Crosscap States Research Articles

Frobenius–Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group π:Gˆ↠{1,−1}, a π-twisted 2-cocycle θˆ on BGˆ and a character λ:Gˆ→U(1). The underlying oriented theory is a twisted Dijkgraaf–Witten theory. The construction is based on a detailed study of the (Gˆ,θˆ,λ)-twisted Real representation theory of ker⁡π. In particular, twisted Real representations are boundary conditions of the unoriented theory and the generalized Frobenius–Schur element is its crosscap state.

Integrable crosscap states: from spin chains to 1D Bose gas

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

Integrable crosscaps in classical sigma models

We study the integrable boundaries and crosscaps of classical sigma models. We show that there exists a classical analog of the integrability condition and KT-relation of the boundary and crosscap states of quantum spin chains. We also classify the integrable crosscaps for various sigma models including examples which are relevant in the AdS/CFT correspondence at strong coupling.

Integrable crosscap states in mathfrak{gl} (N) spin chains

We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the mathfrak{gl} (N) symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the Bethe states. The first part of the derivation is to calculate sum formula for the off-shell overlap. Using this formula we prove that the normalized overlaps of the multi-particle states are ratios of the Gaudin-like determinants. Furthermore we collect the integrable crosscap states which can be relevant in the AdS/CFT correspondence.

Crosscap States in Integrable Field Theories and Spin Chains

We study crosscap states in integrable field theories and spin chains in 1+1 dimensions. We derive an exact formula for overlaps between the crosscap state and any excited state in integrable field theories with diagonal scattering. We then compute the crosscap entropy, i.e. the overlap for the ground state, in some examples. In the examples we analyzed, the result turns out to decrease monotonically along the renormalization group flow except in cases where the discrete symmetry is spontaneously broken in the infrared. We next introduce crosscap states in integrable spin chains, and obtain determinant expressions for the overlaps with energy eigenstates. These states are long-range entangled and their entanglement entropy grows linearly until the size of the subregion reaches half the system size. This property is reminiscent of pure-state black holes in holography and makes them interesting for use as initial conditions of quantum quench. As side results, we propose a generalization of Zamolodchikov’s staircase model to flows between D-series minimal models, and discuss the relation to fermionic minimal models and the GSO projection.

On the topological and crosscap entropies in non-oriented RCFTs

We establish a relation between the boundary and the topological entropies for the conformal minimal models in some of the simplest models of the unitary A–A series. We show that in these models the boundary entropy is a difference of topological entropies. Furthermore, we define the crosscap entropy as the analog to the boundary entropy in non-oriented theories. The crosscap entropy is defined as the logarithm of the degeneracy of the ground state due to the presence of crosscaps, and it can be expressed in terms of the crosscap coefficients. This crosscap entropy has not an explicit relation to the topological entropy as the boundary entropy. However, we propose a new quantity, \(\hat{S} = \text {ln}P_{0i}\) defined in terms of the modular transformation P between the open and closed channel of the Möbius partition function. With this quantity, the crosscap entropy can be regarded as a difference of entropies, very similar to the boundary entropy. We also compute the left-right entanglement entropy (LREE) for crosscap states, and we express it in terms of \(\hat{S}\). An explicit example of the LREE of the crosscap state in Wess–Zumino–Witten is carried out.

From mathcal{N} = 4 Super-Yang-Mills on \u211d\u21194 to bosonic Yang-Mills on \u211d\u21192

We study the four-dimensional mathcal{N} = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge mathcal{Q} , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding mathcal{N} = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.

AGT/\u21242

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional mathcal{N}=2 gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different ℤ2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a ℤ2 quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the ℝℙ4 partition function of four-dimensional mathcal{N}=2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known ℝℙ2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.

Derivation of the Time-Reversal Anomaly for (2+1)-Dimensional Topological Phases.

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2+1)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T^{2} eigenvalues, generalizing the recent analysis by Barkeshli etal. in the bosonic case.

A CFT perspective on gravitational dressing and bulk locality

We revisit the construction of local bulk operators in AdS/CFT with special focus on gravitational dressing and its consequences for bulk locality. Specializing to 2+1-dimensions, we investigate these issues via the proposed identification between bulk operators and cross-cap boundary states. We obtain explicit expressions for correlation functions of bulk fields with boundary stress tensor insertions, and find that they are free of non-local branch cuts but do have non-local poles. We recover the HKLL recipe for restoring bulk locality for interacting fields as the outcome of a natural CFT crossing condition. We show that, in a suitable gauge, the cross-cap states solve the bulk wave equation for general background geometries, and satisfy a conformal Ward identity analo-gous to a soft graviton theorem. Virasoro symmetry, the large N conformal bootstrap and the uniformization theorem all play a key role in our derivations.

Topological phases protected by reflection symmetry and cross-cap states

Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented spacetime, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of spacetime, and can be obtained by rearranging the spacetime geometry and exchanging the role of space and time coordinates. When the system is in a non-trivial SPT phase, we find that the corresponding cross-cap state is non-invariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the Z_8 classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.

Exact partition functions on $ \mathbb{R}{{\mathbb{P}}^2} $ and orientifolds

We consider gauged linear sigma models (GLSM) on $\mathbb{RP}^2$, obtained from a parity projection of $S^2$. The theories admit squashing deformation, much like GLSM on $S^2$, which allows us to interpret the partition function as the overlap amplitude between the vacuum state and crosscap states. From these, we extract the central charge of Orientifold planes, and observe that the Gamma class makes a prominent appearance as in the recent D-brane counterpart. We also repeat the computation for the mirror Landau-Ginzburg theory, which naturally brings out the $\theta$-dependence as a relative sign between two holonomy sectors on $\mathbb{RP}^2$. We also show how our results are consistent with known topological properties of D-brane and Orientifold plane world-volumes, and discuss what part of the wrapped D-brane/Orientifold central charge should be attributed to the quantum volumes.

Nonsupersymmetric Seiberg duality, orientifold QCD, and noncritical strings

We propose an electric-magnetic duality and conjecture an exact conformal window for a class of nonsupersymmetric $U({N}_{c})$ gauge theories with fermions in the (anti)symmetric representation of the gauge group and ${N}_{f}$ additional scalar and fermion flavors. The duality exchanges ${N}_{c}\ensuremath{\rightarrow}{N}_{f}\ensuremath{-}{N}_{c}\ensuremath{\mp}4$ leaving ${N}_{f}$ invariant, and has common features with Seiberg duality in $\mathcal{N}=1$ super QCD (SQCD) with $SU$ or $SO/Sp$ gauge group. At large $N$ the duality holds due to planar equivalence with $\mathcal{N}=1$ SQCD. At finite $N$ we embed these gauge theories in a setup with D-branes and orientifolds in a nonsupersymmetric, but tachyon-free, noncritical type 0B string theory and argue in favor of the duality in terms of boundary and crosscap state monodromies as in analogous supersymmetric situations. One can verify explicitly that the resulting duals have matching global anomalies. Finally, we comment on the moduli space of these gauge theories and discuss other potential nonsupersymmetric examples that could exhibit similar dualities.

D-branes in unoriented non-critical strings and duality in SO(N) and Sp(N) gauge theories

We exhibit exact conformal field theory descriptions of SO(N) and Sp(N) pairs of Seiberg-dual gauge theories within string theory. The N=1 gauge theories with flavour are realized as low energy limits of the worldvolume theories on D-branes in unoriented non-critical superstring backgrounds. These unoriented backgrounds are obtained by constructing exact crosscap states in the SL(2,R)/U(1) coset conformal field theory using the modular bootstrap method. Seiberg duality is understood by studying the behaviour of the boundary and crosscap states under monodromy in the closed string parameter space.

Permutation orientifolds

We consider orientifold actions involving the permutation of two identical factor theories. The corresponding crosscap states are constructed in rational conformal field theory. We study group manifolds, in particular the examples $SU(2) \times SU(2)$ and $U(1)\times U(1)$ in detail, comparing conformal field theory results with geometry. We then consider orientifolds of tensor products of N=2 minimal models, which have a description as coset theories in rational conformal field theory and as Landau Ginzburg models. In the Landau Ginzburg language, B-orientifolds and D-branes are described in terms of matrix factorizations of the superpotential. We match the factorizations with the corresponding crosscap states.

Orientifolds of Gepner models

We systematically construct and study Type II Orientifolds based on Gepner models which have N=1 supersymmetry in 3+1 dimensions. We classify the parity symmetries and construct the crosscap states. We write down the conditions that a configuration of rational branes must satisfy for consistency (tadpole cancellation and rank constraints) and spacetime supersymmetry. For certain cases, including Type IIB orientifolds of the quintic and a two parameter model, one can find all solutions in this class. Depending on the parity, the number of vacua can be large, of the order of 10^{10}-10^{13}. For other models, it is hard to find all solutions but special solutions can be found -- some of them are chiral. We also make comparison with the large volume regime and obtain a perfect match. Through this study, we find a number of new features of Type II orientifolds, including the structure of moduli space and the change in the type of O-planes under navigation through non-geometric phases.

Permutation orientifolds of Gepner models

In tensor products of a left-right symmetric CFT, one can define permutation orientifolds by combining orientation reversal with involutive permutation symmetries. We construct the corresponding crosscap states in general rational CFTs and their orbifolds, and study in detail those in products of affine U(1)2 models or N = 2 minimal models. The results are used to construct permutation orientifolds of Gepner models. We list the permutation orientifolds in a few simple Gepner models, and study some of their physical properties — supersymmetry, tension and RR charges. We also study the action of corresponding parity on D-branes, and determine the gauge group on a stack of parity-invariant D-branes. Tadpole cancellation condition and some of its solutions are also presented.

Lattice realizations of the open descendants of twisted boundary conditions forsl(2) A–D–E models

The twisted boundary conditions and associated partition functions of the conformalsl(2) A–D–E models are studied on the Klein bottle and the Möbius strip. TheA–D–E minimal lattice models give realization to the complete classification of the open descendants of thesl(2) minimal theories. We construct the transfer matrices of these lattice models that areconsistent with non-orientable geometries. In particular, we show that in order to realize allthe Klein bottle amplitudes of different crosscap states, not only the topological flip on thelattice but also the involution in the spin configuration space must be taken into account.This involution is the symmetry of the Dynkin diagrams which corresponds to the simple current of theOcneanu algebra.

Crosscap states in [formula omitted] Liouville theory

We construct crosscap states in the N = 2 Liouville theory from the modular bootstrap method. We verify our results by comparing it with the calculation from the minisuperspace approximation and by checking the consistency with the conformal bootstrap equation. Various overlaps with other known branes are studied. We further discuss the topological nature of the discrete terms in the crosscap wavefunction and their connection with the Landau–Ginzburg approach in a nontrivial dilaton background. We find that it can be mapped to the Landau–Ginzburg theory with a negative power superpotential by a simple change of variables, extending the known duality to the open string sector. Possible applications to the two-dimensional noncritical string theories and supersymmetric orientifolds in the higher dimension are also discussed.

Orientifolds and Mirror Symmetry

We study parity symmetries and crosscap states in classes of N=2 supersymmetric quantum field theories in 1+1 dimensions, including non-linear sigma models, gauged WZW models, Landau-Ginzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of them. The case of the N=2 minimal model are studied in complete detail, from all three realizations -- gauged WZW model, abstract RCFT, and LG models. We also identify mirror pairs of orientifolds, extending the correspondence between symplectic geometry and algebraic geometry by including unorientable worldsheets. Through the analysis in various models and comparison in the overlapping regimes, we obtain a global picture of orientifolds and D-branes.