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Boundary Structure and Module Decomposition of the Bosonic $Z_2$ Orbifold Models with $R^2=1/2k$

Hatzinikitas, A, Smyrnakis, J M

Published Mar 14, 2002• 0 citations
OpenAlex

Abstract

The $Z_2$ bosonic orbifold models with compactification radius $R^2=1/2k$ 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.

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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.
The massless bosonic field compactified on the circle of rational R 2 is reexamined in the presense of boundaries. A particular class of models corresponding to R 2 = 1 2k is distinguished by demanding the existence of a consistent set of Newmann boundary states. The boundary states are constructed explicitly for these models and the fusion rules are derived from them. These are the ones prescribed by the Verlinde formula from the S-matrix of the theory. In addition, the extended symmetry algebra of these theories is constructed which is responsible for the rationality of these theories. Finally, the chiral space of these models is shown to split into a direct sum of irreducible modules of the extended symmetry algebra. 1
We study the space of the allowed boundary states in a particular c=1 conformal field theory, the symmetric orbifold of the level one SU(2) WZW model, where the orbifolding group G is a generic binary finite subgroup of SU(2). We especially construct generic boundary states in the orbifold, which can be extended from the Cardy states decomposed into Virasoro Ishibashi states. Our result involves conformal boundary states parameterized by the quotient SU(2)/G, and the fractional boundary states sitting at every fixed-point in the quotient. We find also that a conformal boundary state at a fixed-point in the quotient becomes a linear combination of the fractional boundary states sitting at the fixed-point. To be complete, we check the mutual consistency among all the boundary states we construct.
We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1) U ( 1 ) , \mathbb{R} ℝ and \mathbb{Z}_2 ℤ 2 symmetry.
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}}}_2 Z2​ 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.
In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the CFT edge modes. Finally, we relate our factorization map to the coproduct formula for the CFT symmetry algebra, which we show is equivalent to a Boguliubov transformation in the case of a free boson.