There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Superstring theoryをめぐって(放談室)

These notes are loosely based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge theories, February 2009, and at the IPM String School in Tehran, April 2009. I have focused on a few concrete topics and also on addressing questions that have arisen repeatedly. Background condensed matter physics material is included as motivation and easy reference for the high energy physics community. The discussion of holographic techniques progresses from equilibrium, to transport and to superconductivity.

https://iopscience.iop.org/article/10.1088/0264-9381/26/22/224002/pdf

Lectures on holographic methods for condensed matter physics

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.

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The frozen phase of F-theory

For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary $\mathbb{Z}$-valued Witten index vanishes. Similarly, for 2d supersymmetric quantum field theories, there are cases when a mod-2 elliptic genus can be defined, even when a more ordinary elliptic genus vanishes. We study such mod-2 elliptic genera in the context of $\mathcal{N}=(0,1)$ supersymmetry, and show that they are characterized by mod-2 reductions of integral modular forms, under some assumptions. For mathematicians: We study the image of the standard homomorphism $\pi_n \mathrm{TMF}\to \pi_n \mathrm{KO}((q))\simeq \mathbb{Z}/2((q))$ for $n=8k+1$ or $8k+2$, by relating them to the mod-2 reductions of integral modular forms.

Remarks on mod-2 elliptic genus

Our next task is to study $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with one hypermultiplet in the doublet representation. This is often called the SU(2) theory with one flavor, or more simply N f = 1. We will see that all the methods employed in the last chapter are readily adapted to this theory, too. We again follow the presentation of the original paper [1], but we use the Seiberg–Witten curve in a form more suitable for the generalization later. Appendix C of [2] is a good source where many different forms of the Seiberg–Witten curves of SU(2) theories are summarized.

SU(2) Theory with One Flavor

Starting with a choice of a gauge group in four dimensions, there is often freedom in the choice of magnetic and dyonic line operators. Different consistent choices of these operators correspond to distinct physical theories, with the same correlation functions of local operators in R^4. In some cases these choices are permuted by shifting the theta-angle by 2pi. In other cases they are labeled by new discrete theta-like parameters. Using this understanding we gain new insight into the dynamics of four-dimensional gauge theories and their phases. The existence of these distinct theories clarifies a number of issues in electric/magnetic dualities of supersymmetric gauge theories, both for the conformal N=4 theories and for the low-energy dualities of N=1 theories.

Reading between the lines of four-dimensional gauge theories

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 SL(2,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.

Yuji Tachikawa

Papers

The frozen phase of F-theory

Remarks on mod-2 elliptic genus

SU(2) Theory with One Flavor

Reading between the lines of four-dimensional gauge theories

Anomaly of the Electromagnetic Duality of Maxwell Theory