Nakahara M. Geometry, Topology and Physics

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Then the Hamiltonian is

H =−

− L =



ijk

. (12.131)

The Poincar´e one-form of this system is

θ =

dψ

. (12.132)

The corresponding symplectic two-form is

ω = dθ =

dψ

∧ dψ

(12.133)

from which we obtain the Poisson bracket

[ψ

, iψ

]

= iδ

. (12.134)

Quantization of the system is achieved by replacing this Poisson bracket by the

anti-commutation relation

{ψ

,ψ

}=δ

. (12.135)

This anti-commutation relation is called the Clifford algebra in

.Letσ

be the

ith component of the Pauli matrices. It is easily veriﬁed from the observation

{σ

,σ

}=2δ

that ψ

= σ

√

2 is the two-dimensional representation of the Clifford algebra.

It is known that the ﬁnite-dimensional irreducible representation of the Clifford

algebra is unique (modulo conjugate transformations). Thus, the Hilbert space of

this system turns out to be

. The Hamiltonian is rewritten in terms of the

Pauli matrices as

H =−

B · σ . (12.136)

This Hamiltonian is known as the Pauli Hamiltonian and describes a spin in a

magnetic ﬁeld.

Similarly, the Clifford algebra deﬁned in

and

2n+1

acts on the Hilbert

space

12.9.2 Supersymmetric quantum mechanics in ﬂat space

The Pauli Hamiltonian is made only of the spin coordinates ψ

and is independent

of the space coordinate x

. Accordingly, it cannot describe a travelling spin. Now

the Hamiltonian is modiﬁed so that the spin may move around the space. This can

be realized by adding a kinetic term to the Hamiltonian. Let us consider a spin in

and put B = 0 to obtain the Hamiltonian

L =

˙x

. (12.137)

The coefﬁcients of this Lagrangian have been chosen so that the system has a

supersymmetry deﬁned later. The canonically conjugate momenta are p

=˙x

and π

=−iψ

/2, from which we obtain the Poisson brackets of the system

, x

]

=[p

, p

]

= 0 [x

, p

]

=[ψ

,ψ

]

= δ

It is easy to derive (anti)commutation relations from these Poisson brackets. The

canonical (anti)commutation relations are

, x

]=[p

, p

]=0 [x

, p

]={ψ

,ψ

}=δ

. (12.138)

The Hamiltonian is

H =˙x

−

− L =

=−

 (12.139)

where  =



k=1

∂

is the d-dimensional Laplacian. The Hilbert space on which

H acts is L

(

) ⊗

,whereL

(

) stands for the set of square-integrable

functions in

and n ≡[d/2] is the integer part of d/2.

Variation of the Lagrangian yields

δL =˙x

δx

δψ

Let us verify that the Lagrangian is invariant under the following supersymmetry

transformation

δx

= iψ

δψ

=− ˙x

(12.140)

where  is an ‘inﬁnitesimal’ real Grassmann constant. In fact,

δL = i ˙x



−

 ˙x

−

 ¨x

= i ˙x



−

 ˙x

−

(ψ

 ˙x

) +

 ˙x

=−

(ψ

 ˙x

) (12.141)

and the action S =



Ldt is left invariant. The corresponding charge (the

generator) is called the supercharge and deﬁned through the Noether’s theorem

 Q ≡ ip

= iψ

˙x

. (12.142)

Exercise 12.8. Show that

δx

=[x

,Q] (12.143)

δψ

={ψ

,Q}. (12.144)

Note that the mass of the particle is set to unity and hence we have p

=˙x

These equations show that Q is the generator of SUSY transformations.

Let us take d = 2n to be an even integer and quantize the system in the

following. We introduce the matrix representation ψ

= γ

√

2, which is the

generalization of the two-dimensional representation introduced in the previous

subsection. Here γ

are the d-dimensional Dirac matrices that satisfy the Clifford

algebra

{γ

,γ

}=2δ

. (12.145)

The Hamiltonian acts on the Hilbert space

= L

(

) ⊗

The supercharge takes the form, upon diagonalizing the coordinate,

Q = iψ

√

∂

∂x

. (12.146)

The operator

∂ ≡ γ

∂

∂x

(12.147)

is nothing but the Dirac operator in Euclidean space

and plays an important

role in the proof of the index theorem.

The hypercharge Q transforms in an interesting way under an SUSY

transformation (12.140)

δ Q = i(δψ

) ˙x

+i ψ

δx

= i(− ˙x

) ˙x

+ iψ

(i

)

=−i ˙x

˙x

+ ψ

=−2i



˙x



=−2iL. (12.148)

Namely, the variation of the supercharge under an inﬁnitesimal SUSY

transformation is the Lagrangian!

We next consider the relation between the supercharge and the Hamiltonian

of the system. Let us consider successive SUSY transformations with Grassmann

parameters 

and 

. If a transformation with 

is applied ﬁrst and then 

next,

we obtain



→ x

+ i



→ x

+ i(

+ 

)ψ

− i



˙x



→ ψ

− 

˙x



→ ψ

− (

+ 

) ˙x

− i



while if the order of the SUSY transformations is reversed,

→ x

+ i(

+ 

)ψ

− i



˙x

→ ψ

− (

+ 

) ˙x

− i



We ﬁnd, from these results, the commutation relation of the SUSY variations:

[δ



,δ



]=δ



− δ



=−2i



∂

∂t

. (12.149)

The observation that the commutation relation of two SUSY transformations

is a time derivative, i.e. the Hamiltonian, suggests that the anti-commutation

relation of the supercharge, the generator of the SUSY transformation, also yields

the Hamiltonian. In fact,

{Q, Q}=2Q

= 2(i p

)(i p

)

=−p

(ψ

+ ψ

) =−p

=−2H.

After all, the SUSY algebra reduces to

=−H. (12.150)

Since Q is anti-Hermitian, the Hamiltonian is a Hermite operator with non-

negative spectrum.

In summary, we proved in equations (12.148) and (12.141) that

δ Q =−2iL δL =



. (12.151)

If these equations are compared with the SUSY transformations (12.140) of

the coordinates x

and ψ

, we readily notice that the roles played by bosonic

quantities (x

and L) and the fermionic quantities (ψ

and Q) are interchanged.

Note that the variation of the supercharge Q in (12.151) is always a time derivative

of the Lagrangian L. This observation is crucial in constructing a SUSY-invariant

Lagrangian out of a supercharge Q.

12.9.3 Supersymmetric quantum mechanics in a general manifold

Let M be a Riemannian manifold with dim M = 2n. The Riemannian metric is

= g

µν

and the inner product of two vectors X and Y with respect to this metric is denoted

X, Y =g

µν

The vector ψ

(t) belongs to TM

x(t )

at each instant of time t. Therefore,

(t) obeys the ordinary transformation rule for a vector under the coordinate

transformation x

→ x

µ

= x

µ

→ ψ

µ

∂x

µ

∂x

. (12.152)

Then, under the SUSY transformation δ ≡ δ



, the coordinates transform as

δx

µ

∂x

µ

∂x

δx

∂x

µ

∂x

iψ

= iψ

µ

and

δψ

µ

∂

µ

∂x

δx

∂x

µ

∂x

δψ

∂

µ

∂x

iψ

∂x

µ

∂x

(−i ˙x

) =− ˙x

µ

where the anti-commutativity of Grassmann numbers has been used to obtain the

last equality. These transformation rules show that the SUSY transformation is

covariant under the coordinate transformation x

→ x

µ

The supercharge Q introduced in the previous subsection should be

generalized on the manifold M as

Q = i˙x,ψ=ig

µν

(x ) ˙x

. (12.153)

The SUSY-invariant Lagrangian on M is constructed from the SUSY variation of

this Q as

δ Q = i∂

µν

δx

˙x

+ ig

µν

δ ˙x

+ ig

µν

˙x

δψ

= i∂

µν

iψ

˙x

+ ig

µν

(i

)ψ

+ ig

µν

˙x

(− ˙x

)

=−2i



µν

˙x

µν

−

˙x



∂

µν

− ∂

µλ

− ∂

λν



=−2i



µν

˙x

µν

˙x

λρ



µν



where



λµ

νρ



∂

ρµ

+ ∂

λρ

− ∂

λν

is the Christoffel symbol associated with the Levi-Civita connection. Note the

symmetry 

µν

= 

νµ

. By comparing this δ Q with (12.151), we read off the

Lagrangian,

L =

µν

(x ) ˙x

˙x

µν

(x )ψ



dψ

+˙x



λκ

(x )ψ



˙x, ˙x+

ψ,

Dψ

. (12.154)

Here Dψ/Dt is the covariant derivative of ψ along the curve x(t).

Exercise 12.9. Show that the SUSY variation of the Lagrangian is proportional to

the time derivative of the supercharge,

δL =



. (12.155)

The quantum version of the supercharge is

Q ∼ g

µν

(12.156)

that is the Dirac operator

∂ on M.

Let us deﬁne some symbols that will be employed in the next section. The

connection one-form is



= dx



λν

(12.157)

while the Riemann curvature two-form is

= d

+ 

∧ 

. (12.158)

The Riemann curvature two-form is expanded in terms of dx

∧ dx

to yield

νρσ

∧ dx

(12.159)

the component of which is the ordinary Riemann curvature tensor. This

component is also written in terms of the connection ∇

λµν

, ∇

∇

∂

∂x

−∇

∇

∂

∂x

= ∂



νλ

− ∂



µλ

+ 

νλ



µη

− 

µλ



νη

. (12.160)

12.10 Supersymmetric proof of index theorem

The proof of the index theorem in its simplest setting will be given in the present

section by making use of the supersymmetric quantum mechanics developed in

the previous section.

12.10.1 The index

Let us consider vector bundles E

−→ M, E = E

⊕E

−

and let be an elliptic

differential operator acting as

: (M, E

) → (M, E

−

It is possible, by using the ﬁbre norm, to deﬁne the adjoint of

†

: (M, E

−

) → (M, E

Assuming that is Fredholm, the index

Ind

= dim ker − dim ker

†

(12.161)

is well deﬁned.

Theorem 12.4. The number ind

is invariant under a ‘small’ deformation of .

Proof.Note,ﬁrst,that

†

and

†

are non-negative and, hence, it follows that

ker

= ker

†

ker

†

= ker

†

Let {φ

} be the orthonormal set of eigensections of

†

: (M, E

) →

(M, E

(

†

)φ

= λ

Deﬁne ψ

≡ φ

√

for λ

> 0, namely φ

∈ (ker )

⊥

. Then we ﬁnd that

is an eigensection with the same eigenvalue λ

, namely ψ

∈ (ker

†

)

⊥

since

(

†

)ψ

= (

†



= λ



= λ

Note also that {ψ

} is an orthonormal eigensection,

ψ

|ψ

=

√

φ

†

|φ

=

√

= δ

Thus, it follows that there is a natural isomorphism between (ker

)

⊥

and

(ker

†

)

⊥

. Note, however, that there exists no such isomorphism between ker

and ker

†

. Suppose N states in ker obtain non-vanishing eigenvalues as a

result of a small perturbation of the operator

and dim ker decreases by N.

Then it follows from this observation that the same number of states must also

leave ker

†

.Otherwise(ker )

⊥

is no longer isomorphic to (ker

†

)

⊥

. Similary,

if dim ker

increases by N ,dimker

†

must also increase by N to keep the

pairing properties of (ker

)

⊥

and (ker

†

)

⊥

. Therefore, ind is invariant under

small perturbations of

Theorem 12.5. Let be a Fredholm differential operator. Then its index is given

ind

= Tr e

−β

†

− Tr e

−β

†

(12.162)

where β>0 is a real constant. In fact, the index is independent of β.

Proof. The traces in (12.162) are over {φ

} and {ψ

}, respectively. Let {φ

}

and {ψ

} be orthonormal eigensections of ker and ker

†

, respectively, and

1 ≤ i ≤ dim ker

and 1 ≤ j ≤ dim ker

†

. Then it follows that

Tr e

−β

†

− Tr e

−β

†



=0

φ

−β

†

|φ

−



=0

ψ

−β

†

|ψ





φ

|φ

−



ψ

|ψ





=0

−βλ

(

φ

|φ

−ψ

|ψ



)



1 −



= dim ker

− dim ker

†

= ind .

Since the summations over i and j are independent of β,ind

thus deﬁned is

independent of β.

The trace that appears in theorem 12.5 is identiﬁed with the heat kernel.

Let E = E

⊕ E

−

and deﬁne a differential operator acting on E by

(cf

equation (12.79))

iQ ≡



†



: E → E. (12.163)

Moreover, deﬁne a ‘Hamiltonian’ and a matrix  by

H = (iQ)



†



 =



0 −1



. (12.164)

Since Q thus deﬁned is anti-Hermitian, the operator H is Hermite and non-

negative. The index of

is rewritten in a compact form by making use of 

ind

= Tr e

−β H

. (12.165)

Let M be a spin manifold, for which the second Stiefel–Whitney class

(M) is trivial. Accordingly, the SO(k) principal bundle over M may be lifted

to the SPIN(k) principal bundle as

SO(k) → SPIN(k).

↓ π

Let E = (M) be this spin bundle. Then, associated with (M) is a Clifford

algebra {γ

,γ

}=2δ

µν

. Let us deﬁne the chirality operator

2n+1

≡ i

...γ

. (12.166)

The operator Q will be identiﬁed with the supercharge later.

It follows from γ

2n+1

= 1 that the eigenvalues of γ

2n+1

are restricted to be ±1,

which we call chirality.

Exercise 12.10. Use the Clifford algebra to show that

2n+1

= 1 {γ

,γ

2n+1

}=0.

The set of sections (M,)for an even k is not an irreducible representation

of SPIN(k) but can be decomposed into two subspaces according to the chirality

(M,) = (M,

) ⊕ (M,

−

) (12.167)

where ψ

∈ (M,

) satisfy γ

2n+1

=±ψ

. We assign the fermion

number F = 0 to sections in (M,

) while F = 1 for those in (M,

−

Then the  deﬁned in (12.164) can be written as

 = (−1)

. (12.168)

It is clear that the operator Q ﬂips the chirality and hence {Q,}=0.

Let Q be the Dirac operator on M and let  = γ

2n+1

. In fact, it follows from

exercise 12.11 that {Q,γ

2n+1

}=0andγ

2n+1

is identiﬁed with (−1)

.When

is diagonalized as in (12.164), the chirality eigensections are expressed as





−



−



. (12.169)

It should be then clear that

: (M,

) → (M,

−

) and

†

: (M,

−

) →

(M,

) are identiﬁed with D and D

†

, respectively, in (12.79). Accordingly,

the index of the Dirac operator is deﬁned as

ind Q = dim ker D − dim ker D

†

. (12.170)

Physicists often call the sections in ker D and ker D

†

zero modes. Then, the

index of the Dirac operator is the difference between the number of positive and

negative chirality zero modes. This index has a path integral expression as we see

in the next subsection.

12.10.2 Path integral and index theorem

Let us consider a Dirac operator Q on a 2n-dimensional spin manifold M.We

employ Euclidean time (t →−it) from now on.

Let H = (iQ)

µν

be the Hamiltonian corresonding to Q.Then

the index of the Dirac operator has a path integral expression

ind Q = Tr e

−β H

= Tr(−1)

−β H



PBC

x ψ e

−



dtL

(12.171)

Note the slight abuse of notations. The symbols ψ

have been used to denote sections in (M, S)

as well as those in (M,

where the Lagrangian L has been introduced in (12.154),

L =

µν

(x ) ˙x

˙x

µν

(x )ψ

Dψ

(12.172)

and PBC stands for the boundary condition in which the path integral is over

functions satisfying a periodic boundary condition over [0,β]. The factor (−1)

disappears if the anti-periodic boundary condition for the fermionic variables is

changed into a periodic one. This can be seen from the following observation. In

the path integral formalism, the trace with (−1)

is (see section 1.5)

tr(−1)

−β H



n|(−1)

−β H

|n



dθ

∗

dθ−θ |(−1)

−β H

|θe

−θ

∗

(12.173)

where F = c

†

c is the Fermion number operator. By noting that

|θ=|0+|1θ(−1)

|θ=|0−|1θ =|−θ

this integral is cast into the form



dθ

∗

dθθ |e

−β H

|θe

−θ

∗

. (12.174)

Thus, by eliminating (−1)

, we have to change the boundary condition to a

periodic one.

This path integral is evaluated in the rest of this section to show that it reduces

to a topological index obtained from the Dirac

A-genus.

The SUSY transformation in Euclidean time is obtained by the replacement

t →−it in (12.140) as

δx

= iψ

δψ

=−i ˙x

As was shown in the previous subsection, the index is independent of β and,

hence, we may consider the limit β ↓ 0 in computing the trace. By rescaling the

time parameter as t = βs, we cast the action into the form





µν

(x ) ˙x

˙x

µν

(x )ψ

Dψ







µν

(x )

µν

(x )ψ

Dψ



. (12.175)

Thus, any path with ˙x = 0 has an exponentially small contribution to the path

integral in the limit β ↓ 0. Accordingly, the contributions to the path integral

come only from paths x (t) = constant in this limit. Clearly, these paths satisfy

the periodic boundary condition.