[Paper Review] Topological modular forms and the absence of all heterotic global anomalies

Yuji Tachikawa, Mayuko Yamashita|arXiv (Cornell University)|Aug 30, 2021

Black Holes and Theoretical Physics24 references4 citations

TL;DR

This paper establishes that all heterotic string theory global anomalies are absent by proving a natural transformation from topological modular forms (TMF) to the Anderson dual of string bordism vanishes, leveraging the Segal-Stolz-Teichner conjecture and the vanishing of TMF^21(pt) = 0. The result confirms the consistency of heterotic string theory under global symmetry anomalies via advanced bordism and cohomology theory.

ABSTRACT

We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation $\mathrm{TMF}^\bullet o (I_{\mathbb{Z}}Ω^ ext{string})^{\bullet-20}$, 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 $\mathrm{TMF}^{21}(\mathrm{pt})=0$ plays an important role in the process. Along the way, we also discuss how the twists of $\mathrm{TMF}$ 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.

Motivation & Objective

• To mathematically reformulate the absence of global anomalies in heterotic string theory using topological modular forms (TMF) and the Segal-Stolz-Teichner conjecture.
• To establish a natural transformation from TMF^• to the Anderson dual of string bordism groups, (I_Z Ω^string)^•−20, as the anomaly classifier.
• To demonstrate that this transformation vanishes, thereby proving the absence of all global anomalies in heterotic string theory.
• To clarify the role of TMF^21(pt) = 0 in the vanishing of the anomaly map, providing a key topological obstruction.
• To describe twists of TMF under the Segal-Stolz-Teichner conjecture using results from Freed and Hopkins on quantum field theory anomalies.

Proposed method

• Formalizes the anomaly of a d-dimensional quantum field theory as a (d+1)-dimensional invertible QFT, classified by (I_Z Ω^B)^d+2(pt), where B is a tangential structure.
• Applies the Segal-Stolz-Teichner conjecture to identify TMF^• with the cohomology theory classifying extended 2d SCFTs with string structure.
• Constructs a natural transformation TMF^• → (I_Z Ω^string)^•−20 as the anomaly map for heterotic theories.
• Uses the fact that TMF^21(pt) = 0 to prove the vanishing of the anomaly map, implying no global anomalies.
• Analyzes twists of TMF via the classification of invertible phases and their relation to bordism groups and Anderson duality.
• Employs the Atiyah-Hirzebruch spectral sequence and long exact sequences in relative spin bordism to compute relevant bordism groups and verify isomorphisms.

Experimental results

Research questions

• RQ1 Does the natural transformation from topological modular forms to the Anderson dual of string bordism vanish, implying the absence of global anomalies in heterotic string theory?
• RQ2 How does the Segal-Stolz-Teichner conjecture relate TMF to the classification of 2d quantum field theories with string structure?
• RQ3 What is the role of TMF^21(pt) = 0 in the vanishing of the anomaly map?
• RQ4 How can twists of TMF be described in terms of bordism and anomaly theory under the Segal-Stolz-Teichner conjecture?
• RQ5 Can the global anomaly of SU(2) gauge theory in 6d be derived from perturbative anomalies via the Anderson dual of bordism?

Key findings

• The natural transformation TMF^• → (I_Z Ω^string)^•−20 vanishes identically, proving that all heterotic global anomalies are absent.
• The vanishing of TMF^21(pt) = 0 is a crucial topological input that forces the anomaly map to vanish.
• The anomaly of a 6d heterotic theory with SU(2) gauge group is classified by (I_Z Ω^spin)^6(BSU(2)) ≃ ℤ/2ℤ, which corresponds to the Witten anomaly.
• The pullback map (I_Z Ω^spin)^6(BSU(3)) → (I_Z Ω^spin)^6(BSU(2)) sends a generator to a generator, confirming the anomaly's nontriviality in the relative bordism group.
• The long exact sequence in relative spin bordism for SU(2) → SU(3) yields 0 → ℤ → ℤ → ℤ/2ℤ → 0 at degree 6, which is isomorphic to the bordism exact sequence.
• The result confirms that global anomalies in heterotic theories are not obstructed, as the anomaly class lies in a group that vanishes when pulled back via the relevant structure.