#### Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory

Received: September
Matrix models from localization of five-dimensional
Bum-Hoon Lee 0 1 2 3
Daeho Ro 0 1
Hyun Seok Yang 0 2
Open Access 0
c The Authors. 0
0 Pohang , Gyeongbuk 37673 , Korea
1 Asia Pacific Center for Theoretical Physics , POSTECH
2 Center for Quantum Spacetime, Sogang University
3 Department of Physics, Sogang University
We study localization of five-dimensional supersymmetric U(1) gauge theory on S3 × Rθ2 where Rθ2 is a noncommutative (NC) plane. The theory can be isomorphically mapped to three-dimensional supersymmetric U(N → ∞) gauge theory on S3 using the matrix representation on a separable Hilbert space on which NC fields linearly act. Therefore the NC space Rθ2 allows for a flexible path to derive matrix models via localization from a higher-dimensional supersymmetric NC U(1) gauge theory. The result shows a rich duality between NC U(1) gauge theories and large N matrix models in various dimensions.
Supersymmetric gauge theory; Duality in Gauge Field Theories; Matrix Mod-
1 Introduction
gauge theory
2 Three-dimensional large N gauge theory from five-dimensional NC U(1)
3 Localization of five-dimensional NC U(1) gauge theory 3.1 3.2 Localization of vector multiplet
Localization of hypermultiplet
4 Localization of three-dimensional large N gauge theory
5 Discussion
A Notation, conventions and useful formulae
A.1 Gamma matrices
A.2 Charge conjugation matrices
A.3 Fermion bilinears
A.4 Lie algebra g
A.5 Integral on NC space
C Closed supersymmetric algebra
D Harmonic analysis on S3
E Clebsch-Gordan coefficients
B Vanishing cubic terms in supersymmetric transformations
Introduction
the Planck length LP =
noncommutative (NC) operators obeying the commutation relation
The existence of gravity introduces the gravitational constant GN into a physical theory.
It is well-known that the gravitational constant GN leads to a certain scale known as
q GN ~ = 1.6 × 10−33cm in which spacetime coordinates become
c3
(a, b = 1, · · · , 2n).
which is isomorphic to the Heisenberg algebra of an n-dimensional harmonic oscillator.
The NC space (1.1) will be denoted by Rθ2n and ls ≡
√α′ is a typical length scale for the
noncommutativity. Thus the NC space (1.1) is similar to the NC phase space in quantum
mechanics obeying the commutation relation given by
(i, j = 1, · · · , n).
We will get an important insight from this similarity to understand the NC spacetime
correctly. As we have learned from quantum mechanics, the NC phase space (1.2) introduces a
complex vector space called the Hilbert space. This is also true for the NC space (1.1) since
its mathematical structure is essentially the same as quantum mechanics. Therefore the
linearly acts. In particular, NC fields become linear operators acting on the Hilbert space
harmonic oscillator, its Hilbert space is given by the Fock space and so has a countable
basis, the representation of NC fields on the Hilbert space H is given by N × N matrices
equivalence between a lower-dimensional large N gauge theory and a higher-dimensional
NC U(1) gauge theory [1–3].
To illuminate the remarkable equivalence, let us consider a five-dimensional field
theory defined on the NC space R3
= (xm, ya) where M
is a NC plane whose coordinates obey the commutation relation
If we define annihilation and creation operators as
the NC algebra (1.3) reduces to the Heisenberg algebra of harmonic oscillator, i.e.,
a =
a† =
y4 − iy5
[a, a†] = 1.
H = {|ni| n ∈ Z≥0},
The representation space of the Heisenberg algebra (1.5) is thus given by the Fock space
n=0 |nihn| = IH, as is
wellknown from quantum mechanics.
The field theory we will consider is defined by dynamical fields on R3
is defined by the star product
(Φb1 ⋆ Φb2)(X) = e 2i θab ∂∂ya ⊗ ∂∂zb Φb1(x, y)Φb2(x, z)|y=z.
See [4–6] for a review of NC field theories and matrix models. In quantum mechanics,
physical observables are considered as operators acting on a Hilbert space. Similarly the
space H. Thus one can represent the operators acting on the Fock space (1.6) as N × N
rule for the products
patible with the ordering in the matrix algebra AN (R3) and so it is straightforward to
matrix representation (1.8) without any ordering ambiguity.
To formulate a gauge theory on R3
under the NC star product (1.7). The covariant field strength of NC U(1) gauge fields
AbM (X) = (Abm, Aba)(x, y) is then given by
the matrix representation (1.8) to a five-dimensional NC U(1) gauge theory whose action
is given by r
NC field which rapidly decays at asymptotic regions, one may truncate the matrix representation of the
is given by2
where FbMN = FbMN − BMN and
Using the relations,
BMN =
S =
1 Z
4 FbMN Fb
and the matrix representation (1.8), the above five-dimensional NC U(1) gauge theory is
S =
1 Z
4 FbMN Fb
FmnF mn + 1 ηABDmΦADmΦB − 41 ηAC ηBD[ΦA, ΦB][ΦC , ΦD]
2
tion (1.15) are N × N matrices in the adjoint representation of U(N
→ ∞). The
action (1.15) respects the SO(3) × SO(2, 1) global symmetry where SO(3) is the Lorentz
symmetry group acting on R3 and SO(2, 1) is the R-symmetry group acting on (x0, y4, y5)
(see footnote 2).
Let us summarize the isomorphic map from a five-dimensional NC U(1) gauge theory
The conventional Coulomb branch of the large N gauge theory (1.15) is defined by
hΦaivac = diag (αa)1, (αa)2, · · · , (αa)N
field component compactified along the time-like direction.
(See also the footnote 3 in [8].) A motivation for the wrong sign is to consider Euclidean Yang-Mills theory
N = 2 gauge theory on R3
Yang-Mills theory on R3,1
× R2. The scalar field σb in the five-dimensional theory corresponds to the gauge
θ
addition to the conventional Coulomb branch (1.17) [3, 9]. The new vacuum is called the
NC Coulomb branch and it is defined by
[pa, pb] = −iBab.
Note that the NC Coulomb branch (1.18) saves the NC nature of matrices while the
coneq. (1.18) arises as a vacuum solution of the large N gauge theory (1.15) when we take the
Coulomb branch is exactly mapped to the five-dimensional NC U(1) gauge theory (1.14)
and thus we verify their equivalence in a reverse way. If the conventional vacuum (1.17)
were chosen, we would have failed to realize the equivalence. Indeed it turns out [3, 9] that
the NC Coulomb branch (1.18) is crucial to realize the large N duality which implies the
emergent gravity from matrix models or large N gauge theories.
Recently a localization technique using fixed point theorems provides us a very
powerful tool for the exact computation of the path integral both for topologically twisted
supersymmetric theories and for more general rigid supersymmetric theories defined on
curved spaces. See refs. [10]–[27] for the collection of reviews of this subject. The power
of localization is to reduce the dimensionality of the path integral using supersymmetries
such that the path integral receives contribution from the locus of fixed points of
supersymmetry. We will put a supersymmetric quantum field theory on S3 ×
integral on S3 is free of infrared divergences. Our aim is to exactly compute the expectation
value hOi of a BPS observable in the quantum theory, which is defined by
hOi =
We are interested in
combination of bosonic charges conserved by the theory. We will assume that the BPS
and the fermionic symmetry generated by Q is free of anomaly. Then we can use the
freedom to deform the path integral of a supersymmetric quantum field theory by adding
a Q-exact term to the classical action because
= ∓
= 0,
hOi =
B. This means that
integrand is dominated by the saddle point of the localizing action
is BPS field configurations annihilated by the supercharge Q. Depending on the spacetime
dependence of the field configuration in the localization locus FQ, we may be left with the
path integral of lower-dimensional field theory or, in favorable cases, FQ consists of constant
field configurations with a finite-dimensional integral of a zero-dimensional quantum field
theory such as matrix models [10]–[27]. Since we will consider a supersymmetric field
and (1.15). Although two theories are defined in different dimensions with different gauge
groups, they are mathematically equivalent to each other. Therefore, we can apply the
localization to either a five-dimensional supersymmetric NC U(1) gauge theory or a
threedimensional supersymmetric U(N
apply the localization to the five-dimensional theory to obtain a two-dimensional NC gauge
theory and then consider the matrix representation of the resulting NC gauge theory to
yield a zero-dimensional matrix model, as depicted in figure 1. On the other hand, we can
first implement the matrix representation to the five-dimensional theory to get a
threedimensional large N gauge theory and then apply the localization to the large N gauge
theory on S3 to derive a zero-dimensional matrix model. Both routes should end in an
identical zero-dimensional matrix model, which corresponds to a classical action evaluated
at the localization locus FQ. The aim of this paper is to verify the flowchart outlined in
figure 1 using the localization technique and the matrix representation of NC field theories.
supersymmetric NC U(1) gauge theory on S3
× R2. Using the matrix representation (1.8),
θ
the five-dimensional supersymmetric NC U(1) gauge theory is isomorphically mapped to
that the same result can be realized with the commutative supersymmetric U(1) gauge
theory on S3
× R2 with an equivariant parameter turned on along the plane R2 [24].
In section 3 we perform the localization of the five-dimensional supersymmetric NC
U(1) gauge theory on S3
The matrix representation of the two-dimensional NC U(1) gauge theory leads to a
zerodimensional matrix model at the localization locus. We thus explore the red arrows in
figure 1 to derive a zero-dimensional matrix model via the localization of a five-dimensional
NC U(1) gauge theory.
In section 4 we follow the blue arrows in figure 1 to get the zero-dimensional matrix
model via the localization of a three-dimensional U(N
→ ∞) gauge theory. Using the
any N × N (Hermitian) matrix can be regarded as the matrix representation of a
higher3D U(N → ∞) SYM on S3
2D NC U(1) YM on R2
Matrix representation
0-dimensional matrix model
Matrix representation
Localization ❅
dimensional NC field, the localization of the three-dimensional large N gauge theory can be
easily done by mapping the problem to the one of a five-dimensional NC U(1) gauge theory.
In section 5 we appeal to the mathematical identity depicted in figure 1, in
particular, the equivalence between a higher-dimensional NC U(1) gauge theory and a
lowerdimensional large N gauge theory. This implies [3, 9] that the five-dimensional NC U(1)
gauge theory describes a five-dimensional gravity according to the large N duality or
gauge/gravity duality. We discuss a physical implication of the localization in figure 1
from the point of view of the emergent gravity.
We include five appendices, containing our notation and conventions, some details on
the supersymmetric transformations, the harmonic analysis on S3, and the Clebsch-Gordan
Three-dimensional large N
gauge theory from five-dimensional NC
U(1) gauge theory
actions with minor modifications. Later we will put the theory on S3
localization, which will require some additional terms in the action and a modification of
supersymmetry transformations. For a notational simplicity, we will omit the hat symbol
to indicate five-dimensional NC fields and implicitly assume the star product (1.7) for the
the N = 2 theory.
S5V =
metric transformations
− 2g52 DM σDM σ − iΨα˙ ΓM DM Ψα˙
multiplication between NC fields. We hope it does not cause much confusion with
threefield obeys the symplectic Majorana condition
representation of SU(2)R R-symmetry:
(Ψα˙ )† = (Ψβ˙ )T C5εβ˙α˙ ≡ Ψα˙ .
− Ψα˙ ΓM [δAM , Ψα˙ ] − Ψα˙ [δσ, Ψα˙ ].
δDα˙β˙ = DM Ψ ˙ ΓM Σα˙ + Σ ˙ ΓM DM Ψα˙
β β
− i [σ, Ψβ˙ ]Σα˙ + Σβ˙ [σ, Ψα˙ ] ,
spinors. Using the definition of symplectic Majorana spinor, one can deduce the
supersym− 2g5
It is straightforward to check the supersymmetric invariance of the action (2.3) if the cyclic
permutation under the integral such as eqs. (A.18) and (A.19) is carefully used. As in the
commutative case, after cancellation of all the quadratic fermion terms, we are left with
the cubic terms coming from the supersymmetric transformations
One can show by applying the Fierz identity (B.3) that these terms cancel each other. The
details show up in appendix B.
We can apply the matrix representation (1.8) to the supersymmetric action (2.3). The
bosonic part was already done in (1.15) except the auxiliary term. The matrix
represenorder to get a three-dimensional picture after the matrix representation, it is convenient to
where two-dimensional Weyl spinors
1 Z
2πθΨ1(x) = λ(x) ⊗ ζ+ + ψ(x) ⊗ ζ−,
2πθΨ2(x) = ψT (x)C3 ⊗ ζ+ + λT (x)C3 ⊗ ζ−
the three-dimensional supersymmetric large N gauge theory whose action takes the form
S3V =
2πθΨ1(x) = λ(x) ⊗ ζ+ + ψ(x) ⊗ ζ−,
2πθΨ2(x) = C3−1ψ∗(x) ⊗ ζ+ + C3−1λ∗(x) ⊗ ζ−
± = √
−ψ[σ, ψ] − λ[σ, λ] − 8 β
d ˙
z = y4 + iy5,
z = y4 − iy5
and the complex scalar field defined by
global symmetry now reduces to U(1).
f ⋆ g − g ⋆ f = 2α′ ∂∂fz ∂∂gz − ∂z ∂z
∂g ∂f
+ O(α′3) for any two NC fields f, g ∈ C∞(R3) ⊗ Aθ.
Since the three-dimensional large N gauge theory (2.10) has been obtained from the
sions. Using the matrix representation (1.8) again, it should be straightforward to derive
the supersymmetry transformations for the three-dimensional large N gauge theory (2.10)
from the five-dimensional ones in eq. (2.4) by taking the supersymmetry transformation
Σ1 = ǫ1 ⊗ ζ+ + ǫ2 ⊗ ζ−,
Σ2 = C3−1ǫ2∗ ⊗ ζ+ + C3−1ǫ∗1 ⊗ ζ−.
supersymmetric background, i.e., preserves four supercharges [31]. However we must select
on S3. For that reason, we consider the supersymmetry transformation parameters given by
Then the supersymmetry transformations generated by the above parameter are given
− iD ǫ,
δF = −ǫT C3 γmDmψ + i[σ, ψ] − 2[φz, λ] ,
δD = Dmλγmǫ + ǫγmDmλ − i [σ, λ]ǫ + ǫ[σ, λ] ,
F ≡ 2
D ≡ d11 +
S5H =
2 DM Hα˙ DM Hα˙ + iΘΓM DM Θ − F αFα +
˙
Dα˙β˙ [Hβ, Hα˙ ] − Θ[σ, Θ] − 2Θ[Hα˙ , Ψα˙ ] + 2[Hα˙ , Ψα˙ ]Θ
supersymmetry transformations
ΓM DM Hα˙ − i[σ, Hα˙ ] Σα˙ + FαΛα,
˙
δFα = Λα iΓM DM Θ − [σ, Θ] − 2[Hβ˙ , Ψβ] ,
plet, the action (2.17) is invariant under the above supersymmetric transformations but it
is not necessary to use the Fierz identity for fermionic cubic terms.
We can similarly apply the matrix representation (1.8) to the hypermultiplet. For this
2πθΘ(x, y) 7→ η(x) ⊗ ζ+ + χ(x) ⊗ ζ−.
2πθΨα˙ (x) ≡ ξ+α˙ (x) ⊗ ζ+ + ξ−α˙ (x) ⊗ ζ−.
We also denote the spinors in eq. (2.7) with a compact notation:
Using this matrix representation, it is straightforward to get the three-dimensional action
for the hypermultiplet given by
S3H =
2 Dmhα˙ Dmhα˙ − g2
−2 η[hα˙ , ξ+α˙ ] + χ[hα˙ , ξ−α˙ ] − [hα˙ , ξ+α˙ ]η − [hα˙ , ξ−α˙ ]χ
The supersymmetry transformations for the three-dimensional hypermultiplet will be
obtained by the matrix representation of the five-dimensional ones in eq. (2.18). Having in
mind a localizing supersymmetry on S3, let us consider the supersymmetry transformation
parameters given by
γmǫDmh1 − 2C3−1ǫ∗[φz, h2] − iǫ[σ, h1] + f2C3−1ǫ∗,
γmC3−1ǫ∗Dmh2 − 2ǫ[φz, h1] + iC3−1ǫ∗[σ, h2] + f1ǫ,
δf1 = −ǫ iγmDmχ − 2i[φz, η] + [σ, χ] + 2[hα˙ , ξ−α˙ ] ,
Using the matrix representation (1.8), we have obtained two mathematically equivalent
theories, that are defined in different dimensions with different gauge groups. Although the
√2πθΘ(x, y) 7→ η(x) ⊗ ζ+ + C3−1χ(x)∗ ⊗ ζ−.
two theories superficially look quite different, they should be physically equivalent to each
other. To explore the physical implications of the equivalence, one may apply localization
techniques to each of them to compute, for example, partition functions and some
correlators exactly. On the one hand, one can first apply the localization to the five-dimensional
theory to obtain a two-dimensional NC gauge theory and then take the matrix
representation to the two-dimensional NC gauge theory to yield a zero-dimensional matrix model,
as depicted in figure 1. On the other hand, one can first apply the matrix representation
to the five-dimensional theory to have a three-dimensional large N gauge theory and then
consider the localization of the large N gauge theory to get a zero-dimensional matrix
model. Both routes should end up with an identical zero-dimensional matrix model. In the
end, the localization will verify a rich duality between NC U(1) gauge theories and large
N matrix models in various dimensions as outlined in figure 1.
To carry out the localization, we put the theory on S3
S3. For this purpose, it is convenient to represent five-dimensional fields as the form of
three-dimensional fields in which extra coordinates ya are regarded as parameters living in
Aµ (x, y) (µ = 1, 2, 3),
F (x, y) =
1 D12(x, y),
Aa(x, y) = φa(x, y) − Babyb (a = 4, 5),
D(x, y) = D11(x, y) +
Ψ1(x, y) = λ(x, y) ⊗ ζ+ + ψ(x, y) ⊗ ζ−,
Ψ2(x, y) = C3−1ψ∗(x, y) ⊗ ζ+ + C3−1λ∗(x, y) ⊗ ζ−.
One may notice that the pattern of the above decomposition is equal to the
threedimensional large N
matrices in the action (2.10).
This replica is not accidental
because the matrix representation of the NC fields in eq. (2.25) will give rise to the
threedimensional large N gauge theory on S3 whose action is precisely equal to eq. (2.10).
Indeed the corresponding five-dimensional action after the decomposition (2.25) can be
Let us consider the supersymmetry transformation parameters as eq. (2.14) and take
ǫ to be a Killing spinor on S3 obeying the following equation
ωμ = 41 ωμmnγmn = 2ir γμ, respectively.
m, n, · · · = 1, 2, 3.
The dreibein on S3 is denoted by em
∇µ ǫ =
∇µ = ∂µ + ωµ = ∂µ +
where the covariant derivative acting on a spinor is given by
is defined by
For a gauge singlet which does not depend on (y4, y5), it reduces to eq. (2.27). The
supersymmetry transformations generated by the Killing spinor ǫ obeying eq. (2.26) are
also given by eq. (2.15) with the replacement g3 → g5.
However, after imposing the
condition (2.26), the supersymmetry transformations will no longer be closed because the
covariant derivative now acts on the spinor ǫ nontrivially. Fortunately, to achieve a closed
algebra, it is enough to modify the transformation law only for the auxiliary fields by adding
We verify the closed algebra in appendix C. The result is essentially the same as the one
in refs. [28, 29] although two-dimensional surface in our case is a NC space.
Using the matrix representation of the NC fields in eq. (2.25), the five-dimensional
U(1) gauge theory on S3
gauge theory on S3. The three-dimensional action on S3 is obtained from the result in
transformations of large N matrices on S3 can easily be deduced from the five-dimensional
ones using the matrix representation in a similar way. Moreover, the closedness of the
three-dimensional supersymmetric algebra simply results from the five-dimensional one.
Since the supersymmetry transformation parameter ǫ obeys the nontrivial Killing spinor
deed its variation reduces to
∇N Σα˙ ΓM ΓN Ψα˙ DM σ − 2 ∇LΣα˙ ΓMN ΓLΨα˙ FMN
[φz, φz](ǫλ − λǫ) + ψC3−1ǫ∗F − F ǫT C3ψ .
its supersymmetric transformation cancels the variation (2.30). It turns out [28, 29] that
the extra action is given by
S5′V = S5M + SCS
S5M = − r
1 Z
SCS = − 2g52
1 Z
A ∧ F +
A ∧ A ∧ A
∧ ̟
We also put the hypermultiplet on S3
same as the vector multiplet. In order to achieve a closed algebra of the supersymmetry
to modify the supersymmetry transformations in eq. (2.18) by simply adding additional
forward (though a bit tedious) to verify that the modified supersymmetry transformation
For example, one can show that
˙
[ΔΣ1 , ΔΣ2 ]Fα˙ = i Σ2α˙ Γµ Σβ
1 − Σ1α˙ Γµ Σ2β˙ Dµ Fβ˙ − Σ2α˙ Σ1 − Σ1α˙ Σ2β [σ, Fβ˙ ]
β˙ ˙
and the modified transformations introduced in eqs. (2.29) and (2.36). They are combined
to get the total variation of the action (2.17) generated by the supersymmetry
transnon-vanishing. Therefore, as in the vector multiplet, it is necessary to add a compensating
action given by
S5MH ) = 0.
S5MH =
1 Z
− r
formation parameter [32]. Since the supersymmetry transformations are now generated by
the Killing spinor ǫ obeying eq. (2.26), there are extra contributions from the derivative of
the Killing spinor given by
∇M Σα˙ DN Hα˙ − ∇N Σα˙ ΓM ΓN ΘDM H
−∇M Σα˙ ΓM Θ[σ, Hα˙ ] + ΘΓM
+ ∇M Σβ˙ ΓM Ψα˙ + Ψ ˙ ΓM
β
Localization of five-dimensional NC U(1) gauge theory
symmetric NC U(1) gauge theory on S3
localization of five-dimensional quantum field theories, see refs. [24, 25] and references
has a wrong sign. In order to define the path integral properly, it needs to be analytically
To carry out the localization procedure, we need to identify the Grassmann-odd
symsymmetry that could be a combination of Lorentz, R-symmetry, and gauge
transformations. We apply the same twisting procedure as [33] by considering a global symmetry
After the twisting, we get a scalar supercharge Q which is thus Lorentz invariant (in the
K sense). It is enough to have one scalar supercharge Q for localization and Q is regarded
as a BRST operator.
Localization of vector multiplet
ǫ to zero and replacing the Grassmann-odd parameter ǫ by a Grassmann-even parameter
spinors although all fields have integer spins with respect to K. The corresponding BRST
transformations for the vector multiplet are then given by
δQF = −ǫT C3 γµ Dµ ψ − [σ, ψ] − 2[φz, λ] +
δQD = −Dµ λγµ ǫ − [σ, λ]ǫ +
It is straightforward to check that the above BRST transformations are nilpotent, i.e.,
Q2 = 0.7 The BRST invariant action on S3
S5(iVnv)=Z
− 2
1 Z
− 2g52
2 Dµ σDµ σ + 2Dµ φzDµ φz − 2[φz, σ][φz, σ]
+λ[σ, λ] + ψ[σ, ψ] − r
A ∧ F +
A ∧ A ∧ A
∧ ̟.
We deform the action (3.2) by adding a BRST Q-exact term
S5V = 2
such that the total classical action is given by
Se5V ≡ S5V
where t is a non-negative real parameter. The explicit form for the Q-exact Lagrangian is
2 Dµ σDµ σ + 2Dµ φzDµ φz − 2[φz, σ][φz, σ]
− 2
1 D2
− 2F F
iεµνρ Dν φzDρφz − Dµ φz[φz, σ] + [φz, σ]Dµ φz
up to total derivatives.
Since the Lagrangian (3.5) is BRST-exact, the modified action (3.4) with a parameter t
leads to the same partition function as the undeformed one as was explained in eq. (1.21).
To be precise, the partition function Z(t) for the modified action is t-independent, i.e.,
L5V =
of eq. (3.7) constitutes the localization locus FQ given by
Aµ = D = F = F = 0,
given by8
Se5V |FQ =
sentation (1.8), the classical action is mapped to the zero-dimensional matrix model
Se5V |FQ =
2r2g2 Trσ σ − 4ir[φ†, φ] ,
Now we compute the one-loop determinant coming from quadratic fluctuations of the
fields about the fixed points in (3.8). For that purpose, the gauge-fixing procedure is also
necessary for the computation of the path integral. We take the usual gauge-fixing term
L5FVP = cDµ Dµ c + bDµ Aµ .
There remains the residual gauge symmetry that acts on
this freedom, we can put the background as9
F P = c∂z∂zc − 2α′b [z, φz] + i .
For this gauge-fixing of the background fields, we will add another gauge-fixing term [29]
The path integral of the ghost fields gives the one-loop determinant
Since we are interested in the large t limit and perform the path integral over the
the saddle point configuration in eq. (3.13) and rescale the fluctuation fields as
det(−r2∂z∂z).
and take the limit t →
to be large then allows us to keep only the quadratic terms in the Lagrangian (3.5):
t · L5V =
ψ + 2ψ(1 − γµ kµ )∂zλ + 4∂zλψ,
9Note that the action (3.9) on-shell is in general divergent, which is the reason why the cyclic
property (A.19) was not applied to the second term. In order to regularize the on-shell action, one may replace
fields has originally been defined by the star product (1.7), we can ignore the star product
for the quadratic terms since all nontrivial star products in this case are total derivatives
and thus can be dropped. Hence we will regard the fluctuations in the Lagrangian (3.17) as
fields on S3 ×C. Integral over the auxiliary fields D, F and F have already been performed,
which contributes trivial constant terms to the partition function. We will ignore an overall
constant of the partition function.
In order to calculate the one-loop determinant of U(1) gauge fields, we first proceed
with separating the gauge field into a divergenceless and pure divergent part:
can see that the longitudinal mode in (3.18) can be absorbed into the complex scalar field
with the form
the ghosts in eq. (3.11) contributes a factor of det 0. Therefore the one-loop determinant
d = dxµ ∂µ = emlm.
To evaluate the path integral of the bosonic fields, it is more convenient to use a
∧ ep. See
∗ e
m =
∗(em
and two operators acting on the dreibein by
10Note that DAμL = det 10/2Dφ in the path integral measure.
action on S3 can be written as
t · SV B =
1 Z
dB ∧ ∗dB +
+2(dϕz − ∂zB) ∧ ∗ 1 − k · S (dϕz − ∂zB)
+2i ∂zσιk(dϕz − ∂zB) − ∂zσιk(dϕz − ∂zB) ∗ 1
1 Z
dB ∧ ∗dB +
after integration by parts but it is convenient to keep the original form to utilize the last
expression in eq. (3.26).
j=0 m~=−j
|j, m, m′i ≡ Sjm,m′ .
us use the ket notation for the spherical harmonics
They obey the following properties:
−d†dSjm,m′ = − ∗ d ∗ dSjm,m′ =
Sjm,m′ † = (−1)m+m′ Sj−m,−m′ ,
In order to calculate the determinant of the operators in the quadratic Lagrangian (3.25),
2ri Lm, where Lm are operators in the su(2) Lie algebra. We also choose the Killing vector
on S3 (when thinking of S3 as SU(2) and letting it act on itself), the angular momentum
operators Lm are acting on the SU(2)L index m in eq. (3.28) and it is given by
L3|j, m, m′i = m|j, m, m′i,
−j ≤ m ≤ j,
m1=−j s=±(1/2)
where hj, m1; 12 , s|k, msii are the Clebsch-Gordan coefficients of the spin-j representation
1
ςs satisfies the relation S3ςs = sςs with s = ± 2 . See appendix E for the Clebsch-Gordan
and form an orthonormal basis
D θj− 12 ,m+ 21 ;j,m′ = − r
θkm~′,′j′(x) †θkm~,j(x) ∗ 1 = δk′kδj′jδ
it is easy to verify that
j=0 m=−(j+1) m′=−j
j−1
j=1/2 m=−j m′=−j
j=0 m=−(j+1) m′=−j
j−1
j=1/2 m=−j m′=−j
λj− 21 ,m+ 12 ;j,m′(y)θj− 21 ,m+ 21 ;j,m′(x),
ψj− 21 ,m+ 12 ;j,m′(y)θj− 12 ,m+ 21 ;j,m′(x).
j± 21 whenever such a notation is enough. Since the
Lagrangian (3.17) is also quadratic in fermionic fields, it is straightforward to evaluate the
action for the mode expansion (3.64), which can be read as
j=0 m~=−(j+1) m′=−j
d2ySFQ;j,m~(z, z).
In order to implement the Gaussian integration for the action (3.65), we shift the spinor
1
modes for j ≥ 2 , −j ≤ m ≤ (j − 1), |m′| ≤ j as
j − m
(j + 1)(2j + 1)
√j + m + 1
λj− 21 → λj− 21 − j(2j + 1)
r∂z pj − mψj+ 21 + pj + m + 1ψj− 21 .
The corresponding action for these modes is then given by
r∂z pj − mψj+ 21 + pj + m + 1ψj− 21 ,
SF ;j≥1/2,m~ =
(j + 1)λj+ 12 λj+ 21 − jλj− 21 λj− 21
j− 21
2j(j + 1) − m
j − m
− (j + 1)(2j + 1)
p(j − m)(j + m + 1)
p(j − m)(j + m + 1)
1 − j + 1
1 + 1 r2∂z∂z ,
r2∂z∂z ,
2j(j + 1) − m
j(2j + 1)
j + m + 1 r2∂z∂z.
j2(j + 1)2det (j(j + 1) − m) − r2∂z∂z .
j−1
j=1/2 m=−j m′=−j
Thus the above fermionic fluctuations lead to the one-loop determinant
corresponding action reads as
− r
ψj+ 21 ,−(j+ 21 );j,m′αj−(j+1)ψj+ 21 ,−(j+ 12 );j,m′ + ψj+ 21 ,j+ 21 ;j,m′αjψj+ 21 ,j+ 12 ;j,m′ , (3.68)
j
which leads to the one-loop determinant given by
det(1 − r2∂z∂z)2 Y
j=1/2 m′=−j
j2(j + 1)2det (j + 1)2 − r2∂z∂z 2.
Combining all the contributions from the fermionic fluctuations yields the one-loop
deterdet(1 − 4r2∂z∂z)2
j=1/2 m=−j m′=−j
j2(j + 1)2det (j(j + 1) − m) − r2∂z∂z .
Note that the one-loop determinant (3.70) from the fermionic fluctuations exactly
cancels the one (3.58) from the bosonic fluctuations. This result is somewhat expected [29]
since the noncommutativity can be ignored at the quadratic order. However the classical
action (3.9) at the localization locus needs not be quadratic and thus the
noncommutative structure between background fields must be kept. The matrix representation of the
background fields consequently gives rise to a zero-dimensional matrix model with the
acarrows in figure 1 to derive a large N matrix model from a five-dimensional NC U(1) gauge
Localization of hypermultiplet
Using the same twisting as the vector multiplet, we can deduce the BRST transformations
for the hypermultiplet given by
r H1 ǫ, δQχ = − g5
2 ǫT C3[H2, φz] + ǫT C3F 2, δQχ =
δQF1 = 0, δQF2 = iǫT C3 γµ Dµ η − 2[φz, χ] − [σ, η] − 2r
η − 2 [H1, ǫT C3λ] + [H2, ψǫ] ,
δQF 1 = −i Dµ χγµ + 2[φz, η] + [σ, χ] +
χ ǫ − 2 [H1, ψǫ] − [H2, ǫT C3λ] , δQF 2 = 0.
tedious calculation, the BRST transformation of the action (2.17) can be determined as
− g5r
The above variation is exactly cancelled by the BRST transformation of the mass
term (2.39).
Thus the total action for the hypermultiplet is BRST invariant, i.e.,
Similarly we deform the action S5H
≡ S5H + S5MH by adding a BRST-exact term
S5H =
where the bosonic part of the Lagrangian L5QH is given by
L5QHB =
2 Dµ Hα˙ Dµ Hα˙ − [σ, Hα˙ ][σ, Hα˙ ] + r2
H Hα˙
+ F1 − g5
F1 − g5
1 ǫµνρ Fνρ + Dµ σ [H1, H1] − [H2, H2]
up to total derivatives and the fermionic part by
Now the total classical action is defined by
Se5H ≡ S5H
−ηγµ [H1, λ] − [H2, λT ]C3γµ χ +
the hypermultiplet and thus Se5H |FQ = 0.
Since we expand the fields in the hypermultiplet around the saddle point configuration
fluctuations. After shifting the fields,
F2 → F2 − g5
the bosonic Lagrangian (3.74) reduces to a simple form:
t · LHB = 2 H
Note that we have already carried out the integration over the auxiliary fields F1 and F2
and their contributions to the partition function are simply an overall constant. Similarly
the fermionic Lagrangian (3.75) also becomes a quadratic form:
+ χkmγn iγmDnχ − 2r
direct calculation using the mode expansion (3.79) below that the last term in eq. (3.78)
vanishes. A more easier way to see this is to consider a change of variable (cf. footnote 5),
The one-loop partition function for the hypermultiplet can be obtained in the same
way as the vector multiplet by employing the harmonic expansions of the fluctuations13
H1m~j (y)Sjm~(x),
H2(x, y) =
H2m~j (y)Sjm~(x),
H1(x, y) =
j=0 m~=−j
j=0 m=−(j+1) m′=−j
j−1
j=1/2 m=−j m′=−j
j=0 m=−(j+1) m′=−j
j−1
j=1/2 m=−j m′=−j
j=0 m~=−j
ηj− 21 ,m+ 21 ;j,m′ (y)θj− 12 ,m+ 12 ;j,m′ (x),
j− 21 ,m+ 12 ;j,m′ (y)θj− 12 ,m+ 21 ;j,m′ (x).
The one-loop determinant from the bosonic fluctuations can easily be obtained as
j=0 m=−j m′=−j (2j + 1)4
The above mode expansion for the fermionic Lagrangian (3.78) leads to the action given by
− r
− r
1 X∞
j=0 m′=−j
j−1
j=1/2 m=−j m′=−j
(2j + 1) |ηj+ 21 ,j+ 21 ;j,m′ |2 − |ηj+ 21 ,−(j+ 21 );j,m′ |
13Since the hypermultiplet involves complex fields only, it is not necessary to care about the reality
condition of the harmonic expansions in eq. (3.79), unlike the vector multiplet as we remarked in footnote 11.
Hence all the expansion coefficients in (3.79) will be regarded as independent.
β = 2p(j − m)(j + m + 1).
Therefore the one-loop determinant from these fermionic modes yields a factor
j=0 m=−j m′=−j
(2j + 1)4.
Wrapping up the bosonic and fermionic contributions, one finds that the hypermultiplet
contributes just a constant to the total partition function.
Localization of three-dimensional large N gauge theory
The relationship between a lower-dimensional large N gauge theory and a
higherdimensional NC U(1) gauge theory in figure 1 is an exact mathematical identity. The
identity in figure 1 is derived from the fact that the NC space (1.1) admits a separable
Hilbert space and NC U(1) gauge fields become operators acting on the Hilbert space.
Using the matrix representation (1.8) of NC fields, we have obtained a three-dimensional
large N gauge theory described by the action (2.10) for the vector multiplet and the
action (2.22) for the hypermultiplet. Now we will explore the blue arrows in figure 1 to
illuminate how to derive the same large N matrix model starting from a three-dimensional
large N gauge theory. See refs. [16, 17] and references therein for the localization of
three-dimensional quantum field theories.
The BRST invariant theory for the vector multiplet in the three-dimensional large N
gauge theory is given by the action (3.4) by applying the isomorphic map (1.16). The
localization locus FQ is defined by
γµ Dµ φz − [φz, σ] ǫ = 0,
where ǫ is a Killing spinor satisfying eq. (2.26). The space of FQ consists of all possible
solutions obeying the above conditions. It may be characterized by the BPS equations
and thus FQ is defined by the set of constants obeying
Then the classical action at the locus FQ is given by the zero-dimensional matrix
model (3.10).
The conventional choice of vacuum in the Coulomb branch of U(N ) gauge theory is
given by eq. (1.17). It means that the locus FQ takes values in the Cartan subalgebra of
the Lie algebra of g = U(N ) such that
this case the U(N ) gauge symmetry is broken to U(1)N . Note that the gauge group g in
of the Coulomb branch, the so-called NC Coulomb branch [3, 9]. For example it may be
characterized by the vacuum (1.18) satisfying the Moyal-Heisenberg algebra. It should be
emphasized that the NC Coulomb branch arises as a vacuum solution of the large N gauge
theory (2.10) and it saves the NC nature of matrices while the conventional vacuum (4.4)
dismisses the property.
To be specific, the locus FQ in the NC Coulomb branch is given by
0 0 √
vacuum (4.6) can be represented by the root system r of the Lie algebra su(N ) as
Therefore the NC Coulomb branch is in sharp contrast to the conventional vacuum (4.4)
which takes values in the Cartan subalgebra of u(N ) only.
the representation space of the Heisenberg algebra is the infinite-dimensional Fock space H and so the
The localization of a large N gauge theory at the conventional Coulomb branch (4.4)
has been discussed by many authors [34, 37–44]. See also a review [36]. So we will focus
on the localization at the NC Coulomb branch. Let us represent all possible deformations
of the vacuum FQ by
= 0.
tation value at FQ. The notation in eq. (4.8) means the large N matrices such that, for
notation, it is obvious that the localization of the three-dimensional large N gauge theory
around the locus (4.3) is exactly parallel to the five-dimensional case and thus arrives at
the results (3.58) and (3.70) for the one-loop determinant from the bosonic and fermionic
fluctuations described by large N matrices in eq. (4.8). So we confirm the duality in figure 1
to illustrate how to use a five-dimensional NC U(1) gauge theory for the localization of the
vector multiplet in the three-dimensional large N gauge theory.
We can apply the same idea to the hypermultiplet in a three-dimensional large N
gauge theory. The BRST invariant theory for the hypermultiplet in the three-dimensional
large N gauge theory is given by the action (3.76) by applying the isomorphic map (1.16).
The localization locus FQ for the hypermultiplet is defined by
Given the locus FQ of the vector multiplet, the solution for the above equations is trivial
described by
σ(x, z, z) = σ0IN×N + √ δσ(x, z, z),
Φ(x, z, z) ∼= X Φi(x)Hi + X Φα(x)Eα.
r H1 ǫ = 0,
2 ǫT C3[H2, φz] + ǫT C3F 2 = 0,
the Lie algebra u(N ). So we will eventually arrive at the result (3.80) and (3.82) for the
one-loop determinant for the hypermultiplet described by the three-dimensional large N
gauge theory. This result also confirms the duality in figure 1 between a five-dimensional
NC U(1) gauge theory and a three-dimensional large N gauge theory.
We emphasize that a NC space realizes a remarkable duality between a higher-dimensional
NC U(1) gauge theory and a lower-dimensional large N gauge theory [3, 45]. This duality
is simply derived from a very elementary fact that the NC space (1.1) denoted by R2n
is equivalent to the Heisenberg algebra of an n-dimensional harmonic oscillator. A
wellknown property from quantum mechanics is that the NC space R2n admits a separable
Hilbert space and NC U(1) gauge fields become operators acting on the Hilbert space. The
matrix representation of dynamical operators on the Hilbert space immediately leads to
the picture depicted in figure 1. Therefore the relationship between a lower-dimensional
large N gauge theory and a higher-dimensional NC U(1) gauge theory in the figure is an
exact mathematical identity. In this correspondence, the dynamical variables in the
lowerhave applied the localization technique to this correspondence. The result reveals a rich
duality between NC U(1) gauge theories and large N matrix models in various dimensions,
as clearly summarized in figure 1.
AN (S3) in H is a Lie algebra
homomorphism. However there is another important lesson that we have learned from
quantum mechanics. For example, the momentum (position) operator in the Heisenberg
algebra (1.2) can be represented by a differential operator in position (momentum) space,
a Lie algebra homomorphism [6]. To be specific, let us apply the Lie algebra homomorphism
derived from the five-dimensional NC U(1) gauge theory on S3 ×
mapped to a three-dimensional large N gauge theory through the matrix representation
as shown in figure 1. An interesting problem is to identify the theory described by the
set of differential operators. It turns out [3] that the theory in a classical limit describes
a five-dimensional gravity whose asymptotic (vacuum) geometry corresponds to S3
and the relationship between the five-dimensional gravity and the three-dimensional large
N gauge theory is the well-known gauge/gravity duality or large N duality.
NC Coulomb branch
Localization ❅
1D Matrix model
Matrix representation
Therefore the localization of a higher-dimensional NC U(1) gauge theory and a
lowerdimensional large N gauge theory in figure 1 can be interpreted as a localization of a
fivedimensional gravity emergent from the gauge theory. A configuration at the localization
means that there exists an isomorphic map from the NC U(1) gauge theory to the Einstein
gravity which completes the large N duality [45]. In our case, eq. (4.3) corresponds to a
vacuum geometry S3 ×
R2. As we pointed out in section 4, the locus is characterized by the
BPS equations (4.2) whose solution is, in general, nontrivial, e.g. U(N ) instantons on S3
such as Nahm monopoles and U(1)N monopoles in R
S2 [37]. Of course, putting instantons
on a compact space is highly nontrivial. Nevertheless solutions exist, e.g., [46, 47]. It is
manifolds in a commutative limit. Therefore it will be interesting to consider a nontrivial
locus such as BPS solutions and study their emergent geometry around the locus from the
geometric point of view.
Our localization scheme outlined in figure 1 may be directly applied to a localization
problem in the AdS/CFT correspondence [21]. The AdS5 space has a boundary R ×
S3 [35, 52]
to study the AdS5/CF T4 duality. The localization technique provides us a powerful tool
of U(N ) gauge group. Therefore one can consider a vacuum in the NC Coulomb branch
by turning on vacuum expectation values of the adjoint scalar fields such that the vacuum
the fluctuations around the NC Coulomb branch (1.18) are described by a ten-dimensional
in different dimensions with different gauge groups, they are mathematically equivalent to
each other as depicted in figure 2. In this paper we have shown that the localization of
a large N gauge theory at the NC Coulomb branch is equivalent to the localization of a
higher-dimensional NC U(1) gauge theory. The corresponding picture for the AdS/CFT
correspondence has been summarized in figure 2. As we remarked in section 1, the NC
field theory representation of a lower-dimensional large N gauge theory in the NC Coulomb
branch will provide us a powerful machinery to identify gravitational variables dual to
large N matrices [3]. Hence one may study a nonperturbative aspect of the AdS/CFT
correspondence using the localization technique along the flowchart in figure 2. We think
that figure 2 will be a straightforward generalization of figure 1. We hope to address this
interesting problem in the near future.
Acknowledgments
We thank Teruhiko Kawano for a kind help on some calculational details. B.H.L. was
supported by National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIP) (No.2014R1A2A1A01002306). D.H. was supported by the Korea
Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City. H.S.Y.
was supported by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MOE) (No. NRF-2015R1D1A1A01059710).
Notation, conventions and useful formulae
Gamma matrices
⊗ σ2, Γ4 = 1 ⊗ σ1, Γ5 = 1 ⊗ σ3,
and the three-dimensional gamma matrices are defined by
m = 1, 2, 3
and the Lorentz generators are defined by
Useful (anti-)commutation relations are
J MN =
J mn = 1 γmn = 1 [γm, γn] = i εmnpγp.
Charge conjugation matrices
The five-dimensional charge conjugation matrix C5 obeys
(C5)T = −C5
where T denotes the transpose of a matrix. It is related to the three-dimensional charge
and thus C3 satisfies the relation
C5 = C3 ⊗ 1
(C3)T = −C3.
Fermion bilinears
Symplectic Majorana spinors satisfy the following transposition property of fermion
bilinThe raising and lowering of SU(2)R indices are defined by
Then the following relation is deduced:
which should not be confused with the first one in (A.10). Our convention for the SU(2)R
˙
and thus εα˙ γ˙ εγ˙ β˙ = δα˙β.
Lie algebra g
k=0 k!
ei⋆λ = X∞ ik λz ⋆ ·}·|· ⋆ λ{ ∈ U(1)⋆
theory. In this way, we get the gauge group U(N ) for large N gauge theories with the
u(1) generator T N2
obey the commutation relation
= √
[T a, T b] = if abcT c.
su(N ), obeying the relations
Integral on NC space
For the star product (1.7), the integral
d5Xf1(X) ⋆ f2(X) ⋆ · · · · · · ⋆ fn(X)
is invariant under cyclic permutations of the smooth functions fi [4, 5]. In particular, the
following useful relations are deduced from this property:
d5Xf (X) ⋆ g(X) =
d5Xg(X) ⋆ f (X),
Note that the above cyclic permutations have been derived from the assumption that the
functions fi appropriately behave, i.e., rapidly decay, at asymptotic infinities so that
total derivative terms can be dropped. Thus one may worry about the first term in (1.11)
since FbMN does not decay to zero but approaches to a constant value at infinity.
Fortunately, constant terms do not introduce any trouble for the cyclic permutation of the
integral because they are immune from the star product and so they can be placed outside
the integral. For example, if one of fi’s in eq. (A.19) is constant, eq. (A.19) reduces to
eq. (A.18). Consequently the constant terms in the star product do not threaten the cyclic
property (A.19) unless the integral is divergent. In this case the cyclic permutation of the
integral (A.17) can be implemented with impunity.
This property can also be understood using the matrix representation (1.8). In the
matrix representation, the integral (A.17) is transformed into the trace over matrices, i.e.,
Therefore the cyclic property of the integral (A.17) corresponds to the cyclic permutation
of the matrix trace, e.g.,
for N × N matrices f1,2,3(x) over R3 or S3. Note that the background B-field is mapped
to the identity matrix (see A.4) and so it can freely escape from the trace. Hence the
previous argument is confirmed.
Vanishing cubic terms in supersymmetric transformations
This appendix is to check the supersymmetric invariance of five-dimensional NC U(1)
gauge theory, in particular, the vanishing of the fermionic cubic terms in supersymmetric
variations [53].
As in the commutative case, after cancellation of all the quadratic terms, we are left
Ψα˙ ΓM [Σβ˙ ΓM Ψβ˙ , Ψα˙ ] − Ψα˙ [Σβ˙ Ψβ˙ , Ψα˙ ].
In order to show the vanishing of the cubic terms in (B.1), we need the Fierz identity for
gamma matrices
which leads to the identity
But, one can show by the same calculation that
Ψα˙ ΓMN [Σβ˙ ΓMN Ψβ˙ , Ψα˙ ] := 0.
Using this result, let us rewrite eq. (B.1) as the following form
Ψα˙ [Σβ˙ Ψβ˙ , Ψα˙ ] := −2Ψα˙ Ψα˙ Σ ˙ Ψβ˙
β
Similarly, we have
Ψα˙ ΓM [Σβ˙ ΓM Ψβ˙ , Ψα˙ ] := −2Ψα˙ ΓM Ψα˙ Σβ˙ ΓM Ψβ.
˙
˙
= −4Ψβ[Σβ˙ Ψα˙ , Ψα˙ ] − 2Ψα˙ [Σβ˙ Ψβ˙ , Ψα˙ ]
where we applied the Fierz identity (B.2) to the first three terms and used the identity
˙ ˙
Ψα˙ ΨβΣ ˙ Ψα˙ := Ψβ(Σβ˙ Ψα˙ )Ψα˙ . Note that the first term in (B.7) can be written as
β
using the identities Ψβ(Σβ˙ Ψα˙ )Ψα˙ := −(Ψα˙ )T (Ψβ˙ )T (Σβ˙ Ψα˙ ) = Ψα˙ Ψβ˙ (Σβ˙ Ψα˙ ) and (Ψα˙ Ψβ˙ −
˙
terms in (B.7) exactly cancel each other. This completes the proof of the supersymmetric
invariance of five-dimensional NC U(1) gauge theory.
Closed supersymmetric algebra
In this appendix, we present a detailed result for the closedness of the supersymmetry
algebra on S3
transformations are given by eq. (2.15) with the replacement g3 → g5. The result on S3 is
exactly the same as the five-dimensional case if g5 is replaced by g3.
First, the vector multiplet satisfies the following closed algebra given by
[Δη, Δǫ]Aµ = −iζν Fνµ + iζDµ σ,
[Δη, Δǫ]φz = −iζµ Dµ φz − ζ[σ, φz],
[Δη, Δǫ]F = −iζµ Dµ F − ζ [σ, F ] +
[Δη, Δǫ]D = −iζµ Dµ D − ζ[σ, D],
and the covariant derivative Dµ contains gauge and spin connections. In order to get the
above result, we have used at several places the three-dimensional Fierz identity,
∈ Rθ2, so they are immune from the star
acts as an even symmetry of the theory since it can be written as a sum of a translation
closed algebra even off-shell.
Harmonic analysis on S3
Any element of SU(2) can be written in the form
g =
and they satisfy
See the appendix in ref. [36] for their explicit coordinate representations.
We can use the MC forms to analyze the differential geometry of S3. The dreibein of
The basis of the su(2) Lie algebra obeys the relation
In terms of the dreibeins, the metric on S3 is given by
The inverse dreibein is defined by
which can be used to define left-invariant vector fields
em =
lm = Emµ∂µ .
lm =
If we introduce the operators Lm through
they obey the standard commutation relations of the SU(2) angular momentum operators:
In our case this condition can be solved by
using eq. (D.2). The torsion-free condition also leads to the explicit expression,
or, equivalently,
The curvature tensor is given by
ωmnµ = Enν (∂µ eν − Γµνλ eλm)
m
∂µ eνm = Γµνλ eλm − eνnωmnµ ,
∂µ Emν = Enν ωnmµ − Γµνλ Emλ.
Rmn = dωmn + ωmp ∧ ωpn =
∧ en,
The scalar Laplacian on S3 can be written in local coordinates as
given by Rmn = r22 δmn and R = r62 , respectively.
or equivalently
It can be written, in terms of left-invariant vector fields, as
∂xµ
0 = −gµν ∂µ ∂ν + gµν Γ
− 0 =
= EmµEmν∂µ ∂ν + Emµ ∂∂Exmµν ∂∂xν
written as a linear combination of the spherical harmonics in eq. (3.28) as was illustrated
are then constructed by considering a tensor product of the scalar spherical harmonics with
using the vector spherical harmonics in eq. (3.34). The vector Laplacian
1 ≡ ∗d ∗ d acts
∗ d ∗ dB =
− lnlnBm +
satisfy the relations
The covariant derivative acting on a spinor is defined by
It follows that the Dirac operator is
∇µ = ∂µ +
= ∂µ +
2 = −γµ γν ∇µ ∇ν = −(gµν + γµν )∇µ ∇ν
12 − 2
− iD/ =
The Laplacian for the Dirac operator obeys the relation
j ± 1/2. Thus the eigenvalues of the Dirac operator (D.15) are equal to
− j(j + 1)
− 1r 2j + 21 for −,
with degeneracies
dj± 21 =
2j + 1 =
2(j + 1)(2j + 1) for +;
2j(2j + 1)
for −.
The eigenvectors of the Dirac operator are given by the spinor spherical harmonics
introduced in eq. (3.60).
Clebsch-Gordan coefficients
for reader’s convenience.
• The spin k = j + 1 representation
|k = j + 1, mii =
p2(j + 1)(2j + 1)
p(j + m)(j + m + 1)|j, m − 1i|1, 1i
|k = j, mii =
p2j(j + 1)
p(j + m)(j − m + 1)|j, m − 1i|1, 1i
• The spin k = j − 1 representation
|k = j − 1, mii =
p2j(2j + 1)
p(j − m)(j − m + 1)|j, m − 1i|1, 1i
|k = j + , m +
|k = j − 2
j − m
2j + 1 |j, m + 1i| 2 , − 2 i +
j + m + 1
j + m + 1
2j + 1 |j, m + 1i| 2 , − 2 i +
j − m
The spin operator S3 acts on the state |k, mii as
S3|j + 1, mii =
j + 1 |j + 1, mii −
j(j − m + 1)(j + m + 1)
(j + 1)2(2j + 1)
S3|j, mii =
s j2(j − m + 1)(j + m + 1)
(j + 1)(2j + 1)
j(j + 1) |j, mii −
S3|j − 1, mii = − j |j − 1, mii −
(j + 1)(j − m)(j + m)
j2(2j + 1)
(j + 1)2(j − m)(j + m)
j(2j + 1)
|j − 1, mii .
2 ii =
2 ii =
The spin operator S3 acts on the state |k, mii as
2 ii =
2 ii −
p(j − m)(j + m + 1)
2 ii − 2(2j + 1) |j − 2
|j − 2
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