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arXiv:1706.03802 · Journal of High Energy Physics

Conformal interfaces between free boson orbifold theories

Melanie Becker, Yaniel Cabrera, Daniel G. Robbins

Published Sep 1, 2017• 9 citations
arXivHTMLPDF

Abstract

We construct a large class of conformal interfaces between two-dimensional c = 1 conformal field theories describing compact free bosons and their $$ {\mathrm{\mathbb{Z}}}_2 $$ orbifolds. The interfaces are obtained by constructing boundary states in the corresponding c = 2 product theories and applying the unfolding procedure. We compute the fusion products for all of these defects, and identify the invertible topological interfaces associated to global symmetries, the interfaces corresponding to marginal deformations, and the interfaces which map the untwisted sector of an orbifold to the invariant states of the parent theory.

DOI
10.1007/jhep09(2017)148

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We construct a large class of conformal interfaces between two-dimensional c = 1 conformal field theories describing compact free bosons and their Z2{\mathrm{\mathbb{Z}}}_2Z2​ orbifolds. The interfaces are obtained by constructing boundary states in the corresponding c = 2 product theories and applying the unfolding procedure. We compute the fusion products for all of these defects, and identify the invertible topological interfaces associated to global symmetries, the interfaces corresponding to marginal deformations, and the interfaces which map the untwisted sector of an orbifold to the invariant states of the parent theory.
We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\ensuremath{\nu}=2/(2M+q)$ with integers MMM and even (odd) integers qqq for fermionic (bosonic) states. They are termed ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ orbifold states, which have a topological order with a neutral sector described by the c=1c=1c=1 orbifold conformal field theory (CFT) at radius Rorbifold=p/2{R}_{\mathrm{orbifold}}=\sqrt{p/2}Rorbifold​=p/2​ with even integers ppp. When p=2p=2p=2, the state can be viewed as two decoupled layers of Moore-Read states, whose neutral sector is described by the Ising $\ifmmode\times\else\texttimes\fi{}$ Ising CFT and contains a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion subalgebra. We demonstrate that orbifold states with p>2p>2p>2, also containing a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion algebra, can be obtained by coupling together an array of $\mathrm{MR}\ifmmode\times\else\texttimes\fi{}\mathrm{MR}$ wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the Z4{Z}_{4}Z4​ orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer ppp and contains a Z4{Z}_{4}Z4​ fusion algebra.
Boundary Structure and Module Decomposition of the Bosonic Z2Z_2Z2​ Orbifold Models with R2=1/2kR^2=1/2kR2=1/2k
The Z2Z_2Z2​ bosonic orbifold models with compactification radius R2=1/2kR^2=1/2kR2=1/2k are examined in the presence of boundaries. Demanding the extended algebra characters to have definite conformal dimension and to consist of an integer sum of Virasoro characters, we arrive at the right splitting of the partition function. This is used to derive a free field representation of a complete, consistent set of boundary states, without resorting to a basis of the extended algebra Ishibashi states. Finally the modules of the extended symmetry algebra that correspond to the finitely many characters are identified inside the direct sum of Fock modules that constitute the space of states of the theory.
Abstract We study the 2D symmetric orbifold CFT of two copies of free bosons. The twist operator can join the two separated copies in the untwisted sector into a joined copy in the twisted sector. Starting with a state with any number of quanta in the untwisted sector, the state in the twisted sector obtained by the action of the twist operator can be computed by using the covering map method. We develop a new method to compute the effect of a twist operator by using the Bogoliubov ansatz and conformal symmetry. This may lead to more efficient tools to compute correlation functions involving twist operators.
Topological phases of matter in (2+1) dimensions are commonly equipped with global symmetries, such as electric-magnetic duality in gauge theories and bilayer symmetry in fractional quantum Hall states. Gauging these symmetries into local dynamical ones is one way of obtaining exotic phases from conventional systems. We study this using the bulk-boundary correspondence and applying the orbifold construction to the (1+1)-dimensional edge described by a conformal field theory (CFT). Our procedure puts twisted boundary conditions into the partition function and predicts the fusion, spin, and braiding behavior of anyonic excitations after gauging. We demonstrate this for the electric-magnetic self-dual ZN{\mathbb{Z}}_{N}ZN​ gauge theory, the twofold symmetric SU(3)1\text{SU}{(3)}_{1}SU(3)1​, and the S3{S}_{3}S3​-symmetric SO(8)1\text{SO}{(8)}_{1}SO(8)1​ Wess-Zumino-Witten theories.
Abstract We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.