Mark Mahowald, Charles Rezk
We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma_0(3)).
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This article is a brief survey of the theory of topological modular forms (TMF) and the theory of topological automorphic forms (TAF). It will be a chapter in forthcoming "Handbook of Homotopy Theory" edited by Haynes Miller.
We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald-Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.
We describe a new link between the theory of topological modular forms and representations of vertex operator algebras obtained by certain lattices. The construction is motivated by the arithmetic Whitehead tower of the orthogonal groups. The tower discloses the role of codes in representation theory.
The spectrum of strict units of topological modular forms
本文在谱代数几何框架下，计算拓扑模形式的严格单位，即从$\boldsymbol{\boldsymbol{Z}}$到单位谱的映射谱的连通覆盖，为该领域研究提供了关键计算结果。
Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E-infinity genera valued in topological modular forms with level structure. We give examples of such series for tmf_0(N) and show that the Ochanine genus comes from an E-infinity ring map. We also show that, away from 6, certain tmf orientations of String cobordism descend to orientations of Spin cobordism.
Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.
We compute the homotopy groups of the $C_2$ fixed points of equivariant topological modular forms at the prime $2$ using the descent spectral sequence. We then show that as a $\mathrm{TMF}$-module, it is isomorphic to the tensor product of $\mathrm{TMF}$ with an explicit finite cell complex.
The descent spectral sequence for topological modular forms
本文通过发展合成谱技术，证明了拓扑模形式Tmf的间隙定理，确认了Hopkins和Mahowald二十多年前的π_* tmf计算。
Connective Models for Topological Modular Forms of Level $n$
本文构造并研究高阶拓扑模形式的连接版本，并利用它们将 Hirzebruch 的 $n$ 级亏格实现为环谱映射。
It has been observed that certain localizations of the spectrum of topological modular forms are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves $ \M $, yet is only true in the derived setting. When $ 2 $ is inverted, a choice of level $ 2 $ structure for an elliptic curve provides a geometrically well-behaved cover of $ \M $, which allows one to consider $ Tmf $ as the homotopy fixed points of $ Tmf(2) $, topological modular forms with level $ 2 $ structure, und…
The algebraic Novikov spectral sequence for topological modular forms
本文研究拓扑模形式的代数Novikov谱序列，通过计算mmf/τ的C-动机Adams谱序列并分析τ乘法长正合列中的映射，以纯代数方法独立得到其Adams-Novikov谱序列E2页。
Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new…
The $\mathbb C$-motivic Adams-Novikov spectral sequence for topological modular forms
本文分析了 $\mathbb{C}$-motivic 模形式谱 $\mathit{mmf}$ 及经典拓扑模形式谱 $\mathit{tmf}$ 的 Adams-Novikov 谱序列，主要采用纯代数技术，并解决了 $\mathit{tmf}$ 同伦群乘法结构的一个遗留细节。
Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theor…
The element $P_2^1$ of the mod 2 Steenrod algebra has the property $(P_2^1)^2=0$. This property allows one to view $P_2^1$ as a differential on $H_*(X, \mathbb{F}_2)$ for any spectrum $X$. Homology with respect to this differential, $\mathcal{M}(X, P_2^1)$, is called the $P_2^1$ Margolis homology of $X$. In this paper we give a complete calculation of the $P_2^1$ Margolis homology of the 2-local spectrum of topological modular forms $tmf$ and identify its $\mathbb{F}_2$ basis via an iterated algorithm. We apply the same techniques to calculate $P_2^1$ Margolis homology for any smash power of $…
In this paper, we study the elliptic spectral sequence computing $tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, $η$, and $ν$. For $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$, we also compute the effect of the $v_1$-self maps of $\mathbb{R} P^2 \wedge \mathbb{C} P^2$ on $tmf$-homology.
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.
Spacetime theories obtained from perturbative string theory constructions are automatically free of perturbative anomalies, but it is not settled whether they are always free of global anomalies. Here we discuss a possible $\mathbb{Z}_{24}$-valued pure gravitational anomaly of heterotic compactifications down to two spacetime dimensions, and point out that it can be shown to vanish using the theory of topological modular forms, assuming the validity of the Stolz-Teichner conjecture.
The present article studies decompositions of vector bundles on the moduli stack of elliptic curves that are pushforwards of vector bundles on moduli of elliptic curves with level structure. These imply decomposition results for rings of modular forms and also for topological modular forms. We give explicit formulas for these decompositions and also apply them to equivariant topological modular forms. Moreover, we study the dualizing sheaf on $\overline{\mathcal{M}}_1(n)$ and characterize the numbers $n$ such that $Tmf_1(n)$ is Anderson self-dual.
We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage t…
We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.
Constructing and calculating Adams operations on dualisable topological modular forms
本文构造了拓扑模形式理论Tmf上的Adams操作，并利用降维谱序列和Anderson对偶计算其在球体上的cohomology，应用于构建连接高度2的Adams分量和J谱像。
In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.
We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.
On equivariant topological modular forms
本文遵循Lurie的思路，构建了不限制特征零的等变椭圆上同调通用构造，应用于泛椭圆曲线得到拓扑模形式的等变谱，并计算了其在圆周群及更一般环面群下的不动点。
High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories
该研究利用Weiss的酉微积分将向量丛问题转化为稳定同伦计算，借助色与等变同伦理论中的广义上同调理论，推导出球面与复射影空间上非平凡稳定平凡向量丛的存在性。
The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf_0(3) consistent with properties…
We construct deformation invariants of $2|1$-dimensional Euclidean field theories valued in a cohomology theory approximating topological modular forms. This implies several results anticipated by Stolz and Teichner and gives the first torsion invariants of field theories valued in $π_*{\rm TMF}$. The framework leads to a version of the elliptic Euler class described in terms of field-theoretic data, as well as a partial construction of the supersymmetric sigma model with target a string manifold.
Hecke operators on topological modular forms
本文将经典模形式的Adams运算、Hecke算子和Atkin-Lehner对合精化为拓扑模形式TMF上的稳定算子，利用代数几何与同伦论技术证明其性质，推导Ramanujan经典同余式并给出满足Maeda猜想的新Hecke算子族。
We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a fe…
The theory of topological modular forms (TMF) predicts that elliptic genera of physical theories satisfy a certain divisibility property, determined by the theory's gravitational anomaly. In this note we verify this prediction in Duncan's Supermoonshine module, as well as in tensor products and orbifolds thereof. Along the way we develop machinery for computing the elliptic genera of general alternating orbifolds and discuss the relation of this construction to the elusive "periodicity class" of TMF.
This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspe…
Holomorphic CFTs and topological modular forms
本文利用拓扑模形式理论约束玻色全纯CFT，验证了配分函数常数项的整除性猜想，并排除了包含中心荷为48、72、96和120的无限极值CFT集合的存在。
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 cl…
Anderson duality of topological modular forms and its differential-geometric manifestations
本文构造了实现拓扑模形式（TMF）Anderson对偶性的谱态射及其微分版本，利用RO(G)分次TMF理论识别了负度同伦群元素，并探讨了与顶点算子代数相关的猜想。
Topological modular forms and the absence of all heterotic global anomalies
本研究利用Segal-Stolz-Teichner猜想将杂化弦整体反常缺失问题转化为拓扑模形式到弦配边群Anderson对偶的自然变换问题，证明该变换为零从而论证杂化整体反常始终不存在，还讨论了TMF扭结的描述方式。
A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms
本研究受非超对称杂弦与拓扑模形式关联的进展启发，证明特定中心荷的世界面理论与$(E_8)_2$理论在RG轨迹意义下连续连通，结合已有工作为TMF中对应非平凡挠动元的结论提供物理推导。
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
Chiral Spin(7) Sigma Models and Topological Modular Forms
本文在 Spin(7) 流形上构造手性 sigma 模型，通过计算拓扑模形式类，证明其非零性等价于靶欧拉示性数为奇数，并给出实现非零类的 Fermat 六次曲面商实例。
A new approach to (3+1)-dimensional TQFTs via topological modular forms
本文提出一种位于低维拓扑与同伦论等交叉点的新型(3+1)维TQFT构造，通过导出代数几何定义TMF-模并赋予流形，推广了格的theta函数，并展示了其在区分量子场论相态等物理问题中的应用。
Among topological modular forms with level structure, $TMF_0(7)$ at the prime $3$ is the first example that had not been understood yet. We provide a splitting of $TMF_0(7)$ at the prime 3 as $TMF$-module into two shifted copies of $TMF$ and two shifted copies of $TMF_1(2)$. This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level $n$ and introduce cubical versions of moduli stacks of elliptic curves with level structure.
We show that the Atiyah-Patodi-Singer reduced $η$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin$^c$-manifolds with a deep topological theorem due to Hopkins.
Modular functors from conformal blocks of rational vertex operator algebras
本文证明强有理顶点算子代数的共形块空间构成模函子，进而导出模融合范畴结构，并延拓为三维拓扑场论。
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