arXiv subjects

Yuji Tachikawa

Publications and source records attributed to Yuji Tachikawa.

At least 55 records · Page 3Linked to original sources

arXiv2021

Some comments on 6d global gauge anomalies

Global gauge anomalies in $6d$ associated with non-trivial homotopy groups $π_6(G)$ 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 groups $Ω^\text{spin}_7(BG)$, 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 $π_6(G)$. In this paper, we revisit and clarify the relation between these two different approaches.

hep-th↗

arXiv2020

More on $\mathcal{N} =2$ S-folds

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.

hep-th↗

arXiv2020

$SL(2,\mathbb{Z})$ action on QFTs with $\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants

We consider an analogue of Witten's $SL(2,\mathbb{Z})$ 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 multiplication 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.

hep-th↗

arXiv2020

Revisiting Wess-Zumino-Witten terms

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.

hep-th↗

arXiv2020

Topological Superconductors on Superstring Worldsheets

We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible phases, or equivalently topological superconductors, on the worldsheet. This allows us to find that the unoriented Type $\rm 0$ string theory with $Ω^2=(-1)^{\sf f}$ admits different GSO projections parameterized by $n$ mod 8, depending on the number of Kitaev chains on the worldsheet. The presence of $n$ boundary Majorana fermions then leads to the classification of D-branes by $KO^n(X)\oplus KO^{-n}(X)$ in these theories, which we also confirm by the study of the D-brane boundary states. Finally, we show that there is no essentially new GSO projection for the Type $\rm I$ worldsheet theory by studying the relevant bordism group, which classifies corresponding invertible phases. This paper provides the details for the results announced in the letter \cite{Kaidi:2019pzj}.

hep-th↗

arXiv2020

General Lieb-Schultz-Mattis type theorems for quantum spin chains

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 generally, 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.

math-ph↗

arXiv2020

Generalized symmetries and holography in ABJM-type theories

We revisit the N=6 superconformal Chern-Simons-matter theories and their supergravity duals in the context of generalized symmetries. This allows us to finally clarify how the $SU(N)\times SU(N)$ and $(SU(N)\times SU(N))/\mathbb{Z}_N$ theories, as well as other quotient theories that have recently been discussed, fit into the holographic framework. It also resolves a long standing puzzle regarding the di-baryon operator in the $U(N)\times U(N)$ theory.

hep-th↗

arXiv2020

The super Frobenius-Schur indicator and finite group gauge theories on pin$^-$ surfaces

It is well-known that the value of the Frobenius-Schur indicator $|G|^{-1} \sum_{g\in G} χ(g^2)=\pm1$ of a real irreducible representation of a finite group $G$ determines which of the two types of real representations it belongs to, i.e. whether it is strictly real or quaternionic. We study the extension to the case when a homomorphism $φ:G\to \mathbb{Z}/2\mathbb{Z}$ gives the group algebra $\mathbb{C}[G]$ the structure of a superalgebra. Namely, we construct of a super version of the Frobenius-Schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in [Wall1964] it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin$^-$ surfaces.

math.RT↗

arXiv2019

On gauging finite subgroups

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $Γ$-symmetric theory. Depending on how anomalous $Γ$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=Γ/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

hep-th↗

arXiv2019

Anomaly of the Electromagnetic Duality of Maxwell Theory

We consider the ($3{+}1$)-dimensional Maxwell theory in the situation where going around nontrivial paths in the spacetime involves the action of the duality transformation exchanging the electric field and the magnetic field, as well as its $\mathrm{SL}(2,\mathbb{Z})$ generalizations. We find that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. This result is derived in two independent ways: one is by using the bulk symmetry protected topological phase in $4{+}1$ dimensions characterizing the anomaly, and the other is by considering the properties of a ($5{+}1$)-dimensional superconformal field theory known as the E-string theory. This anomaly of the Maxwell theory plays an important role in the consistency of string theory.

hep-th↗

arXiv2019

Why are fractional charges of orientifolds compatible with Dirac quantization?

Orientifold $p$-planes with $p\le4$ have fractional D$p$-charges, and therefore appear inconsistent with Dirac quantization with respect to D$(6-p)$-branes. We explain in detail how this issue is resolved by taking into account the anomaly of the worldvolume fermions using the $η$ invariants. We also point out relationships to the classification of interacting fermionic symmetry protected topological phases. In an appendix, we point out that the duality group of type IIB string theory is the pin+ version of the double cover of $GL(2,\mathbb{Z})$.

hep-th↗

arXiv2019

GSO projections via SPT phases

We point out that the choice of phases in GSO projections can be accounted for by a choice of fermionic SPT phases on the worldsheet of the string. This point of view not only easily explains why there are essentially two type II theories, but also predicts that there are unoriented type 0 theories labeled by n mod 8, and that there is an essentially unique choice of the type I worldsheet theory. We also discuss the relationship between this point of view and the K-theoretic classification of D-branes.

hep-th↗

arXiv2019

Reflection groups and 3d $\mathcal{N}\ge $ 6 SCFTs

We point out that the moduli spaces of all known 3d $\mathcal{N}=$ 8 and $\mathcal{N}=$ 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form $\mathbb{C}^{4r}/Γ$ where $Γ$ is a real or complex reflection group depending on whether the theory is $\mathcal{N}=$ 8 or $\mathcal{N}=$ 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases $H_{3,4}$. Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to-be-discovered 3d $\mathcal{N}=$ 8 theories for $H_{3,4}$. We also show that all known $\mathcal{N}=$ 6 theories correspond to complex reflection groups collectively known as $G(k,x,N)$. Along the way, we demonstrate that two ABJM theories $(SU(N)_k\times SU(N)_{-k})/\mathbb{Z}_N$ and $(U(N)_k\times U(N)_{-k})/\mathbb{Z}_k$ are actually equivalent.

hep-th↗

arXiv2019

Classification of 4d N=2 gauge theories

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

hep-th↗

arXiv2019

On gapped boundaries for SPT phases beyond group cohomology

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

cond-mat.str-el↗

arXiv2019

autoboot: A generator of bootstrap equations with global symmetry

We introduce autoboot, a Mathematica program which automatically generates mixed-correlator bootstrap equations of an arbitrary number of scalar external operators, given the global symmetry group and the representations of the operators. The output is a Python program which uses Ohtsuki's cboot which in turn uses Simmons-Duffin's sdpb. In an appendix we also discuss a simple technique to significantly reduce the time to run sdpb, which we call hot-starting.

hep-th↗

arXiv2019

Compactifications of 6d $N = (1, 0)$ SCFTs with non-trivial Stiefel-Whitney classes

We consider compactifications of very Higgsable 6d $N =(1,0)$ SCFTs on $T^2$ with non-trivial Stiefel-Whitney classes (or equivalently 't Hooft magnetic fluxes) introduced for their flavor symmetry groups. These systems can also be studied as twisted $S^1$ compactifications of the corresponding 5d theories. We deduce various properties of the resulting 4d $N=2$ SCFTs by combining these two viewpoints. In particular, we find that all 4d rank-1 $N =2$ SCFTs with a dimension-6 Coulomb branch operator with flavor symmetry $\mathfrak{e}_8$, $\mathfrak{usp}(10)$, $\mathfrak{su}(4) $ and $\mathfrak{su}(3)$ can be uniformly obtained by starting from a single-tensor theory in 6d. We also have a mostly independent appendix where we propose a rule to determine the Coulomb branch dimensions of 4d $N =2$ theories obtained by $T^2$ compactifications of 6d very Higgsable theories with and without Stiefel-Whitney twist.

hep-th↗

arXiv2018

The frozen phase of F-theory

We study the interpretation of O7+ planes in F-theory, mainly in the context of the six-dimensional models. In particular, we study how to assign gauge algebras and matter content to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+ planes. By including O7+ planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of su. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively.

hep-th↗