We construct Adams operations \psi^{k} on the cohomology theory \mathrm{Tmf} of dualisable topological modular forms after inverting k ; the first such multiplicative stable operations on this cohomology theory. These Adams operations are then calculated on the homotopy groups of \mathrm{Tmf} using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height 2 analogues of Adams summands and image-of- J spectra .

A uniform construction of non-supersymmetric 0-, 4-, 6- and 7-branes in heterotic string theory was announced and outlined in our letter [61]. In this full paper, we provide details on their properties. Among other things, we discuss the charges carried by the branes, their topological and dynamical stability, the exact worldsheet descriptions of their near-horizon regions, and the relationship of the branes to the mathematical notion of topological modular forms.

Lurie’s theorem states that there exists a sheaf of ring spectra on the site of formally étale Deligne–Mumford stacks over the moduli stack of p-divisible groups of height n, which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie’s theorem. Applications of this theorem to Lubin–Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.

Motivated by recent developments connecting non-supersymmetric heterotic string theory to the theory of Topological Modular Forms (TMF), we show that the worldsheet theory with central charge (17,32) obtained by fibering the (E8)1 × (E8)1 current algebra over the two N = (0, 1) sigma model on S1 with antiperiodic spin structure (such that the E8 factors are exchanged as we go around the circle), is continuously connected to the (E8)2 theory in the Gaiotto Johnson-Freyd Witten sense of going “up and down the RG trajectories”. Combined with the work of Tachikawa and Yamashita, this furnishes a physical derivation of the fact that the (E8)2 theory corresponds to the unique nontrivial torsion element [(E8)2] of TMF31 with zero mod-2 elliptic genus.

We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a \mathbb{Z}_3 ℤ 3 -valued discrete topological term even after we turn off the E_8×E_8 E 8 × E 8 and Spin(32)/\mathbb{Z}_2 S p i n ( 32 ) / ℤ 2 gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. This result will be used to show further that the Spin(16)×Spin(16) S p i n ( 16 ) × S p i n ( 16 ) non-tachyonic non-supersymmetric heterotic string has the same non-zero \mathbb{Z}_3 ℤ 3 -valued discrete topological term. We will also explain the relation of our findings to the theory of topological modular forms. The paper is written as a string theory paper, except for an appendix translating the content in mathematical terms. We will explain there that our finding identifies a representative of the \mathbb{Z}/3 ℤ / 3 -torsion element of \pi_{-32}\mathrm{TMF} π − 32 T M F as a particular self-dual vertex operator superalgebra of c=16 c = 16 and how we utilize string duality to arrive at this statement.

A classification of complex rank 3 vector bundles on [formula omitted]

Abstract We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group.

Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

Organizational manageability is a crucial aspect of business management, requiring a combination of forecasting, planning, organizing, implementing, controlling and decision-making. Topological modular forms study the properties of objects that are invariant under certain types of transformations and the authors search for and identify a set of key factors that are essential to the organizational manageability (both stable and unstable) and create a framework that captures these factors. Organizational manageability is highly complex and multifaceted field that requires the integration of mentioned elements. In order to simplify incommensurable complexity, authors offer the hypothesis that differentiating the approach to manageability in the two distinct situations "steady and familiar condition" and "unsteady and with considerable uncertainty condition" is effective. Discrimination between those two situations is essential for business success and requires a deep understanding of market trends, customer needs, design of organization and usage of resources. By mastering the principles of organizational manageability based on mentioned classification of situations, businesses can improve their performance, increase their competitiveness and achieve their goals more effectively.

We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation text {TMF} ^bullet rightarrow (I_mathbb {Z}Omega ^{text {string} })^{bullet -20}, from topological modular forms to the Anderson dual of string bordism groups, using the Segal–Stolz–Teichner conjecture. We will show that this natural transformation vanishes, implying that heterotic global anomalies are always absent. The fact that text {TMF} ^{21}(text {pt} )=0 plays an important role in the process. Along the way, we also discuss how the twists of text {TMF} can be described under the Segal–Stolz–Teichner conjecture, by using the result of Freed and Hopkins concerning anomalies of quantum field theories. The paper contains separate introductions for mathematicians and for string theorists, in the hope of making the content more accessible to a larger audience. The sections are also demarcated cleanly into mathematically rigorous parts and those which are not.

Local Gorenstein duality in chromatic group cohomology

Abstract John Rognes developed a notion of Galois extension of commutative ring spectra, and this includes a criterion for identifying an extension as unramified. Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André–Quillen homology. In the classical algebraic context, it is important to distinguish between tame and wild ramification. Noether’s theorem characterizes tame ramification in terms of a normal basis, and tame ramification can also be detected via the surjectivity of the trace map. For commutative ring spectra, we suggest to study the Tate construction as a suitable analog. It tells us at which integral primes there is tame or wild ramification, and we determine its homotopy type in examples in the context of topological K-theory and topological modular forms.

Let $A_1$ be any spectrum in the class of finite spectra whose mod-2 cohomology is isomorphic to $\mathcal{A}(1)$ as a module over the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra; let $tmf$ be the connective spectrum of topological modular forms. In this paper, we prove that the $tmf$-Hurewicz image of $A_1$ is surjective.

Abstract 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.

We use the theory of topological modular forms to constrain bosonic holomorphic CFTs, which can be viewed as (0, 1) SCFTs with trivial right-moving supersymmetric sector. A conjecture by Segal, Stolz and Teichner requires the constant term of the partition function to be divisible by specific integers determined by the central charge. We verify this constraint in large classes of physical examples, and rule out the existence of an infinite set of extremal CFTs, including those with central charges $$c=48, 72, 96$$ and 120.

Abstract Homotopy theory folklore tells us that the sheaf defining the cohomology theory $\operatorname {\mathrm {Tmf}}$ of topological modular forms is unique up to homotopy. Here we provide a proof of this fact, although we claim no originality for the statement. This retroactively reconciles all previous constructions of $\operatorname {\mathrm {Tmf}}$ .

We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional mathcal{N} = (0, 1) supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the B-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQFTs suggested by topological modular forms, and is expected to be zero for the cases that are relevant to actual heterotic string theories.

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.

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.