Topological modular forms and the absence of all heterotic global anomalies

Abstract

We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation TMF • → (I Z Ω string ) •−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 TMF 21 (pt) = 0 plays an important role in the process. Along the way, we also discuss how the twists of 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.

Looking for structure in the cobordism conjecture

“…As such, the latter is a codimension-3 object in ten dimensions. Its space dimensions are however special: they split into five dimensions plus one isometry direction, 16 where the latter is fibred over the transverse dimensions. The KKm is also a supergravity solution, and it admits no H-flux.…”

“…Indeed, even if the conjecture is quite recent, already a number of works appeared studying some of its aspects and consequences, including [5][6][7][8][9][10][11]. Bordisms are relevant in several areas in mathematics and physics, a nonexhaustive list of related works is [12][13][14][15][16] for anomalies, [17,18] in relation to bubbles of nothing, [19] for the conjecture in a holographic context. An introduction to some of the mathematical material, in a physical context, can be found e. g. in [20,21], that appeared prior to [4].…”

Global anomalies in 8d supergravity

“…the case of E 8 with level 2 and the case of Sp(n)) for future work. Some of the rank 10 theories are constructed in heterotic string theories [CHL95], and the results of [TY21] suggests that fermion anomalies may be zero as long as the 2-form field is regarded as a background field imposing the (twisted) string structure (4.12). The explicit path integral over the 2-form field is subtle, but the appropriate action for the 2-form field may be obtained by the construction along the lines of [KOT19,Yon20].…”

Differential models for the Anderson dual to bordism theories and invertible QFT's, II

“…In the examples in this subsection, we used the Anderson selfduality elements in IE n (pt) for E = HZ, K. However, the results in this subsection do not use the self-duality, and indeed there are many other interesting examples given by non-self-duality elements in IE n (pt). For example, in the analysis of anomalies of the heterotic string theories in [TY21], we encounter such examples when E = TMF and E = KO((q)) with the Witten genus G = Wit : M T String → TMF and G = Wit Spin : M T Spin → KO((q)).…”