Global Symmetry Breaking:- A perspective on Symmetry

Inconsistencies

Soumyadip Nandi

August 2025

∗

1 Abstract

Global Symmetries are inconsistent with Quantum Gravity. Most global symmetries are

broken/gauged away at lower energies . These kind of symmetries, even though are derived

well out of Quantum Field Theories and General Relativity, it undergoes spontaneous

symmetry breaking at low energy levels and remain lacking at uniﬁcation. Spontaneous

Symmetry Breaking(SSB) gives rise to Goldstone modes which signals the presence of

massless excitations. A great example of SSB is the Higgs Field. The problem has been

approached from a few diﬀerent perspectives- namely - global symmetries on the string

worldsheet, the consequences of symmetry breaking that happens on a conserved Noether’s

charge and how divergences occur when the vacuum state does not remain invariant and

ﬁnally a topological perspective to how gauge symmetry is broken and how it leads to the

rise of both broken and unbroken generators and mass terms.

Major focus has been devoted towards understanding how a non-triviality approach in the

context of conserved Noether current as well as a Holonomy group can give us deep insight

on how divergences lead to inconsistencies(this is very apparent in case of Black Holes). It

also provides information as to how compactiﬁcation of higher dimensional gauge group

leads to the breakdown of gauge symmetry.

Contents

1 Abstract 1

2 Basic Introduction for Symmetries and Gauge theory 3

2.1 Symmetries and Gauge theory, an overview . . . . . . . . . . . . . . . . . . . 3

2.2 Gauge Theory , an overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

∗

doi = ”https://doi.org/10.5281/zenodo.16814396”, inspire hep =”https://inspirehep.net/literature/2960119”

3 Global Symmetries on the string worldsheet 5

3.1 Fixing a Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 BRST Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Continuous Global Symmetry Breaking 10

4.1 Symmetries in Classical Field Theories . . . . . . . . . . . . . . . . . . . . . 10

4.2 Symmetries in QFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 The Non-triviality of Noether’s current in symmetry breaking . . . . . . . . 12

4.3.1 The rise of massless poles . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Symmetry Breaking: A mathematical proof 13

5.1 Gauge Symmetry Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.1 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.2 Why Higher Dimensions ? . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.3 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.4 Adjoint on scalar S

. . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1.5 The mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.1.6 Field Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2 Goldstone Excitations, Spontaneous symmetry breaking . . . . . . . . . . . . 19

6 Conclusion 20

7 Appendix A 20

8 Appendix B 21

2 Basic Introduction for Symmetries and Gauge theory

2.1 Symmetries and Gauge theory, an overview

Considering symmetries , in quantum ﬁeld theory there are two kinds of symmetry, local

symmetry and global symmetry.

A global symmetry is a transformation that acts uniformly on all points in spacetime.

Mathematically, a global symmetry transformation is represented by an operator U.

U is an unitary operator that commutes with the ﬁeld operators of the theory. For a scalar

ﬁeld theory, a global symmetry transformation can be expressed as:

ψ(x) → ψ

′

(x) = Uψ(x)ϕ(x) where ϕ(x) is the ﬁeld operator at spacetime point x , and U is

the global symmetry operator. Global symmetries lead to conservation laws through

Noether’s theorem which states that for every symmetry there is a conserved quantity,

whether it be momentum, angular momentum or energy.

For example, a global U(1) symmetry leads to the conservation of electric charge in QED.

A local symmetry, also known as gauge symmetry, is a transformation that varies from

point to point in spacetime. In gauge theories, such as Quantum Electrodynamics (QED)

and Quantum Chromodynamics (QCD), local symmetries are associated with gauge ﬁelds

and gauge bosons. Mathematically, a local symmetry transformation is represented by a

gauge transformation

U(x) that depends on spacetime coordinates: ψ(x) → ψ

′

(x) = U(x)ψ(x)

Here, ψ(x) represents the ﬁeld operator of the fermion ﬁeld, and

U(x) is the gauge transformation. Gauge symmetries introduce redundancy in the

description of the theory, which manifests as gauge degrees of freedom. Gauge theories are

constructed to be invariant under local gauge transformations, leading to the emergence of

gauge ﬁelds and ensuring that physical observables are independent of the choice of gauge.

2.2 Gauge Theory , an overview

The gauge principle is a fundamental concept in theoretical physics that states that the

laws of physics should be invariant under local transformations of a certain group. In the

context of gauge theories, such as electromagnetism and the weak and strong nuclear forces,

the gauge principle underlies the symmetries and interactions of elementary particles.

Let’s consider a complex scalar ﬁeld ψ(x) as an example. Under a gauge transformation,

the ﬁeld ψ(x) undergoes a local phase transformation:ψ(x) → ψ

′

(x) = e

iα(x)

ψ(x). Here,

α(x) is an arbitrary real-valued function of spacetime x.

The gauge principle demands that the physical predictions of the theory remain unchanged

under such local gauge transformations. Mathematically, this can be expressed as:

L(ψ, ∂

ψ, A

) = L(ψ

′

, ∂

′

, A

) where the gauge ﬁeld A

is representing the interaction.

To ensure gauge invariance, we introduce a gauge ﬁeld A

(x), that transforms under gauge

transformations such that the gauge-invariant derivative is preserved. This is done by

replacing ordinary derivatives with covariant derivatives:

= ∂

− iqA

where q is a coupling constant associated with the interaction.

Under a gauge transformation, the gauge ﬁeld A

can be transformed under an Unitary

transformation, which gives us the covariant derivative as,

ψ(x) = d

ψ(x) −iqA

ψ(x)

Gauge theory ensures the consistency and invariance of physical laws under local

transformations, leading to the introduction of gauge ﬁelds and the covariant derivatives

that preserve gauge invariance. The gauge principle is associated with a gauge symmetry

group, such as U(1) for electromagnetism or SU(2) for the weak force. The choice of gauge

group depends on the speciﬁc theory being considered.

2.3 Gauge Symmetry

Now, to ensure gauge invariance, the electromagnetic potential A

needs to transform

under gauge transformations. It transforms as:

(x) → A

′

(x) = A

(x) + ∂

Λ(x)

for a function Λ(x), we will ask only that x dies suitably quickly at spatial inﬁnity . We

call this a gauge symmetry. The ﬁeld strength remains locally invariant. ψ(x) = e

iα(x)

ψ(x)

which remains phase invariant under α(x) rotations.

So what are we to make of this? We have a theory with an inﬁnite number of symmetries,

one for each function x. Previously we only encountered symmetries which act the same at

all points in spacetime. Noether’s theorem told us that these symmetries give rise to

conservation laws. Do we now have an inﬁnite number of conservation laws? . The answer

is no! Gauge symmetries have a very diﬀerent interpretation than the global symmetries

that we make use of in Noether’s theorem. While the latter take a physical state to another

physical state with the same properties, the gauge symmetry is to be viewed as a

redundancy in our description. That is, two states related by a gauge symmetry are to be

identiﬁed: they are the same physical state.

As we get deepen our investigation into gauge and global symmetries, we are going to draw

analogies from diﬀerent theories and contexts. The motivation as mentioned in the

beginning , is to give a symmetry treatment to Quantum Gravity. A fundamental question

which needs exploration with regards to Quantum Gravity is why are global symmetries

inconsistent with it, speciﬁcally when the theory of General Relativity and QFTs derive

well deﬁned consistencies . Let’s begin.

3 Global Symmetries on the string worldsheet

The main question here is - Does global symmetries apply everywhere in spacetime ? YES.

Then what happens in Quantum Gravity. ?

The most mathematically consistent theory which we have right now, when it comes to the

uniﬁcation of both Quantum mechanics and General Relativity is string theory. Gravity, as

described by Einstein’s ﬁeld equations is a consequence of string theory. If we quantize a

string in 11 dimensions, Einstein’s ﬁeld equations comes out of it as a consequence. In

string theory, most global symmetries are gauged at lower energies. How exactly does it

happen ?. We will see now.

The idea that any theory of quantum gravity cannot have global symmetries has a long

history [1].

The lack of such ordinary global continuous symmetries is known to be satisﬁed in all

descriptions of quantum gravity. It was shown in the paper [2], to be satisﬁed in

perturbative string theory - global symmetries on the string world-sheet lead to gauge

symmetries in spacetime, and there is no way to have global symmetries in spacetime.

Let’s look at it mathematically,

from the Polyakov Action on the worldsheet, [3]

S = −

4πα

′

∂

√

−gg

αβ

∂

µν

where g = det(g

αβ

) , is the determinant of the metric tensor, g

αβ

is the world-sheet metric.

The world-sheet metric can also be expressed as:- g

αβ

= 2f(σ)∂

X.∂

The equation of motion for X

, which is a bunch of scalar ﬁelds X , coupled to 2d gravity

will be:-

∂

(

√

−gg

αβ

∂

) = 0

The Polyakov action still has the two symmetries of the Nambu-Goto action. -

Poincare Invariance.

- Reparameterization invariance.

1) Poincare Invariance:- This is a global symmetry on the worldsheet.

→ Λ

+ c

2) Reparameterization invariance:- also known as diﬀeomorphisms. This is a gauge

symmetry on the worldsheet. The ﬁelds X

transform as worldsheet scalars, while g

αβ

transforms in the manner appropriate for a 2d metric.

Along with these symmetries, there is a new kind of symmetry called Weyl invariance.

Weyl invariance:- Under this transformation of the scalar ﬁeld X

, the metric transforms

as equation (1)

αβ

(σ) = Ω

(σ).g

αβ

(σ) (1)

Or, inﬁnitesimally, we can also write, Ω

(σ) = e

2ϕ

for small ϕ such that,

δg

αβ

(σ) = 2ϕ(σ)g

αβ

(σ)

It is simple to see that the Polyakov action is invariant under this transformation.

This means that two metrics which are related by a Weyl transformation as given in

equation (1) are to be considered as the same physical state.

So, one component for poincare invariance and two components for Reparameterization

invariance. So total, three components for the world-sheet metric g

αβ

. This means that we

expect to be able to set any two of the metric components to a value of our choosing. We

will choose to make the metric locally conformally ﬂat.

*Note:- Although, if we encounter a dynamical metric, reparameterization invariance wouldn’t lead to the diﬀeomorphisms

which lead to gauge redundancy, we need an invariant operator which allows us to integrate over the whole wordsheet . These

are called vertex operators.

3.1 Fixing a Gauge

We can use,

αβ

= e

2ϕ

.η

αβ

(2)

to make the metric conformally ﬂat. Choosing a metric of this form is called conformal

gauge., where ϕ(σ, τ) is some function on the world-sheet. This is equation (2).

We have only used reparameterization invariance to get to equation (2). We still have Weyl

transformations to play with. Clearly, we can use these to remove the last independent

component of the metric and set ϕ = 0 such that,

αβ

= η

αβ

We end up with the ﬂat metric on the worldsheet in Minkowski coordinates.

As mentioned in this case, the metric has three independent components [4] namely,

αβ





where, g

= g

Reparametrization invariance allows us to choose two of the components ofg , so that only

one independent component remains. But this remaining component can be gauged away

by using the invariance of the action under Weyl rescalings.

So in the case of the string there is suﬃcient symmetry to gauge ﬁx g

αβ

completely.

Therefore, the resultant metric can be chosen as, so as to end up with a ﬂat worldsheet.

αβ

= η

αβ



−1 0

0 1



3.2 BRST Operations

After Gauge ﬁxing, we are left with the gauge redundancy and these redundancies need to

be handled carefully when we talk about the symmetries. This is where Faddeev- poppov

ghosts come in, Ghosts in the string worldsheet arises as a consequence of gauge-ﬁxing the

worldsheet diﬀeomorphism. [See Appendix A]

To deal with these kinds of Gauge redundancies, we introduce BRST quantization. On a

very basic level, BRST is an approach to performing anomaly free perturbative calculations

in a non-abelian Gauge theory. Its signiﬁcance is for rigorous canonical quantization of

Yang-Mills theory.

The BRST charge for a closed string, in the context of string theory, is a crucial operator

used for quantizing the theory and identifying physical states. It’s a combination of

left-moving and right-moving BRST charges and is nilpotent, meaning its square is zero.

Physical states are deﬁned as those that are annihilated by this charge. [5]

The BRST charge, Q

BRST

acts on both matter and ghost ﬁelds. As we mentioned, it will

satisfy,

- Q

= 0 , nilpotent

- Physical states must be invariant under BRST transformations, meaning they are

annihilated by the BRST charge, Q

BRST

= |ψ⟩ = 0 , closed.

- |ψ⟩ ≈ |ψ⟩ + Q

BRST

|X⟩ (exact)

This deﬁnes the BRST cohomology, which picks out the physical states of the string (i.e.,

those that survive gauge ﬁxing and are unitary).

Now, if you recall from the previous overview section of symmetries and gauge theory, you

will see that the global symmetries give rise to Noether currents and their associated

conserved charges. Hence, the worldsheet should deﬁnitely have conserved current. The

continuity equation can be checked by ,

∂

µν

= 0

This corresponds to a worldsheet global symmetry (e.g. SU(N), SO(N), etc.).

3.3 Vertex Operators

The BRST cohomology describes the physical states of the string, but there are still

constraints on the physical state |ψ⟩ described by Virasoro algebra.[6] Sometimes , in order

BRST invariant operators that connects both open and close strings

to handle certain issues like singularities in Conformal Field theory, a normal ordering of

operators are required . This is done to deﬁne operator products consistently.

In our case, there are ambiguities that arise from the constraints on the physical state.

Hence, we need to ﬁx the normal ordering of ambiguities. These ambiguities are usually

represented by the conformal weight c, c

representing both left and right moving sectors in

CFT[two independent sets of degrees of freedom that arise when dealing with

two-dimensional spacetime]. How do we do this ?

A simple way is to ﬁrst replace the states with operator insertions on the worldsheet using

the state operator map: |ψ⟩ → O. [3]

But we have a further requirement on the operators O: gauge invariance.

There are two gauge symmetries as mentioned before: reparameterization invariance and

Weyl symmetry. Both restrict the possible states.

Let’s start by considering reparameterization invariance. in a theory with a dynamical

metric, this does not give rise to a diﬀeomorphism invariant operator. To make an object

that is invariant under reparameterizations of the worldsheet coordinates, we should

integrate over the whole worldsheet. Our operator insertions (in conformal gauge) are

therefore of the form,

V ∼

∂

Here the ∼ sign reﬂects an overall normalization constant. Integrating over the worldsheet

takes care of diﬀeomorphisms. But what about Weyl symmetries? The measure ∂

z has

weight (-1, -1) under rescaling.

From here, we can construct a Vertex operator which includes the Noether’s current that

arises from Global symmetries, represented by equation(3).

V ∼

∂

∂X

∂˜X

O(c, c˜) (3)

In Equation(3) , the measure ∂

z has weight (-1, -1) under rescaling, J

is the conserved

current, X are a bunch of scalar ﬁelds , represented on the worldsheet, O is the operator.

∂X

and ∂˜X

gives us two diﬀerent excitations and if we check using Weyl invariance, we

will get the ﬁrst excited states.[3]

And this operator corresponds to a massless gauge boson. A

µν

in spacetime.

It’s a BRST-closed operator Q

BRST

= |ψ⟩ = 0 , proving that the ﬁrst excited states of the

string are massless, thus part of the physical spectrum of the string. Shown below.

As mentioned, ∂X

and ∂˜X

gives us two diﬀerent excitations, α

−1

and α˜

. If we

consider an open bosonic string , the ﬁrst excited state which is level 1 gives us,

|ψ⟩ = Λ

−1

|k⟩

where k

is the momentum. Applying closed BRST condition on the physical state,

BRST

|ψ⟩ = 0 ⇒ k

= 0

= 0 proves that the ﬁrst excited states of the string are indeed massless.

The gauge symmetry appears from redundancies in the deﬁnition of the vertex operator.

(From gauge transformation A

(x) → A

′

(x) = A

(x) + ∂

Λ(x) , we get the gauge

symmetry, δA

= ∂

Λ)

Hence, we can conclude that Every global symmetry current J

µν

on the worldsheet leads to

a gauge boson in spacetime → symmetry is gauged, not global.

4 Continuous Global Symmetry Breaking

The question here is - why do continuous global symmetries break ?

There are multiple reasons why. Perturbative approaches fail when it comes to QCD. In

the Hadronization phase of QCD where the coupling constant is strong, perturbative

approaches do not work. At the critical point (, around 150-170 MeV , boundary between

the conﬁned and the unconﬁned phases) , we move away from conﬁnement. The chiral

symmetry breaks as conﬁnement breaks the hadron composition.

When a system’s ground state does not match the underlying symmetry that the theory

possesses, global spontaneous symmetry is broken. A great example is the Higgs

mechanism , where a U(1) symmetry is spontaneously broken, giving mass to the W and Z

bosons. Whenever a continuous global symmetry is spontaneously broken, there will be

massless scalar ﬁelds. These ﬁelds are called Goldstone bosons, which we are going to get

into in more detail. But ﬁrst, let’s look at Symmetries that are available to us in classical

as well as Quantum Field theories.

4.1 Symmetries in Classical Field Theories

Continuous symmetries are connected to conservation laws by Noether’s theorem as we

have mentioned previously. Consider an action where S =

xLϕ , involving ﬁelds.

Assuming that there’s some inﬁnitesmal transformation on the ﬁeld ϕ, we are going to get

ϕ → ϕ + ϵ

δϕ

, where a can be considered as a set of global parametres.[7]

Under the set of global parametres, the action remains invariant. We need the derivative of

the global parametres which will help us recover the invariance under global transformation

in case of constant parametres. To recover the equations of motion, we need the derivative

of the action given by:-

S =

∂

(x) = 0

where the derivative of the global parametres is give by, ∂

(x) . now, under the equation

of motion, the continuity equation under Noether’s current remains conserved and this is

denoted by, ∂

= 0. The Noether charge denoted by Q

remains constant under time

evolution , it is give by Q

D−1

4.2 Symmetries in QFTs

As suggested by Wigner’s theorem, the symmetry of operations of a quantum system is

induced by an unitary or an anti-unitary transformation. Therefore, for continuous

symmetries , the unitary transformation can be constructed by Noether’s charge. Given by

, U = e

iϵ

and as mentioned previously, they act on the ﬁelds as :- ϕ → ϕ´= UϕU

†

. This

explicit form of U is written in terms of creation and annihilation operators. The

correlation function can be deﬁned by the path integral, and the path integral can be

written as:-

Z =

Dϕϵ

We can ﬁnd out the ward identities which are the quantum counterpart of the classical

laws. The correlation function can be deﬁned as :- ⟨X⟩ =

DϕXe

, where X is the

product of the ﬁelds and is denoted by :- X =

ϕ(x

). The ward identities of the

quantum counterpart of Noether’s current J

will be ⟨J

(x)X⟩ .

We can also consider a set of relations among the correlation functions and conservation

laws. It can be denoted as:- ∂

⟨J

(x)X⟩ and the ward identities would be:-

∂

⟨J

(x)X⟩ = −i

(

D)(x − x

)⟨ϕ(x

).....δϕ

)....ϕ(x

)⟩ (4)

This will be useful in understanding divergences a little bit later.

Upon integrating both sides over spacetime from equation (4) we will obtain

δ⟨ϕ(x

).....(x

)⟩ = 0. This is the reﬂection of the symmetry on the correlation function and

it is also a non-trivial statement (undergoes transformation) about symmetry as it is

computed as:-

Dϕδ[ϕ(x

).......ϕ(x

)]e

= 0

As we can see this is time ordered, so there will be equal time commutators for ϵ which

tends to 0.

When continuous global symmetries are spontaneously broken, Noether’s current even

though it is conserved (∂

= 0)under symmetry transformations, it starts to exhibit

divergences in the set of relations . This divergence, leads to massless Goldstone bosons ,

which are excitations.

4.3 The Non-triviality of Noether’s current in symmetry breaking

Noether’s current even though it is a physical observable , produces divergences , because

the symmetry is still present in the underlying Lagrangian whether or not the vacuum

state is invariant. It produces divergences in the set of relations (in a distribution sense)

which gives us the ward identities.

For an unbroken symmetry, Noether’s current is conserved . Now, suppose this symmetry

is broken but vacuum |0⟩ is not invariant under Noether’s charge, we will have :-

Q|0⟩ =

(x)|0⟩ = 0

Q for a symmetry group G. The set of relations for the vacuum state is going to be:-

⟨0|∂

(x)|0⟩ .

Considering a scalar ﬁeld ϕ at vacuum , this will be :- ⟨0|∂

(x)ϕ(0)|0⟩ .

From the previous equation(), this will lead to the vacuum term,

iδ(x

))[∂

, ϕ(0)] + constant

Now for a broken symmetry, the vacuum state does not remain invariant, so we can say

that the commutator will be non zero, [∂

, ϕ(0)] = 0 =⇒ ⟨0|∂

(x)ϕ(0)|0⟩ = 0

If the commutator is non-zero, the current J

is non-trivial. This is how divergences

appear in the set of relations even if current is conserved.

4.3.1 The rise of massless poles

Let’s assume that our current term which is J

, associated with a broken symmetry

appears between two arbitrary states α and β. With respect to an Unitary transformation,

this can be represented as[8]:-

⟨β|J

(x)|α⟩ = e

iqx

⟨β|J

(0)|α⟩

where the diﬀerence between the four momentum for the Goldstone bosons and the four

momentum for the arbitrary states is denoted by:- q

≡ p

− p

Since, we know that the current term is non-trivial (i.e, it has non zero matrix element), it

has a pole at q

= 0. This creates a goldstone boson. The more elaborate proof is given in.

The S-matrix element will emit a Goldstone boson of state |β, q⟩ of four momentum q in

the transition between α and β. Roughly speaking, for each broken symmetry, there will be

atleast one Goldstone Boson.

Let’s take a look at non-triviality from a mathematical perspective.

5 Symmetry Breaking: A mathematical pro of

So far, we have seen that global symmetries can be spontaneously broken and we have done

the rigourous mathematical proof behind how massless Gauge bosons arise as a

consequence of symmetry breaking from SU(2) → U(1). We know that from the property

of Global symmetries, Ward identities emerge and from there a path towards symmetry

breaking is formulated.

Now, looking at our problem through the lens of non-trivial holonomy and therefore

breaking the resultant gauge symmetry , this approach provides us with a deep insight into

Wilson loops, boundary conditions as well as compactiﬁcations.

But ﬁrst a few fundamental deﬁnitions are in order:-

- Fibre-bundle:- A general structure on a space projected onto another space where each

”ﬁbre” represent a set of points which can be considered equivalent in some way.

- Priniciple-bundle:- A speciﬁc kind of bundle where the ﬁbres are copies of lie groups.

- Holomorphic function:- A Holomorphic function can be deﬁned as a complex valued

function of one or more complex variable which is diﬀerentiable to the neighbourhood of

each point in a complex space C

. (include diagram).

- Homotopy :- A continuous deformation of one function or geometric object into another.

- Holonomy :- Holonomy is a geometric consequence of the curvature of a connection( a

structure which is deﬁned on a ﬁbre bundle) . The holonomy of a connection on a smooth

manifold is the extent to which a parallel transport along a closed loop fails to preserve the

geometric data which is being transported. (give diagram)

- Wilson Line :- Consider a manifold M , take a point on the manifold x

and then project

into an identity g

on a p = 2 , (p + 1 dimensional brane ) and see how it changes to g

which are points on a horiontal subspace , considered after projecting . The change

happens on a spacetime curve ˜γ . where, γ : [0, 1] → M between x

and x

.[9] The curve

γ(0) = g

and its tangent vectors lie on the horiontal subspace. The ﬁbre bundle represents

a gauge theory between two horiontal subspaces H

and H

. This gauge theory can be

represented by : - A

(x) = A

(x). Therefore, the Wilson line can be given by: -

) = W[x

, x

] = Pexp(i

∂x

)

where P is the path -ordering operator. W is gauge invariant under local gauge

transformations. The trace of a closed Wilson line is called a Wilson loop. Wilson lines are

used to describe particles(charges) along gauge groups.

Consider a principle G bundle , given by: - P(M, G) , which is connected over a manifold

M with a non-trivial Holonomy representation is denoted by :- ρ = π

(M) → G.

This representation assigns to each Holonomy class of loops on element G. So, in order to

describe a Holonomy representation associated with the loop γ as mentioned before, we can

deﬁne the representation as:-

P([γ]) = Pexp(i

where [γ] denotes the Homotopy class of loop γ.

An example of a Holonomy group

We have to remember that a loop γ , deﬁnes the transformation on the ﬁbre.

Let’s consider an R bundle over M where R

= M. The connection one-form , which is

given by ω and the loop [γ] deﬁne a map ρ : π

(M) → G. Take a point h , which belongs

on the group G. and consider a set of loops π

(M) which commutes with ρ(π

(M)) , such

that π

(M) ≡ γ[0, 1] → M [9]

If we wish to deﬁne a subgroup that reduces to H from G, we will deﬁne the subgroup as:-

H = {h ∈ G|hρ([γ]) = ρ([γ])h∀[γ] ∈ π

(M)}

Now, the holonomy imposes the condition , for a section along the associated bundle where

the section corresponds to a G equivariant function for g ∈ G. Since ρ(γ) is ﬁxed, only the

elements h ∈ G commuting with ρ(γ) preserve the section. Therefore, the structure group

reduces to H, which is what we are looking for when we talk about group structure

decomposition in symmetry breaking. [check Appendix B]

Considering a Gauge group with G = SU(2), on a spacetime manifold represented by

M = S

∗ R

. S

represents a compact spatial structure which can arise from

compactiﬁcation of the extra dimensions.[10]

SU (2) is motivated by the electroweak sector and U can be given by U = exp(iQσ

) where

are the generators of SU(2) and Q is a constant angle parametrerization of the

Holonomy, which represents the non-trivial ﬁeld conﬁguration that exists due to the

topology of S

. We are now going to prove how to perform a reduction of the gauge

symmetry from SU(2) → U(1).

5.1 Gauge Symmetry Reduction

The subgroup, as we know H ⊂ SU(2), which commutes with U. Now since U is generated

by the form exp(iQσ

) , we can say that U(1) subgroup of SU(2) is generated by σ

and as

a result the group SU(2) reduces to U(1). [The Holonomy group reduces to a subgroup H ,

then only gauge transformations on H acts non-trivially]

The Gauge ﬁeld A

can be decomposed into broken and unbroken generators and can be

given by:- g = h ⊕ m where g is the Lie algebra decomposition.

h is the unbroken generators and m is the broken generator. We have to prove that the

massless A

can be decomposed into a combination of both broken and unbroken

generators. Let’s see how this can be accomplished.

Aim:- From the Boundary conditions, we are going to derive both broken and unbroken

generators. The goal is to compactify the higher dimensions and by ﬁxing a gauge , we can

come up with the unbroken generators and the non-trivial holonomy which cannot be

gauged away produces the broken generators and eventually the mass term will emerge

under mode expansion of the constant gauge. Only the subgroup which can commute with

the non-trivial holonomy remains unbroken.

5.1.1 Boundary Condition

As a continuation from the Holonomy group, let’s assume that we are compactifying one

spatial direction , let’s say x ∼ x + L , thereby making spacetime topologically

(

d − 1) ∗ S

The holonomy along S

can be described by the Wilson loop Ω = Pexp(i

∂x) ∈ C.

or along the circumference we can write, Ω = Pexp(i

∂x) ∈ G. Now, a ﬁeld ϕ(x)

which is charged under the gauge group G must satisfy:- ϕ(x + L) = Ωϕ(x). This is a

twisted boundary condition, where the twist is governed by the Holonomy. Now, if the

Holonomy is trivial i.e, Ω = 1, the ﬁeld can be said as periodic.

We are interested in SU(2) decomposing into U(1) through compactiﬁcation on a circle S

with a non-trivial Holonomy. Now, since compactiﬁcation is introduced, we are interested

in compactifying a higher dimensional gauge theory which is a very powerful way to realize

Spontaneous Symmetry breaking via Holonomy.

5.1.2 Why Hi gher Dimensions ?

The reason being , we can obtain Eﬀective 4D theories because gauge ﬁelds are vector

ﬁelds , A

with no extra scalar-like components unless we introduce them by hand (like the

Higgs ﬁeld). But in higher dimensions, gauge ﬁelds have extra components. For eg:- in 5D

[11], the gauge ﬁeld can be written as A

, x

) with M = 0, 1, 2, 3, 5 .

behaves like a scalar ﬁeld from a 4D perspective. So, x

, when it is compactiﬁed on a

circle S

, it would be:-

Ω = Pexp(i

∂x

)

This holonomy cannot be gauged away globally due to the nontrivial topology of S

. It

leads to symmetry breaking: only the subgroup that commutes with Ω remains unbroken.

This holonomy cannot be gauged away globally due to the nontrivial topology of S

. It

leads to symmetry breaking: only the subgroup that commutes with Ω remains unbroken.

This method of breaking gauge symmetry spontaneously by a Wilson loop W is also known

as the Hosotani mechanism.

Note:- It’s actually the index for the ﬁfth coordinate, but in many Kaluza - Klein or extra-dimensional setups, the numbering

skips 4 to avoid confusion with the Lorentz index range , µ = 0, 1, 2, 3

5.1.3 The Set-Up

We are considering a Gauge theory on R

∗ S

. Let’s start by compactifying one spatial

dimension, let’s say x

∼ x

+ L, so the spacetime is R

∗S

within the Gauge group SU(2)

So, the Holonomy representation can be written as:-

Ω = Pexp(i

)∂x

) ∈ SU(2)

5.1.4 Adjoint on scalar S

The aim is to compactify A

(5D) → S

. Introduce a Wilson loop, choose a Wilson line

gauge which is a constant and derive the generators.

Let ϕ(x) ∈ g (adjoint scalar) . Under Gauge transformation, this will be:-

ϕ(x + L) = g(x + L)ϕ(x)g(x + L)

−1

If the gauge ﬁeld has a non-trivial Wilson line Ω, the periodicity [12] will be twisted as

mentioned in the previous section. This gives us :- ϕ(x + L) = Ωϕ(x)Ω

−1

. If we choose a

gauge where A

is constant, the Wilson loop(Holonomy) along the compact direction can

be written as:-

Ω(x

) = Pexp(i

, x

)∂x

) ∈ G

Therefore, the Gauge invariant using ϕ(x + L) = g(x + L)ϕ(x)g(x + L)

−1

will be

Ω(x

) → g(x

)Ω(x

−1

So, from here we can choose a gauge which is constant, which will give us both the broken

and the unbroken generators. From there, we can derive the mass terms.

We can choose A

which is a constant and the gauge can be written as:-

, x

) =

ϕ(x

)

, ϕ ∈ g ∈ SU(2)

Let’s choose the ﬁeld ϕ which will give us the generators and thereby, the Holonomy.

ϕ = ασ



1 0

0 −1



⇒ Ω = exp(iQ

)

where Q

is the charge.

If we consider the Wilson loop i.e, Ω(x

) = exp(iϕ(x

)),

- The non-local Wilson line operator becomes a local ﬁeld ϕ(x

)

- ϕ behaves like a scalar ﬁeld in 4 Dimensions.

So, we can say that the generator, σ

∈ SU(2) commutes with Ω , while σ

and σ

do not.

Thus,

- σ

generates an unbroken U(1)

- σ

and σ

, generate broken direction.

Therefore, SU(2) → U(1) . A

basically becomes a scalar. In Wilson line gauge, the entire

content of the gauge-invariant Wilson loop is encoded in a constant background value of A

. This value cannot be gauged away globally due to the nontrivial topology of S

and it

can break gauge symmetry.

5.1.5 The mass term

If we perform mode expansion(Fourier) on the Gauge A

, x

) , we are going to get:-

, x

) =

n∈Z

(

2πinx

/L)

The covariant derivative acting on charged modes is shifted by A

. The eﬀective mass

denoted by m

becomes -

2πn

where, m

2πn

is the eﬀective 4D mass ( From Kaluza–Klein derivation)[13].

So, the oﬀ-diagonal component becomes massive , while the diagonal component remains

unchanged.

- A

∼ σ

1,2

becomes massive while,

- A

∼ σ

remains unchanged.

which is consistent with what we had formulated previously, h = broken generators, m =

unbroken generators.

5.1.6 Field Decomposition

So, the ﬁnal ﬁeld decomposition from g = h ⊕ m is going to look like:- SU(2) = U(1) ⊕ m ,

where U(1) is generated by:- σ

and m is generated by :- σ

and σ

Therefore, from here we can conclude that : -

- Fields in h = U(1) - massless gauge bosons (photon-like).

- Fields in m:- massive bosons due to non-trivial holonomy, which cannot be gauged away.

This is structurally more or less the same in the Higgs mechanism, but geometrically we

are describing it using the boundary conditions via the Wilson line.

The Wilson line is going to be useful a little bit later.

5.2 Goldstone Excitations, Spontaneous symmetry breaking

Continuing from section (4.2) , we have the correlation function for the current J

(x)

which is ⟨J

(x)X⟩ from the Ward identity ⟨X⟩ . From equation (4) , if we picked a single

ﬁeld X, ϕ(y) then equation (4) would be reduced to:-

∂

⟨J

(x)ϕ(y)⟩ = −iδ

(x − y)⟨δ

ϕ(y)⟩

we can take the fourier transform with respect to x, and it will give us:-

ipx

∂

⟨J

(x)ϕ(y)⟩ = −i

ipx

(x − y)⟨δ

ϕ(y)⟩

−i

ipx

⟨J

(x)ϕ(y)⟩ = −ie

ipy

⟨δ

ϕ(y)⟩

⟨J

(x)ϕ(y)⟩ = e

ipy

⟨δ

ϕ(y)⟩



(x)e

−ipy

ϕ(y)



= ⟨δ

ϕ(y)⟩

where, the Fourier transform can be identiﬁed as: - J

(p) =

ipx

(x). Then, we can

integrate both sides with respect to y and get ,

⟨J

(p)ϕ(−p)⟩ =

y⟨δ

ϕ(y)⟩ = ⟨δ

ϕ(p = 0)⟩

The term , ⟨δ

ϕ(p = 0)⟩ characterizes the phases and when ⟨δ

ϕ(p = 0)⟩ = 0, it indicates

spontaneous symmetry breaking. When the phase is broken, the correlation function must

have a massless pole at zero-momentum ⟨J

(p)ϕ(−p)⟩ ∼

This indicates the presence of massless excitations and these excitations can be termed as

the Goldstone Bosons which is consistent with what we had found in section(4.3.1)

6 Conclusion

A string theory, QFT and a topological route , all of these have led to the existence of

Goldstone Bosons. We ﬁgure out how massless poles arise as a consequence of symmetry

being either gauged/broken. A broader direction has been considered for this paper and

many other approaches can also be considered as we try and investigate how global

symmetries work at low energies as well as at the Planck scale. At ordinary as well as

higher form symmetries, topological operators start behaving like symmetry operators and

using such operators , we can perform analysis on the behavior of symmetries and also

come up with a formulation on how Eﬀective Field Theories (EFTs) behave when coupled

with gravity. These are interesting and emergent areas of theories which will be discussed

as we explore Global Symmetry behavior in Quantum Gravity.

7 Appendix A

One can consider a compact Gauge Group G. The corresponding representation of the

matrices of G can be denoted T

and the structure constants of G in the basis can be

written as:- f

, T

] = f

This is the Yang Mills type. The ﬁeld strengths are the covariant derivative of the matter

ﬁelds , which are denoted by:- f

µ,ν

and D

where ψ

is the matter ﬁeld, which can be

fermionic or bosonic.

In BRST formalism, the gauge parameters are replaced by anticommutating ﬁelds which

are called the ”ghost ﬁelds”. BRST cohomologies capture important physical information

about the system. The reason for BRST symmetries and cohomologies being important is

to become a substitute for gauge redundancies , which would otherwise become obscure.

[14]

8 Appendix B

In general when we are given a subgroup G, one choice that we can make is to take the

manifold to be the group itself. G = M. In this case, each element , g ∈ G gives us the

natural map M → M given by g´∈ M → g.g´.

Another choice is to take a subgroup(coset space). This is the manifold M =

. where

H ⊂ G is a group of G , which is exactly what we were trying to prove in section (5)

A point g in the coset

is deﬁned by the equivalence relation among the elements of G.

g ≡ g.h. for all h ∈ H. Again, any element of g ∈ G gives us a natural map M →

→

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