sine-Liouville Theory Research Articles

Instantons in sine-Liouville theory

We compute instanton corrections to the partition function of sine-Liouville (SL) theory, which provides a worldsheet description of two-dimensional string theory in a non-trivial tachyon background. We derive these corrections using a matrix model formulation based on a chiral representation of matrix quantum mechanics and using string theory methods. In both cases we restrict to the leading and subleading orders in the string coupling expansion. Then the CFT technique is used to compute two orders of the expansion in the SL perturbation parameter λ, while the matrix model gives results which are non-perturbative in λ. The matrix model results perfectly match those of string theory in the small λ expansion. We also generalize our findings to the case of perturbation by several tachyon vertex operators carrying different momenta, and obtain interesting analytic predictions for the disk two-point and annulus one-point functions with ZZ boundary condition.

FZZ-triality and large [formula omitted] super Liouville theory

We examine dualities of two dimensional conformal field theories by applying the methods developed in previous works. We first derive the duality between SL(2|1)k/(SL(2)k⊗U(1)) coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large N=4 super Liouville theory and a coset of the form Y(k1,k2)/SL(2)k1+k2, where Y(k1,k2) consists of two SL(2|1)ki and free bosons or equivalently two U(1) cosets of D(2,1;ki−1) at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden SL(2)k′ in SL(2|1)k or D(2,1;k−1)1. The relation of levels is k′−1=1/(k−1). We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with SL(n|1)k/(SL(n)k⊗U(1)) for arbitrary n>2.

String correlators in AdS3 from FZZ duality

Motivated by recent works in which the FZZ duality plays an important role, we revisit the computation of correlation functions in the sine-Liouville field theory. We present a direct computation of the three-point function, the simplest to the best of our knowledge, and give expressions for the N-point functions in terms of integrated Liouville theory correlators. This leads us to discuss the relation to the {H}_3^{+} WZW-Liouville correspondence, especially in the case in which spectral flow is taken into account. We explain how these results can be used to study scattering amplitudes of winding string states in AdS3.

Higher rank FZZ-dualities

We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the mathfrak{sl}(2)/mathfrak{u}(1) coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from mathfrak{sl}(2) Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type mathfrak{sl}left(N+1right)/left(mathfrak{sl}(N)times mathfrak{u}(1)right) and investigate the equivalence to a theory with an mathfrak{sl}left(N+left.1right|Nright) structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for mathfrak{sl}(N) and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

LCFT and Sine-Liouville theory

LCFT is special conformal field theory with logarithmic terms in operator product expansion OPE. It is well-known that it is possible to construct operators with logarithmical terms in OPE at Liouville model. Here we discuss this topic for some generalization of Liouville model, such as Sine-Liouville model.

Stringy horizons and generalized FZZ duality in perturbation theory

We study scattering amplitudes in two-dimensional string theory on a black hole bakground. We start with a simple derivation of the Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality, which associates correlation functions of the sine-Liouville integrable model on the Riemann sphere to tree-level string amplitudes on the Euclidean two-dimensional black hole. This derivation of FZZ duality is based on perturbation theory, and it relies on a trick originally due to Fateev, which involves duality relations between different Selberg type integrals. This enables us to rewrite the correlation functions of sine-Liouville theory in terms of a special set of correlators in the gauged Wess-Zumino-Witten (WZW) theory, and use this to perform further consistency checks of the recently conjectured Generalized FZZ (GFZZ) duality. In particular, we prove that n-point correlation functions in sine-Liouville theory involving n − 2 winding modes actually coincide with the correlation functions in the mathrm{S}mathrm{L}left(2,mathrm{mathbb{R}}right)/mathrm{U}(1) gauged WZW model that include n − 2 oscillator operators of the type described by Giveon, Itzhaki and Kutasov in reference [1]. This proves the GFZZ duality for the case of tree level maximally winding violating n-point amplitudes with arbitrary n. We also comment on the connection between GFZZ and other marginal deformations previously considered in the literature.

The FZZ duality with boundary

The Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality relates Witten's cigar model to sine-Liouville theory. This duality was proven in the path integral formulation and extended to the case of higher genus closed Riemann surfaces by Schomerus and one of the authors. In this note we further extend the duality to the case with boundary. Specifically, we relate D1-branes in the cigar model to D2-branes in the sine-Liouville theory. In particular, the boundary action for D2-branes in the sine-Liouville theory is constructed. We also consider the fermionic version of the FZZ duality. This duality was proven as a mirror symmetry by Hori and Kapustin, but we give an alternative proof in the path integral formulation which directly relates correlation functions. Also here the case with boundary is investigated and the results are consistent with those for branes in N=2 super Liouville field theory obtained by Hosomichi.

Remarks on free field realization of SL(2,R)/U(1) U(1) WZNW model

Free field representations of vertex algebra in SL(2,R)/U(1) x U(1) WZNW model are constructed by considering a twisted version of the Bershadsky-Kutasov free field description of discrete states in the two-dimensional black hole CFT. These correspond to conjugate representations describing primary states in the model on SL(2,R)/U(1) x U(1). A particular evaluation of these leads to identities due to the spectral flow symmetry of sl(2)_k algebra. The computation of correlation functions is discussed and, as an application, these are compared with analogous results known for the sine-Liouville theory. Exact agreement is observed between both analytic structures.

A Holographic View on Matrix Model of Black Hole

We investigate a deformed matrix model proposed by Kazakov et.al. in relation to Witten's two-dimensional black hole. The existing conjectures assert the equivalence of the two by mapping each to a deformed c=1 theory called the sine-Liouville theory. We point out that the matrix theory in question may be naturally interpreted as a gauged quantum mechanics deformed by insertion of an exponentiated Wilson loop operator, which gives us more direct and holographic map between the two sides. The matrix model in the usual scaling limit must correspond to the bosonic SL(2,R)/U(1) theory in genus expansion but exact in \alpha'. We successfully test this by computing the Wilson loop expectation value and comparing it against the bulk computation. For the latter, we employ the \alpha'-exact geometry proposed by Dijkgraaf, Verlinde, and Verlinde, which was further advocated by Tseytlin. We close with comments on open problems.

Three-point functions in sine-Liouville theory

We calculate the three-point functions in the sine-Liouville theory explicitly. The same calculation was done in the (unpublished) work of Fateev, Zamolodchikov and Zamolodchikov to check the conjectured duality between the sine-Liouville and the SL(2,R)/U(1) coset CFTs. The evaluation of correlators boils down to that of a free-field theory with a certain number of insertion of screening operators. We prove that the winding number conservation is violated up to (+-)1 in three-point functions, which is in agreement with the result of FZZ that in generic N-point correlators the winding number conservation is violated up to N-2 units. A new integral formula of Dotsenko-Fateev type is derived, using which we write down the generic three-point functions of tachyons explicitly. When the winding number is conserved, the resultant expression is shown to reproduce the correlators in the coset model correctly, including the group-theoretical factor. As an application, we also study the superstring theory on linear dilaton background which is described by super-Liouville theory. We obtain the three-point amplitude of tachyons in which the winding number conservation is violated.