Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, ${\mathbb{RP}}^{2}$ ). Through a cut-and-sew process, ${\mathbb{RP}}^{2}$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{1}{2}lnk$ related to a nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and $c$ the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, ${\mathbb{RP}}^{2}$). Through a cut-and-sew process, ${\mathbb{RP}}^{2}$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{1}{2}lnk$ related to a nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and $c$ the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.

arXiv:1801.07635Physical review. B./Physical review. BJun 28, 20187 citations

The logarithmic Cardy case: Boundary states and annuli

Jürgen Fuchs, Terry J. Gannon, Gregor Schaumann +1 more

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

arXiv:1712.01922Nuclear Physics BMar 28, 20189 citations

Boundary conditions in rational conformal field theories

Roger E. Behrend, Paul A. Pearce, V. B. Petkova +1 more

arXiv:hep-th/9908036Nuclear Physics BMar 1, 2000215 citations

Exceptional boundary states at c = 1

Romuald A. Janik

We consider the CFT of a free boson compactified on a circle, such that the compactification radius $R^2$ is an irrational multiple of $R^2_{selfdual}$ . Apart from the standard Dirichlet and Neumann boundary states, Friedan suggested [1] that an additional 1-parameter family of boundary states exists. These states break U(1) symmetry of the theory, but still preserve conformal invariance. In this paper we give an explicit construction of these states, show that they are uniquely determined by the Cardy-Lewellen sewing constraints, and we study the spectrum in the `open string channel', which is given here by a continous integral with a nonnegative measure on the space of conformal weights.We consider the CFT of a free boson compactified on a circle, such that the compactification radius $R^2$ is an irrational multiple of $R^2_{selfdual}$. Apart from the standard Dirichlet and Neumann boundary states, Friedan suggested [1] that an additional 1-parameter family of boundary states exists. These states break U(1) symmetry of the theory, but still preserve conformal invariance. In this paper we give an explicit construction of these states, show that they are uniquely determined by the Cardy-Lewellen sewing constraints, and we study the spectrum in the `open string channel', which is given here by a continous integral with a nonnegative measure on the space of conformal weights.

Jan 1, 200147 citations

{\mathbb{RP}}^{2}

). Through a cut-and-sew process,

{\mathbb{RP}}^{2}

is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term

\frac{1}{2}lnk

related to a nonorientable genus, with

k

a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and

c

the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.

We consider the CFT of a free boson compactified on a circle, such that the compactification radius

R^2

is an irrational multiple of

R^2_{selfdual}

. Apart from the standard Dirichlet and Neumann boundary states, Friedan suggested [1] that an additional 1-parameter family of boundary states exists. These states break U(1) symmetry of the theory, but still preserve conformal invariance. In this paper we give an explicit construction of these states, show that they are uniquely determined by the Cardy-Lewellen sewing constraints, and we study the spectrum in the `open string channel', which is given here by a continous integral with a nonnegative measure on the space of conformal weights.