Coulomb branches with complex singularities

We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of $\mathcal N=4$ superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as $\mathcal N=4$ sYM theories with disconnected gauge groups.Comment: 43 page

4d N\mathcal{N}=2 theories with disconnected gauge groups

In this paper we present a beautifully consistent web of evidence for the existence of interacting 4d rank-1 $\mathcal{N}=2$ SCFTs obtained from gauging discrete subgroups of global symmetries of other existing 4d rank-1 $\mathcal{N}=2$ SCFTs. The global symmetries that can be gauged involve a non-trivial combination of discrete subgroups of the $U(1)_R$, low-energy EM duality group $SL(2,\mathbb{Z})$, and the outer automorphism group of the flavor symmetry algebra, Out($F$). The theories that we construct are remarkable in many ways: (i) two of them have exceptional $F_4$ and $G_2$ flavor groups; (ii) they substantially complete the picture of the landscape of rank-1 $\mathcal{N}=2$ SCFTs as they realize all but one of the remaining consistent rank-1 Seiberg-Witten geometries that we previously constructed but were not associated to known SCFTs; and (iii) some of them have enlarged $\mathcal{N}=3$ SUSY, and have not been previously constructed. They are also examples of SCFTs which violate the Shapere-Tachikawa relation between the conformal central charges and the scaling dimension of the Coulomb branch vev. We propose a modification of the formulas computing these central charges from the topologically twisted Coulomb branch partition function which correctly compute them for discretely gauged theories.Comment: 45 pages, 3 figure

Quantum Fields on Noncommutative Spacetimes

The 20th century has been defined by many the century of physics. When it started, 111 years back, our understanding of the world was still constrained within the context of what is now called classical physics. The two major revolutions in physics, which later would be called General Relativity and Quantum Mechanics, were yet to come. Physicists were still thinking in terms of just three flat dimensions and in a complete deterministic manner. Not more than 30 years later the warm and convenient setting of classical physics every physicist was comfortable with, was completely turned up side down. By the thirties Quantum Mechanics and General Relativity had already become almost universally accepted theories which constituted the new frame where physics speculations could find place. The transition from the three dimensional, ether-filled, static and eternal flat universe where everything happened in a deterministic and uniquely predictable way, to a strange four dimensional, curved, shape changing object, now appropriately called space-time, where notion of simultaneity and present-past-future became much fuzzier concepts and with a totally messy and chaotic behaviour at the microscopic level, was long and painful. At least for the physicists that from the beginning embraced the new ideas of the universe. Only incontrovertible evidences could defeat an extremely conservative old school physics community which for years laughed at the ones presenting the ideas which later on revolutionized completely our understanding of how nature works. Such new ideas affected not only physics, but the whole approach of human thinking. Countless are the philosophical implications of Special Relativity, but probably none is as deeply disturbing as the ones brought up by quantum mechanics. Once and for all, human ambitions to be able to understand the world as a clock going forward in time, where everything is a uniquely determined effect of the cause who produced it, were killed by the theory of quanta. And it was quite of a brutal murder. Once the new theories were accepted and the new ideas pushed forward it was time to let them bloom. And the spring of physics was a very pleasant one. For the remaining 60/70 years it was a non-stop flow of wonderful insights which let the seeds of General Relativity and Quantum Mechanics flower in what is today known as the Standard Model of Particle Physics and Cosmology. We now understand physics all the way down to almost 10^{-18} cm and could reconstruct the history of the universe from 10^{-36} s after a yet not understood event which has been called Big Bang but of which we really have little idea about. Despite how pleasant the status of High Energy physics might appear, very few physicists have chosen to step back and spend the rest of their time proudly celebrating their achievements. The theory is still missing few ingredients (or maybe many more than we now think) which will hopefully lead us to fill up the holes still present in our current understanding. Discontent and desire to better understanding, easily prevailed on pride. Already Einstein in the last few decades of his life felt the urge of a theory which would unify General Relativity and Quantum Mechanics and would push the limits of our understanding, both in time and distance, all the way down to what needed to enlighten the darkness surrounding the Big Bang, the beginning of the whole. Although Einstein was not very successful in his attempts, many have decided to follow his steps and to embark in the challenging trail which might lead to what Einstein liked to call ``The theory of everything". Such a theory is now called ``Quantum Gravity" and this thesis is aimed to give a modest small contribution in the direction of its development. Do we have any ideas on what Quantum Gravity should look like? Will it be a quantum or a classical theory? And how do we know that such a theory should exist to begin with? Any of these questions has hardly a firm and solid answer but easily many speculative ones. We hope to soon be able to answer all. And as it happens in physics it will only come from experiments. But let's try to unroll a few ideas here to get a taste of how the challenge looks like. The questions are likely in decreasing order of difficulty to be answered. It is fair to say that we really have little idea on what the features of Quantum Gravity might be. More and more physicists have convinced themselves that the new pillar of physics which will be taken down is the number of spacetime dimensions. Extra-dimensions, whose number varies a lot depending on the specific proposal going from 1, in Randall-Sundrum type of model, all the way up to 26 in the case of bosonic String Theory, have become almost ubiquitous in the realm of Beyond the Standard Model physics proposals. But other physicists prefer to get rid of something else feeling particularly comfortable in a four-dimensional spacetime. Most of these theories are not less deeply disturbing. The notion to be abandoned is our perception of spacetime as a continuous. Noncommutative Geometry, which it is the subject of the present work, falls into this set of theories. As we will have time to explain below, we will get rid of the notion of a point altogether. Possibly the second question has a more grounded answer. Although we do not have a certainty, nor has anybody come up with a no-go theorem, there is common agreement that quantum effects should have their appearance in Quantum Gravity. There exist arguments which strongly sustain the inconsistency of a unified theory of Gravity and Quantum Mechanics where the former is still treated classically but yet interacts with quantized matter fields. We don't want to get deeper into such arguments. We just want to mention that most of the problems with a non-quantized theory of gravity which interacts with particle described quantum mechanically, arise because of the way a gravity mediated measurement would affect the wave function. It can be easily proven that, under a wide range of assumption, such a possibility opens to exchange of superluminal signal. We will refer to the literature for more details. In the treatment of Noncommutative Geometry which will be presented here, in fact, Quantum Fields will still represent the mathematical objects which we will be using to construct our formalism. Part of the treatment will be in fact devoted on how to construct quantum fields on a spacetime which does not carry a meaningful notion of points, a noncommutative spacetime. The answer to the last question is probably the most important one and at the same time likely the most solid. Although we know quite little about what a theory of ``Quantum Gravity" might look like, there are plenty of insights on its existence. Thus far what physicists have exploited to circumvent the absence of a unified theory, is the huge scale separation between the three ``quantum forces'' and gravity. Thanks to this seemingly lucky coincidence, for the most part we can carry out extremely precise calculations completely neglecting one of the two side of the whole picture. The really problematic side of the story is that there is no sharp distinction in nature between what can be treated classically and what quantum mechanically. And there are cases where there is absolutely no possibility to favor one approach over the other (black holes or the Big Bang are two cases where gravity should be as important as quantum effects). If from the theoretical point of view seems to be obvious that a theory of Quantum Gravity should exist and we should soon abandon our vision of a two-sided world made of either classical or quantum object (and often the same object acquires different connotations depending on what it is interacting with or which one of its features is under study), the experiments have coexisted quite well with such a fuzzy distinction. Despite lack of experimental results that quantum gravity proposals can attempt to explain, there are reasons to believe that things can change drastically in the years to come. We have recently entered an extremely promising age of physics with many running experiments aiming at probing our current frontiers of knowledge. It is therefore not exceedingly optimistic to hope that soon we could also appeal to experimental evidences of a theory of quantum gravity. Let us keep our hopes well alive, further major revolutions in our understanding of physics might be waiting for us around the corner, and their secrets might be unveiled much sooner than we expect

The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 $\mathcal{N}{=}2$ superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special K\"ahler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the $\rm\, U(1)_R$ symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.Comment: 2 references added, submitted to JHE