arXiv subjects

Yuji Tachikawa

Publications and source records attributed to Yuji Tachikawa.

At least 73 records · Page 4Linked to original sources

arXiv2018

Comments on the twisted punctures of $A_\text{even}$ class S theory

We point out that the $\text{USp}$ symmetry associated to a full twisted puncture of a class S theory of type $A_\text{even}$ has the global anomaly associated to $π_4(\text{USp})=\mathbb{Z}_2$. We discuss manifestations of this fact in the context of the superconformal field theory $R_{2,2N}$ introduced by Chacaltana, Distler and Trimm. For example, we find that this theory can be thought of as a natural ultraviolet completion of an infrared-free $\text{SO}(2N+1)$ gauge theory with $2N$ flavors, whose $\text{USp}(4N)$ symmetry clearly has the global anomaly.

hep-th↗

arXiv2018

A study of time reversal symmetry of abelian anyons

We perform a study of time reversal symmetry of abelian anyons $\mathcal{A}$ in 2+1 dimensions, in the spin structure independent cases. We will find the importance of the group $\mathcal{C}$ of time-reversal-symmetric anyons modulo anyons composed from an anyon and its time reversal. Possible choices of local Kramers degeneracy are given by quadratic refinements of the braiding phases of $\mathcal{C}$, and the anomaly is then given by the Arf invariant of the chosen quadratic refinement. We also give a concrete study of the cases when $|\mathcal{A}|$ is odd or $\mathcal{A}=(\mathbb{Z}_2)^N$.

hep-th↗

arXiv2018

4d N=1 from 6d N=(1,0) on a torus with fluxes

Compactifying N=(1,0) theories on a torus, with additional fluxes for global symmetries, we obtain N=1 supersymmetric theories in four dimensions. It is shown that for many choices of flux these models are toric quiver gauge theories with singlet fields. In particular we compare the anomalies deduced from the description of the six-dimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compactifications of M5-branes probing C_2/Z_k singularities.

hep-th↗

arXiv2018

Anomalies of Duality Groups and Extended Conformal Manifolds

A self-duality group $\cal G$ in quantum field theory can have anomalies. In that case, the space of ordinary coupling constants $\cal M$ can be extended to include the space $\cal F$ of coefficients of counterterms in background fields. The extended space $\cal N$ forms a bundle over $\cal M$ with fiber $\cal F$, and the topology of the bundle is determined by the anomaly. For example, the ${\cal G}=SL(2,\mathbb{Z})$ duality of the 4d Maxwell theory has an anomaly, and the space ${\cal F}=S^1$ for the gravitational theta-angle is nontrivially fibered over ${\cal M}=\mathbb{H}/SL(2,\mathbb{Z})$. We will explain a simple method to determine the anomaly when the 4d theory is obtained by compactifying a 6d theory on a Riemann surface in terms of the anomaly polynomial of the parent 6d theory. Our observations resolve an apparent contradiction associated with the global structure of the Kahler potential on the space of exactly marginal couplings of supersymmetric theories.

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arXiv2018

Anomalies involving the space of couplings and the Zamolodchikov metric

The anomaly polynomial of a theory can involve not only curvature two-forms of the flavor symmetry background but also two-forms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the R-symmetry and the topology of the space of exactly marginal couplings of class S theories. Using supersymmetry, we translate this anomaly to the Kähler class of the Zamolodchikov metric. We compare the result against a holographic computation in the large N limit. In an appendix, we explain how to obtain the bosonic components in the supergravity completion of the curvature squared terms using compatibility with the topological twist.

hep-th↗

arXiv2018

First-order conformal perturbation theory by marginal operators

We perform conformal perturbation theory by marginal operators to first order. A suitable renormalization method is needed that makes the conformal invariance of the deformed correlation functions manifest. Combining the embedding space formalism with the dimensional regularization, we explicitly check that the deformed and renormalized two and three point functions are conformally invariant.

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arXiv2017

On 'categories' of quantum field theories

We give a rough description of the 'categories' formed by quantum field theories. A few recent mathematical conjectures derived from quantum field theories, some of which are now proven theorems, will be presented in this language.

math-ph↗

arXiv2017

On finite symmetries and their gauging in two dimensions

It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

hep-th↗

arXiv2017

8d gauge anomalies and the topological Green-Schwarz mechanism

String theory provides us with 8d supersymmetric gauge theory with gauge algebras $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, $\mathfrak{sp}(N)$, $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$, but no construction for $\mathfrak{so}(2N{+}1)$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$ is known. In this paper, we show that the theories for $\mathfrak{f}_4$ and $\mathfrak{so}(2N{+}1)$ have a global gauge anomaly associated to $π_{d=8}$, while $\mathfrak{g}_2$ does not have it. We argue that the anomaly associated to $π_d$ in $d$-dimensional gauge theories cannot be canceled by topological degrees of freedom in general. We also show that the theories for $\mathfrak{sp}(N)$ have a subtler gauge anomaly, which we suggest should be canceled by a topological analogue of the Green-Schwarz mechanism.

hep-th↗

arXiv2017

More on time-reversal anomaly of 2+1d topological phases

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in 2+1 dimensional fermionic topological quantum field theories. The crucial step is to determine the crosscap state in terms of the modular S matrix and $\mathsf{T}^2$ eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

hep-th↗

arXiv2017

$E_8$ instantons on type-A ALE spaces and supersymmetric field theories

We consider the 6d superconformal field theory realized on M5-branes probing the $E_8$ end-of-the-world brane on the deformed and resolved $\mathbb{C}^2/\mathbb{Z}_k$ singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the $T^2$ compactification, and the star-shaped quiver obtained as the mirror of the $T^3$ compactification.

hep-th↗

arXiv2017

On hydrogen-like bound states in $\mathcal{N}=4$ super Yang-Mills

Using relativistic quantum mechanics, we study the spectrum of a non-BPS two-particle bound state in the massive phase of $\mathcal{N}=4$ super Yang-Mills, in the limit when one of the particles is infinitely heavier than the other. We find that the spectrum shows the exact $n^2$ degeneracy for each principal quantum number $n$, just as in the strict non-relativistic limit. This is in line with the findings of Caron-Huot and Henn, who studied the same system in the large $N$ limit with the technique of integrability and the dual conformal symmetry.

hep-th↗

arXiv2017

Smallest 3d hyperbolic manifolds via simple 3d theories

We provide strong pieces of evidence that the mathematics of the three-dimensional hyperbolic manifolds of the first, second and third smallest volume is captured by the physics of the three-dimensional theories composed of a complex boson and a Dirac fermion, both of unit charge, coupled to a U(1) gauge field with the Chern-Simons level $-5/2$, $-7/2$ and $-3/2$, respectively.

hep-th↗

arXiv2017

Frozen

We revisit the duality between ALE singularities in M-theory and 7-branes on a circle in F-theory. We see that a frozen M-theory singularity maps to a circle compactification involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and Kodaira's classification of singular elliptic fibrations. Our analysis strongly suggests that the O7+ plane is the only completely frozen F-theory singularity.

hep-th↗

arXiv2017

Anomaly matching on the Higgs branch

We point out that we can almost always determine by the anomaly matching the full anomaly polynomial of a supersymmetric theory in 2d, 4d or 6d if we assume that its Higgs branch is the one-instanton moduli space of some group G. This method not only provides by far the simplest method to compute the central charges of known theories of this class, e.g. 4d E_{6,7,8} theories of Minahan and Nemeschansky or the 6d E-string theory, but also gives us new pieces of information about unknown theories of this class.

hep-th↗

arXiv2016

Anomaly of strings of 6d $\mathcal{N}{=}(1,0)$ theories

We obtain the anomaly polynomial of strings of general 6d $\mathcal{N}{=}(1,0)$ theories in terms of anomaly inflow. Our computation sheds some light on the reason why the simplest 6d $\mathcal{N}{=}(1,0)$ theory has $E_8$ flavor symmetry, and also partially explains a curious numerology in F-theory.

hep-th↗

arXiv2016

On time-reversal anomaly of 2+1d topological phases

We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter $ν$ mod 16 of the bulk 3+1d topological superconductor with $\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.

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arXiv2016

A brief review of the 2d/4d correspondences

An elementary introduction to the 2d/4d correspondences is given. After quickly reviewing the 2d q-deformed Yang-Mills theory and the Liouville theory, we will introduce 4d theories obtained by coupling trifundamentals to SU(2) gauge fields. We will then see concretely that the supersymmetric partition function of these theories on S^3 x S^1 and on S^4 is given respectively by the q-deformed Yang-Mills theory and the Liouville theory. After giving a short discussion on how this correspondence may be understood from the viewpoint of the 6d N=(2,0) theory, we conclude the review by enumerating future directions. Most of the technical points will be referred to more detailed review articles.

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