arXiv subjects

V. B. Petkova

Publications and source records attributed to V. B. Petkova.

At least 19 recordsLinked to original sources

arXiv2026

On some Coulomb gas integrals in higher dimensions

We point out that there is a generalization to higher dimensions $d=2h>2$ of the two-dimensional Dotsenko-Fateev formula \DF\ for particular Coulomb gas conformal invariant integrals. These expressions represent structure constants of 3-point functions of vertex operators related to a higher dimensional generalization of the Liouville theory. The Coulomb gas formulae admit two different analytic continuations generalizing the DOZZ formula, i.e., the theory is not selfdual for $d>2$.

hep-th↗

arXiv2016

$W_4$ Toda example as hidden Liouville CFT

We construct correlators in the $W_4$ Toda 2d conformal field theory for a particular class of representations and demonstrate a relation to a $W_2$ (Virasoro) theory with different central charge. The relevance of the classical limits of the constructed 3-point functions and braiding matrices to problems in 4d conformal theories is discussed.

hep-th↗

arXiv2015

On some 3-point functions in the $W_4$ CFT and related braiding matrix

We construct a class of 3-point constants in the $sl(4)$ Toda conformal theory $W_4$, extending the examples in Fateev and Litvinov. Their knowledge allows to determine the braiding/fusing matrix transforming 4-point conformal blocks of one fundamental, labelled by the 6-dimensional $sl(4)$ representation, and three partially degenerate vertex operators. It is a $3 \times 3$ submatrix of the generic $6 \times 6$ fusing matrix consistent with the fusion rules for the particular class of representations. We check a braiding relation which has wider applications to conformal models with $sl(4)$ symmetry. The 3-point constants in dual regions of central charge are compared in preparation for a BPS like relation in the $\hat{sl}(4)$ WZW model.

hep-th↗

arXiv2012

On the semiclassical 3-point function in AdS_3

We reconsider the problem of determining the semiclassical 3-point function in the Euclidean AdS_3 model. Exploiting the affine symmetry of the model we use solutions of the classical Knizhnik-Zamolodchikov (KZ) equation to compute the saddle point of the action in the presence of three vertex operators. This alternative derivation reproduces the "heavy charge" classical limit of the quantum 3-point correlator. It is different from the recently proposed expression obtained by generalised Pohlmeyer reduction in AdS_2

hep-th↗

arXiv2010

On the crossing relation in the presence of defects

The OPE of local operators in the presence of defect lines is considered both in the rational CFT and the $c>25$ Virasoro (Liouville) theory. The duality transformation of the 4-point function with inserted defect operators is explicitly computed. The two channels of the correlator reproduce the expectation values of the Wilson and 't Hooft operators, recently discussed in Liouville theory in relation to the AGT conjecture.

hep-th↗

arXiv2008

Non-critical string pentagon equations and their solutions

We derive pentagon type relations for the 3-point boundary tachyon correlation functions in the non-critical open string theory with generic c_{matter} < 1 and study their solutions in the case of FZZ branes. A new general formula for the Liouville 3-point factor is derived.

hep-th↗

arXiv2007

Non-Rational 2D Quantum Gravity: I. World Sheet CFT

We address the problem of computing the tachyon correlation functions in Liouville gravity with generic (non-rational) matter central charge c<1. We consider two variants of the theory. The first is the conventional one in which the effective matter interaction is given by the two matter screening charges. In the second variant the interaction is defined by the Liouville dressings of the non-trivial vertex operator of zero dimension. This particular deformation, referred to as "diagonal'', is motivated by the comparison with the discrete approach, which is the subject of a subsequent paper. In both theories we determine the ground ring of ghost zero physical operators by computing its OPE action on the tachyons and derive recurrence relations for the tachyon bulk correlation functions. We find 3- and 4-point solutions to these functional equations for various matter spectra. In particular, we find a closed expression for the 4-point function of order operators in the diagonal theory.

hep-th↗

arXiv2006

Non-Rational 2D Quantum Gravity II. Target Space CFT

We explore the formulation of non-rational 2D quantum gravity in terms of a chiral CFT on a Riemann surface associated with the target space. The CFT in question is constructed as the collective theory for a matrix chain, which is dual to a statistical height model on dynamical triangulations. The heights are associated with the sheets of the Riemann surface, which represents an infinite branched cover of the spectral plane. We consider two examples of height models: the SOS model and the semi-restricted SOS (SRSOS) model, in which the heights are restricted from below. Both models are described in the continuum limit by theories of 2D quantum gravity with conformal matter, perturbed by a thermal operator (1,3). We give a compact operator expression for the n-loop amplitudes as a collection of target space Feynman rules. The n-point functions of local fields are obtained by shrinking the loops. In particular, we show that the 4-point function of order operators in the SRSOS model coincides with the 4-point function of the ``diagonal'' world sheet CFT studied in [1].

hep-th↗

arXiv2005

Bulk correlation functions in 2D quantum gravity

We compute bulk 3- and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c<1, following and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground ring generators deformed by Liouville and matter ``screening charges''. A by-product general formula for the matter 3-point OPE structure constants is derived. We also consider a ``diagonal'' CFT of 2D quantum gravity, in which the degenerate fields are restricted to the diagonal of the semi-infinite Kac table. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points.

hep-th↗

arXiv2004

Virasoro Symmetry in a 2h-dimensional Model and Its Implications

The set of two partial differential equations for the Appell hypergeometric function in two variables F_4(a,b,c,a+b-c+2-h,x,y) is shown to arise as a null vector decoupling relation in a 2h-dimensional generalisation of the Coulomb gas model. It corresponds to a level two singular vector of an intrinsic Virasoro algebra.

hep-th↗

arXiv2002

Boundary conditions in charge conjugate sl(N) WZW theories

We compute the representations (``nimreps'') of the fusion algebra of affine sl(N), which determine the boundary conditions of sl(N) WZW theories twisted by the charge conjugation. This is done following two procedures, one of general validity, the other specific to the problem at hand. The problem is related to the classical problem of decomposition of the fundamental representations of sl(N) onto representations of $B_l=so(2l+1)$ or $C_l =sp(2l)$ algebras. The relevant nimreps and their diagonalisation matrix are thus expressed in terms of modular data of the affine B or C algebras

hep-th↗

arXiv2001

Characters of $\hat{sl}(4)_k$ fusion algebra at non-rational level

We construct the fusion ring of a quasi-rational $\hat{sl}(4)_k$ WZNW theory at generic level $k \not\in Q$. It is generated by commutative elements in the group ring $Z[\tilde{W}]$ of the affine Weyl group $\tilde{W}$ which extend polynomially the formal characters of finite dimensional representations of $sl(4)$.

hep-th↗

arXiv2001

The many faces of Ocneanu cells

We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A}, reappear in several guises. With {\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.

hep-th↗

arXiv2001

Conformal Boundary Conditions and what they teach us

The question of boundary conditions in conformal field theories is discussed, in the light of recent progress. Two kinds of boundary conditions are examined, along open boundaries of the system, or along closed curves or ``seams''. Solving consistency conditions known as Cardy equation is shown to amount to the algebraic problem of finding integer valued representations of (one or two copies of) the fusion algebra. Graphs encode these boundary conditions in a natural way, but are also relevant in several aspects of physics ``in the bulk''. Quantum algebras attached to these graphs contain information on structure constants of the operator algebra, on the Boltzmann weights of the corresponding integrable lattice models etc. Thus the study of boundary conditions in Conformal Field Theory offers a new perspective on several old physical problems and offers an explicit realisation of recent mathematical concepts.

hep-th↗

arXiv2000

Generalised twisted partition functions

We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.

hep-th↗

arXiv2000

BCFT: from the boundary to the bulk

The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory ``in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden ``quantum symmetry''. The Ocneanu triangular cells -- the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.

hep-th↗

arXiv2000

Fusion Rings Related to Affine Weyl Groups

The construction of the fusion ring of a quasi-rational CFT based on $\hat{sl}(3)_k$ at generic level $k\not \in {\Bbb Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring ${\Bbb Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\hat{sl}(3)_k$. Some partial results towards the $\hat{sl}(4)_k$ generalisation of this character ring are presented.

hep-th↗

arXiv1999

A note on decoupling conditions for generic level $\hat{sl}(3)_k$ and fusion rules

We find the solution of the $\hat{sl}(3)_k$ singular vector decoupling equations on 3-point functions for the particular case when one of the fields is of weight $w_0\cdot kΛ_0$. The result is a function with non-trivial singularities in the flag variables, namely a linear combination of 2F1 hypergeometric functions. This calculation fills in a gap in [1] and confirms the $\hat{sl}(3)_k$ fusion rules determined there both for generic $κ\not \in \IQ$ and fractional levels. We have also analysed the fusion in $\hat{sl}(3)_k$ using algebraic methods generalising those of Feigin and Fuchs and again find agreement with [1]. In the process we clarify some details of previous treatments of the fusion of $\hat{sl}(2)_k$ fractional level admissible representations.

hep-th↗