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Type IIA String Theory and tmf with Level Structure

Arun Debray, Matthew Yu

Abstract

We look at a new string$^h$ tangential structure first introduced by Devalapurkar and relate it to the $W_7=0$ condition of Diaconescu-Moore-Witten for type IIA string theory and M-theory. We show that a string$^h$ structure on the target space automatically satisfies the $W_7=0$ condition and we also explain when the $W_7=0$ condition gives rise to a string$^h$ structure. Devalapurkar initially constructed $MString^h$ in such a way that it orients $tmf_1(3)$; we extend Devalapurkar's result, showing that $MString^h$ orients $tmf_1(n)$. We compute the homotopy groups of $MString^h$ in the dimensions relevant for physical applications, and apply them to anomaly cancellation applications for certain compactifications of type IIA string theory.

Type IIA String Theory and tmf with Level Structure

Abstract

We look at a new string tangential structure first introduced by Devalapurkar and relate it to the condition of Diaconescu-Moore-Witten for type IIA string theory and M-theory. We show that a string structure on the target space automatically satisfies the condition and we also explain when the condition gives rise to a string structure. Devalapurkar initially constructed in such a way that it orients ; we extend Devalapurkar's result, showing that orients . We compute the homotopy groups of in the dimensions relevant for physical applications, and apply them to anomaly cancellation applications for certain compactifications of type IIA string theory.

Paper Structure

This paper contains 12 sections, 86 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The String$^h$ tangential structure
  3. String$h$ structures and characteristic classes
  4. Rigidifying to $E_\infty$-structures
  5. Relation between string$h$ structures and spin$c$ structures on loop spaces
  6. Orienting $\mathit{tmf}_1(n)$
  7. Real-equivariance
  8. Low-degree homotopy groups of MStringh
  9. Does $\sigma_1(n)$ split?
  10. String$^h$ and the Diaconescu-Moore-Witten Anomaly
  11. Relating Diaconescu-Moore-Witten anomaly cancellation with string$h$
  12. Applications of string$h$ for type IIA compactifications

Key Result

Theorem 1

There is a map of $E_\infty$-ring spectra $\sigma_D\colon \mathit{MString}^h_{(2)}\to\mathit{tmf}_1(3)_{(2)}$.

Figures (2)

  • Figure 1: Homotopy Long Exact Sequence for computing the homotopy groups of $B\mathrm{String}^h$ in degrees up to 10.
  • Figure 2: Homotopy Long Exact Sequence for computing the homotopy groups of $B\mathrm{String}^h$ in degrees up to 10.

Theorems & Definitions (199)

  • Definition 1.2: Devalapurkar Dev
  • Theorem : Devalapurkar Dev
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Deb24
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.8
  • Definition 2.10
  • ...and 189 more