Lecture Notes on Topology in Quantum Physics by Yuxin Zhao

Lecture 1: Topological Spaces in Physics

Gang Jiang

(Dated: September 11, 2019)

Contents

1. Euclidean Space 1

2. Torus 2

3. Riemann Surface 2

4. Mobius Strip 3

5. Projective Space 3

6. Projective Complex Space 4

7. Grassmannian Space 5

8. Lie Group 5

9. Symmetric Space G/K 7

10. Bundles 7

References 7

1. EUCLIDEAN SPACE

Euclidean space Rd, d = 0, 1, 2, 3 . . ., they are real space.

Sphere Sd−1:∑d

i=1 x2i = 1.

2

2. TORUS

The 2-dimension torus T 2, T n = S1 × S1 × . . .× S1︸ ︷︷ ︸n times

.

If we give the periodic boundary conditions to a square, we get the T 2(Fig. 1). So the 2D

Brillouin zone is a T 2. It’s similar to 3D condition.

FIG. 1: 2D brillouin zone (BZ) and torus T 2.

3. RIEMANN SURFACE

This section is talking about compact and orientable Riemann Surface.

When g = 0 (genus), it’s a sphere; When g = 1, it’s a torus; When g = 2, it’s Fig. 2.

FIG. 2: g = 2.

Also, g can be taken as 3, 4, 5 . . . , n, . . ., see them in Fig. 3 and Fig. 4.

FIG. 3: g = 3.

3

FIG. 4: g = n.

4. MOBIUS STRIP

A example of Mobius strip, the space of configurations of two identical particles on a circle.

It is a typical unorientible space.

We can use θ and φ to describe the position of 2 identical particles on a circle, θ, φ ∈ [0, 2π].

Make θ x-axis and φ y-axis, we get a square {(θ, φ) ∈ R2 | θ, φ ∈ [0, 2π]}. Note that we

should identify the points (θ, 0) and (θ, 2π), the points (0, φ) and (2π, φ), the points (θ, φ)

and (φ, θ) shown in Fig. 5. After the identification, we know the sapce is homeomorphic to

the Mobius strip.

FIG. 5: The space of configurations of two identical particles on a circle.

5. PROJECTIVE SPACE

Projective Space RP n.

Beginning with the sphere, we can remove a single disc, and add a Mobius strip in its place,

then we get RP 2.

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Also, begin with the unit sphere S2 and identify antipodal points of the sphere, then we get

RP 2. We can show it in in Fig. 6.

FIG. 6: RP 2.

In the same way, we can begin with Sn and get RP n. RP 1 ∼= S1.

And RP n are unorientable. If we begin with the orientable Riemann Surface, remove a

single disc, and add a Mobius strip in its place, we can get unorientable Riemann surface.

More picture.

6. PROJECTIVE COMPLEX SPACE

Hilbert space H = Cd, and a quantum state |ψ〉 is a point of H.

We know 〈ψ|ψ〉 = 1 and eiφ |ψ〉 is the same as |ψ〉 in quantum physics. So the space of

quantum state |ψ〉 is CP d−1. CP 1 ∼= S2

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7. GRASSMANNIAN SPACE

In mathematics, the Grassmannian is a space which parametrizes all k-dimensional linear

subspaces of the n-dimensional vector space V .

Gr(N, n,R/C)

8. LIE GROUP

O(N), U(N) and Sp(N)

First, O(N) and Sp(N) can be regarded as subgroups of U(N). For the ND complex

Hilbert space with the usual inner product, a general linear transformation is given by

η → Uη, ξ → Uξ. (1)

A unitary transformation U preserves the inner product

〈η|ξ〉 = η†χ. (2)

In other words,

U †U = 1, U ∈ U(N). (3)

O(N) is a subgroup of U(N) preserving the symmetric inner product

ηT ξ. (4)

Equivalently, O ∈ O(N) if and only if

OTO = 1, O = O∗. (5)

Sp(N) is a subgroup of U(N), for N even, that preserves the anti-symmetric inner product

ηTJξ, with J =

0 −1N/2

1N/2 0

. (6)

Equivalently, Sp ∈ Sp(N) if and only if

Sp†Sp = 1, SpTJSp = J. (7)

Second, U(N) and Sp(N) can be subgroups of O(2N). Consider a 2ND real Euclidean

space. O ∈ O(N) if and only if

OtO = 1. (8)

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Here, we distinguish the matrix transpose in real space from that in complex space.

For a complex matrix M = MR + iMI , we can convert it into a real one,

M = 1N ⊗MR + I ⊗MI , (9)

where

I =

0 −1

1 0

, with I t = −I, I2 = −12. (10)

Accordingly,

M † = M t, MT = KM tK, (11)

where

K =

0 1

1 0

, with Kt = K, K2 = 12. (12)

By the above construction, we find that U(N) is a subgroup of O(2N) with [U, I] = 0,

i.e.,

[U, I] = 0, U tU = 1, I2 = −1, I t = −I. (13)

Sp(N) is a subgroup of U(N) with N even, and Sp ∈ Sp(N) if and only if

SpTJSp = KSptKJSp = J. (14)

In other words, Sp(N) is a subgroup of O(2N) specified by

[I, Sp] = 0, [JK, Sp] = 0, {I, JK} = 0, (15)

with

(JK)2 = −1, I2 = −1, (JK)t = −JK, I t = −I. (16)

From above discussions, we have the sequence,

O(N)→ U(N)→ O(2N), (17)

which gives

O 7→

O 0

0 O

(18)

according to our conventions.

Situation 1, Hamiltonian H, diagonalize it, H = U †EU

time-reversal symmetry, T 2 = ±1

Spinless T = K, Spinful T = iσ2K, where σi are Pauli’s matrices.

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9. SYMMETRIC SPACE G/K

H = U †

+1

. . .0

0−1

. . .

U

The space isU(N +M)

U(N)× U(M)

Situation 2, symmetry breaking

Helium-3, G = SO(3)L × SO(3)S × U(1)N

Order parameter2λ

V〈∑~p

~p a~pα a−~pβ〉vac = ~eµ(σµiσ2)αβ

eµi : µ, rotation of spin, SO(3)S; i, rotation of orbits, SO(3)L.

U(1) eµi → e2iαeµi

3He-B:

eµi = ∆0δµi ⇒ HB = SO(3)S+L

Mvacuum∼=

G

HB

∼=SO(3)L × SO(3)S × U(1)N

SO(3)S+L∼= SO(3)× U(1)N

3He-A:

eµi = ∆0Zµ(xi + iyi)⇒ HA = U(1)Lz−N2× U(1)Sz

Mvacuum∼=

SO(3)L × SO(3)S × U(1)NU(1)Lz−N

2× U(1)Sz

∼=SO(3)× S2

Z2

10. BUNDLES

Vector bundles & Principle bundles.

[1] Topology definition point-set topology, Sec.3 of G.Moore’s Notes

[2] A motivation example: Gaussian linking number , Sec.2 of G.Moore’s Notes