conformal symmetry | master's dissertation · conformal symmetry do not posses intrinsic...

CONFORMAL SYMMETRY

Dissertation Reportsubmitted in partial fulfillment of the requirements for the degree of

Master of Science

in

Physics

Session: 2013 – 2014

Under the Supervision of Submitted by

Dr. Sanjay Siwach Saurav Dwivedi

Nuclear Physics Section

Department of Physics

Banaras Hindu University

Varanasi – 221005, INDIAmwww.geocities.ws/dwivedi/data/mth.pdf

Enrollment No. 300411 Roll No. 12481SC078

http://www.geocities.ws/dwivedi/data/mth.pdf

Certificate

This is to certify that the dissertation report entitled “Conformal Symmetry”,

submitted by Saurav Dwivedi for the award of the degree of M.Sc. in Physics

is based on original studies carried out by him under my supervision. The dis-

sertation or any part thereof has not been previously submitted for any other

degree or diploma.

Date: 15/05/2014

Place: B.H.U., Varanasi, INDIA

Dr. Sanjay Siwach

Assistant Professor

Nuclear Physics Section

Department of Physics

Banaras Hindu University

Acknowledgments

I am grateful to all my teachers, friends, and technical staff for their kind sup-

port, and encouragement.

Lists of Abbreviations

A,B, C, . . . Arbitrary sets.

a, b, c, . . . Elements of sets.

F, #, . . . Binary operations on sets.

eF, e#, . . . Identity elements corresponding to binary

operationsF, #, . . . .

A, B, C, . . . Algebraic objects, such as groups, rings, modules,. . . .

A,B,C, . . . Lie algebras.

G(A;F; eF) A Group, with underlying set A , binary operationF

on it, and identity eF .

R(A;F, #; eF, e#) A Ring, with underlying set A , binary operationsF, #

on it, and identities eF, e# .

C Set of conformal generators.

Con(C) Conformal algebra.

LW Set of Witt generators.

Wit(LW ) Witt algebra.

LV Set of Virasoro generators.

Vir(LV ) Virasoro algebra.

GS Set of fermionic generators.

SVAN=1(LV ,GS) N = 1 super Virasoro algebra.

SVAN=2(LV ,J ,G±S ) N = 2 super Virasoro algebra.

Contents

1 Introduction 1

2 Conformal Symmetry 4

2.1 Conformal Symmetry in d Dimensions . . . . . . . . . . . . . . . . 5

2.1.1 Infinitesimal Conformal Transformations in d Dimensions 6

2.1.2 Conformal Algebra in d Dimensions . . . . . . . . . . . . . 7

2.2 Conformal Symmetry in d = 2 Dimensions . . . . . . . . . . . . . 9

2.2.1 Witt Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Virasoro Algebra . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Primary Field . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Infinitesimal Conformal Transformations of Primary Field 13

2.2.5 Highest Weight Representations (HWR) of Virasoro Algebra 14

2.2.6 Kac Determinant . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Superconformal Symmetry 17

3.1 Suppersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

v

CONTENTS vi

3.3 N = 1 Super Virasoro Algebra . . . . . . . . . . . . . . . . . . . . 21

3.4 HWR of N = 1 Super Virasoro Algebra . . . . . . . . . . . . . . . 21

3.5 N = 2 Super Virasoro Algebra . . . . . . . . . . . . . . . . . . . . 22

3.6 HWR of N = 2 Super Virasoro Algebra . . . . . . . . . . . . . . . 23

4 Summary 25

Chapter1Introduction

There are two prevalent regimes in the quest of mathematization of physics,

“Symmetries” [35] and “Deformation Quantization” [4]. Former entails dynamical

invariance upon certain transformations, later deals with transmutation of theo-

ries upon deformation of their algebras. This report explores the former one,

and summarizes (Lie) algebraic aspects of conformal symmetry in d–dimensions,

and restricts it to 2–dimensions. Two-dimensional conformal symmetry entails

infinite-dimensional Lie algebras. Further, I introduce suppersymmetry, and sum-

marize Lie algebras entailing superconformal symmetry in 2–dimensions, for

minimal (N = 1) and extended (N = 2) supersymmetry, which are infinite-

dimensional as well.

Symmetry is invariance under certain structure preserving transformations1.

Simplectic systems are more symmetric than complexes. Symmetry reduces the

order of hierarchy, and thus structural organization of the theory (or model).

In mathematical argot, symmetries correspond to transformations which are

1Here, the term structure is entailed in somewhat generic sense. It could be an algebraic,

topological or a physical structure/entity, e.g. group, connectedness, mass or charge.

1

2

infinitesimal (continuous) or finite (discrete and continuous); finite transforma-

tions are also constituted from infinitesimal ones, which lead to regular sym-

metries.

Let f be a dynamical variable parametrized by x . An infinitesimal transfor-

mation x 7→ x + ε has Taylor expansion

f (x) 7−→ f (x + ε) = f (x) + ε∂ f (x)

∂x+O(ε2) . (1.1)

It is customary to write it in operator language for unitary representations (es-

pecially for quantum theories) as,

f (x) 7−→ f (x + ε) = (1 + iGε) f (x) , (1.2)

where

Gε = −iε∂

∂x(1.3)

is termed generator of the infinitesimal transformation x 7→ x + ε . For a se-

quence of N number of infinitesimal transformations x 7→ x + ε , we get

f (x) 7−→ f (x + Nε) =

(1 + ε

∂

∂x

)Nf (x) . (1.4)

In the limit N → ∞ , it leads to finite transformation x 7→ x + ∆x , with ∆x =

limN→∞

Nε , and

f (x) 7−→ f (x + ∆x) =(

1 + ∆x∂

∂x

)f (x) , (1.5)

with generator of finite transformation

G = −i∆x∂

∂x. (1.6)

3

Definition 1.0.1 (Continuous Symmetry). If the dynamics of any physical system

remains invariant under infinitesimal transformation of the form (1.1), or finite

transformations which are constituted from infinitesimal ones, then the system

entails continuous symmetry, and the underlying transformation is termed con-

tinuous symmetry transformation.

Examples of continuous transformations are rotation, translation, time evolu-

tion, scale transformations, gauge transformations etc.

Definition 1.0.2 (Discrete Symmetry). If the dynamics of any physical system

remains invariant under finite transformations, which are not constituted from

infinitesimal ones of the form (1.1), the system entails discrete symmetry, and the

underlying transformation is termed discrete symmetry transformation.

Examples of discrete transformations are parity transformation, charge conju-

gation, time reversal, etc.

Continuous symmetry transformations generate a class of algebras studied

by Marius Sophus Lie, called Lie algebras.

Chapter2Conformal Symmetry

Conformal symmetry is algebro-geometric symmetry. Certain transformations on

manifolds leave the geometry locally intact, termed conformal transformations.

Definition 2.0.3 (Conformality). Certain deformations of an arbitrary manifold

lead to deformed manifolds, which are locally equivalent to former’s dilatation.

Such resemblances between manifolds is termed conformality.

Definition 2.0.4 (Conformal Transformation). Transformations on a manifold,

leading to conformally equivalent manifolds, are termed conformal transfor-

mations.

Conformally equivalent manifolds are locally equivalent-upto-scaling. Ac-

tions of string models entail conformal symmetry [26, 27]. Systems entailing

conformal symmetry do not posses intrinsic length, mass, or energy scales, mak-

ing physics scale invariant. However, I restrict my approach to algebraic aspects

of conformal symmetry, rather than field theoretic.

4

2.1. CONFORMAL SYMMETRY IN D DIMENSIONS 5

2.1 Conformal Symmetry in d Dimensions

Let C be the set of transformations on a d-dimensional manifold Md , with gen-

eral metric gµν(x) (µ, ν = 0, 1, 2, . . . , d− 1) .

Definition 2.1.1 (Conformal Symmetry). Let f ∈ C . The transformation f :

Md −→M′d , leading to gµν(x) 7→ Ω(x)gµν(x) is termed conformal transforma-

tion. Manifolds Md and M′d are locally equivalent up-to-scaling, where Ω(x) is

termed scale factor. The symmetry entailed by conformal transformations C is

termed conformal symmetry.

In the argot of differential geometry, gρσ(x) transforms under x 7→ x′ , as

g′ρσ(x′) =∂xµ

∂x′ρ∂xν

∂x′σgµν(x) . (2.1)

The condition for x 7→ x′ to be conformal (i.e., local dilatation), [13, 6]

g′µν(x′) = Ω(x)gµν(x) , (2.2)

leads to,

g′ρσ(x′)∂x′ρ

∂xµ

∂x′σ

∂xν= Λ(x)g′µν(x′) , (2.3)

which is general condition for conformal symmetry, where Λ(x) = Ω−1(x) . For

flat manifolds with Lorentz metric ηµν = diag(−1, . . . ,+1, . . . ) ,

ηρσ∂x′ρ

∂xµ

∂x′σ

∂xν= Λ(x)ηµν (2.4)

is the condition for conformal symmetry on flat manifolds.


2.1.1 Infinitesimal Conformal Transformations in d Dimensions

Let Cε be set of infinitesimal conformal transformations on flat manifold Md ,

such that

x′ρ = xρ + ερ(x) +O(ε2) , (2.5)

where ε(x) 1 . Imposing condition for conformal symmetry (2.4), we get [6]

ηρσ∂x′ρ

∂xµ

∂x′σ

∂xν= ηµν +

∂εµ

∂xν+

∂εν

∂xµ +O(ε2) = Λ(x)ηµν . (2.6)

By setting

∂µεν + ∂νεµ = K(x)ηµν , (2.7)

where ∂µ := ∂∂xµ , we get

Λ(x) = 1 + K(x) +O(ε2) . (2.8)

Further, by tracing (2.7), we obtain

ηµν(∂µεν + ∂νεµ) = K(x)ηµνηµν , or (2.9)

K(x) =2d(∂ · ε) , (2.10)

where ∂ · ε = ∂µεµ . Thus, the condition for infinitesimal conformal symmetry

in d-dimensions is

∂µεν + ∂νεµ =2d(∂ · ε)ηµν , (2.11)

with scale factor Λ(x) = 1 + 2d (∂ · ε) +O(ε

2) .

We operate (2.11) by ∂ν , and sum it over ν , getting

∂ν(∂µεν + ∂νεµ

)=

2d

∂ν (∂ · ε) ηµν ,

∂µ(∂ · ε) +εµ =2d

∂µ (∂ · ε) ,(2.12)


where, = ∂µ∂µ . Operating it by ∂ν , we get

∂µ∂ν (∂ · ε) +∂νεµ =2d

∂µ∂ν (∂ · ε) . (2.13)

By permuting µ, ν , and adding resulting equation to previous one, we get

2∂µ∂ν (∂ · ε) +(∂µεν + ∂νεµ

)=

4d

∂µ∂ν (∂ · ε) . (2.14)

Contracting it with ηµν, using (2.11), we get

(d− 1) (∂ · ε) = 0 . (2.15)

Furthermore, we operate (2.11) by ∂ρ , getting

∂ρ∂µεν + ∂ρ∂νεµ =2d

ηµν∂ρ(∂ · ε) . (2.16)

Permuting ρ, µ, ν , we obtain

∂ν∂ρεµ + ∂µ∂ρεν =2d

ηρµ∂ν(∂ · ε) ,

∂µ∂νερ + ∂ν∂µερ =2d

ηνρ∂µ(∂ · ε) .(2.17)

Adding these, and subtracting from (2.16), we get

2∂µ∂νερ =2d(−ηµν∂ρ + ηρµ∂ν + ηνρ∂µ

)(∂ · ε) . (2.18)

2.1.2 Conformal Algebra in d Dimensions

The conformality condition (2.15) pertains that (∂ · ε) is at most linear in xµ ,

(∂ · ε) = A + Bµxµ , (2.19)

with arbitrary infinitesimal constants A, Bµ . (2.19) pertains that εµ is at most

quadratic in xµ ,

εµ = aµ + bµνxν + cµνρxνxρ , (2.20)


where, aµ, bµν, cµνρ are infinitesimal constants, and cµνρ = cµρν. Thus,

xµ 7−→ xµ + aµ + bµνxν + cµνρxνxρ , (2.21)

is the infinitesimal conformal transformation in d-dimensions.

(2.21) is interpreted term-by-term using (1.3):

1. The first contribution is infinitesimal translation xµ 7→ xµ + aµ , generated

by momentum operator Pµ = −i∂µ .

2. The linear term has symmetric and anti-symmetric contributions; bµν is

decomposed into symmetric and anti-symmetric parts,

bµν = αηµν + mµν ,

with infinitesimal α, mµν , and mµν = −mνµ . The symmetric contribu-

tion is infinitesimal dilatation xµ 7→ (1 + α)xµ , generated by dilatation

operator D = −ixµ∂µ . The anti-symmetric contribution is infinitesimal

transformation xµ 7→ (δµν + mµ

ν )xν , generated by Lorentz generator Lµν =

i(xµ∂ν − xν∂µ) .

3. The quadratic term is calculated using (2.20) and (2.18), which leads to

xµ 7−→ xµ + 2(x · b)xµ − (x · x)bµ , (2.22)

termed infinitesimal special conformal transformation (SCT), generated by SCT

operator Kµ = −i(2xµxν∂ν − (x · x)∂µ

).

The generators of conformal transformation (2.21) satisfy commutation rela-

2.2. CONFORMAL SYMMETRY IN D = 2 DIMENSIONS 9

tions

[D, Pµ] = iPµ ,

[D, Kµ] = −iKµ ,

[Kµ, Pν] = 2i(ηµνD + Lµν) ,

[Kρ, Lµν] = i(ηρµKν − ηρνKµ) ,

[Pρ, Lµν] = i(ηρµPν − ηρνPµ) ,

[Lµν, Lρσ] = i(ηνρLµσ + ηµσLνρ − ηµρLνσ − ηνσLµρ) ,

(2.23)

which constitute the Lie algebra termed conformal algebra Con(C) , where C =

Pµ, D, Lµν, Kµ is set of generators of conformal transformations in d-dimensions.

2.2 Conformal Symmetry in d = 2 Dimensions

We consider 2-dimensional euclidean space with metric ηµν = diag(+1,+1), (µ, ν =

0, 1) with following setting,

z = σ0 + iσ1, ε = ε0 + iε1, ∂z =12(∂0 − i∂1) ,

z = σ0 − iσ1, ε = ε0 − iε1, ∂z =12(∂0 + i∂1) .

(2.24)

The condition for infinitesimal conformal symmetry (2.11) implies

∂0ε0 = +∂1ε1, ∂0ε1 = −∂1ε0 , (2.25)

which are Cauchy-Riemann equations, that ascertain ε(z) to be analytic, regular

or holomorphic. Holomorphicity of ε(z) also suffices infinitesimal transforma-

tions z 7→ z + ε(z) to be conformal.

Let f , f ∈ Cε be infinitesimal conformal transformations

f : z 7−→ f (z) = z + ε(z) ,

f : z 7−→ f (z) = z + ε(z) .(2.26)


Conformality condition (2.25) implies that Wirtinger derivatives are zero,

∂ f∂z

= 0 ,∂ f∂z

= 0 , (2.27)

which ascertain that f and f are holomorphic; f is functionally independent of z ,

and f is independent of z . Here, f (z) may not be confused with f (z) ; f (z) and

f (z) are holomorphic, while f (z) and f (z) are anti-holomorphic.

Conformal symmetry entails autonomy of z and z , which pertains that con-

formal transformations (2.26) act on C2 rather than C .

Lemma 2.2.1. When a 2-dimensional manifold C admits conformal symmetry, z

and z are treated independently, and symmetry transformations (2.26) act on C2 .

Under conformal transformations (2.26),

ds2 = dzdz 7−→ ds′2 =∂ f∂z

∂ f∂z

dzdz , (2.28)

with scale factor

Λ(z) = 1 + ∂0ε0 + ∂1ε1 =∂ f∂z

∂ f∂z

=

∣∣∣∣∂ f∂z

∣∣∣∣2 . (2.29)

2.2.1 Witt Algebra

We have two types of infinitesimal conformal transformations on C2 , f and f

that require ε(z) and ε(z) to be holomorphic in some open sets. However, it is

inevitable that ε(z) and ε(z) may be meromorphic, having isolated singularities

beyond these open sets. More general infinitesimal conformal transformations

in 2-dimensions are give by Laurent expansions of ε(z) and ε(z) about z = 0

and z = 0 respectively,

ε(z) = ∑n∈Z

εn

(−zn+1

),

ε(z) = ∑n∈Z

εn

(−zn+1

),

(2.30)


where εn and εn are infinitesimal constants. The infinitesimal conformal trans-

formations (2.26) read

f : z 7−→ f (z) = z + ∑n∈Z

εn

(−zn+1

),

f : z 7−→ f (z) = z + ∑n∈Z

εn

(−zn+1

),

(2.31)

with Witt generators (in the sense of (1.3)) ln, ln ∈ LW ,

ln = −zn+1∂z , and ln = −zn+1∂z , ∀ n ∈ Z . (2.32)

However, number of Witt generators is infinite, which is contrary to conformal

symmetry in higher dimensions. The Witt generators (2.32) satisfy commuta-

tion relations,

[lm, ln] = zm+1∂z

(zn+1∂z

)− zn+1∂z

(zm+1∂z

)= (n + 1)zm+n+1∂z − (m + 1)zm+n+1∂z

= −(m− n)zm+n+1∂z

= (m− n)lm+n ,

[lm, ln] = (m− n)lm+n ,

[lm, ln] = 0 , ∀ m, n ∈ Z .

(2.33)

which form copies of infinite dimensional Lie algebras [6] termed Witt alge-

bra Wit(LW ) . The first two Lie brackets form Witt algebras, and third entails

autonomy of lm and ln .

2.2.2 Virasoro Algebra

The Witt Algebras (2.33) entail central extension. Centrally extended generators

Ln, Ln ∈ LV of infinitesimal conformal transformations (2.31), termed Virasoro


generators, satisfy commutations relations

[Lm, Ln] = (m− n)Lm+n + cp(m, n) ,

[Lm, Ln] = (m− n)Lm+n + c p(m, n) ,

[Lm, c] = 0 ,

[Lm, c] = 0 ,

[Lm, c] = 0 ,

[Lm, c] = 0 ,

[Lm, Ln] = 0 ,

(2.34)

∀ m, n ∈ Z . Here, p : LV × LV −→ C is bilinear, and c is termed central

charge or conformal charge. The anti-symmetry of Lie bracket implies p(m, n) =

−p(n, m) and p(m, n) = −p(n, m) .

For first of (2.34), p(m, n) is obtained to be

p(m, n) =112

(m3 −m)δm+n,0 , (2.35)

which leads to Virasoro Algebra Vir(LV)

[Lm, Ln] = (m− n)Lm+n +c

12(m3 −m)δm+n,0 . (2.36)

However, for m = (−1, 0,+1) , p → 0 , and Virasoro Algebra reduces to Witt

Algebra.

2.2.3 Primary Field

Let a field φ be parametrized by space (σ0, σ1) ∈ R2 in a 2-dimensional man-

ifold. We could parametrize the field φ by (z, z) ∈ C2 in the sense of Lemma

(2.2.1). This complexification process φ(σ0, σ1) 7→ φ(z, z) is R2 → C2 .


Definition 2.2.1 (Chiral Field). A holomorphic field φ(z) , depending on z alone

is termed chiral. Respectively, an anti-holomorphic field φ(z) , depending on z

alone is termed anti-chiral.

Definition 2.2.2 (Conformal Dimension). When a field φ(z, z) transform under

scaling z 7→ λz , as

φ(z, z) 7−→ φ′(z, z) = λhλhφ(λz, λz) , (2.37)

it has conformal dimension (h, h) .

Definition 2.2.3 (Primary Field). When a field φ(z, z) transforms under confor-

mal transformation z 7→ f (z) , as

φ(z, z) 7−→ φ′(z, z) =(

∂ f∂z

)h(

∂ f∂z

)h

φ( f (z), f (z)) , (2.38)

it is termed primary field of conformal dimension (h, h) .

For global conformal transformations of (2.38) type, φ(z, z) is termed quasi-

primary field. All primary fields are quasi-primary, but not conversely. Fields

which are neither primary, nor quasi-primary are termed secondary fields.

2.2.4 Infinitesimal Conformal Transformations of Primary Field

For infinitesimal conformal transformations (2.26), we have(∂ f∂z

)h= 1 + h∂zε(z) +O(ε2) ,(

∂ f∂z

)h

= 1 + h∂zε(z) +O(ε2) .

(2.39)


The primary field transforms as

φ(z + ε(z), z + ε(z))

= φ(z, z) + ε(z)∂zφ(z, z) + ε(z)∂zφ(z, z) +O(ε2, ε2) .(2.40)

The primary field transforms under infinitesimal conformal transformation as

φ(z, z) 7−→(

1 + h∂zε(z) + ε(z)∂z + h∂zε(z) + ε(z)∂z

)φ(z, z) . (2.41)

2.2.5 Highest Weight Representations (HWR) of Virasoro Alge-

bra

Analogous to su(2) , highest weight representations (HWR) of Virasoro algebra

Vir(LV) can be constructed. The state |h〉 corresponding to a primary field with

conformal dimension h , satisfying

Ln |h〉 = 0 , ∀ n > 0 ,

L0 |h〉 = h |h〉 ,(2.42)

is termed highest weight state. However, the action of Ln on |h〉 for n < 0 gener-

ates new states. All such states

HV = Ln |h〉 |∀ n < 0 , (2.43)

form Verma module VER(HV) .

Verma module VER(HV) includes states

L−1 |h〉 , L−2 |h〉 , L−1L−1 |h〉 , L−3 , . . . (2.44)

from top-down. The underlying set HV of Verma module corresponds to con-

formal family [φ(z)] of primary fields φ(z) (2.38) with conformal dimension h .


The combination (h, c) entails whether the states in Verma module have posi-

tive, vanishing, or negative norm. Whether there exist states of vanishing norm

in Verma module is entailed by Kac determinant.

2.2.6 Kac Determinant

The Kac determinant is calculated using the matrix MN(h, c) at level N , whose

entries are inner product of elements of Verma module

〈h|∏i

L+ki ∏j

L−mj |h〉 , ki, mj ≥ 0 . (2.45)

The condition

∑i

ki = ∑j

mj = N (2.46)

ascertains that the level is N . For states at different levels, i.e. ∑i

ki 6= ∑j

mj , the

matrix MN(h, c) vanishes.

We use Virasoro algebra (2.34) and (2.42) to obtain MN(h, c) for N = 1 and

N = 2 . For N = 1 , Kac determinant is

M1(h, c) = 〈h| L1L−1 |h〉 = 2 〈h| L0 |h〉 = 2h . (2.47)

Thus for for N = 1 , Kac determinant is

det M1(h, c) = 2h , (2.48)

which pertains that for h = 0 , level N = 1 has one null state.


For N = 2 , M2(h, c) has entries

〈h| L2L−2 |h〉 = 〈h|c2+ 4L0 |h〉 = 4h +

c2

,

〈h| L1L1L−2 |h〉 = 〈h| L1 · 3L−1 |h〉 = 6h ,

〈h| L2L−1L−1 |h〉 = 〈h| 3L1 · L−1 |h〉 = 6h ,

〈h| L1L1L−1L−1 |h〉 = 〈h| L1[L1, L−1]L−1 |h〉+ 〈h| L1L−1L1L−1 |h〉 ,

= 〈h| L1 · 2L0L−1 |h〉+ 〈h| [L1, L−1][L1, L−1] |h〉 ,

= 2 〈h| L1[L0, L−1] |h〉+ 4h2 + 4h2 ,

= 4h + 8h2 .

(2.49)

It pertains that for h = 0 , level N = 2 has three null states.

However, for N = 2 , Kac determinant is

det M2(h, c) = det

4h + c2 6h

6h 4h(2h + 1)

, (2.50)

= 32h(

h2 − 58 h + 1

8 hc + 116 c)

, (2.51)

with roots

h1,1 = 0, h1,2 , h2,1 =5− c

16∓ 1

16

√(1− c)(25− c) . (2.52)

The Kac determinant has form

det M2(h, c) = 32 (h− h1,1(c)) (h− h1,2(c)) (h− h2,1(c)) . (2.53)

Victor G. Kac generalized it for arbitrary level N . Kac determinant for level

N is

det MN(h, c) = αN ∏p,q≤N

(h− hp,q(c)

)P(N−pq) , (2.54)

where αN is a positive constant, p, q > 0 , and

hp,q(m) =((m + 1)p−mq)2 − 1

4m(m + 1), m = −1

2± 1

2

√25− c1− c

. (2.55)

Chapter3Superconformal Symmetry

3.1 Suppersymmetry

Suppersymmetry has its roots in work of Miyazawa (1966) [21]. Suppersym-

metry unifies matter with mediators, and statistics of bosonic and fermionic

sectors1.

Supersymmetry reduces bosonfermion and interactionmatter hier-

archies (diads), and simplifies them to susyon and supersystem (monads).

bosonfermion is compositional containment hierarchy; fermions constitute

bosons, but not conversely.

Bosonic and fermionic sectors of QFTs entail (ultraviolet) divergences. Sup-

persymmetry entails fermions and bosons on equal footing, that cancels these

infinities via pairing-off divergences of superpartners. Supersymmetry is a

semi-finitistic prescription [25]. A more radical finitistic prescription is met in

1Provisionally, 1024 bosons collectively behave as fermions in Palev statistics [24]; Palev sets

an upper bound (provisionally 1024) to the degeneracy of bosons from algebraic regularization

scheme. How do supersymmetric models entail palevons?

17

3.1. SUPPERSYMMETRY 18

string models, which entail supersymmetry as well as extra-dimensions. Nev-

ertheless, supersymmetry appears to be symptom of finitism for quantum field

theories, which remains sole motivation for making QFTs suppersymmetric.

The generators of suppersymmetry Qr ∈ QS , also termed supercharges, trans-

form bosons (systems with integral-spin 0, 1, 2, . . . ) into fermions (systems with

half-integral-spin 1/2, 3/2, 5/2, . . . ), and vice versa

Qr |boson〉 = | f ermion〉 , Qr | f ermion〉 = |boson〉 . (3.1)

Particles so related by (3.1) are termed superpartners. It is customary to suffix

fermionic superpartners of bosons with -ino, and prefix bosonic superpartners

of fermions with s-. It is anticipated that every fundamental particle found in

nature has a superpartner. Superpartner of electron would be spin-0 selectron,

and superpartner of photon would be spin-1/2 photino.

The generators concerned here are operators in Hilbert space, that annihilate

a particle, and create another one in the state space of the system-under-study,

leaving physics intact. The generator has a general form [29],

G = ∑ij

∫d3pd3q a†

i (p)Kij(q, p)aj(q) , (3.2)

which has symbolic form as convolutions,

G = a† ∗ K ∗ a . (3.3)

The operator G is termed generator of a symmetry if it commutes with the

S-matrix

[S, G] = 0 , (3.4)

which leaves the physics invariant under transformations of G on system-under-

study. The symmetry generator G is decomposed into bosonic (even) and fermionic

3.1. SUPPERSYMMETRY 19

(odd) parts

G = B + F . (3.5)

B transforms bosons-to-bosons and fermions-to-fermions, while F transforms

bosons-to-fermions and vice-versa. For bosonic and fermionic creators b†, f † and

annihilators b, f , B and F entail representations

B = b† ∗ Kbb ∗ b + f † ∗ K f f ∗ f ,

F = f † ∗ K f b ∗ b + b† ∗ Kb f ∗ f .(3.6)

Acting on a state |j〉 ,

B |j〉 = |j± n〉 , F |j〉 = |j± n/2〉 , ∀ n ∈ Z∗ . (3.7)

Thus fermionic generators F (3.6) are essentially supersymmetric ones (3.1).

Virasoro algebra (2.34) holds for both bosonic and fermionic sectors, which are

autonomous to each other. However, in the sense of (3.5), Virasoro generators

Ln decompose into bosonic and fermionic ones,

Ln := Lbos.n + L f erm.

n , (3.8)

where, Lbos.n and L f erm.

n separately satisfy Virasoro algebra (2.34), with central

charges c = 1 and c = 1/2 , while Ln (3.8) satisfies Virasoro algebra with central

charge c = 3/2 . The bosonic and fermionic Virasoro generators satisfy

[Lbos.m , L f erm.

n ] = 0 . (3.9)

In field-theoretic-regime [6], energy-momentum tensor T(z) has Laurent ex-

pansion

T(z) = ∑n∈Z

Lnz−n−2 , Ln =1

2πi

∮dz zn+1T(z) , (3.10)

3.2. SUPERSPACE 20

having fermionic superpartner G(z) , with Laurent expansion

G(z) = ∑r∈Z+ 1

2

Grz−r− 32 , Gr = ∑

s∈Z+ 12

jr−sψs , (3.11)

where, j(z) and ψ(z) are primary fields (2.38) of conformal dimensions h = 1 ,

and h = 1/2 , having Laurent expansions

j(z) = ∑n∈Z

jnz−n−1 ,

ψ(z) = ∑r

ψrz−r− 12 ,

(3.12)

However, G(z) (3.11) is a fermionic field, constructed from bosonic field j(z) and

fermionic field ψ(z) .

3.2 Superspace

The superspace is 2-dimensional vector space (analogous to C) with basis Z =

(z, Θ) , where z ∈ C , and Θ (analogous to i) is Grassmann variable satisfying

Θ, Θ = 0 , (3.13)

implying Θ2 = 0 . The derivative acting on a superfield Φ(Z) parametrized by

Z is defined as

D := ∂Θ + Θ∂z , (3.14)

with D2 = ∂z . The superfield Φ(Z) has bosonic and fermionic components φ(z)

and ψ(z) , with

Φ(Z) = φ(z) + Θψ(z) . (3.15)

3.3. N = 1 SUPER VIRASORO ALGEBRA 21

3.3 N = 1 Super Virasoro Algebra

Minimal supersymmetry (N = 1) entails one superpartner to each system (par-

ticle or field). The super Virasoro generators Ln, Lm ∈ LV , and Gr, Gs ∈ GS

satisfy commutation relations


12(m3 −m)δm+n,0 ,

[Lm, Gr] =(m

2− r)

Gm+r ,

Gr, Gs = 2Lr+s +c3

(r2 − 1

4

)δr+s,0 ,

(3.16)

which form infinite dimensional Lie algebra, termed N = 1 Super Virasoro Alge-

bra SVAN=1(LV ,GS) .

3.4 Highest Weight Representations ofN = 1 Super

Virasoro Algebra

The state |h〉 satisfying

Ln |h〉 = 0 , for n > 0 , (3.17)

Gr |h〉 = 0 , for r > 0 , (3.18)

is termed superconformal highest weight state. However, for 0 < c < 3/2 , unitary

highest weight representations of super Virasoro algebra entail discrete values

of central charge,

c =32

(1− 8

(m + 2)(m + 4)

), m ∈ Z . (3.19)

3.5. N = 2 SUPER VIRASORO ALGEBRA 22

3.5 N = 2 Super Virasoro Algebra

N = 2 supersymmetric models entail two superpartners to each system (particle

or field). We define complex free bosonic field

Φ(z, z) =1√2

(X(1)(z, z) + i X(2)(z, z)

). (3.20)

However, free boson X(z, z) has vanishing conformal dimension, and thus, is

not a conformal field. Instead, j(z) = i∂X(z, z) , or Vertex operator V(z, z) =:

eiαX(z,z) : is used as conformal free bosonic field [6]. Thus, we have complex free

bosonic and fermionic fields

j(z) =1√2

(j(1)(z) + i j(2)(z)

),

Ψ(z) =1√2

(Ψ(1)(z) + i Ψ(2)(z)

).

(3.21)

The Laurent modes of bosonic current j(z) are

jn = −i ∑s∈Z+ 1

2

ψ(1)n−sψ

(2)s , (3.22)

In analogy to N = 1 case (3.11), we construct a fermionic field G(z) having

bosonic and fermionic contributions, which, for N = 2 has two parts G+(z)

and G−(z) ,

G(z) = G+(z) + G−(z) =1√2

(G(1)(z) + G(2)(z)

), (3.23)

having Laurent modes,

G±r =1√2

∑s∈Z+ 1

2

(j(1)r−s ∓ i j(2)r−s

) (ψ(1)s ± i ψ

(2)s

), (3.24)

Unlike N = 1 , here we have two generators for each bosonic and fermionic

field. Thus, the Laurent mode (3.8) for N = 2 satisfies Virasoro algebra (2.34)

3.6. HWR OF N = 2 SUPER VIRASORO ALGEBRA 23

with central charge c = 3 . The N = 2 super Virasoro generators Ln, Lm ∈ LV ,

jn, jm ∈ J (3.22) and G±r , G±s ∈ G±S (3.24) satisfy commutation relations


12(m3 −m)δm+n,0 ,

[Lm, jn] = −njm+n ,

[Lm, G±r ] =(m

2− r)

G±m+r ,

[jm, jn] =c3

mδm+n,0 ,

[jm, G±r ] = ±G±m+r ,

G+r , G−s = 2Lr+s + (r− s)jr+s +

c3

(r2 − 1

4

)δr+s,0 ,

G+r , G+

s = G−r , G−s = 0 .

(3.25)

which form infinite dimensional Lie algebra, termed N = 2 super Virasoro alge-

bra SVAN=2(LV ,J ,G±S )

3.6 Highest Weight Representations ofN = 2 Super

Virasoro Algebra

The N = 2 super Virasoro algebra entails two fermionic (or supersymmetric)

generators for each bosonic and fermionic field. Laurent modes Lbos.n and L f erm.

n

separately satisfy Virasoro algebra (2.34), with central charges c = 1 and c =

1/2 . However, for N = 2 , we have twice contributions of these, leading to

mode Ln satisfying Virasoro algebra (2.34), with central charge c = 3 .

However, there exists a discrete series of rational unitary models for 0 < c <

3 , with

c =3k

k + 2, k ≥ 1 . (3.26)

3.6. HWR OF N = 2 SUPER VIRASORO ALGEBRA 24

For each value of k in unitary series (3.26), there exists a finite number of highest

weight representation φlm,s , which are specified by conformal weight,

hlm,s =

l(l + 2)−m2

4(k + 2)+

s2

8, qm,s = −

mk + 2

+s2

. (3.27)

The integers l, m, s are constrained by

0 ≤ l ≤ k , 0 ≤ |m− s| ≤ l . (3.28)

Chapter4Summary

This report summarizes algebraic aspects of conformal symmetry, suppersym-

metry, superconformal symmetry, and their corresponding algebras. In the In-

troduction, I defined infinitesimal symmetry, and constructed the notion of gen-

erator of symmetry. I formalized infinitesimal symmetry and finite continuous

symmetry, and discussed their relevance in physics.

In chapter 2, I introduced conformal symmetry in d-dimensions, and devel-

oped conformal symmetry in 2-dimensions. Section 2.1 summarizes conformal

algebra in d-dimensions. Section 2.2 summarizes Witt algebra and its central

extension, called Virasoro algebra, and its highest weight representations. The

highest weight representation of Virasoro algebra entails Verma module. Verma

module has states with vanishing norm, which were ascertained using Kac de-

terminant. Section 2.2.6 summarizes Kac determinant for levels N = 1 and

N = 2 , and it was further generalized for arbitrary level N .

Chapter 3 introduces suppersymmetry and supperconformal symmetry, and

summarizes super Virasoro algebra for N = 1 and N = 2 , and their highest

weight representations.

25

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