# Lectures notes on compact Riemann surfaces

@article{Eynard2018LecturesNO, title={Lectures notes on compact Riemann surfaces}, author={Bertrand Eynard}, journal={arXiv: Mathematical Physics}, year={2018} }

This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann--Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and…

## 12 Citations

### Aspects of Scattering Amplitudes and Moduli Space Localization

- Mathematics
- 2020

We propose that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with $n$ punctures, $\mathcal{M}_{0,n}$, compute tree-level scattering amplitudes…

### Riemann surfaces for KPZ with periodic boundaries

- MathematicsSciPost Physics
- 2020

The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume.…

### Riemann surface for TASEP with periodic boundaries

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

The Bethe ansatz solution of periodic TASEP is formulated in terms of a ramified covering from a Riemann surface to the sphere. The joint probability distribution of height fluctuations at n distinct…

### Multi-trace correlators from permutations as moduli space

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe study the n-point functions of scalar multi-trace operators in the U(Nc) gauge theory with adjacent scalars, such as N$$ \mathcal{N} $$ = 4 super Yang-Mills, at tree-level by using finite…

### On KdV characters in large c CFTs

- Physics
- 2019

Two-dimensional conformal ﬁeld theories with just Virasoro symmetry are endowed with integrable structure. We review how to construct the integrable charges in a two-dimensional conformal ﬁeld theory…

### Korteweg–de Vries characters in large central charge CFTs

- Physics
- 2020

Two-dimensional conformal field theories with just Virasoro symmetry are endowed with integrable structure. We review how to construct the integrable charges in a two-dimensional conformal field…

### Scattering and Strebel graphs

- MathematicsSciPost Physics
- 2022

We consider a special scattering experiment with n particles in
\mathbb{R}^{1,n-3}ℝ1,n−3.
The scattering equations in this set-up become the saddle-point
equations of a Penner-like matrix model,…

### Riemann surfaces for integer counting processes

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2022

Integer counting processes increment the integer value at transitions between states of an underlying Markov process. The generator of a counting process, which depends on a parameter conjugate to…

### Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks

- Mathematics
- 2020

In this paper we study height fluctuations of random lozenge tilings of polygonal domains on the triangular lattice through nonintersecting Bernoulli random walks. For a large class of polygons which…

### Edge Universality for Nonintersecting Brownian Bridges.

- Mathematics
- 2020

In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq…

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