arXiv subjects

Yuji Tachikawa

Publications and source records attributed to Yuji Tachikawa.

At least 19 recordsLinked to original sources

arXiv2026

Bosonic SPT and invertible phases and its relation to Steenrod's problem

Bosonic invertible and symmetry-protected topological (SPT) phases are well-known to be described by ordinary cohomology groups in low dimensions, but `beyond-cohomology' phases appear in higher dimensions. We make a systematic study of them, and find that the first major non-triviality is a mod-3 phenomenon, and not a mod-2 phenomenon as in the case of fermionic phases. We also point out that this is a dual manifestation of the classic question of Steenrod, namely the issue of the existence of homology cycles without manifold representatives. Thom developed the theory of cobordisms to answer this question, and we explain how the same analysis leads to Dijkgraaf-Witten phases which are nontrivial on general simplicial complexes but become trivial on manifolds.

hep-th↗

arXiv2026

On Generalised Discrete Torsion

For a 2d gauged sigma model with target space $M$ and discrete gauge group $G$, we consider a generalisation of Vafa's discrete torsion $H^2(BG; U(1))$ that assigns different local discrete torsion phases to different singular loci of the orbifold $M/G$. Our generalised discrete torsion lives in $H^2_G(M; U(1))$, and gives a consistent implementation of Gaberdiel and Kaste's prescription for inserting such local discrete torsion phases by hand at higher genus. We revisit the original application to $T^6/\mathbb{Z}_2^2$ and $T^7/\mathbb{Z}_2^3$ orbifold CFTs, and determine what smooth Calabi-Yau and $G_2$ geometries result from different choices of the generalised discrete torsion. We find that the local discrete torsion phases can be different from each other, but are not completely independent either; in the $T^7/\mathbb{Z}_2^3$ case for example, the orbifold CFTs only realise 3 out of the 9 possible Betti numbers of $G_2$ resolutions constructed by Joyce.

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arXiv2026

On the trivalent junction of three non-tachyonic heterotic string theories

Recently, Altavista, Anastasi, Angius and Uranga discussed a method to construct junctions and bouquets of different perturbative string theories. Following this analysis, we here argue that three non-tachyonic ten-dimensional heterotic string theories can be joined together at a nine-dimensional junction. This is done by creating a two-dimensional non-conformal $\mathcal{N}{=}(0,1)$ supersymmetric quantum field theory with three asymptotic ends, each equipped with one of the worldsheet theories of the supersymmetric $E_8\times E_8$ theory, the supersymmetric $SO(32)$ theory, and the non-supersymmetric $SO(16)\times SO(16)$ theory, respectively. It is actually a special case of a more general construction involving an arbitrary $\mathbb{Z}_2$-symmetric theory $T$, its $\mathbb{Z}_2$-orbifold $T/\mathbb{Z}_2$, and the modified $\mathbb{Z}_2$-orbifold $(T\times q)/\mathbb{Z}_2$, where $q$ is a certain $\mathbb{Z}_2$-symmetric spin invertible theory.

hep-th↗

arXiv2026

What happens to wavepackets of fermions when scattered by the Maldacena-Ludwig wall?

We study wavepackets of exotic excitations after two-dimensional fermions are scattered by the boundary condition constructed by Maldacena and Ludwig, which turns elementary excitations into exotic fractionally-charged objects. They are of interest in the s-wave approximation of the fermion-monopole scattering in four-dimensional QED and of the multi-channel Kondo effect. We in particular give an explicit expression of the outgoing state of a pair of such particles; we then examine its properties, such as the charge density $\langle J(x)\rangle$ and the expectation value $\langle N\rangle$ of the number of fermions and anti-fermions in the state. The charge density $\langle J(x)\rangle$ is found to be localized with its integral finite and fractional, while the expectation value $\langle N\rangle$ diverges when the wavepacket is localized to a point.

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arXiv2025

On holography with ADE singularities

We study aspects of the AdS/CFT correspondence for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory on $S^3/\Gamma$, where $\Gamma \subset SU(2)$ is a finite subgroup, leading to an ADE singularity in the bulk AdS geometry. We show that a large vacuum degeneracy arises from the choice of gauge holonomy on $S^3/\Gamma$. On the gravity side, we argue that the bulk ADE singularity supports topological degrees of freedom responsible for this degeneracy. We then provide a holographic derivation of a corresponding large vacuum degeneracy for class S theories of type $U(N)$, showing that these topological degrees of freedom admit an effective description in terms of a three-dimensional level-$N$ Chern-Simons theory, whose gauge group $\mathsf{G}$ is determined by $\Gamma$. Finally, we discuss how the one-form symmetries of the $\mathcal{N}=4$ super Yang-Mills theory are realized on the Chern-Simons theory side.

hep-th↗

arXiv2025

On a long exact sequence of groups of equivalence classes of 2d $\mathcal{N}{=}(0,1)$ SQFTs

Strongly motivated by a mathematical result by Lin and Yamashita (arXiv:2412.02298), we describe a long exact sequence formed by groups of equivalence classes of two-dimensional $\mathcal{N}{=}(0,1)$ supersymmetric quantum field theories (SQFTs) with and without $SU(2)$ symmetry. As an application, we study chiral fermions in heterotic compactifications with $SU(2)$ symmetry of level one to four dimensions, and show that each even-dimensional irreducible representation of $SU(2)$ appears even times, assuming the conjectural relation between topological modular forms and SQFTs. This implies the absence of the Witten anomaly, but contains more information than that.

hep-th↗

arXiv2025

On anomalies and fermionic unitary operators

We point out that fermionic unitary operators which anticommute among themselves appear in various situations in quantum field theories with anomalies in the Hamiltonian formalism. To illustrate, we give multiple derivations of the fact that position-dependent $U(1)$ transformations of two-dimensional theories with $U(1)$ symmetry of odd level are fermionic when the winding number is odd. We then relate this mechanism to the anomalies of the discrete $\mathbb{Z}_N \subset U(1)$ symmetry, whose description also crucially uses unitary operators which are fermionic. We also show that position-dependent $SU(2)$ transformations of four-dimensional theories with $SU(2)$ symmetry with Witten anomaly are fermionic and anticommute among themselves when the winding number is odd.

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arXiv2025

On invariants of two-dimensional minimally supersymmetric field theories

We construct a new invariant of two-dimensional $\mathcal{N}{=}(0,1)$ supersymmetric quantum field theories (SQFTs), under a couple of assumptions on the general properties of such SQFTs motivated by considerations in heterotic string theory. This new invariant is in addition to the known 'primary' invariants, i.e. the ordinary elliptic genus and the mod-2 elliptic genus, and the known 'secondary' invariant, which can be considered as an SQFT analog of the Atiyah-Patodi-Singer $\eta$-invariant when its continuous variation vanishes. We also define a spectral invariant of supercharge more generally by the integration of the expectation value of the supercharge over the fundamental region of $\mathrm{ SL}(2,\mathbb{Z})$. With this spectral invariant, we construct a spectral pairing which can be used to systematically detect all the known invariants, including the new one we introduce in the first half of this paper.

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arXiv2025

Type I anomaly cancellation revisited

We revisit the issue of how the perturbative and global fermion anomaly of Type I string theory in ten dimensions is cancelled by the Green-Schwarz mechanism using the RR fields. This will be done by realising the RR fields as boundary modes of an eleven-dimensional bulk theory described in terms of a quadratic refinement of the differential KO-theory pairing. We will then generalise this analysis to Sugimoto's $usp(32)$ string and Sagnotti's $u(32)$ string. We also discuss in a more general setting the procedures which need to be followed when we try to cancel fermion anomalies in terms of $p$-form fields based on differential K-theory classes. This we illustrate by performing an analysis of the mod-2 anomaly cancellation in nine dimensions arising from the $S^1$ compactification of the Type I theory.

hep-th↗

arXiv2025

On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups

Let $N(\Gamma,G)$ be the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by $G$. Physics of four-dimensional $\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(\Gamma,G)=N(\Gamma , \tilde G)$ when $\Gamma$ is a finite subgroup of $SU(2)$, $G$ is a connected compact simple Lie group and $\tilde G$ is its Langlands dual. This statement is known to be true when $\Gamma=\mathbb{Z}_n$, but the statement for non-Abelian $\Gamma$ is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a couple of examples, namely $(G,\tilde G)=(SU(n),PU(n))$ and $(Sp(n),SO(2n+1))$ for arbitrary $\Gamma$, and $(PSp(n),Spin(2n+1))$ for exceptional $\Gamma$. A more refined version of the conjecture, together with proofs of some concrete cases, will also be presented. The authors would like to ask mathematicians to provide a more uniform proof applicable to all cases.

math.RT↗

arXiv2024

Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu\nu}$

We study if and when mod-2 anomalies can be canceled by the Green-Schwarz mechanism with the introduction of an antisymmetric tensor field $B_{\mu\nu}$. As explicit examples, we examine $SU(2)$ and more general $Sp(n)$ gauge theories in four and eight dimensions. We find that the mod-2 anomalies of 8d $\mathcal{N}=1$ $Sp(n)$ gauge theory can be canceled, as expected from it having a string theory realization, while the mod-2 Witten anomaly of 4d $SU(2)$ and $Sp(n)$ gauge theory cannot be canceled in this manner.

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arXiv2024

On non-supersymmetric heterotic branes

A uniform construction of non-supersymmetric 0-, 4-, 6- and 7-branes in heterotic string theory was announced and outlined in our letter \cite{Kaidi:2023tqo}. In this full paper, we provide details on their properties. Among other things, we discuss the charges carried by the branes, their topological and dynamical stability, the exact worldsheet descriptions of their near-horizon regions, and the relationship of the branes to the mathematical notion of topological modular forms.

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arXiv2024

Non-invertible symmetries act locally by quantum operations

Non-invertible symmetries of quantum field theories and many-body systems generalize the concept of symmetries by allowing non-invertible operations in addition to more ordinary invertible ones described by groups. The aim of this paper is to point out that these non-invertible symmetries act on local operators by quantum operations, i.e. completely positive maps between density matrices, which form a natural class of operations containing both unitary evolutions and measurements and play an important role in quantum information theory. This observation will be illustrated by the Kramers--Wannier duality of the one-dimensional quantum Ising chain, which is a prototypical example of non-invertible symmetry operations.

hep-th↗

arXiv2024

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

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 $\pi_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.

hep-th↗

arXiv2024

On a class of selection rules without group actions in field theory and string theory

We discuss a class of selection rules which i) do not come from group actions on fields, ii) are exact at tree level in perturbation theory, iii) are increasingly violated as the loop order is raised, and iv) eventually reduce to selection rules associated with an ordinary group symmetry. We start from basic field-theoretical examples in which fields are labeled by conjugacy classes rather than representations of a group, and discuss generalizations using fusion algebras or hypergroups. We also discuss how such selection rules arise naturally in string theory, such as for non-Abelian orbifolds or other cases with non-invertible worldsheet symmetries.

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arXiv2023

Anderson duality of topological modular forms and its differential-geometric manifestations

We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{TMF}$). Its differential version will then be introduced, allowing us to pair elements of $\pi_d\mathrm{TMF}$ with spin manifolds whose boundaries are equipped with string structure. A few negative-degree elements of $\pi_d\mathrm{TMF}$ will then be constructed using the theory of $\mathrm{RO}(G)$-graded $\mathrm{TMF}$, and will be identified using the differential pairing. We also discuss a conjecture relating vertex operator algebras and negative-degree elements of $\pi_d\mathrm{TMF}$, underlying much of the discussions of this paper. The paper ends with a separate appendix for physicists, in which the contents of the paper are summarized and translated into their language.

math.AT↗

arXiv2023

Non-supersymmetric heterotic branes

The common statement that any consistent quantum gravity theory contains dynamical objects with all possible charges suggests that there are still a number of hitherto-unidentified branes in string theory. Here we give the exact worldsheet description of near-horizon limits of non-supersymmetric $p$-branes in ten-dimensional $\mathrm{Spin}(32)/\mathbb{Z}_2$ or $(E_8 \times E_8) \rtimes \mathbb{Z}_2$ heterotic superstring theories for $p=7,6,4,0$.

hep-th↗

arXiv2023

Classification of chiral fermionic CFTs of central charge $\le 16$

We classify two-dimensional purely chiral conformal field theories which are defined on two-dimensional surfaces equipped with spin structure and have central charge less than or equal to 16, and discuss their duality webs. This result can be used to confirm that the list of non-supersymmetric ten-dimensional heterotic string theories found in the late 1980s is complete and does not miss any exotic example.

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