Topological modular forms
Encyclopedia
In mathematics
, the spectrum
of topological modular forms (tmf) describes a generalized cohomology theory whose coefficient ring
is related to the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting 6.
tmf is constructed as the global sections of a sheaf
of E-infinity
ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory
, the homotopy groups of spheres
, and conjectural index theories
on loop spaces of manifold
s. tmf was first constructed by Mike Hopkins
and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Mike Hopkins, Mark Mahowald, Haynes Miller, Charles Rezk, and Tilman Bauer.
of Hopkins
, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf O top ("top" stands for topological) of multiplicative cohomology theories on the etale
site
of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf
of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology
theory) whose associated formal group
is the formal group of that elliptic curve.
A second construction, due to Jacob Lurie
, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor
that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted. These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne-Mumford compactification
of this moduli stack, in which elliptic curves with nodal singularities are included. TMF is the spectrum that results from the global sections over the moduli stack of smooth curves, and tmf is the spectrum arising as the global sections of the Deligne-Mumford compactification.
TMF is a periodic version of the connective tmf. While the ring spectra used to construct TMF are periodic with period 2, TMF itself has period 576. The periodicity is related to the modular discriminant.
and conformal field theory
. Graeme Segal
first proposed in the 1980s to provide a geometric construction of elliptic cohomology
(the precursor to tmf) as some kind of moduli space of conformal field theories, and these ideas have been continued and expanded by Stephan Stolz and Peter Teichner. Their program is to try to construct TMF as a moduli space of supersymmetric Euclidean field theories.
In work more directly motivated by string theory, Edward Witten
introduced the Witten genus, a homomorphism from the string bordism ring to the ring of modular forms, using equivariantindex theory on a formal neighborhood of the trivial locus in the loop space of a manifold. This associates to any spin manifold with vanishing half first Pontryagin class a modular form. By work of Hopkins, Matthew Ando, Charles Rezk and Neil Strickland, the Witten genus can be lifted to topology. That is, there is a map from the string bordism spectrum to tmf (a so called orientation) such that the Witten genus is recovered as the composition of the induced map on the homotopy groups of these spectra and a map of the homotopy groups of tmf to modular forms. This allowed to prove certain divisibility statements about the Witten genus. The orientation of tmf is in analogy with the Atiyah-Bott-Shapiro map from the spin bordism spectrum to classical K-theory
, which is a lift of the Dirac equation to topology.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the spectrum
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
of topological modular forms (tmf) describes a generalized cohomology theory whose coefficient ring
Eilenberg-Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common...
is related to the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting 6.
tmf is constructed as the global sections of a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of E-infinity
Highly structured ring spectrum
In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring...
ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the homotopy groups of spheres
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
, and conjectural index theories
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
on loop spaces of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s. tmf was first constructed by Mike Hopkins
Michael J. Hopkins
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Northwestern University in 1984 under the direction of Mark Mahowald. In 1984 he also received his D.Phil...
and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Mike Hopkins, Mark Mahowald, Haynes Miller, Charles Rezk, and Tilman Bauer.
Construction
The original construction of tmf uses the obstruction theoryObstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.-In homotopy theory:...
of Hopkins
Michael J. Hopkins
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Northwestern University in 1984 under the direction of Mark Mahowald. In 1984 he also received his D.Phil...
, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf O top ("top" stands for topological) of multiplicative cohomology theories on the etale
Étale
In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts:* Étale morphism** Formally étale morphism* Étale cohomology* Étale topology* Étale fundamental group* Étale space* Étale group scheme...
site
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.-History and motivation:Historically, elliptic cohomology arose from the study of elliptic genera...
theory) whose associated formal group
Formal group
In mathematics, a formal group law is a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between...
is the formal group of that elliptic curve.
A second construction, due to Jacob Lurie
Jacob Lurie
Jacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...
, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted. These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne-Mumford compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of this moduli stack, in which elliptic curves with nodal singularities are included. TMF is the spectrum that results from the global sections over the moduli stack of smooth curves, and tmf is the spectrum arising as the global sections of the Deligne-Mumford compactification.
TMF is a periodic version of the connective tmf. While the ring spectra used to construct TMF are periodic with period 2, TMF itself has period 576. The periodicity is related to the modular discriminant.
Relations to other parts of mathematics
Some interest in tmf comes from string theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
. Graeme Segal
Graeme Segal
Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...
first proposed in the 1980s to provide a geometric construction of elliptic cohomology
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.-History and motivation:Historically, elliptic cohomology arose from the study of elliptic genera...
(the precursor to tmf) as some kind of moduli space of conformal field theories, and these ideas have been continued and expanded by Stephan Stolz and Peter Teichner. Their program is to try to construct TMF as a moduli space of supersymmetric Euclidean field theories.
In work more directly motivated by string theory, Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
introduced the Witten genus, a homomorphism from the string bordism ring to the ring of modular forms, using equivariantindex theory on a formal neighborhood of the trivial locus in the loop space of a manifold. This associates to any spin manifold with vanishing half first Pontryagin class a modular form. By work of Hopkins, Matthew Ando, Charles Rezk and Neil Strickland, the Witten genus can be lifted to topology. That is, there is a map from the string bordism spectrum to tmf (a so called orientation) such that the Witten genus is recovered as the composition of the induced map on the homotopy groups of these spectra and a map of the homotopy groups of tmf to modular forms. This allowed to prove certain divisibility statements about the Witten genus. The orientation of tmf is in analogy with the Atiyah-Bott-Shapiro map from the spin bordism spectrum to classical K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
, which is a lift of the Dirac equation to topology.