Finiteness of conformal blocks over the projective line

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

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Logarithmic Intertwining Operators and the Space of Conformal Blocks Over the Projective Line

Letters in Mathematical Physics ◽

10.1007/s11005-012-0546-9 ◽

2012 ◽

Vol 101 (1) ◽

pp. 13-26 ◽

Cited By ~ 1

Author(s):

Yusuke Arike

Keyword(s):

Projective Line ◽

Intertwining Operators ◽

Conformal Blocks ◽

Logarithmic Intertwining Operators

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Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

Complex Manifolds ◽

10.1515/coma-2017-0005 ◽

2017 ◽

Vol 4 (1) ◽

pp. 43-72 ◽

Cited By ~ 1

Author(s):

Martin de Borbon

Keyword(s):

Vector Field ◽

Einstein Metrics ◽

Projective Line ◽

Tangent Cones ◽

Reeb Vector ◽

Reeb Vector Field ◽

Complex Curves ◽

Complex Dimensions ◽

Irregular Case ◽

Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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Towards Feynman rules for conformal blocks

Journal of High Energy Physics ◽

10.1007/jhep01(2021)005 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sarah Hoback ◽

Sarthak Parikh

Keyword(s):

Field Theory ◽

Effective Field Theory ◽

Hypergeometric Functions ◽

Power Series Expansion ◽

Effective Field ◽

Conformal Block ◽

Tree Level ◽

Conformal Blocks ◽

Simple Set ◽

Feynman Rules

Abstract We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

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Rational functions of degree four that permute the projective line over a finite field

Communications in Algebra ◽

10.1080/00927872.2021.1906887 ◽

2021 ◽

pp. 1-16

Author(s):

Xiang-dong Hou

Keyword(s):

Finite Field ◽

Rational Functions ◽

Projective Line

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Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

Journal of High Energy Physics ◽

10.1007/jhep12(2020)019 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

Yifei He ◽

Jesper Lykke Jacobsen ◽

Hubert Saleur

Keyword(s):

Potts Model ◽

Correlation Functions ◽

Liouville Theory ◽

Analytical Determination ◽

Conformal Weight ◽

Two Dimensional ◽

Conformal Blocks ◽

Conformal Bootstrap ◽

Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

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Dispersion formulas in QFTs, CFTs and holography

Journal of High Energy Physics ◽

10.1007/jhep05(2021)098 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

David Meltzer

Keyword(s):

Fourier Transform ◽

Correlation Functions ◽

Momentum Space ◽

Conformal Blocks ◽

Dispersion Formula ◽

Space Dispersion

Abstract We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At n-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

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