Chiral Algebra Research Articles

Parafermionic chiral algebra [formula omitted] with the dimension of the principal parafermion fields [formula omitted], [formula omitted], [formula omitted

We analyze, and prove, the associativity of the new Z3 parafermionic chiral algebra which has been announced some time ago, with principal parafermionic fields having the conformal dimension Δψ=8/3. In doing so we have developed a new method for analyzing the associativity of a given chiral algebra of parafermionic type, the method which might be of a more general significance than a particular conformal field theory studied in detail in this paper. Still, even in the context of our particular chiral algebra, of Z3 parafermions with Δψ=8/3, the new method allowed us to give a proof of associativity which we consider to be complete.

Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule Fω and a bimodule morphism Zω from Lyubashenkoʼs Hopf algebra object K for the bimodule category to Fω. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to Fω. We further show that the bimodule Fω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.The bimodules K and Fω can both be characterized as coends of suitable bifunctors. The morphism Zω is obtained by applying a monodromy operation to the coproduct of Fω; a similar construction for the product of Fω exists as well.Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.

Microstates at the boundary of AdS

The bound states of the D1D5 brane system have a known gravitational description: flat asymptotics, an anti-de Sitter region, and a 'cap' ending the AdS region. We construct perturbations that correspond to the action of chiral algebra generators on Ramond ground states of D1D5 branes. Abstract arguments in the literature suggest that the perturbation should be pure gauge in the AdS region; our perturbation indeed has this structure, with the nontrivial deformation of the geometry occurring at the 'neck' between the AdS region and asymptotic infinity. This 'non-gauge' deformation is needed to provide the nonzero energy and momentum carried by the perturbation. We also suggest implications this structure may have for the majority of microstates which live at the cap.

On CY–LG correspondence for [formula omitted] toric models

We conjecture a description of the vertex (chiral) algebras of the (0,2) nonlinear sigma models on smooth quintic threefolds. We provide evidence in favor of the conjecture by connecting our algebras to the cohomology of a twisted chiral de Rham sheaf. We discuss CY/LG correspondence in this setting.

Species doublers as super multiplets in lattice supersymmetry: chiral conditions of Wess-Zumino model for D = N = 2

We propose an algebraic lattice supersymmetry formulation which has an exact supersymmetry on the lattice. We show how lattice version of chiral conditions can be imposed to satisfy an exact lattice supersymmetry algebra. The species doublers of chiral fermions and the corresponding bosonic counterparts can be accommodated to fit into chiral supermultiplets of lattice supersymmetry and thus lattice chiral fermion problem does not appear. We explicitly show how N=2 Wess-Zumino model in one and two dimensions can be formulated to keep exact supersymmetry for all super charges on the lattice. The momentum representation of N=2 lattice chiral sypersymmetry algebra has lattice periodicity and thus momentum conservation should be modified to a lattice version of sine momentum conservation, which generates nonlocal interactions and leads to a loss of lattice translational invariance. It is shown that the nonlocality is mild and the translational invariance is recovered in the continuum limit. In the coordinate representation a new type of product is defined and the difference operator satisfies Leibnitz rule and an exact lattice supersymmetry is realized on this product.

On the six-dimensional origin of the AGT correspondence

We argue that the six-dimensional (2,0) superconformal theory defined on M \times C, with M being a four-manifold and C a Riemann surface, can be twisted in a way that makes it topological on M and holomorphic on C. Assuming the existence of such a twisted theory, we show that its chiral algebra contains a W-algebra when M = R^4, possibly in the presence of a codimension-two defect operator supported on R^2 \times C \subset M \times C. We expect this structure to survive the \Omega-deformation.

Commutative algebras in Fibonacci categories

By studying NIM-representations we show that the Fibonacci category and its tensor powers are completely anisotropic; that is, they do not have any non-trivial separable commutative ribbon algebras. As an application we deduce that a chiral algebra with the representation category equivalent to a product of Fibonacci categories is maximal; that is, it is not a proper subalgebra of another chiral algebra. In particular the chiral algebras of the Yang–Lee model, the WZW models of G2 and F4 at level 1, as well as their tensor powers, are maximal.

Chiral algebras of (0, 2) models

We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of K\ahler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the H\ohn-Stolz conjecture.

Chiral algebras of (0, 2) models

We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of K\"ahler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the H\"ohn-Stolz conjecture.

Quasi-topological gauged sigma models, the geometric Langlands program, and knots

We construct and study a closed, two-dimensional, quasi-topological (0, 2) gauged sigma model with target space a smooth G-manifold, where G is any compact and connected Lie group. When the target space is a flag manifold of simple G, and the gauge group is a Cartan subgroup thereof, the perturbative model describes, purely physically, the recently formulated mathematical theory of “Twisted Chiral Differential Operators”. This paves the way, via a generalized T -duality, for a natural physical interpretation of the geometric Langlands correspondence for simply-connected, simple, complex Lie groups. In particular, the Hecke eigensheaves and Hecke operators can be described in terms of the correlation functions of certain operators that underlie the infinite-dimensional chiral algebra of the flag manifold model. Nevertheless, nonperturbative worldsheet twisted-instantons can, in some situations, trivialize the chiral algebra completely. This leads to a spontaneous breaking of supersymmetry whilst implying certain delicate conditions for the existence of BeilinsonDrinfeld D-modules. Via supersymmetric gauged quantum mechanics on loop space, these conditions can be understood to be intimately related to a conjecture by Hohn-Stolz [1] regarding the vanishing of the Witten genus on string manifolds with positive Ricci curvature. An interesting connection to Chern-Simons theory is also uncovered, whence we would be able to (i) relate general knot invariants of three-manifolds and Khovanov homology to “quantum” ramified D-modules and Lagrangian intersection Floer homology; (ii) furnish physical proofs of mathematical conjectures by Seidel-Smith [2] and Gaitsgory [3, 4] which relate knots to symplectic geometry and Langlands duality, respectively. ∗E-mail: tan@ias.edu ar X iv :1 11 1. 06 91 v1 [ he pth ] 2 N ov 2 01 1

Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004), to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004) on chiral enveloping algebras of \({\star}\) -Lie algebras.

A universal inequality for CFT and quantum gravity

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with central charges c_L, c_R > 1) contains a primary operator with dimension Delta_1 that satisfies 0 < Delta_1 < (c_L + c_R)/12 + 0.473695. Translated into gravitational language using the AdS_3 /CFT_2 dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D gravity with cosmological constant Lambda < 0 can be no heavier than 1/(4 G_N) + o(|Lambda|^(1/2)). In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation of the bound applies at finite central charge for the CFT, and does not rely on an asymptotic expansion at large central charge. Neither does our proof rely on any special property of the CFT such as supersymmetry or holomorphic factorization, nor on any bulk interpretation in terms of string theory or semiclassical gravity. Our only assumptions are unitarity and modular invariance of the dual CFT. Our proof demonstrates for the first time that there exists a universal center-of-mass energy beyond which a theory of "pure" quantum gravity can never consistently be extended.

On duality and extended chiral symmetry in the WZW model

Two chiral aspects of the WZW model in an operator formalism are investigated. First, the meaning of duality, or conjugation, of primary fields is clarified. On a class of modules obtained from the discrete series it is shown, by looking at spaces of two-point conformal blocks, that a natural definition of contragredient module provides a suitable notion of conjugation of primary fields, consistent with known two–point functions. We find strong indications that an apparent contradiction with the Clebsch–Gordan series of , and proposed fusion rules, is explained by non-semi-simplicity of a certain category. Second, results indicating an infinite cyclic simple current group, corresponding to spectral flow automorphisms, are presented. In particular, the subgroup corresponding to even spectral flow provides part of a hypothetical extended chiral algebra resulting in proposed modular invariant bulk spectra.

RG flow of magnetic brane correlators

The magnetic brane solution to five-dimensional Einstein-Maxwell-Chern-Simons theory provides a holographic description of the RG flow from four-dimensional Yang-Mills theory in the presence of a constant magnetic field to a two-dimensional low energy CFT. We compute two-point correlators involving the U(1) current and the stress tensor, and use their leading IR behavior to confirm the existence of a single chiral current algebra, and of left-and right-moving Virasoro algebras in the low energy CFT. The common central charge of the Virasoro algebras is found to match the Brown-Henneaux formula, while the level of the current algebra is related to the Chern-Simons coupling. The coordinate reparametrizations produced by the Virasoro algebras on the AdS3 near-horizon geometry arise from physical non-pure gauge modes in the asymptotic AdS5 region, thereby providing a concrete example for the emergence of IR symmetries. Finally, we interpret the infinite series of sub-leading IR contributions to the correlators in terms of certain double-trace interactions generated by the RG flow in the low energy CFT.

A heterotic sigma model with novel target geometry

We construct a (1,2) heterotic sigma model whose target space geometry consists of a transitive Lie algebroid with complex structure on a Kaehler manifold. We show that, under certain geometrical and topological conditions, there are two distinguished topological half--twists of the heterotic sigma model leading to A and B type half--topological models. Each of these models is characterized by the usual topological BRST operator, stemming from the heterotic (0,2) supersymmetry, and a second BRST operator anticommuting with the former, originating from the (1,0) supersymmetry. These BRST operators combined in a certain way provide each half--topological model with two inequivalent BRST structures and, correspondingly, two distinct perturbative chiral algebras and chiral rings. The latter are studied in detail and characterized geometrically in terms of Lie algebroid cohomology in the quasiclassical limit.

Hawking radiation and near horizon universality of chiral Virasoro algebra

We show that the diffeomorphism anomaly together with the trace anomaly reveal a chiral Virasoro algebra near the event horizon of a black hole. This algebra is the same irrespective of whether the anomaly is covariant or consistent, thereby manifesting its universal character and the fact that only the outgoing modes are relevant near the horizon. Our analysis therefore clarifies the role of the trace anomaly in the diffeomorphism anomaly approach [Robinson and Wilczek in Phys. Rev. Lett. 95:011303, 2005; Iso et al. in Phys. Rev. Lett. 96:151302, 2006; Banerjee and Kulkarni in Phys. Rev. D 77:024018, 2008; Gangopadhyay and Kulkarni in Phys. Rev. D 77:024038, 2008] to the Hawking radiation.

Formula omitted] and the triplet model

Conformal field theories with sl ˆ ( 2 ) − 1 / 2 symmetry are studied with a view to investigating logarithmic structures. Applying the parafermionic coset construction to the non-logarithmic theory, a part of the structure of the triplet model is uncovered. In particular, the coset theory is shown to admit the triplet W-algebra as a chiral algebra. This motivates the introduction of an augmented sl ˆ ( 2 ) − 1 / 2 -theory for which the corresponding coset theory is precisely the triplet model. This augmentation is envisaged to lead to a precise characterisation of the “logarithmic lift” of the non-logarithmic sl ˆ ( 2 ) − 1 / 2 -theory that has been proposed by Lesage et al.

Baryon fields withUL(3)×UR(3)chiral symmetry: Axial currents of nucleons and hyperons

We use the conventional F and D octet and decimet generator matrices to reformulate chiral properties of local (non-derivative) and one-derivative non-local fields of baryons consisting of three quarks with flavor SU(3) symmetry that were expressed in SU(3) tensor form in Ref. [12]. We show explicitly the chiral transformations of the [(6,3)\oplus(3,6)] chiral multiplet in the "SU(3) particle basis", for the first time to our knowledge, as well as those of the (3,\bar{3}) \oplus (\bar{3}, 3), (8,1) \oplus (1,8) multiplets, which have been recorded before in Refs. [4,5]. We derive the vector and axial-vector Noether currents, and show explicitly that their zeroth (charge-like) components close the SU_L(3) \times SU_R(3) chiral algebra. We use these results to study the effects of mixing of (three-quark) chiral multiplets on the axial current matrix elements of hyperons and nucleons. We show, in particular, that there is a strong correlation, indeed a definite relation between the flavor-singlet (i.e. the zeroth), the isovector (the third) and the eighth flavor component of the axial current, which is in decent agreement with the measured ones.

Non-conformal limit of AGT relation from the 1-point torus conformal block

Given a 4d N=2 SUSY gauge theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require regularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformal blocks in 2d theories with W_{N_c} chiral algebra. For $N_c=2$ and one adjoint multiplet it coincides with a torus 1-point Virasoro conformal block. We check the AGT relation between conformal dimension and adjoint multiplet's mass in this case and investigate the limit of the conformal block, which corresponds to the large mass limit of the 4d theory e.i. the asymptotically free 4d N=2 supersymmetric Yang-Mills theory. Though technically more involved, the limit is the same as in the case of fundamental multiplets, and this provides one more non-trivial check of AGT conjecture.

Module Categories For Permutation Modular Invariants

We show that a braided monoidal category C can be endowed with the structure of a right (and left) module category over C \times C. In fact, there is a family of such module category structures, and they are mutually isomorphic if and only if C allows for a twist. For the case that C is premodular we compute the internal End of the tensor unit of C, and we show that it is an Azumaya algebra if C is modular. As an application to two-dimensional rational conformal field theory, we show that the module categories describe the permutation modular invariant for models based on the product of two identical chiral algebras. It follows in particular that all permutation modular invariants are physical.