Mixed 't Hooft anomalies between 0-form discrete chiral symmetries and 1-form center symmetries and their physical implications in simple SU(N) models

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Tesi di Laurea Specialistica

Department of Physics, ”E. Fermi”

Mixed ’t Hooft anomalies between 0-form discrete

chiral symmetries and 1-form center symmetries and

their physical implications in simple SU(N) models

Candidato:

Andrea Luzio

Relatore:

prof. Kenichi Konishi

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Abstract

The infrared (IR) behavior of non-abelian gauge theories is one of the most interesting, open problems in physics, from both theoretical and phenomenological points of view. Among these, strongly-coupled Yang-Mills theories with matter chiral fermions are especially difficult to analyze, because of the lack of a viable lattice formulation, and because most of the known powerful theorems or methods of analysis are applicable to vectorlike gauge theories only.

One of the few tools that apply equally well to these cases (chiral gauge theories) is the ’t Hooft anomaly matching, a non-perturbative argument that constrains the low energy symmetry realization of the theory, at or below the confinement mass scale. Sometimes ’t Hooft anomaly matching condition can predict the spontaneous symmetry breaking of some global symmetry, but often the argument is not stringent enough, leaving several possible, consistent dynamical alternatives.

Recently, a remarkable development has taken place in exploiting the more general concepts of symmetries, such as those acting on extended objects (lines, surfaces, etc.) but not on the local fields as in conventional symmetries. In particular, the idea of gauging these higher symmetries (especially, 1-form, discrete symmetries such as the center symmetry in the SU (N ) Yang-Mills theory), and of considering the associated 0-form-1-form mixed ’t Hooft anomalies, was found to provide us with a more powerful constraint on the symmetry realizations in the IR than before. In the first part of the thesis we review these new developments, starting from the basic concept of abelian and nonabelian gauge theories, ideas of asymptotic freedom and confinement, dynamical Higgs phenomenon, and of the anomalies, preparing the artillery needed in the discussion of the original part of the thesis.

The original part in fact consists of the applications of these new, generalized ’t Hooft anomaly matching argument to some class of simple vectorlike and chiral SU (N ) gauge theories. Models discussed include: the SU (N ) gauge theory with Nf copies of Weyl fermions in self-adjoint

single-column antisymmetric representation, the well-discussed adjoint QCD, QCD-like theories in which the quarks are in a two-index representation of SU (N ), and a chiral SU (N ) theory with fermions in the symmetric as well as in anti-antisymmetric representations but without fundamentals. Mixed ’t Hooft anomalies between the 1-formZC

k symmetry and some 0-form

(standard) discrete symmetry provide us with useful information about the infrared dynamics of the system. In some cases they give decisive indication to select only few possibilities for the infrared phase of the theory.

The hope is that some of these new results could turn out to be useful in future attempts of a realistic model building, such as the construction of the correct electroweak symmetry breaking sector or of the candidate dark-matter sector, within the theory of the fundamental interactions.

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List of Figures

2.1 Tree different possible running for the coupling constant. The arrow point towards low energies. The blue case realizes when there are few matter fields, therefore both b0 and b1 are negative. It’s the asymptotically free scenario. The red case realizes

when there are enough matter fields such that b1 > 0, but still b0 < 0. In the

limit b0→ 0 the IR fixed point (known as Banks-Zaks fixed point) is perturbative,

otherwise it is difficult so say what happen. In green case there are enough matter field, so that b0> 0 the system is IR free, but strongly coupled in UV. . . 17

3.1 Tricycle condition on triple overlaps between patches. . . 25

3.2 Dirac monopole as simple example of nontrivial bundle. This choice of cover for Σ explain the relation between Π1_{(U (1)) =}_{Z and the possible presence of abelian}

monopoles. In blue the Dirac string for the north pole.. . . 27

3.3 Compactification forR4_. _{. . . .} ₂₈

3.4 The action of a global 1-form center symmetry can be seen directly on bundle transition functions. Clearly it acts non-trivially on fundamental Wilson loops. . 32

3.5 The path γ(θ) ∈ SU (N ) that goes from the identity to an element of Zk, and

parametrizes the gauge transformation with a wrong periodicity. . . 32

3.6 The action of the 1-form global symmetry on the lattice. Only the yellow links get transformed. On contractible Wilson loop, the action is trivial, wheres it is nontrivial on non-contractible ones.. . . 33

3.7 ’t Hooft twisted boundary conditions. . . 35

5.1 Basic cell of the nonanomalousZ8×Z20 lattice. In red, group elements that lie in

Z40. In blue, group elements that lie inZ4. . . 63

5.2 The basic cell of theZ8×Z20lattice. In red, group elements that lie inZ40. In blue,

group elements that lie inZ4. In cyan other nonanomalous points. Other (black)

points are anomalous. Elements in blue squares do not act on ψψ condensate. Elements in green diamond do not act on χχ condensate. Elements in red five points stars do not act on ψχ condensate. Elements in six pointed star, (1, 0), (0, 1) and (2, 15), are anomalous, and each putative condensate breaks a couple of

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Chapter 1

Introduction

One of the central but not yet satisfactorily solved questions in theoretical high-energy physics is the fate of gauge theories when they become strongly coupled in the infrared. This task is of fundamental importance, because gauge theories play a unique role in physics.

• From the experimental point of view, gauge theories have proven of extraordinary efficacy to describe the real world: Quantum Electrodynamics (QED), Quantum Chromodynamics (QCD), and the Standard Model (SM) of fundamental interactions based on SU (3)QCD×

(SU (2)L× U (1))GW S are all gauge theories. The fact that three different kind of forces,

strong, weak and electromagnetic interactions, are all described by the same type of theories - nonabelian gauge theories - shows that a deep principle of unification underlies the working

of Nature.

• From the mathematical point of view, nonabelian gauge theories (with not too many matter particles) are the only types of theories which are asymptotically free, i.e., where the coupling constants, running toward the ultraviolet (UV), approach the trivial fixed point. Only such theories are well defined. Quantum Chromodynamics (QCD) is an example. • At the same time, these asymptotically-free gauge theories in general become strongly

coupled at long distances (i.e., in the infrared), and any possible application of these theories hinges upon our understanding of the behavior of these strongly-coupled theories.

In spite of an impressive success of the standard model, and after many years of theoretical studies of four dimensional gauge theories, our understanding of strongly-coupled gauge theories is basically limited to vectorlike gauge theories, i.e. theories in which the matter fermions are in a real representation of the gauge group. Indeed, an almost half-century of studies of vectorlike gauge theories like SU (3) quantum chromodynamics (QCD), based on lattice simulations with ever more powerful computers, and roughly ∼ 25 years of beautiful theoretical developments in models with N = 2 supersymmetries, all concern vectorlike theories. In contrast, surprisingly little is known today about the strongly-interacting chiral gauge theories, i.e., theories with left-handed Weyl fermions in a complex representation of the gauge group1_{. Perhaps it is not}

senseless to make some more efforts to understand this class of gauge theories, which Nature might be making use of, in an as yet unknown way to us.

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10 Introduction

There are at least two reasons that motivate this kind of work, a pragmatic, phenomenological one and a more general theoretical one. From the pragmatic point of view, such an investigation can be crucial to guide the model building. It is true that, electroweak (EW) sector of the standard model (SM), a theory whose chiral character is well understood and is responsable for observed Parity violation, experiences the EW symmetry breaking before the theory becomes strongly coupled. Thus the electroweak theory is well under control by means of perturbation theory. At the same time, however, there are still unsatisfactory aspects in the SM, such as, among others, the radiative stability of the Higgs boson mass: so-called Naturalness problem. Moreover, a true explanation of the EW symmetry breaking is still lacking, apart from the phenomenological Weinberg-Salam Higgs Lagrangian. Even though a Higgs boson of 125 Gev/c2_{has been discovered}

experimentally, its true nature (elementary? composite? Nambu-Goldstone boson? dilaton?) is still covered by mysteries. Several authors indeed proposed that the EW breaking could be triggered by a new strong dynamics. Models of this kind include technicolor (TC) models, and more general, composite Higgs (CH) models. Most of the models studied in this context so far are vectorlike theories, in order to exploit the theoretical and computational arsenal at our disposal. But Nature could well be making use of some unknown, strongly interacting chiral gauge theory. Therefore, we must enrich our theoretical arsenal to comprehend the dynamics, such as confinement, dynamical Higgs mechanism, possible infrared conformal fixed points, symmetry breaking, etc., in strongly-coupled chiral gauge theories, better. Of particular importance in this context is understanding the fate of global symmetries such as the pattern of spontaneous symmetry breaking, in the given models.

The other reason is driven by curiosity. In the context of strongly-coupled chiral gauge theories one of the most important tools that are in our possession is ’t Hooft’s anomaly matching conditions. In this argument a RG-flow invariant quantity, the ’t Hooft anomaly, is compared between the UV phase, well above the putative symmetry breaking and where the theory is weakly coupled, where perturbative calculations are reliable, and in all possible IR phases. It’s possible that several, a priori possible IR scenarios need to be abandoned because of their failure to meet these constraints. Furthermore, not only continuous global symmetries but also discrete symmetries generate nontrivial consistency conditions, which must all be taken into account properly. Unfortunately, in many cases studied, it turns out that ’t Hooft’s anomaly UV-IR matching conditions alone are not quite strong enough, leaving more than one possible dynamical scenarios for the infrared, in general. More powerful, new theoretical tools need to be developed, to make further progress.

Recently, the concept of generalized symmetries [3,4] has been applied to Yang-Mills theories and QCD like theories, to yield new, stronger, version of ’t Hooft anomaly matching constraints [5]-[6], involving "0-form" and "1-form" symmetries together. The generalized symmetries do not act on local field operators, as in conventional symmetry operations, but only on extended objects, such as closed line or surface operators2_{. A key aspect of these development is the idea}

of "gauging a discrete symmetry", i.e., identifying the field configurations related by the 1-form (or a higher-form) symmetries, and eliminating the consequent redundancies, effectively modifying the path-integral summation rule over gauge fields [7,8]. Since these generalized symmetries are symmetries of the models considered, even though they act differently from the conventional ones, it is up to us to decide to "gauge" these symmetries. Anomalies we encounter in doing so, are indeed obstructions of gauging a symmetry, i.e., a ’t Hooft anomaly by definition. And as in the conventional applications of ’t Hooft anomalies such as the "anomaly matching" between UV and

2_{A familiar example of a 1-form symmetry is the Z}

Ncenter symmetry in SU (N ) Yang-Mills theory, acting on

closed Wilson loops or on Polyakov loops in Eulidean formulation. As is well-known, a vanishing (nonvanishing) VEV of the Polyakov loop can be used as a criterion for detecting confinement (Higgs) phase of the theory.

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IR theories, a similar constraint arises in considering the generalized symmetries together with a conventional ("0-form") symmetry, which has come to be called in recent literatures as a "mixed ’t Hooft anomaly". Another term of "global inconsistency" was also used to describe a related

phenomenon.

The main aim of this thesis work is to take a few gauge theories, either chiral or vectorlike, as an exercise grounds, and to examine whether these new theoretical tools can be usefully applied to them, and whether they provide us with new insights into the infrared dynamics and global symmetry realizations of these models. There will be no attempts to build a realistic model, nor to apply it to any actual physical problem. Our goal is to understand how this new theoretical tool works and to sharpen it as much as possible. We leave the eventual phenomenological exploitation of these chiral gauge theories for future studies. In particular, vectorlike symmetries will serve as a consistency check for our new method, as generally their behaviour is much more understood, whereas chiral ones are our final target.

The presentation of this thesis is organized as follows. In Chapter2 we review the main features of the gauge theories, and recall some generalities of them. This chapter serves to introduce the family of theories within this works applies, and the main questions that such kind of analysis aims to tackle.

In Chapter3we discuss some subtleties that arises on gauge theories when put in topolog-ically nontrivial spacetime. Here the concept of higher-form symmetries (in particular 1-form symmetries), and their gauging, is discussed.

In Chapter 4, the concept of anomalous symmetries and the ’t Hooft anomaly matching argument, our main investigation tool, are briefly reviewed. Then it is explained how the 1-form gauging modify (strengthen) those conditions.

Chapter5 contains the main original content of this thesis work, where we apply this renewed argument to simple models.

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Chapter 2

Gauge theories

In this chapter we remember briefly what Yang-Mills theories are, classically and quantum mechanically, focusing on those whose gauge group Lie algebra is su(N ) (section2.1). Then we summarize the notion of asymptotic freedom and the related of strong coupling in IR regimes that causes confinement or dynamical Higgs mechanism (section2.2). Within this framework we list the possible IR behaviors that gauge theories can show (section 2.3) In particular we classify briefly which classes of global symmetries our models (Yang-Mills theory with fermionic matter) can manifest, and we stress the importance of understanding their fate in IR regimes, when the strong force confines, to describe the low-energy physics (section2.4). At this point we can describe the problem that this work wants to tackle, i.e. to constrain the symmetry breaking pattern of some discrete symmetries, and, from them, the form of the condensate that implies this breaking, in some simple SU (N ) models.

2.1 Yang-Mills theories

A gauge theory is a theory where the coordinates (field values at each space-time position) used to describe the physical state (or its history) are redundant, and the physical state itself (or its history) can be represented in different ways, related each other by a gauge symmetry. In general gauge symmetries can be continuous or discrete, local or non-local.

A central role in the modern high-energy particle physics is played by gauge theories where the gauge symmetry is local, i.e. the gauge transformations are compact Lie-group valued functions gs that act locally on fields,

(

g : M → G ,

g(x) : φ(x) → g(x)φ(x) . (2.1.1)

Here M is a manifold which represents the spacetime, and the gauge group G is a Lie-Group which acts, through a linear unitary1 representation R : G → U (n), on fields φ (seen as, generally complex, vectors).

Within this framework, there is a straightforward procedure to build up a Lagrangian invariant under the local gauge symmetry, called minimal coupling. We add a new Lie-algebra valued 1_{For each compact Lie-group each linear representation can be seen as unitary respect a properly chosen}

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14 Gauge theories

field, the gauge field, or gauge potential Aµ, that transforms under the gauge transformation as

Aµ→ gAµg−1− i(∂µg)g−1, (2.1.2)

and we promote the standard derivative to the covariant derivative

∂µ→ ∇Aµ = ∂µ− iAµ, (2.1.3)

where Aµ is evaluated in the same representation as φ.

The introduction of a gauge potential has a great geometrical meaning, as it allows to connect the values of the field at one point to that at another. Therefore Aµ is also called connection.

Fixed a connection and given a curve γ(s), one can define the Wilson line (also known as parallel-transport) W [γ] = P exp Z γ A (2.1.4) (P is the path-ordered product), a G-valued functional which gives the general solution to the

Cauchy problem (_dφ ds = ˙γ µ_∇ µφ φ(0) = φ0 =⇒ φ(1) = R(W [γ])φ0. (2.1.5)

W [γ] is not gauge invariant, but the Wilson loop

WR[γ] = trR(W [γ]) , (2.1.6)

the object that measure the impossibility to choose a covariatly constant φ, it is. This object will play a central role in the chapter 3, as 1-form symmetries will act naturally on it 3.3. Locally the same impossibility is measured by Fµν (called field-strength or curvature), the

Lie-algebra-valued commutator

Fµν = [∇µ, ∇ν] . (2.1.7)

Now one can construct a local, gauge invariant, Lagrangian for matter field, coupled with an external fixed connection Aµ, writing down terms that are invariant under a constant gauge

transformation and that are built using fields and covariant derivative. In this thesis we will focus on theories with fermions, but without (fundamental) scalars. Schematically the kinetic term for fermions reads Sfermions[ψ, ¯ψ] = Z ¯ ψγµ∇µψ = Z ¯ ψ /∇µψ , (2.1.8)

where Aµcouple differently the two chiralities of the Dirac fermion (then we have a chiral theory)

or in the same manner (in a vectorlike theory). Depending if the gauge group allows to write down a gauge invariant bi-linear, also a mass term can be allowed2_{. We will say little more about}

fermions, and the flavor symmetries of those models, in section2.4.

In general (but not always, e.g. not in section4.4) one aims to promote also Aµto a dynamical

degrees of freedom. If the Lie-algebra is simple the easiest kinetic action for this new field is the

Yang-Mills action, i.e.

S[A] = Z ₁ 2g2tr(FµνF µν_{) =}Z 1 4g2F a µνF µν a , (2.1.9)

2_{Wheres higher dimensional operators (e.g. four fermions operator) are forbidden if one requires renormalizability,}

i.e. one is defining the high energy description of our theory, whereas should be allowed if the theory is an effective description of the low energy dynamics.

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Asymptotic freedom and confinement 15

where the trace is evaluated in the defining representationof the gauge group. Here we have introduced a single coupling constant, g; while, whenever the gauge group is a product of different groups, we have a different coupling constant for each factor.

One can notice that the Yang-Mills action is not free, as packed into eqn (2.1.9) there are cubic and quartic

A A A A A A A interaction vertices3_.

In principle one can add to the Yang-Mills action another term, θ 8π2 Z tr(F ∧ F ) = θ 32π2 Z tr(FµνFρσ)µνρσ, (2.1.10)

which breaks the CP in-variance (unless θ ∈ πZ or there are massless fermions). As this term is a total derivative, it does not modify the equation of motion, and its value is completely fixed by the boundary conditions at infinity. Its physical consequences are not obvious, and related to topics that will be discussed lately, in particular in section3.1.2and in chapter4.

The canonical quantization of Yang-Mills theories is a complex task, because of some technical issues, but it is possible to recover the formal Feynman’s path integral expressions

Z = X

G-bundles

Z

DADφ eiSY M[A]+iSM[A,φ] _and _(2.1.11)

< O1...On> = 1 Z X G-bundles Z

DA Dφ O1[A, φ] . . . On[A, φ]eiSY M[A]+iSM[A,φ], (2.1.12)

for the partition function and for correlation functions. Here the path integral is performed on gauge equivalence classes of A and φ, while Oiare gauge invariant functionals. The formal nature

of these expressions are problematic in some context, e.g. in perturbation theory, where however one can modify them, lifting the gauge symmetry, and still not modify the correlation functions between gauge invariant operators.

2.2 Asymptotic freedom and confinement

In order to evaluate an amplitude or a correlation function in a Yang-Mills theory, as in any other QFT, one has to go through the well known process of renormalization: one must regularize the theory introducing an explicit cut-off ΛU V, define a new renormalized coupling constant g that is

3_{One could say that the beauty of the theory is that its geometry (gauge in-variance) fixes completely the}

coupling of a highly nontrivial, interacting theory. Moreover this structure is rigid enough that it cannot be deformed by renormalization, through the RG-flow!

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16 Gauge theories

related to the original bare coupling constant g0 by a formal power series with coefficients that

diverges in ΛU V, re-express the amplitude in term of this renormalized coupling constant, and

then remove the cut-off. In this process a new energy scale, the renormalization scale µ, needs to be introduced.

The typical renormalized amplitude (the amplitude in terms of the renormalized coupling constant) is schematically

A(Q) ∼ g(µ) +C1g(µ)

2

16π2 log(Q/µ) , (2.2.1)

where Q is the energy scale of the physical process, whereas C1 is an unimportant numerical

coefficient. From this expression is clear that, whenever the typical energy scale of the experiment is far from the renormalization scale µ, large logarithmic terms will appear.

This is the sign that the real perturbation expansion parameter is g(µ)log(Q/µ), which becomes big disregarding the value of g(µ). Fortunately, imposing that the physics is invariant under an arbitrary change of the fictitious renormalization scale, one gets a simple differential equation

µ d

dµg(µ) = β(g) = −b0g − b1g

2_{− . . . ,} _(2.2.2)

which re-sums all the power series in [glog(Q/µ)]n _{(at 1-loop order, if one goes a 2-loops it}

re-sum also terms like g[glog(Q/µ)]n_{). The process of evolving the coupling constant through}

this differential equation is called running the coupling constant, and, because the scattering amplitudes do not exhibit large logarithms if one chooses the renormalization equal to the energy scale relevant for the process, the running coupling g(µ) is a good indicator of the interaction strength, at the given energy scale.

The beta-function can be evaluated perturbatively µ d

dµg(µ) = β(g) = −b0g

3_{− b}

1g5− . . . , (2.2.3)

and at 2-loop we keep only the first two terms [9] ( b0= _16π12 _{11N −2}P iT (Rfi)Nfi 3 b1= (16π12)2 34 3C2(adj.) − 4Pi 5 3C2(adj.) + C2(Rfi) T (Rfi)Nfi) , (2.2.4)

In those expressions Nfi is the number of Weyl fermions in the representation Rfi, T (Rfi) is its

Dynkin index, and C2(Rfi) its quadratic Casimir. Varying the number of fermions, we can have

three possible scenarios, summarized in fig. 2.1:

• With few enough fermion both b0 and b1are negative, and the theory is asymptotically

free, thus well defined in terms of its perturbative expansion4_.

• Rising the Nf, b1 becomes positive while b0keep being negative. The theory flow (in UV)

towards another fixed point (BanksZaks fixed point), i.e. it is defined as a small perturbation around this nontrivial conformal field theory.

• Rising again Nf both b0 and b1 become positive, the theory is strongly coupled in UV.

4_{Asymptotically free (non-abelian) gauge theories are the unique theory well defined (perturbatively) as a}

small perturbation around the trivial UV fixed point, in 4 dimensions ([10]), a fact that we will explain a little more carefully in the next section.

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Confinement vs dynamical Higgs 17

g β(g)

Figure 2.1: Tree different possible running for the coupling constant. The arrow point towards low energies. The blue case realizes when there are few matter fields, therefore both b0 and b1

are negative. It’s the asymptotically free scenario. The red case realizes when there are enough matter fields such that b1> 0, but still b0< 0. In the limit b0→ 0 the IR fixed point (known as

Banks-Zaks fixed point) is perturbative, otherwise it is difficult so say what happen. In green case there are enough matter field, so that b0> 0 the system is IR free, but strongly coupled in UV.

In this thesis we are interested in the first scenario. Here, running the coupling constant towards IR, makes the coupling constant grow and, at some point, ΛIR,

g(ΛIR) ' 4π , (2.2.5)

and perturbation theory breaks down. Here all becomes less understood. Experimentally it is seen that the strong forces is screened, and the whole particle spectrum is color-less. There are two different mechanism that could explain this phenomena: confinement and Higgs mechanism (in this case dynamical Higgs mechanism). Those two behaviors, depending on the theory matter content, can describe two truly different phases, or they can only be two different languages that describe the same physics (therefore there are no phase transition between the two phases).

2.3 Confinement vs dynamical Higgs

In a strongly coupled gauge theory there is two main routes that the system can follow to screen the strong force: confinement and dynamical Higgs mechanism.

The latter is more easy to understand, because it is related to the weakly coupled Higgs mechanism. This is commonly rephrased as the spontaneous symmetry breaking of a gauge

symmetry, and happen when the the path integral is controlled by configurations where a scalar

field φ sits in a nontrivial gauge orbit. This means that, performing a gauge fixing (let us say, imposing unitary gauge), this scalar field gets a (gauge variant) not-null expectation value5_.

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18 Gauge theories

Depending on the φ representation, there could be a subgroup H of the gauge group which is not gauge fixed by the choice of unitary gauge for φ, i.e.

G−−−−−−→ H .broken to (2.3.1)

If the condensate is a fundamental field (i.e. is one of the field averaged in path integral), and the non-abelian force is still weakly coupled, the phenomena is under perturbative control and it makes sense to evaluate the particle masses at tree level. This analysis leads the well known result that gauge bosons corresponding to generator of h, the lie algebra of H, remain massless (as the photon in electroweak symmetry breaking), whereas whose corresponding to generators of G/H get a mass (as the W± and Z bosons). If one is interested in low energy dynamics one can write an effective action, where the symmetry G is explicitly broken (and the corresponding would be gauge fields are massive) but H is preserved.

If the condensate is a composite field and the effective potential that imposes the gauge

sym-metry breaking (the fact that nontrivial orbits are favorable) is generated by the strong dynamics

itself, we have the dynamical Higgs mechanism. Still, depending on the particularities of the condensate, some gauge in-variance can remain unbroken, whereas some global symmetries can be broken. Whenever this happens when the theory is strongly coupled, and aside from perturbative control, there are no simple interpretations of the new low energy theory in term of the high energy variables.

When there are no stronger theoretical inputs at our disposal, a useful heuristic argument that suggests which gauge variant symmetry breaking (bi-linear) condensate actually forms, is the most attractive channel (MAC) criterion [1]. It states that the preferred condensate maximizes the t-channel scattering amplitude

(2.3.2) which is proportional6_{to C}

2(R)−C2(R1)−C2(R2), where R ⊂ R1⊗R2is the putative condensate

representation, and R1 and R2 are the representations in which the two fundamental fields lives.

The logic behind this principle is that this amplitude is proportional to the potential between the particles, therefore the condensate should be the one that maximizes this (attractive) potential. Clearly MAC is a perturbative statement thus it should be seen only as a guiding principle, and not trusted at strong coupling.

The confined phase is more difficult to understand, and to properly define. Sometime it overlaps with the dynamical Higgs phase, sometime there are fundamental difference between the two. In general we talk about a confining force whenever there is a range of distances between two static color sources where the potential energy grows linearly.

the other hand the gauge fixing is required in this discussion, as otherwise there is no clear notion of averaging upon fields.

6_{This can quickly found noticing that the amplitude is proportional to}

∼X

α

TabαTcdα|R∼

1

2{C2(R) − C2(R1) − C2(R2)} . (2.3.3) where α run along color index, a, b in R1, and c, d in R2.

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Chiral symmetries and chiral gauge theories 19

In an SU (N ) Yang-Mills theory without fundamental matter there is a sharp separation between confining phase and Higgs phase.These phases are characterized as follows:

• If the potential between a fundamental particle and an anti-particle scales linearly

V (∆x) = λ∆x, (2.3.4)

or equivalently the asymptotic scaling of a fundamental Wilson loop expectation value follow an area law

< W [γ] >→ e−λA[γ], (2.3.5) the theory is in the confining phase.

• If the potential between static sources scales as: V (∆X) = e

−M ∆X

r (2.3.6)

then the theory is in a Higgs phase. In this case

< W [γ] >→ e−M P [γ], (2.3.7) where P [γ] is the perimeter of the curve, therefore a perimeter law.

This phenomena can be understood as the spontaneous symmetry breaking of a generalized symmetry3, whose symmetry breaking parameter is the Wilson loop (the Polyakov loop) itself.

2.4 Chiral symmetries and chiral gauge theories

One crucial point that must be addressed in the study a quantum field theory is the individuation of its global symmetries and the characterization of their fate in IR. Often this fixes completely the low energy dynamics, guiding to write an effective action which describes it.

Exact global symmetries7 can be either realized (à la WignerWeyl) or spontaneously broken (SSB). In the former case the vacuum is unique, the symmetry operators act trivially on it, and the particle spectrum arranges in representations of the symmetry. In the latter there are different vacua which transform each other under the symmetry transformation, and are labeled by the expectation value of a symmetry variant operator (the condensate). Moreover, if the symmetry is continuous, for each broken generator a massless scalar particle, called Nabu-Goldstone boson, appears.

Given the importance of identifying the symmetries, let us classify more precisely the ones that are relevant in our models, and, more generally, in a gauge theory with fermionic matter. The aim of this thesis will be to characterize and to constrain their spontaneous breaking, with our improved methods.

Let us rewrite all the fermions of our theories in terms of left-handed Weyl spinors8_{. When}

they transform in a real or in a pseudoreal representation of a symmetry, it is called vectorlike. 7_{We focus on flavor symmetries, i.e. (0-form) global symmetries that acts point-wise on field, i.e. that commute}

with the Poincaré group.

8_{In 4-dimensions this is always possible, because, given a right-handed spinor ψ}

R, we can always perform a

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20 Gauge theories

Otherwise, the fermions transform in a complex representation, the symmetry is not vectorlike, and it is said chiral. When a symmetry is real, it is possible to write a symmetry preserving mass (either Dirac or Majorana) term without breaking it, whereas when the symmetry is chiral, it is impossible. The other way around, if it is possible to rearrange Weyl spinors within Dirac spinors that transform in a given representation of the symmetry group, then the symmetry is vectorlike. For more details on fermions, and real/pseudoreal representations, see7.3.

By extension, if the gauge symmetry is vectorlike, the model is a vectorlike gauge theory, whereas if the gauge symmetry is chiral, the model is a chiral gauge theory. Clearly it is possible that a vectorlike theory posses a chiral flavor symmetry, or that a chiral theory posses a vectorlike flavor symmetry. Now we provide two famous clarifying examples.

The prototypical vector-like gauge theory is QCD. Here there are Nf = 2, 3, ..., 6 flavors (i

index) of Dirac fermions (quarks) Q, that transform in the fundamental representation of SU (3)c,

i.e.

qi_L∈ , q_Ri ∈ Q =qL qR

∈ for i ∈ 1, ..., Nf . (2.4.1)

qL(qR) are the left-handed (right-handed) Weyl components that compose Q, and transform in

the SU (3) fundamental too9. The dynamics is given by the Lagrangian

LQCD = LY.M.+ ¯Qiγµ(∂µ− igAµ)Qi+ MijQ¯iQj; (2.4.3)

where Mij is a symmetric matrix in flavor space, which gives a Dirac mass term for quarks. Given

that the masses of three quarks, up, down and strange, are much lower than the confinement scale ΛQCD, it is convenient to consider Nf = 3 and to suppress this term (its effect can be taken

in account as a small perturbation).

The flavor symmetry group of this action is10

SU (3)V × SU (3)A× U (1)V × U (1)A, (2.4.4)

where SU (3)V and U (1)V are vector-like and act as

(

SU (3)V : Q → eiαiT

i

Q ,

U (1) : Q → eiα_{Q ;} (2.4.5)

whereas U (1)Aand SU (3)A are chiral

( SU (3)A: Q → eiαiT i_γ5 Q , U (1)A: Q → eiαγ 5 Q , (2.4.6)

and are called axial symmetries. As the strong force confines, QCD breaks spontaneously all the axial symmetries. This implies that some Nabu-Goldstone boson must appears. From the breaking of SU (3)A are required eight Goldstone-bosons, which are π0, π±, K0, ¯K0, η. They can

be described through the effective chiral Lagrangian, and acquire a small mass from the explicit symmetry breaking quark mass term. The other Nabu-Goldstone boson, which corresponds to

9_{Thus the conjugate of q}

R, which is left-handed, transforms in the anti-fundamental, and

⊕ ¯ (2.4.2)

is real.

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Chiral symmetries and chiral gauge theories 21

the breaking of U (1)A, gets ∼ Λ mass by quantum effects (anomaly) which we will describe in

chapter4.

More in general, the dynamics of vector-like theories is basically understood, as there are interesting theoretical result (e.g. the Vafa-Witten theorem that states that the Vectorial subgroup cannot be spontaneously broken, and the QCD sum rules), and powerful numerical methods.

Also chiral gauge theories have important phenomenological application, as the

Weinberg-Salam model (SU (2)W × U (1)) is chiral, thus also the Standard-Model is a chiral gauge

theory. Indeed left-handed quarks (leptons) transforms under Qup Qdown l ∈ , 1/6 and e − νe ∈ , 1/2 of SU (2)W × U (1)Y . (2.4.7)

whereas right-handed particles are singlet of SU (2)W (and have different charges under U (1)Y),

thus couple with gauge bosons in a chiral manner. The combination of SU (2)W and U (1)Y that

survive the electroweak symmetry breaking, U (1)em, is vector-like.

In the case of chiral theories we still lack a trustable lattice formulation because of the doubling

problem: for each left-handed fermion field which reduces to a left-handed fermion performing the

continuum limits, it arises a right-handed copy, which transforms in the same representation of the flavor/gauge group, and does not get diverging mass as the lattice spacing is sent to 0, thus does not decouple in the continuum limit. In other words, chiral symmetries become vector-like symmetries (either flavor or gauge), and chiral gauge theories become vector-like. This behavior is enforced by the Nielsen-Ninomiya theorem [11], and has a physical explanation in terms of anomalies, which we will give in the chapter4. In the case of flavor symmetries, the doubling problem can be overcome introducing a flavor breaking term (as Wilson term), but in case of chiral theories it is still unsolved.

As we still lack such lattice formulation for chiral gauge theories, we need to sharpen as much as possible any analytical tools that we have in our hand. In the case of this work we revisit the well known ’t Hooft anomaly matching, including the consequences of a new (generalized) symmetry has been recognized in (some) Yang-Mills theories. This gives improved constraints on the IR behavior (e.g. it can constrain the possible condensates that it forms) of our theories. Thus we have applied those improved tool to some simple models, either vector-like Yang-Mills theories and chiral one, to see if sharper consequences can be achieved.

A reader interested only on consequences on the dynamics of gauge models (i.e. the application of the tool, which is the original part of this thesis) can simply jump to 5, while if one wishes to understand how the tool works one should read also the chapter4and the sections3.3,3.4and

4.5. Other sections of chapter3gives technicalities on the new generalized symmetries and their gauging.

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Chapter 3

Nontrivial field topologies and

1-form symmetries

Generalized higher-form symmetries are a straightforward generalization of the concept of symme-try, where the symmetry transformation acts naturally and unambiguously on higher dimensional objects, e.g. on Wilson line operators [4]. The aim of this chapter is to identify a discrete 1-form symmetry (known as center symmetry) within SU (N ) gauge theories, and describe its gauging. The physical consequences of this procedure is the main topic of this thesis, and will be described in the section4.5of the following chapter.

Higher symmetries arise only on topologically nontrivial manifolds, thus to deal with them it is convenient to discuss about gauge theories with finer mathematical language, i.e. to describe the gauge field as a connection on a principal G-bundle. We do so in section 3.1. Then we provide two examples of topologically nontrivial configurations, the Dirac monopole (3.1.1) and the instanton (3.1.2), which will prove themselves really important in subsequent considerations.

In section3.2we explain how to construct a ZN (0-form) gauge theory, either as a network of

ZN defects, or as a U (1) theory Higgsed down toZN by a charge N compact scalar. The second

construction will be generalized to higher symmetries in section 3.4.

In section3.3we show that, on topologically nontrivial manifolds SU (N ) gauge theories enjoy a 1-formZN symmetry1. Those symmetries act naturally on each non-contractible Wilson loop

(thus the introduced transformations are labeled by elements of H1(M,Zk)2).

In section3.4we perform the gauging of these 1-form symmetries, i.e. we deform the theory such that it possesses a local version of the 1-form symmetry, and then we consider this localized symmetry as a redundancy. Within this procedure we must introduce new B ∈ H2_(M,_Z

k) data,

represented by the 2-form gauge connections B(2)_{. If we sum up over all the possible B values}

(thus the external 2-form gauge field becomes dynamical) the theory becomes an SU (N )/Zk

gauge theory. Then we recover the discrete 1-form gauging by means of a U (1) 1-form gauge symmetry, with a 1-form U (1) gauge field, charged under the 1-form U (1) symmetry, which Higgs the 1-form symmetry down to a discrete subgroup. This procedure, which generalizes what is usually done to describe discrete (0-form) gauge theories, will be very useful in the following. The consequences of this gauging on the instanton charge (i.e. its fractionalization), and thus on the

1_{For review on generalized higher form symmetries in a more broad context see [4] or [12].} 2_{Within the text we describe briefly what those cohomology groups are.}

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24 Nontrivial field topologies and 1-form symmetries

chiral anomaly, are delayed on section4.5of the following chapter.

As this chapter is somehow more mathematically oriented than the rest of work, we try to present the construction in an easy and down-to-earth way. A reader which is not interested in details, or which is already familiar with those concepts, can freely jump to the next chapter.

3.1 Gauge theories and principal G bundles

In this section we generalize the construction of gauge theories on manifolds M more general thanR4_{. This is useful because:}

• Even if the spacetime is formallyR4_{, we usually impose the most general finite action}

bound-ary condition, which effectively compactify it. This is what allows instantons, fundamental pieces of our understanding of the QCD dynamics.

• It is often useful to describe the nature through an effective theory, which holds only at energies below some scale. In this case it is possible that some configurations, totally well behaved inRN _{from the point of view of the fundamental theory, are effectively described,}

at long distance, as defined in topologically nontrivial manifold. A simple example is the Nielsen-Olesen vortex.

• Even if we are ultimately interested in what happens inRN_{, we can study our theory in a}

different manifold to gain some insight, or to derive some quantities. E.g., studying our QFT in the euclidean cylinder allows to evaluate its thermal partition function.

Given the importance of studying gauge theories on nontrivial manifolds we sketch the principal bundle construction, where the description of such topological non-trivial configuration is straightforward. For a compact but much more detailed and mathematically rigorous presentation see [13].

Let us suppose that our theory lives on a differential manifold M. This means that we can reconstruct our manifold M as a union of open sets, Ui, that cover the whole manifold,

[

i

Ui= M , (3.1.1)

introduce local coordinates in each of them

φi: Ui→RN , (3.1.2)

and describe local physics in these patches, as we would do if we where in RN_{. Clearly the}

functions that allow us to move from a local coordinate system into another one are change of coordinates.

Similarly, if our theory is a gauge theory, we describe our physics locally (within the patch Ui)

providing also a local gauge fixing fi. This allows to represent the connection as a Lie-algebra

valued 1-form Ai, and the matter fields as function valued in a (usually linear) representation of

G, ψi. In this framework local gauge transformations, gij: Ui∩ Uj → G, allow us to change the

local gauge fixing: they bridge from the patch Uj where we have chosen the local gauge fixing fj

to the patch Ui where we have chosen the local gauge fixing Ui. This imposes the gluing formula

for fields that represents locally the gauge connection

Aj= gjiAigji−1+ idgjig−1ji = A gji

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Gauge theories and principal G bundles 25

and for the matter fields

ψj= R(gij)ψi. (3.1.4)

This construction is consistent for every choice of gij if we impose two constraints:

• If one goes from a local gauge fixing to another, and the other way back, nothing changes, i.e.

gji(x) = g−1ij (x) for x ∈ Ui∩ Uj . (3.1.5)

• It is equivalent to go from Ui to Uj and then to Uk, or directly from Ui to Uk, i.e.

gij(x)gjk(x)gki(x) = 1 for x ∈ Ui∩ Uj∩ Uk. (3.1.6)

This last condition is called the cocycle condition.

g12

g31 g23

g12g23g31=1

Figure 3.1: Tricycle condition on triple overlaps between patches.

As we have fixed the global structure of the principal bundle, let us define the Wilson line in this context. Here the only subtle point is that, changing chart, we have to insert the transition function gij. Concretely, to compute the Wilson line from the point p in the chart i and the point

q in the chart j we must split it

W [q, p]ji= W [q, t]jjgij(t)W [t, p]ii (3.1.7)

where t ∈ Ui∩ Uj and Wii is defined as inRN. The Wilson loop generalizes in the same sense.

From now on we will use the language of differential forms, which is more convenient and compact. Thus we will write the gauge connection as a 1-form A = Aµdxµ, and the field-strength

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as the two form F = Fµνdxµ∧ dxν. Clearly these forms are defined only locally, within a given

patch, as a gauge transformation acts non-trivially on them. For more details on these conventions, see appendix7.1.

The classification of all the possible G-bundles over a manifold is a complex task. Pragmatically, it is useful to define some functions, called characteristic classes, that are integrals of closed (as differential forms) gauge invariant forms, made out the field-strength. They can discriminate

between globally different G-bundle but are insensitive to the actual connection chosen on it. The prototypical example of characteristic class is the n-th Chern number

1 (2π)n_n!

Z

M

tr(Fn) , (3.1.8)

and has really intriguing physical consequences in term of anomalies, which we will review in the next chapter (4). It can be proven that this functional depends only on the G-bundle (the set of transition functions), and not on the connection.

Let us now describe two simple example, where the concept of G-bundle is helpful, and that will be two necessary ingredients for our applications.

3.1.1 Example 1: the Dirac monopole

An extremely simple example of a nontrivial principal bundle is given by the Dirac monopole, a static solution of the Maxwell equations in the puncturedR3 space (i.e. in R3 where it has been removed a tiny open set around the origin). As here our manifold is nontrivial3, nontrivial U(1)-bundles are allowed, and, to describe them properly, we have to choose at least two patches to cover our entire space. We pick one that cover the north pole, another that cover the south pole. Each of them can be extended in all the space, but the axis that goes from the origin through the opposite pole, called the Dirac string. Inside the south patch we can write the connection as

ASP = m(1 + cos(θ))

4π2_sin(θ)r ϕ ,ˆ (3.1.9)

whereas, inside the north one, as

AN P = m(1 − cos(θ))

4π2_sin(θ)r ϕ.ˆ (3.1.10)

The gauge field is well behaved inside each patch, i.e. away from the opposite Dirac string, and the magnetic field,

B = mˆr

4π2_r2 , (3.1.11)

has nontrivial flux (m) trough a sphere around the origin.

For consistency, one has to check that the gauge transformation λ that bridges the connection from one patch to the opposite one

AN P − ASP ₌ 2 · m

4π2_sin(θ)ϕ = dλˆ (3.1.12)

in a single valued, thus

λ(φ + 2π) = λ(φ) + 2πk k ∈Z (3.1.13)

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Gauge theories and principal G bundles 27

Monopole AN P

AN P _{− A}SP _{= dλ(ϕ)}

ASP

Figure 3.2: Dirac monopole as simple example of nontrivial bundle. This choice of cover for Σ explain the relation between Π1_{(U (1)) =}_{Z and the possible presence of abelian monopoles. In}

blue the Dirac string for the north pole.

or equivalently (let us say that the smallest U (1) charge is e)

em = 2πk k ∈Z . (3.1.14)

This is the well known Dirac quantization condition. We remark that, if k 6= 0, this nontrivial field configuration cannot be removed by a further gauge transformation: the integer k labels the principal bundle, and there is a different U (1)-bundle (on the puncturedR3_{) for each integer.}

More in general, in a U (1) theory, it is simple to check that the integral of the characteristic class (which reduces to the magnetic flux in the previous construction) is quantized

Z Σ F = 2πZ ∂Σ = ∅ , (3.1.15) as4 Z Σ F =X i Z Ui dAi= X ij Z Uij Ai− Aj= X ij Z Uij dλij= =X ijk Z Uijk λij+ λjk+ λki= X ijk Z Uijk λijk= 2πZ . (3.1.16)

where Uij = Ui∩ Uj and Uijk= Ui∩ Uj∩ Uk.

3.1.2 Example 2: the SU (N ) instanton

Now we can provide another example, the instanton, really relevant here because of its thigh relation with the chiral anomaly4.2, the central topic of this thesis.

Let us consider an SU (N ) gauge theory onR4_{. In order to guarantee finite action, one can}

prescribe that

Aµ∼ ig∂µg† as |x| → ∞ . (3.1.17)

4_{Here we have chosen a good cover for our manifold, i.e. a cover where each n-fold overlaps between patches is}

empty, or contractible. Notice that, given a surface, it is always possible to find a good cover for the manifold which induces a good cover for the surface by restriction.

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28 Nontrivial field topologies and 1-form symmetries r < r0, instanton r > r0 g(ˆr) (a)R4 compactify −−−−−−−→ g(ˆr) North pole, r < r0 South pole, r > r0 (b) S4 Figure 3.3: Compactification for R4_.

This defines a map, g(ˆx), from the asymptotic S3_{sphere to SU (N ). As the third homotopy group,}

Π3(SU (N )) =Z, there are infinite g(ˆx)i, labeled by the integers i called winding numbers or

instanton numbers, which cannot be contracted to the identity. In other words it is impossible

to gauge away the boundary conditions3.1.17by a globally well defined gauge transformation g(ˆx, r)i, such that

(

g(ˆx, 0)i=1

g(ˆx, ∞)i= g(ˆx)i.

(3.1.18)

The other way around, whenever two boundary condition, gi and hi, are labeled by the same

instanton number, there is a globally defined gauge transformation that transform gi into hi: the

instanton number classify the possible boundary condition.

This discussion can be recast in the gauge-bundle language studying the compactification of the theory, i.e. noticing that there are different SU (N )-bundles on an S4 sphere, one for each instanton number. As resumed in figure3.3, the region ofRN that lies outside a (diverging) radius r0 corresponds to the south patch of the S4 sphere, whereas the region inside r0 to the

north patch. In this setting the transition function lives on the S3overlaps between those patches, and is exactly gi.

One can compute the winding number in terms of the transition functions, and re-express it as gauge invariant function of the field strength,

I = 1 24π2 Z S3 tr(gdg†gdg†gdg†) = 1 32π2 Z µνρσ∂µtr(Aν∂ρAσ+ i2/3AνAρAσ) = 1 8π2 Z tr(F ∧ F ) . (3.1.19)

Here one can recognize that the instanton number is a characteristic class, called second-Chern

number. This integer class will play a central role in the chiral anomaly, and will imply its

quantization.

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Discrete gauge theories 29

sector. Indeed, one can bound the action itself completing the square5 1 2g2 Z tr(Fµν∧ Fµν) = Z ₁ 4g2tr((F ± ∗F )µν∧ (F ± ∗F )µν) | {z } first piece ∓ 1 2g2tr(Fµν∧ ∗Fµν) ≥ ∓ 1 g2 Z tr(F ∧ F) = 8π 2 g2 |I| , (3.1.20)

thus one recovers the absolute minima of the action, within each topological sector, solving the linear PDE

F = ∓ ∗ F , (3.1.21)

which clearly saturates the bound6_.

3.2 Discrete gauge theories

In this section we reviewZN gauge theories, which are really relevant in this work per se, and

because they are useful to define 1-form symmetries.

First of all, let us spend few words about flat abelian G (let us say U (1)) connections. In this case the gauge invariant content of our fields configuration can be simply given by the values of the Wilson loops on each non-contractible loops, as for contractible ones they need to vanish. Similarly, wherever γ1and γ2 can be smoothly connected to each other, i.e. there is a surface S12

such that ∂S12= γ1∪ γ2, then

W [γ1] = W [γ2] . (3.2.1)

This implies that to choose a flat G-connection is equivalent to choose a functional, the Wilson loop itself, from the space of loop (up to smooth deformation) to G; which is exactly the definition of the first cohomology group H1_{(M, G)}7_.

Restricting to the discrete case, the most straightforward description of aZN gauge connection

can be given in loop space, as an element of H1(M,ZN), i.e. again by the values of its Wilson

loops. Now we provides two other equivalent description of such data, which can be useful to highlight different features of this class of theories.

The first alternative description is similar to the principal bundle language we used to describe Yang-Mills theories. Let us pick a good cover of our manifold, i.e. a cover {Ui} where each n-fold

overlaps between n patches is empty, or contractible. Then the most generalZN bundle can be

thought as a set of constant zij ∈ZN -valued transition functions, one for each double overlap

Uij. Those functions, as in the continuous case, must merge properly on triple overlaps, therefore

the cocycle condition,

zij+ zjk+ zki= 0 mod N , (3.2.2)

must hold. Being a gauge theory, one has always the freedom to change the local gauge choice, thus to choose aZN valued functional λi, and to transform

zij→ zij+ λi− λj. (3.2.3)

5_{This manipulation is known as the Bogomolny equation.} 6_{Explicit solution of this equation are known.}

7_{Notice that here the cohomology group ’+’ operation is defined as the product of the Wilson loops (or}

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This clearly does not spoil3.2.2.

A good cover induces a triangulation of the manifold, where n-fold intersections corresponds to n-codimensional simplices. From this picture is easy to understand that contractible Wilson loops are always trivial, where non contractible ones condense all the gauge invariant data of the theory. Being the group abelian, we are saying8 _{that to give a gauge connection on a}_Z

N bundle

is equivalent to specify an element of H1(M,ZN).

Now we provide the second alternative description ofZN gauge theories. Here we look at a

discrete gauge theory as a low energy phase of a U (1) gauge theory Higged down toZN by a

charge N compact scalar [7]. The equivalence between the first definition and this alternative one is easy to see. Indeed the Higgs condition

N A = dφ (3.2.4) implies I γ dφ = 2πZ =⇒ I A = 2πZ N . (3.2.5)

i.e. holonomies areZNvalued and dA = 0, and Wilson loops are nontrivial only on non contractible

curves. This matches our first definition ofZN connection as an element of H1(M, ZN). The

direct equivalence between the first alternative description and the second one can be recovered performing an unitary gauge fixing (e.g fixing φ = 0 mod 2π).

3.3 1-form global symmetries

It has been recently recognized that gauge theories often enjoy generalized higher-form

symmetries. Those global symmetries in general behave ambiguously on fundamental fields, as

their local action overlaps with the action of a lower form gauge transformation, but, globally, are not gauge transformations, as they act unambiguously on (gauge invariant) higher dimensional objects, e.g. on Wilson loops. Moreover they share a lot of properties with their more traditional 0-form cousins, as they induce conserved charges, can be gauged, and can even be spontaneously broken.

Instead of discussing higher form symmetries in their full generality, which it is already done in good review articles (e.g. in [4]), let us provide a clarifying example. The simplest theory that enjoys a higher-form (1-form) global symmetry is (pure-gauge) electrodynamics9_{. The Maxwell}

Lagrangian is invariant under a shift of the electromagnetic potential A

A → A + λ dλ = 0 , (3.3.1)

by another flat U (1) connection λ. This transformation should be seen as a (new type of) global symmetry because, whereas its expression on local field eqn (3.3.1) partially overlaps with the action of a standard gauge transformation, it acts non-trivially on gauge invariant objects, as Wilson loops that lie on non-contractible curves

W [γ] → eiHγλ_{W [γ] ,} _(3.3.2)

8_{More formally speaking, the definition of a functional z}

ij, which associates an element of the groupZN for

each double overlaps, and such that satisfies3.2.2, is the the definition of an exact co-chain on the Čeck cohomology theory. In this language the eq.3.2.3becomes the statement that a gauge transformation shifts the data by an exact co-chain, thus the gauge invariant content is only the (fist) Čeck cohomology class, H1

C(M,ZN).

9_{There is another 1-form symmetry, which act on the dual, magnetic potential ˜}_{A, d ˜}_{A = ∗F , and is unbroken as}

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1-form global symmetries 31

whenever the spacetime manifold M has a topology which support them. This means that, strictly specking, the group elements of a U (1) 1-form symmetry are labeled by the elements of the cohomology group10H1_{(M, U (1)), while, more in general, elements of an G p-form symmetries,}

are identified by the elements of Hp_{(M, G).}

Now we are ready to find the 1-form global symmetry that some SU (N ) gauge theories manifest.

3.3.1 1-form symmetry in Yang Mills theory

Let us consider an SU (N ) Yang-Mills theory, with matter which does not transform under aZk

subgroup of theZN center of the gauge group (thus N is a multiple of k). This system enjoys a

Zk 1-form global symmetry, also called 1-form center symmetry. Let us see in detail.

Formally, the 1-form center symmetry transformation can be seen as a shift

A → A + λ, (3.3.3)

of the SU (N ) connection A by a flat Zk gauge field λ. This expression is clearly formal, and

simply means that the a fundamental Wilson loop W which lies on a non-contractible curve γ transforms as

W [γ] → eiRγλ_{W [γ] .} _(3.3.4)

We can provide a more pragmatical definition of this transformation within the principal-bundle construction, directly on the SU (N ) transition functions gij, as

gij→ zijgij (3.3.5)

The requirement that well defined SU (N ) bundles are transformed into well defined SU (N ) bundles imposes that zij∈Zk satisfies the trivial co-cycle condition

zijzjkzki= 1 , (3.3.6)

on triple overlaps Uijk. Clearly zij data that obeys eqn (3.3.6) defines aZk bundle, as reviewed in

section3.2. Local fields transform in a trivial way under eqn (3.3.5), otherwise line operators, if they are valued in representations nontrivial underZk (e.g. fundamental Wilson loops), transform

non-trivially.

As a simple example, we can study a SU (N ) theory on a cylinder 11_{. Here one needs at}

least two different patches, which overlap in 1-codimensional surfaces, where the SU (N ) valued transition functions are defined (see figure 3.4). Then one acts with eqn (3.3.5). This clearly transforms unambiguously (the un-ambiguity comes from the gauge 0-form gauge in-variance of the loop-operator) a fundamental Wilson loop that winds around the compactified dimension, as

W [γ] → z21W [γ]. (3.3.7)

10_{Which are exactly e}iH

γλ_factors.

11_{Passing by, we notice that this geometrical setting applies almost (fermions must be anti-periodic in the}

compactified dimension) directly when one evaluates the thermal partition function of the theory. In this case all the dimensions are rotated in the euclidean signature and the length of the compact dimension is the inverse temperature β = 1/T . One can interpret the partition function with the insertion of a fundamental Wilson line as the partition function of an external probe particle. Therefore an expectation value for this Wilson line, that would signal the breaking of the 1-form global symmetry, would also signal a perimeter low in its expectation value, i.e. a deconfined phase2.3.

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g

12

g

21

W [γ]

→

g

12

z

21

g

21

zW [γ]

Figure 3.4: The action of a global 1-form center symmetry can be seen directly on bundle transition functions. Clearly it acts non-trivially on fundamental Wilson loops.

The 1-form center symmetry transformation can be stated equivalently as a gauge transforma-tion with wrong periodicity. Let us call θ the angular variable of our cylinder, and consider a path

γ : [0, 2π] → SU (N ) , (3.3.8)

such that

γ(0) =1 and γ(2π) = z ∈ Zk. (3.3.9)

We can recover eqn (3.3.4) as a transformation which acts locally as g(t, θ) = γ(θ), but which is not really a gauge transformation exactly because of its wrong periodicity. Clearly the two definitions that we have provided, the one in terms of transition functions and the one in terms of a wrongly periodic gauge transformation, act in a different way on local fields, e.g. on the connection; however this difference is illusory, as it can be erased within each patch via local gauge transformation.

γ[θ]

1

z

Figure 3.5: The path γ(θ) ∈ SU (N ) that goes from the identity to an element of Zk, and

parametrizes the gauge transformation with a wrong periodicity.

1-form center symmetry transformations can be easily discussed on lattice, as in figure3.6. To be as concrete as possible let us suppose that our lattice topologically is T4. Then one can simply pick a 1-codimensional surface at a definite t (let’s say, t = 0, therefore geometrically S1× S1× S1) and act on each links that crosses this surface, multiplying the link variable by a

constant element ofZk.

U0(t = 0) → zU0(t = 0) . (3.3.10)

Clearly this transformation does not modify the lattice action, as it does not modify the any plaquette terms (any other contractible Wilson loop), but it is still a nontrivial transformation.

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Gauging the 1-form symmetry 33

z ∈

Z

N

W → W

W → zW

Figure 3.6: The action of the 1-form global symmetry on the lattice. Only the yellow links get transformed. On contractible Wilson loop, the action is trivial, wheres it is nontrivial on non-contractible ones.

3.4 Gauging the 1-form symmetry

In this section we aim to gauge theZk 1-form center symmetry, i.e. we deform the SU (N ) theory

such that it becomes invariant under a local version of the symmetry and then we consider the 1-form symmetry itself as a redundancy of our description, rather than a true symmetry. As for the global center symmetry, we can define the action of a 1-form gauge transformation directly on the transition functions of the principal SU (N )-bundle. In particular we extend (make it local) the action of eqn (3.3.5) relaxing the cocycle condition for the zij data i.e.

(

gij→ zijgij

gijgjkgki=1 → zijgijzjkgjkzkigki= h ∈Zk

(3.4.1)

This spoils the SU (N ) principal-bundle cocycle condition, which now holds only up to someZk

defects. This means that we enlarge the set ofZk connection in the shift

A → A + λ, (3.4.2)

and now λ can be almost flat.

Clearly, having acted on a SU (N )-gauge bundle with a 1-form gauge transformation in this way, we have spoiled its cocycle conditions, i.e. we have mapped SU (N )-bundles into things that are not SU (N )-bundles anymore. However, if we consider only representation of SU (N )/Zk,

all we have done is trivial: SU (N ) bundles, up toZk 1-form gauge symmetries, are SU (N )/Zk

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It is important to notice that not all the SU (N )/Zk gauge bundles can be seen as the 1-form

gauge equivalence class of a properly defined SU (N ) bundle; as in general we should permit from the beginning someZk errors in each triple overlaps. Let us parameterize this via

Bijk= gijgjkgki∈Zk, (3.4.3)

new data. B transforms non-trivially under 1-form gauge transformations

Bijk → Bijkzijzjkzki, (3.4.4)

and consistency on quadruple overlaps requires B123B243B341B142= g12g23g31 | {z } B123 g24g43g32 | {z } B243 g34g41g13 | {z } 341 g14g42g21 | {z } B142 = · · · = 1, (3.4.5)

an higher form of cocycle condition. Mathematically speaking, an object that satisfies eqn (3.4.5) and which is defined up to eqn (3.4.1) transformations is an element of the Čeck cohomology group H2

c(M,Zk).

Let us choose our cover such that it is a good cover, and that it induces a good cover on a surface Σ, whose ends are two closed loops, γ1 and γ2. We can construct

B[Σ] =X

ijk

Bijk (3.4.6)

where the sum is on the the triple overlaps on the cover induced on the surface (these triple overlaps are generically 2-codimensional thus can be seen as points on our surface). Then we can define the parallel transport of a Wilson loop W [γ1] trough the surface as12

W [γ2] = ei

2πB[Σ]

k _{W [γ}₁_{] .} _(3.4.7)

If the surface Σ is closed, then B[Σ] define an higher analogous of Wilson loop, thus B is a discrete version of a 2-form connection.

This route brings us back to the original definition of (0-form)Zk gauge theories given in3.2,

i.e. in loop space. Here we have trade loops with close surfaces, but the rest it is identical. In this case, 2-surfaces do not define parallel transport for point-like objects, but for 1-dimensional objects, Wilson loops.

Within this renewed framework, we can review the old twisted boundary conditions given by ’t Hooft [14]. This should render more evident that this 1-form gauging is nothing more than a new fancy language13 _{to describe this older construction. In his analysis ’t Hooft chose to work}

with a four-torus T4_{= T}2_{× T}2_{= S}1_{× S}1_{× S}1_{× S}1_{, and to study a pure SU (N ) Yang-Mills}

theory on it. Here ’t Hooft singled out aT2_{and recognized the possibility to choose boundary}

condition on it (see figure3.7), i.e. (

A[x, 0] = Ag(x)_{[x, 1]}

A[0, y] = Ah(y)_{[1, y] ,} (3.4.8)

12_{Keeping care of the orientation of the surface, and of its two boundaries.}

13_{But we think that fancy languages can be really helpful to understand fancy thing, in this case new mixed}

(35)

Gauging the 1-form symmetry 35

h(0)

g(1)

h(1)

g(0)

g(x)

h(y)

Figure 3.7: ’t Hooft twisted boundary conditions.

parameterized by g(x) and h(y), SU (N ) valued transition function (x, y ∈ (0, 1) are coordinates onT2_{). As he was working in pure Yang-Mills he imposed twisted boundary conditions}

g(1)h(0) = z01h(1)g(0) z01∈ZN; . (3.4.9)

This defines a topological number z01= −z10∈ZN relative to the compact directions i = 0 and

j = 1. Therefore, in the 4-dimensional case, there are threeZN electric fluxes (when one of the

indices i or j is 0), and threeZN magnetic fluxes (when both indices are spatial). Those fluxes

are called ’t Hooft fluxes, and corresponds to the B data that we have just introduced (in this case there is no matter field which partially breaks the center symmetry, thus k = N ).

3.4.1 Continuous construction

Now we can provide an equivalent continuous description for theZk 1-form gauge symmetry as a

low-energy phase of a U (1) 1-form gauge symmetry, Higgsed down to its discrete subgroup by a charge k matter field. This is somehow similar to what we have done in the 0-form setting in section 3.2. Within this higher form case the natural candidate for the role of Higgs matter field is itself a U (1) 1-form gauge field, which we call B(1)_{, and Higgs condition simply becomes (here}

the B(2) is a 2-form which substitutes B data)

k B(2)= dB(1). (3.4.10)

In this construction, theZk1-form gauge symmetry clearly is enhanced to a U (1) 1-form symmetry,

i.e. the fields transform as

(

B(1)→ B(1)_{+ kλ ,}

B(2)_{→ B}(2)_{+ dλ ,} (3.4.11)

where now the gauge field λ is, in general, not flat.

Following the steps in section3.2, we must check that integrals of B are Zk valued, and are

topological (i.e. they depends only on the homology class of S). The first is true because of the Dirac quantization condition (3.1.1) of the U (1) connection B(1) _{and of the U (1) connection λ}

that parameterizes the 1-form transformation (_R S dB(1) k = R SB (2)₌2πn k n ∈Z , R SB (2)_→R SB (2)_{+ dλ =}R SB (2)_{+ 2πN} (3.4.12)

Mixed 't Hooft anomalies between 0-form discrete chiral symmetries and 1-form center symmetries and their physical implications in simple SU(N) models

Tesi di Laurea Specialistica

Department of Physics, ”E. Fermi”