We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation $mathrm{TMF}^bulletto (I_{mathbb{Z}}Omega^text{string})^{bullet-20}$, 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 $mathrm{TMF}^{21}(mathrm{pt})=0$ 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.
We study higher symmetries and anomalies of 4d $mathfrak{so}(2n_c)$ gauge theory with $2n_f$ flavors. We find that they depend on the parity of $n_c$ and $n_f$, the global form of the gauge group, and the discrete theta angle. The contribution from t
he fermions plays a central role in our analysis. Furthermore, our conclusion applies to $mathcal{N}=1$ supersymmetric cases as well, and we see that higher symmetries and anomalies match across the Intriligator-Seiberg duality between $mathfrak{so}(2n_c)leftrightarrowmathfrak{so}(2n_f-2n_c+4)$.
Spacetime theories obtained from perturbative string theory constructions are automatically free of perturbative anomalies, but it is not settled whether they are always free of global anomalies. Here we discuss a possible $mathbb{Z}_{24}$-valued pur
e gravitational anomaly of heterotic compactifications down to two spacetime dimensions, and point out that it can be shown to vanish using the theory of topological modular forms, assuming the validity of the Stolz-Teichner conjecture.
In the last few years it was realized that every fermionic theory in 1+1 dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous $mathbb{Z}_2$ symmetry. In this note we determine how the boundary states are mapped
under this correspondence. We also interpret this mapping as the fusion of the original boundary with the fermionization interface.
Global gauge anomalies in $6d$ associated with non-trivial homotopy groups $pi_6(G)$ for $G=SU(2)$, $SU(3)$, and $G_2$ were computed and utilized in the past. In the modern bordism point of view of anomalies, however, they come from the bordism group
s $Omega^text{spin}_7(BG)$, which are in fact trivial and therefore preclude their existence. Instead, it was noticed that a proper treatment of the $6d$ Green-Schwarz mechanism reproduces the same anomaly cancellation conditions derived from $pi_6(G)$. In this paper, we revisit and clarify the relation between these two different approaches.
We carry out a systematic study of 4d $mathcal{N} = 2$ preserving S-folds of F-theory 7-branes and the worldvolume theories on D3-branes probing them. They consist of two infinite series of theories, which we denote following the original papers by $
mathcal{S}^{(r)}_{G,ell}$ for $ell = 2,3,4$ and $mathcal{T}^{(r)}_{G,ell}$ for $ell = 2,3,4,5,6$. Their distinction lies in the discrete torsion carried by the S-fold and in the difference in the asymptotic holonomy of the gauge bundle on the 7-brane. We study various properties of these theories, using diverse field theoretical and string theoretical methods.
We consider an analogue of Wittens $SL(2,mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,mathbb{Z})$ action only closes up to a multiplicatio
n by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $mathbb{Z}_2$ gauge theory.
We revisit various topological issues concerning four-dimensional ungauged and gauged Wess-Zumino-Witten (WZW) terms for $SU$ and $SO$ quantum chromodynamics (QCD), from the modern bordism point of view. We explain, for example, why the definition of
the $4d$ WZW terms requires the spin structure. We also discuss how the mixed anomaly involving the 1-form symmetry of $SO$ QCD is reproduced in the low-energy sigma model.
We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds $mathcal{M}_d$, including the contribution of the isometries of $mathcal{M}_d$. We t
hen use the result to perform a counting of microstates for electrically charged and rotating supersymmetric black strings in AdS$_5times S^5$ and AdS$_7times S^4$ with horizon topology BTZ$ ltimes S^2$ and BTZ$ ltimes S^2 times Sigma_mathfrak{g}$, respectively, where $Sigma_mathfrak{g}$ is a Riemann surface. We explicitly construct the latter class of solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula.
We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more gen
erally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.
