Holmstrom Topological modular forms

Topological modular forms

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Topological modular forms

AT (Algebraic topology), AAG (Arithmetic algebraic geometry)

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Topological modular forms

An appendix of Husemoller: Elliptic curves, has an introduction

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Topological modular forms

See maybe Toen and Vezzosi: Brave new algebraic geometry. File Toen web publ del.pdf.

arXiv:1103.4187 Topological modular forms and conformal nets from arXiv Front: math.AT by Christopher L. Douglas, André G. Henriques 1 person liked this We describe the role conformal nets, a mathematical model for conformal field theory, could play in a geometric definition of the generalized cohomology theory TMF of topological modular forms. Inspired by work of Segal and Stolz-Teichner, we speculate that bundles of boundary conditions for the net of free fermions will be the basic underlying objects representing TMF-cohomology classes. String structures, which are the fundamental orientations for TMF-cohomology, can be encoded by defects between free fermions, and we construct the bundle of fermionic boundary conditions for the TMF-Euler class of a string vector bundle. We conjecture that the free fermion net exhibits an algebraic periodicity corresponding to the 576-fold cohomological periodicity of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we establish a lower bound of 24 on this periodicity of the free fermions.

arXiv:1101.3897 Topological modular forms of level 3 and $E_\infty$-structures on truncated Brown-Peterson spectra from arXiv Front: math.AT by Tyler Lawson, Niko Naumann A generalized truncated Brown-Peterson spectrum of height $2$ at the prime $2$ is constructed as an $E_\infty$-ring spectrum by methods of chromatic homotopy theory. The first portion of the paper introduces the machinery necessary to reduce this to finding an appropriate formal group law as input data. The second portion of the paper shows that the compactified moduli of elliptic curves with level $\Gamma_1(3)$-structure provides this data at the prime 2.

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Topological modular forms

A reading list

arXiv:0910.5130 Topological modular forms (aftern Hopkins, Miller, and Lurie) from arXiv Front: math.AT by Paul G. Goerss This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie theorem on topological modular forms, from which we learn that the Deligne-Mumford moduli stack for elliptic curves is canonically realized as an object in derived algebraic geometry.

Some notes by Behrens on the construction

http://mathoverflow.net/questions/283/what-is-a-tmf-in-topology includes many references, for example a Bourbaki talk of Goerss.

http://ncatlab.org/nlab/show/tmf

nLab page on Topological modular forms