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TFT CONSTRUCTION OF RCFT CORRELATORS II: Unoriented World Sheets
, 2003
"... A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that th ..."
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Cited by 20 (6 self)
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A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion σ on A – an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes. We study the classification of reversions, work out the construction of the annulus, Möbius strip and Klein bottle partition functions, and discuss properties of defect lines on nonorientable world sheets. As an illustration, the Ising model is treated in detail.
Binomial Formula For Macdonald Polynomials And Its Applications
"... . We generalize the binomial formula for Jack polynomials proved in [OO2] and consider some applications. x1 Binomial formula Binomial type theorems (that is Taylor and Newton interpolation expansions about various points) are powerful tools for handling special functions. The highschool binomial f ..."
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Cited by 19 (3 self)
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. We generalize the binomial formula for Jack polynomials proved in [OO2] and consider some applications. x1 Binomial formula Binomial type theorems (that is Taylor and Newton interpolation expansions about various points) are powerful tools for handling special functions. The highschool binomial formula is the Taylor expansion of the function f(x) = x l about the point x = 1. Its qdeformation is the Newton interpolation with knots x = 1; q; q 2 ; : : : ; which reads (1.1) x l = X m l m q (x \Gamma 1) : : : (x \Gamma q m\Gamma1 ) ; where l m q = (q l \Gamma 1) : : : (q l \Gamma q m\Gamma1 ) (q m \Gamma 1) : : : (q m \Gamma q m\Gamma1 ) is the qbinomial coefficient. Denote the Newton interpolation polynomials in the RHS of (1.1) by P k (x; q) = (x \Gamma 1) \Delta \Delta \Delta (x \Gamma q k\Gamma1 ) ; k = 0; 1; 2; : : : : The formula inverse to (1.1) is the Taylor expansion of P l (x; q) about the point x = 0 (1.2) (x \Gamma 1) : : : ...
Correlation functions and boundary conditions in RCFT and threedimensional topology
, 1999
"... We give a general construction of correlation functions in rational conformal field theory on a possibly nonorientable surface with boundary in terms of 3dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions ..."
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Cited by 18 (7 self)
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We give a general construction of correlation functions in rational conformal field theory on a possibly nonorientable surface with boundary in terms of 3dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular invariance and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.
6j symbols for Uq(sl2) and nonEuclidean tetrahedra. math.QA/0305113
"... In this paper we study the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra Uq(sl2) for q a primitive root of unity. Because of the work of Finkelberg [7], these 6j symbols can also be defined in terms of fusion product of representations ..."
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Cited by 13 (1 self)
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In this paper we study the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra Uq(sl2) for q a primitive root of unity. Because of the work of Finkelberg [7], these 6j symbols can also be defined in terms of fusion product of representations of the affine Lie algebra ̂ sl2, defined using
Quantum groups at roots of unity and modularity
 J. Knot Theory Ramifications
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On Algebraic Singularities, Finite Graphs and DBrane Gauge Theories: A String Theoretic Perspective
, 2002
"... In this writing we shall address certain beautiful interrelations between the construction of 4dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such ..."
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Cited by 11 (8 self)
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In this writing we shall address certain beautiful interrelations between the construction of 4dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, HananyWitten setups and Dbrane probes. We investigate aspects of worldvolume gauge dynamics using Dbrane resolutions of various CalabiYau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NSNS Bfield, as well as the Tduals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay’s Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author’s PhD thesis which was written under the direction of Prof. A. Hanany to whom he is much indebted at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.
Correlation functions and boundary conditions in rational conformal field theory and threedimensional topology
 Compositio Math
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Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator
"... Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in te ..."
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Cited by 8 (5 self)
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Abstract. We consider the space of elliptic hypergeometric functions of the sl2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl2 type.