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On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories

Yuji Tachikawa, Hao Y. Zhang

TL;DR

This work identifies a $Z_3$-valued discrete topological term that differentiates the low-energy gravitational sectors of the two ten-dimensional heterotic strings, even when gauge fields are switched off. The authors compute this term via two complementary routes: anomaly inflow to NS5-branes in the bulk and an alternative description using instanton realizations of NS5-branes, establishing a nontrivial global anomaly phase of order three. They further show that the non-tachyonic Spin(16)×Spin(16) theory shares the same $Z_3$ discrete topological term, and they connect these physical findings to the framework of topological modular forms (TMF), arguing that the relevant TMF class sits in $A_{-32} \,\cong \, Z_3$ and is detected by the $Sp(2)$ configuration with unit $H$-flux. Collectively, the results illuminate how discrete topological terms and global anomalies interface with TMF and string duality, offering a refined homotopical viewpoint on heterotic string distinctions with potential implications for symmetry-protected topological phases in string theory.

Abstract

We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a $\mathbb{Z}_3$-valued discrete topological term even after we turn off the $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$ gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. We also find that the $Spin(16)\times Spin(16)$ non-tachyonic non-supersymmetric heterotic string has the same non-zero $\mathbb{Z}_3$-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 $\mathbb{Z}/3$-torsion element of $π_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.

On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories

TL;DR

This work identifies a -valued discrete topological term that differentiates the low-energy gravitational sectors of the two ten-dimensional heterotic strings, even when gauge fields are switched off. The authors compute this term via two complementary routes: anomaly inflow to NS5-branes in the bulk and an alternative description using instanton realizations of NS5-branes, establishing a nontrivial global anomaly phase of order three. They further show that the non-tachyonic Spin(16)×Spin(16) theory shares the same discrete topological term, and they connect these physical findings to the framework of topological modular forms (TMF), arguing that the relevant TMF class sits in and is detected by the configuration with unit -flux. Collectively, the results illuminate how discrete topological terms and global anomalies interface with TMF and string duality, offering a refined homotopical viewpoint on heterotic string distinctions with potential implications for symmetry-protected topological phases in string theory.

Abstract

We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a -valued discrete topological term even after we turn off the and gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. We also find that the non-tachyonic non-supersymmetric heterotic string has the same non-zero -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 -torsion element of as a particular self-dual vertex operator superalgebra of and how we utilize string duality to arrive at this statement.
Paper Structure (30 sections, 2 theorems, 109 equations, 2 figures)

This paper contains 30 sections, 2 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. $\mathbb{Z}_3$-valued discrete topological term:
  3. Discrete topological term and the global anomaly of NS5-branes:
  4. NS5-branes as gauge instantons:
  5. Reconstructing 10D Topological Terms
  6. Relation to topological modular forms:
  7. Structure of the paper:
  8. Anomaly inflow to NS5-branes
  9. Anomaly inflow to NS5-branes
  10. Anomaly polynomials in ten dimensions
  11. NS5-branes in the $Spin(32)/\mathbb{Z}_2$ heterotic string
  12. NS5-branes in the $E_8\times E_8$ heterotic string
  13. Difference of the two cases
  14. Evaluation of the global anomaly
  15. Strategy
  16. ...and 15 more sections

Key Result

Theorem A.10

Suppose that $x \in \pi_{-d-22}{\mathrm{TMF}}={\mathrm{TMF}}^{d+22}(pt)$ lifts to $\tilde{x} \in {\mathrm{TMF}}^{d+22+k}(BG)$ and that the string manifold $M_d$ is such that it is null when sent to ${\mathrm{MString}}_{d+k}(BG)$, where $k: BG\to K(\mathbb{Z},4)$ specifies the twist. Let $N_{d+1}$ be such that it only depends on $\Phi(\tilde{x}) \in {\mathrm{KO}}((q))^{d+22}(BG)$ and the $(BG,k)$-t

Figures (2)

  • Figure 1: The anomaly inflow from the ten dimensional bulk to the NS5-branes.
  • Figure 2: Summary of the main idea in this appendix. On the left, we have the hard problem of taking the VOSA and identifying a pairing as given in Theorem \ref{['thm:pairing']}. This would rigorously show that the VOSA represents a torsional element in $\pi_{-2c}{\mathrm{TMF}}$ for $c = 16$, as explained in Conjecture \ref{['conj']}. However, the direct mathematical implementation is not tractable by the authors. Therefore, the author take the indirect, physical approach, where the steps that do not yet admit full mathematical formulations are written in magenta and labeled differently.

Theorems & Definitions (4)

  • Conjecture A.7
  • Theorem A.10
  • Conjecture A.12
  • Proposition A.15