We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix.
We study the modular Hamiltonians of an interval for the massless Dirac fermion on the half-line. The most general boundary conditions ensuring the global energy conservation lead to consider two phases, where either the vector or the axial symmetry is preserved. In these two phases we derive the corresponding modular Hamiltonian in explicit form. Its density involves a bi-local term localised in two points of the interval, one conjugate to the other. The associated modular flows are also established. Depending on the phase, they mix fields with different chirality or charge that follow differ…
Every 4d $\mathcal{N} = 2$ SCFT $\mathcal{T}$ corresponds to an associated VOA $\mathbb{V}(\mathcal{T})$, which is in general non-rational with a more involved representation theory. Null states in $\mathbb{V}(\mathcal{T})$ can give rise to non-trivial flavored modular differential equations, which must be satisfied by the refined/flavored character of all the $\mathbb{V}(\mathcal{T})$-modules. Taking some $A_1$ theories $\mathcal{T}_{g,n}$ of class-$\mathcal{S}$ as examples, we construct the flavored modular differential equations satisfied by the Schur index. We show that three types of surf…
We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massless Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the higher dimensional Legendre differential operator. The quadratic form of the massless modular Hamiltonian is expressed in terms of an integral of the energy density with the parabolic distribution. We then get the formula for the local entropy of a wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free…
The method of bosonization is extended to the case when a dissipationless point-like defect is present in space-time. Introducing the chiral components of a massless scalar field, interacting with the defect in two dimensions, we construct the associated vertex operators. The main features of the corresponding vertex algebra are established. As an application of this framework we solve the massless Thirring model with defect. We also construct the vertex representation of the sl(2) Kac-Moody algebra, describing the complex interplay between the left and right sectors due to the interaction wit…
We study massive and massless conical defects in Minkowski and de Sitter spaces in various spacetime dimensions. The energy-momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its spacetime metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal abeli…
For all 4d $\mathcal{N} = 4$ SYM theories with simple gauge groups $G$, we show that the residues of the integrands in the $\mathcal{N} = 4$ Schur indices, which are related to Gukov-Witten type surface defects in the theories, equal the vacuum characters of rank$G$ copies of $bc βγ$ systems that provide the free field realization of associated $\mathcal{N} = 4$ VOAs. This result predicts that these residues, as module characters, are additional solutions to the flavored modular differential equations satisfied by the original Schur index. The prediction is verified in the $G = SU(2)$ case, wh…
We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature $β$ and electric potential $V$, is explicitly constructed both for finite and for semi-infinite wires. In the …
The partition function of a massless scalar field on a Euclidean spacetime manifold $\mathbb{R}^{d-1}\times\mathbb{T}^2$ and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL$(2,\mathbb{Z})$ Eisenstein series with $s=(d+1)/2$. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier…
We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We then present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. We also describe the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massless, free QFT in two spacetime dimensions, thus complementing the recent results in higher space dimensions. In particular, we have the f…
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…
We derive Cardy-like formulas for the growth of operators in different sectors of unitary $2$ dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval $[Δ-δ,Δ+δ]$ for large $Δ$ at fixed $δ$. Consequently we prove that given any unitary modular invariant $2$D CFT symmetric under finite global symmetry $G$ (acting faithfully), all the irreducible representations of $G$ appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the ''square'' of the dime…
We study instant conformal symmetry breaking as a holographic effect of ultrarelativistic particles moving in the AdS3 spacetime. We give the qualitative picture of this effect probing it by two-point correlation functions and the entanglement entropy of the corresponding boundary theory. We show that within geodesic approximation the ultra-relativistic massless defect due to gravitational lensing of the geodesics, produces a zone structure for correlators with broken conformal invariance. Meanwhile, the holographic entanglement entropy also exhibits a transition to the non-conformal behaviour…
Defects are a useful tool in the study of quantum field theories. This is illustrated in the example of two-dimensional conformal field theories. We describe how defect lines and their junction points appear in the description of symmetries and order-disorder dualities, as well as in the orbifold construction and a generalisation thereof that covers exceptional modular invariants.
We investigate varies correlation functions of modular Hamiltonians defined with respect to spatial regions in quantum field theories. These correlation functions are divergent in general. We extract finite correlators by removing divergent terms for two dimensional massless free scalar theory. We reproduce the same correlators in general two dimensional conformal field theories.
The celebrated exactly solvable "Schwinger" model, namely massless two-dimensional QED, is revisited. The solution presented here emphasizes the non- perturbative relevance of the topological sector through large gauge transformations whose role is made manifest by compactifying space into a circle. Eventually the well-known non-perturbative features and solution of the model are recovered in the massless case. However the fermion mass term is shown to play a subtle role in order to achieve a physical quantization that accounts for gauge invariance under both small and large gauge symmetries. …
We study the 2D vertex operator algebra (VOA) construction in 4D $\mathcal{N}=2$ superconformal field theories (SCFT) on $S^3 \times S^1$, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus $\mathbb{T}^2\subset S^3\times S^1$, it is $\frac12\mathbb{Z}$-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $\mathbb{Z} +\frac12$-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserve the VOA, we find that it admits discrete deformations, which…
Zero modes of modular Hamiltonian of one interval are found in momentum space for two dimensional massless free scalar theory. Finite correlators are extracted from separate region connected correlation functions with the insertion of zero modes. Correlators of $(n,1)$-type are claimed to be conformal block up to a set of theory dependent constants. We fix the correlators of $(2,1)$-type with the coefficients of three point function in 2d CFTs.
We model the back-reaction of a static observer in four-dimensional de Sitter spacetime by means of a singular $\mathbb Z_q$ quotient. The set of fixed points of the $\mathbb Z_q$ action consists of a pair of codimension two minimal surfaces given by 2-spheres in the Euclidean geometry. The introduction of an orbifold parameter $q>1$ permits the construction of an effective action for the bulk gravity theory with support on each of these minimal surfaces. The effective action corresponds to that of Liouville field theory on a 2-sphere with a finite vacuum expectation value of the Liouville fie…
We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from symmetrized open-string integrals through a tentative generalization of the single-valued projection known from genus zero.
We study quantum fluctuations of a free massless scalar field during inflation in the presence of a point, line, or plane defect such as a black hole, cosmic string, or domain wall, using a perturbative expansion in powers of small defect parameters. We provide results for the scalar two-point correlation functions that show explicitly a small violation of translational invariance during inflation.
We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several …
We study massless Dirac fermions in the background of a specific planar topologically nontrivial configuration in the three-dimensional spacetime. The results show the presence of massive bound states, phase shifts and the consequent differential cross section for the scattering of fermions in the weak coupling regime. Despite the nontrivial topology of the background field, no fermionic zero mode is found.
The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell's theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.
We evaluate the vacuum polarization tensor (VPT) for a massless Dirac field in 1+1 and 3+1 dimensions, in the presence of a particular kind of defect, which in a special limit imposes bag boundary conditions. We also show that the chiral anomaly in the presence of such a defect is the same as when no defects are present, both in 1+1 and 3+1 dimensions. This implies that the induced vacuum current in 1+1 dimensions due to the lowest order VPT is exact.
Entanglement entropy~(EE) contains signatures of many universal properties of conformal field theories~(CFTs), especially in the presence of boundaries or defects. In particular, {\it topological} defects are interesting since they reflect internal symmetries of the CFT, and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, very few ab-initio, lattice computations of these predictions have been available. Here, we present an ab-initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE…
We present a systematic study of the constraints coming from target-space duality and the associated duality anomaly cancellations on orbifold-like 4-D strings. A prominent role is played by the modular weights of the massless fields. We present a general classification of all possible modular weights of massless fields in Abelian orbifolds. We show that the cancellation of modular anomalies strongly constrains the massless fermion content of the theory, in close analogy with the standard ABJ anomalies. We emphasize the validity of this approach not only for (2,2) orbifolds but for (0,2) model…
We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincare' group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and cano…
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley-Lieb algebra extended by the shift operators Omega and Omega^{-1}. In this enlarged algebra, the non-contractible loop fugacity is alpha and the contractible loop fugacity is beta. The torus is formed by gluing the top and bott…
We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representation and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discus…
We construct and investigate quantum fields induced on a d-dimensional dissipationless defect by bulk fields propagating in a (d+1)-dimensional space. All interactions are localized on the defect. We derive a unitary non-canonical quantum field theory on the defect, which is analyzed both in the continuum and on the lattice. The universal critical behavior of the underlying system is determined. It turns out that the O(N)-symmetric phi^4 theory, induced on the defect by massless bulk fields, belongs to the universality class of particular d-dimensional spin models with long-range interactions.…
We study massless fermions interacting through a particular four fermion term in four dimensions. Exact symmetries prevent the generation of bilinear fermion mass terms. We determine the structure of the low energy effective action for the auxiliary field needed to generate the four fermion term and find it has an novel structure that admits topologically non-trivial defects with non-zero Hopf invariant. We show that fermions propagating in such a background pick up a mass without breaking symmetries. Furthermore pairs of such defects experience a logarithmic interaction. We argue that a phase…
Machine Learning Topological Order from Defect Partition Functions
Shahrood University of Technology（沙赫鲁德理工大学）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We present explicit mathematical structures that allow for the reconstruction of the field content of a full local conformal field theory from its boundary fields. Our framework is the one of modular tensor categories, without requiring semisimplicity, and thus covers in particular finite rigid logarithmic conformal field theories. We assume that the boundary data are described by a pivotal module category over the modular tensor category, which ensures that the algebras of boundary fields are Frobenius algebras. Bulk fields and, more generally, defect fields inserted on defect lines, are give…
We study analytically the dynamical formation of lower dimensional branes at the endpoint of brane-antibrane inflation through the condensation of topological defects of the tachyon field which describes the instability of the initial state. We then use this information to quantify the efficiency of the reheating which is due to the coupling of time dependent tachyon background to massless gauge fields which will be localized on the final state branes. We improve upon previous estimates indicating that this can be an efficient reheating mechanism for observers on the brane.
We study some consequences of dimensionally reducing systems with massless fermions and Abelian gauge fields from 3+1 to 2+1 dimensions. We first consider fermions in the presence of an external Abelian gauge field. In the reduced theory, obtained by compactifying one of the coordinates `a la Kaluza-Klein, magnetic flux strings are mapped into domain wall defects. Fermionic zero modes, localized around the flux strings of the 3+1 dimensional theory, become also zero modes in the reduced theory, via the Callan and Harvey mechanism, and are concentrated around the domain wall defects. We also st…
We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines.
