It predicts a partner particle for each particle in the Standard Model. These new particles would solve a major problem with the Standard Model — fixing the mass of the Higgs boson. If the theory is correct, supersymmetric particles should appear in collisions at the LHC.

At first sight, the Standard Model seems to predict that all particles should be massless, an idea at odds with what we observe around us. Theorists have come up with a mechanism to give particles masses that requires the existence of a new particle, the Higgs boson. However, it is a puzzle why the Higgs boson should be light, as interactions between it and Standard-Model particles would tend to make it very heavy. The extra particles predicted by supersymmetry would cancel out the contributions to the Higgs mass from their Standard-Model partners, making a light Higgs boson possible.

The new particles would interact through the same forces as Standard-Model particles, but they would have different masses.

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If supersymmetric particles were included in the Standard Model, the interactions of its three forces — electromagnetism and the strong and weak nuclear forces — could have the exact same strength at very high energies, as in the early universe. There will be included a training week with introductory lectures to some of these topics.

The goal will be to describe the basics of derived categories of sheaves associated to algebraic varieties with a physically motivated point of view. We will start from the beginning and go on to describe various categorical invariants, such as Hochschild homology, which play an important role in field theory. Time permitting, we will explain integration in the context of derived algebraic geometry and describe its relationship to Feynman integrals. These lectures will give an introduction to sigma models for mathematicians and physicists. In the first lecture I will introduce sigma models and the various concepts needed to understand them.

I will give several simple examples and in particular discuss the intriguing relation between properties of the field theory that defines the sigma model and the geometrical characteristics of its target space. The second lecture will be focussed on supersymmetric sigma models and their target space geometry.

The third lecture will give some applications such as quotient constructions reductions and T-duality and also discuss the quantum properties of sigma models in more detail. In these lectures we will introduce and play with the basic notions of vertex algebra theory like Poisson vertex algebras, quasi-classical limits, etc.

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## The Basics of String Theory

We will give several examples reproducing the well known examples in the theory like universal enveloping vertex algebras, lattice vertex algebras and W-algebras. We will then explain how to construct a sheaf of vertex algebras attached to any smooth manifold M, known as the chiral de Rham complex of M. Click here to find out! Our world has many fermions — all the matter particles — and many bosons — all the force carriers. But none of them have the right properties to be super-partners of one other.

So therefore, if supersymmetry were a symmetry of nature, every type of elementary particle that we know of in nature would have to have partners we have not discovered yet. Since there are over two dozen particles known, that would mean we have a lot of work left to do! What is this symmetry, really?

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In a bosonic dimension — the ones we are used to — you can move as far as you want, taking step after step to the left, say. Moving in a fermionic dimension, it is as though you can only take one step. The only thing you can do is go back. Ok, that sounds really weird, and it is; in the end, you have to define that kind of dimension using math, not words or analogies.

His theory consists of a set of equations that obey a certain set of symmetries. So supersymmetry is, in this sense, very special. Were supersymmetry an exact symmetry of nature, we would already have found many superpartners. But fermions have boson superpartners, and vice versa. One other addition: extra Higgs particles are necessary compared with the Standard Model: 5 instead of 1. In Fig. But this exactly supersymmetric world is obviously not our world.

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As indicated in the figure, in such a world particles and their superpartners. For one thing, we would have atoms with electrons in them, and atoms with selectrons in them, and atoms with both. Data and daily life completely exclude this possibility. There are no selectrons with the mass of the electron, period. So exact supersymmetry is not a correct theory of nature, and we knew that before supersymmetry was first thought of.

Despite this seeming disaster, the original theoretical proposal of supersymmetry allows a very simple and even plausible way out. It is a common notion in physics that symmetries may be hidden from view physicists often say spontaneously broken , but really this is not a good image to have intuitively — the symmetry is still there, it has just been made difficult to recognize. But on earth, the rotational symmetry is hidden; right-side up, up-side down and sideways take on meaning.

One example involves rotational symmetry on the earth. The laws of nature are the same no matter how you rotate your experiment see Fig.

## PHY573 Quantum Field Theory III

This is true, but hard to recognize on earth, where it does matter whether your experiment is right-side-up, up-side-down, or tilted sideways. But far out in space, far away from all the planets, moons and stars, the laws of nature are rotationally symmetric. Your experiment will give the same answer no matter how you orient it. Incidentally, measurements of light emitted from very distant atoms do indeed confirm this.

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What the earth does is confuse us. It makes us think that down is a very different direction from up or from left. But that apparent difference is not intrinsic to the laws of nature. The difference arises because the earth is nearby, hiding rotational symmetry from our view. What would happen then?

It turns out that it is quite easy to get a world that looks just like ours, in which the superpartners of the known particles exist but have become heavy… too heavy for us yet to have produced them in experiments. This effect would generate mass for precisely those particles that we haven't yet discovered, pushing the squarks, sleptons, sneutrinos, gluinos, charginos and neutralinos beyond previous experiments, but perhaps as it is hoped by many, for strong reasons within range of the Large Hadron Collider.

To reduce clutter, I have removed the lines that were present in Figure 1, which indicated which particles interact directly. The basic interactions are not changed by supersymmetry breaking, and are still the same as shown in Figure 1. You see that supersymmetry breaking the hiding of the symmetry from easy view , whose details are not specified, has pushed all the superpartners up to a mass scale around or above the top quark mass.

This is not as arbitrary or silly as it sounds — the mathematics readily accomodates this effect.

## UZH - Physik-Institut - PHY Quantum Field Theory III

There are many precise examples of how this could happen… but far too many for us to guess which of them is most likely. This is by no means the unique pattern that could occur through supersymmetry breaking!

There are many, many other possibilities, which I will refer to as different variants of supersymmetr y.