms of sets of generalized `quantum' annihilation and creation operators. The principal examples analysed are based on the $a_2^{(2)}$ and $a_n^{(1)}$ affine Toda models bu…
We advance a correspondence between the topological defect operators in Liouville and Toda conformal field theories - which we construct - and loop operators and domain wall operators in four dimensional N=2 supersymmetric gauge theories on S^4. Our computation of the correlation functions in Liouville/Toda theory in the presence of topological defect operators, which are supported on curves on the Riemann surface, yields the exact answer for the partition function of four dimensional gauge theories in the presence of various walls and loop operators; results which we can quantitatively substa…
Twisted algebra R-matrices and S-matrices for $b_n^{(1)}$ affine Toda solitons and their bound states
构造了新的$U_q(a^{(2)}_{2n-1})$和$U_q(e^{(2)}_6)$不变$R$ -矩阵，并讨论了扭代数的$R$ -矩阵的一般构造.我们用前者构造了$b^{(1)}_n$仿射Toda孤子及其束缚态的$S$矩阵，确定了$b^{(1)}_n$粒子的最低呼吸子.
Comments on fusion matrix in N=1 super Liouville field theory
本文研究了 N=1 超刘维尔场论，证明了 Neveu-Schwarz 扇区融合矩阵元素与结构常数的关系，并收集证据表明其在 Ramond 扇区也成立，进而利用该关系研究了缺陷的 Cardy-Lewellen 方程并构造了缺陷。
The computation of the effect of a simple monodromy defect in the case of a sphere with twisted boundary conditions is revisited and streamlined using earlier calculations for a similar system. Compact and explicit expressions are found for arbitrary integer dimensions. Comments on other work are made.
Using U_q(a_n^(1))- and U_q(a_2n^(2))-invariant R-matrices we construct exact S-matrices in two-dimensional space-time. These are conjectured to describe the scattering of solitons in affine Toda field theories. In order to find the spectrum of soliton bound states we examine the pole structure of these S-matrices in detail. We also construct the S-matrices for all scattering processes involving scalar bound states. In the last part of this paper we discuss the connection of these S-matrices with minimal N=1 and N=2 supersymmetric S-matrices. In particular we comment on the folding from N=2 to…
The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $Ω$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical so…
We critically assess a recent assertion (arXiv:1508.01124) concerning using delta-functions to analyze how higher-codimension branes back-react on their environment. We also briefly summarize the state of the art: describing how stress-energy balance dictates the components of off-brane stress energy in terms brane tension; how this can modify the standard tension/defect-angle relation for codimension-two sources when dilatons are present; and how it all relates to extra-dimensional searches for a small cosmological constant.
We consider topological defect networks with junctions in $A_{N-1}$ Toda CFT and the connection to supersymmetric loop operators in $\mathcal{N} = 2$ theories of class S on a four-sphere. Correlation functions in the presence of topological defect networks are computed by exploiting the monodromy of conformal blocks, generalising the notion of a Verlinde operator. Concentrating on a class of topological defects in $A_2$ Toda theory, we find that the Verlinde operators generate an algebra whose structure is determined by a set of generalised skein relations. These relations encode the represent…
We study localization of bosonic bulk fields on a string-like defect with codimension 2 in a general space-time dimension in detail. We show that in cases of spin 0 scalar and spin 1 vector fields there are an infinite number of massless Kaluza-Klein (KK) states which are degenerate with respect to the radial quantum number, but only the massless zero mode state among them is coupled to fermion on the string-like defect. It is also commented on interesting extensions of the model at hand to various directions such as 'little' superstring theory, conformal field theory and a supersymmetric cons…
In this Reply I present some arguments in favor of the stability of the topological defect composed by global and magnetic monopoles.
There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue - a2(1) and a2(2). By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory, (with or without RSOS restriction), indeed has some fundamental problems, but that these problems are of two different sorts. For a2(1…
We initiate the study of intersecting surface operators/defects in four-dimensional quantum field theories (QFTs). We characterize these defects by coupled 4d/2d/0d theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the four-dimensional QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N = 2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4d/2d/0d QFT. We identify the 4d/2d/0d QFTs that …
Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The d…
Quantum field theories (QFT) in the presence of defects exhibit new types of anomalies which play an important role in constraining the defect dynamics and defect renormalization group (RG) flows. Here we study surface defects and their anomalies in conformal field theories (CFT) of general spacetime dimensions. When the defect is conformal, it is characterized by a conformal $b$-anomaly analogous to the $c$-anomaly of 2d CFTs. The $b$-theorem states that $b$ must monotonically decrease under defect RG flows and was proven by coupling to a spurious defect dilaton. We revisit the proof by deriv…
In this paper we deal with defects inside defects in systems of two scalar fields in 3+1 dimensions. The systems we consider are defined by potentials containing two real scalar fields, and so we are going to investigate domain ribbons inside domain walls. After introducing some general comments on the possibility of finding defects that support internal structure in two specific systems, we introduce thermal effects to show how the picture for domain walls hosting domain ribbons appears at high temperature.
Universal sectors in superconformal defects
研究超对称缺陷CFT中缺陷关联函数的通用特性，通过强耦合展开确定四点函数等价条件，并提取共形数据。
The concept of point-like "jump" defects is investigated in the context of affine Toda field theories. The Hamiltonian formulation is employed for the analysis of the problem. The issue is also addressed when integrable boundary conditions ruled by the classical twisted Yangian are present. In both periodic and boundary cases explicit expressions of conserved quantities as well as the relevant Lax pairs and sewing conditions are extracted. It is also observed that in the case of the twisted Yangian the bulk behavior is not affected by the presence of the boundaries.
We explore the phase structure for defect theories in full generality using the gauge/gravity correspondence. On the gravity side, the systems are constructed by introducing M (probe) D(p+4-2k)-branes in a background generated by N Dp-branes to obtain a codimension-k intersection. The dual gauge theory is a U(N) Supersymmetric Yang-Mills theory on a (1+p-k)-dimensional defect with both adjoint and fundamental degrees of freedom. We focus on the phase structure in the chemical potential versus temperature plane. We observe the existence of two universality classes for holographic gauge theories…
We construct the holographic dual of an electrically charged, localised defect in a conformal field theory at strong coupling, by applying a spatially dependent chemical potential. We find that the IR behaviour of the spacetime depends on the spatial falloff of the potential. Moreover, for sufficiently localized defects with large amplitude, we find that a new gravitational phenomenon occurs: a spherical extremal charged black hole nucleates in the bulk: a hovering black hole. This is a second order quantum phase transition. We construct this new phase with several profiles for the chemical po…
We develop both the gravity and field theory sides of the Karch-Randall conjecture that the near-horizon description of a certain D5-D3 brane configuration in string theory, realized as AdS_5 x S^5 bisected by an AdS_4 x S^2 "brane", is dual to N=4 Super Yang-Mills theory in R^4 coupled to an R^3 defect. We propose a complete Lagrangian for the field theory dual, a novel "defect superconformal field theory" wherein a subset of the fields of N=4 SYM interacts with a d=3 SU(N) fundamental hypermultiplet on the defect preserving conformal invariance and 8 supercharges. The Kaluza-Klein reduction …
We study the properties of a D-brane in the presence of $k$ NS5 branes. The Dirac-Born-Infeld action describing the dynamics of this D-brane is very similar to that of a non-BPS D-brane in ten dimensions. As the D-brane approaches the fivebranes, its equation of state approaches that of a pressureless fluid. In non-BPS D-brane case this is considered as an evidence for the decay of the D-brane into ``tachyon matter''. We show that in our case similar behavior is the consequence of the motion of the D-brane. In particular in the rest frame of the moving D-brane the equation of state is that of …
Current theories of particle physics lead to the unavoidable conclusion that there must have been several phase transitions in the early universe. Further, in the context of these theories, it is possible that cosmological phase transitions would have produced topological defects that may be roaming our heavens today. A finding of these fossils from the early universe would provide a direct confirmation of the thermal history of the cosmos, insight into astrophysical phenomena, and, vital information about particle physics. The elimination of unobserved topological defects provides important c…
On the Physics of Higher Condensation Defects
本文从物理角度论证有限阿贝尔对称性的拓扑缺陷满足Johnson-Freyd高阶融合范畴定义，重点研究Karoubi完备性，并通过四维例子中的连续高阶规范化构造缺陷树加以验证，同时讨论反常对凝聚缺陷性质的影响。
Holographic interpolations of defect CFTs
本文提出了一种新的全息对偶性，连接特定非超对称缺陷共形场论（dCFT）及其引力对偶。通过插值1/2-BPS D3-D3系统与其场论对偶，以及arXiv:2506.14505中的全息对偶，研究了D5探针膜在AdS5×S5几何中的实现及其对称性。
We outline the construction of the Atiyah-Hitchin metric on the moduli space of SU(2) BPS monopoles with charge 2, first as an algebraic curve in C^3 following Donaldson and then as a solution of the Toda field equations in the continual large N limit. We adopt twistor methods to solve the underlying uniformization problem, which by the generalized Legendre transformation yield the Kahler coordinates and the Kahler potential of the metric. We also comment on the connection between twistors and the Seiberg-Witten construction of quantum moduli spaces, as they arise in three dimensional supersym…
We study the full out-of-thermal-equilibrium dynamics of a relativistic classical scalar field through a symmetry breaking phase transition. In these circumstances we determine the evolution of the ensemble averages of the correlation length and topological defect densities. This clarifies many aspects of the non-perturbative dynamics of fields in symmetry breaking phase transitions and allows us to comment on a quantitative basis on the canonical pictures for topological defect formation and evolution. We also compare these results to those obtained from the field evolution in the Hartree app…
In this paper, we study for the first time topological defects in the context of nonlocal field theories in which Lagrangians contain infinite-order differential operators. In particular, we analyze domain walls. Despite the complexity of non-linear infinite-order differential equations, we are able to find an approximate analytic solution. We first determine the asymptotic behavior of the nonlocal domain wall close to the vacua. Then, we find a linearized nonlocal solution by perturbing around the well-known local 'kink', and show that it is consistent with the asymptotic behavior. We develop…
The topological defect scenario of origin of the observed highest energy cosmic rays is reviewed. Under a variety of circumstances, topological defects formed in the early Universe can be sources of very massive particles in the Universe today. The decay products of these massive particles may be responsible for the observed highest energy cosmic ray particles above $10^{20}$ eV. Some massive particle production processes involving cosmic strings and magnetic monopoles are discussed. We also discuss the implications of results of certain recent numerical simulations of evolution of cosmic stri…
Topological 4D gravity and gravitational defects
本研究结合AdS₃引力的Chern-Simons表述与CWY理论，构建了具有SL(2,ℂ)规范对称性的新型拓扑4D引力，推导了引力拓扑缺陷的可观测量，并探讨了其对量子可积性与AdS₃/CFT₂对应拓展的意义。
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