2D quantum gravity on compact Riemann surfaces and two-loop partition function: A first principles approach

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

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2D quantum gravity on compact Riemann surfaces with non-conformal matter

Journal of High Energy Physics ◽

10.1007/jhep01(2017)089 ◽

2017 ◽

Vol 2017 (1) ◽

Cited By ~ 3

Author(s):

Adel Bilal ◽

Lætitia Leduc

Keyword(s):

Quantum Gravity ◽

Riemann Surfaces ◽

Compact Riemann Surfaces

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Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors

Journal of High Energy Physics ◽

10.1007/jhep08(2020)135 ◽

2020 ◽

Vol 2020 (8) ◽

Author(s):

Eric D’Hoker ◽

Carlos R. Mafra ◽

Boris Pioline ◽

Oliver Schlotterer

Keyword(s):

First Principles ◽

Riemann Surfaces ◽

Companion Paper ◽

Type Ii ◽

Pure Spinor ◽

Perfect Agreement ◽

Brst Cohomology ◽

Spinor Formulation ◽

Compact Riemann Surfaces ◽

Genus Two

Abstract The full two-loop amplitudes for five massless states in Type II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The α′ → 0 limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type II supergravity. Investigations of the α′ expansion of the Type II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

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ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES

Modern Physics Letters B ◽

10.1142/s0217984993000199 ◽

1993 ◽

Vol 07 (03) ◽

pp. 171-182 ◽

Cited By ~ 2

Author(s):

ACHILLES D. SPELIOTOPOULOS ◽

HARRY L. MORRISON

Keyword(s):

Field Theory ◽

Riemann Surface ◽

Partition Function ◽

Riemann Surfaces ◽

Compact Riemann Surface ◽

Constant Density ◽

Two Dimensional ◽

Coulombic Interaction ◽

Density Limit ◽

Compact Riemann Surfaces

A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4 He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

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A Hodge theoretic projective structure on compact Riemann surfaces

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2021.02.005 ◽

2021 ◽

Vol 149 ◽

pp. 1-27

Author(s):

Indranil Biswas ◽

Elisabetta Colombo ◽

Paola Frediani ◽

Gian Pietro Pirola

Keyword(s):

Riemann Surfaces ◽

Projective Structure ◽

Compact Riemann Surfaces

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Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-021-09394-2 ◽

2021 ◽

Vol 24 (3) ◽

Author(s):

Alexander I. Bobenko ◽

Ulrike Bücking

Keyword(s):

Error Estimate ◽

Branch Point ◽

Riemann Surfaces ◽

Riemann Sphere ◽

Holomorphic Functions ◽

Convergence Result ◽

Edge Length ◽

Ramified Covering ◽

Compact Riemann Surfaces ◽

Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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