TMYM - ASU seminar 03:27:2015

CHERN-SIMONS DECOMPOSITION OF 3D GAUGE THEORIES AT LARGE DISTANCES

Tuna Yıldırım(UIOWA, ASU)

Arizona State UniversityMarch 27, 2015

• Int.J.Mod.Phys.A, 30(7):1550034, 2015, arXiv:1311.1853• arXiv:1410.8593 (preprint)

Outline

Wilson Loops and Knot Theory

Geometric Quantization of Chern-Simons Theory

Quantization of Topologically Massive Yang-Mills Theory- Chern-Simons Splitting

Quantization of PureYang-Mills Theory- Chern-Simons Splitting

Wilson Loops and Chern-Simons Splitting

Wilson Loops and Knot Theory

Wilson LoopsArea Law

hW (C)i / e��AC

(Mass gap, confined)

Perimeter Law

hW (C)i / e�mLC

(Mass gap, not confined)

Ex: Yang-Mills in 2+1 D (and hopefully 3+1 D)

Ex: Yang-Mills + Chern-Simons

Ex: Chern-Simons

Link Invariants

hW (C)i !

(No mass gap, not confined)

. . .

Knot Theory

A knot is a smooth embedding of a circle in a 3 or higher dimensional space.

6 l . Introduction

that 5o does not depend on the metric at all. In fact, SQ can be understood as theintegral of a three-form on a three-manifold.

Gauge invariance and general covariance are the real reasons for the propertiesof the expectation value (1.17) that we have observed. Gauge invariance forced usto choose the external source to be expressed in terms of closed paths (conservedexternal currents), since only gauge-invariant quantities have an intrinsic mean-ing in gauge theories. Because of general covariance, the final result (1.17) onlydepends on the topological structure of the closed contours. This is why there isinvariance under smooth deformations of the paths in E3.

In the previous section, the source term was represented by the simple two-component link shown in Fig. 1.1. But one can consider more complicated links,of course; an example is shown in Fig. 1.2.

Figure 1.2.

Exercise. Consider the Abelian CS theory with a source term corresponding tothe link shown in Fig. 1.2. In this case, what is the expression (neglecting self-interactions) of the vacuum expectation value {e J * J " ) ?

1.3 Non- Abelian Chern-Simons actionThe action (1.20) can be generalized L 1,3,4] to the case in which the gauge groupG is a non-Abelian. The corresponding CS action reads

= A t ß? (1-21)

where Áì = Ááì Ô", { Ô" } are the Hermitian generators of a compact simple

Lie group G in its defining representation and the real parameter k is the coupling

Authenticated | [email protected] Date | 1/4/12 9:33 PM

A link is a union of non-intersecting knots.

A 3D Knot

Jones Polynomial and Skein Relations

t�1 VL+(t)� t VL�(t) = (t1/2 � t�1/2) VL0(t)

Skein relation of Jones Polynomials

The normalization condition is(the polynomial for the unknot) V0(t) = 1

VL+(t) VL�(t) VL0(t)

Jones Polynomial of the Trefoil KnotWe start with two unknots

t�1 �t = (t1/2 � t�1/2)

= �t1/2 � t�1/2= 1 = 1

t�1 �t = (t1/2 � t�1/2)

= �t1/2 � t�1/2 = 1= �t5/2 � t1/2

Now we can calculate the Jones polynomial of the trefoil knot

t�1 �t = (t1/2 � t�1/2)

= 1 = �t5/2 � t1/2

= t+ t3 � t4

Jones Polynomial of the Trefoil Knot

The Wilson loop integral is

WR(C) = TrR

✓Pexp i

I

cAµdx

µ

◆

A link L is a union of non-intersecting knots Ci

< WR1(C1) . . .WRn(Cn) >⌘< W (L) >

[1] E.Witten, Quantum Field Theory and the Jones Polynomial, Comm. Math. Phys.,121:351, 1989. [2] P. Cotta-Ramussino, E. Guadagnini, M. Martellini, M. Mintchev, "Quantum Field Theory and Link Invariants", Nucl. Phys. B330 (1990) 557-574

Wilson Loops and Skein Relations[1,2]

�SL+ � ��1SL� = zSL0

Generalized Skein Relation

[1] E.Witten, Quantum Field Theory and the Jones Polynomial, Comm. Math. Phys.,121:351, 1989. [2] P. Cotta-Ramussino, E. Guadagnini, M. Martellini, M. Mintchev, "Quantum Field Theory and Link Invariants", Nucl. Phys. B330 (1990) 557-574

Wilson Loops and Skein Relations[1,2]

(HOMFLY polynomial)

� ��1 = z

� = 1� 2⇡

k

1

2N+O

✓1

k2

◆z = �i

2⇡

k+O

✓1

k2

◆Where

Here, SL is a polynomial of β and z=z(β). For CS theory (in fundamental representation)

�hWL+i � ��1hWL�i = z(�)hWL0i

k: level number of CS

Topologically MassiveAdS Gravity

Topologically Massive AdS Gravity[3,4]

The action is

S =

Zd

3x

�p��(R� 2⇤) +

1

2µ✏

µ⌫⇢

✓�↵µ�@⌫�

�⇢↵ +

2

3�↵µ��

�⌫��

�⇢↵

◆�

can be written as

S[e] = �1

2

✓1� 1

µ

◆SCS

⇥A+[e]

⇤+

1

2

✓1 +

1

µ

◆SCS

⇥A�[e]

⇤

A±µab[e] = !µ

ab[e]± ✏abceµ

c

SCS [A] =1

2

Z✏µ⌫⇢

✓Aµ

ab@⌫A⇢

ba +

2

3Aµ

acA⌫

cbA⇢

ba

◆

where

and

[3] S. Deser, R. Jackiw, and S. Templeton, 1982.[4] A. Achu carro and P.K. Townsend, 1986.

Topologically Massive AdS Gravity

For small values of μ (near CS limit)

S[e] ⇡ 1

2µSCS

⇥A+[e]

⇤+

1

2µSCS

⇥A�[e]

⇤

We will see that this is analogous to TMYM at large distances (near CS limit)

For infinite μ

Analogous to YM at large distances

S[e] =1

2SCS

⇥A�[e]

⇤� 1

2SCS

⇥A+[e]

⇤

Geometric Quantization of Chern-Simons Theory

Chern-Simons TheorySCS = � k

4⇡

Z

⌃⇥[ti,tf ]

d

3x ✏

µ⌫↵Tr

✓Aµ@⌫A↵ +

2

3AµA⌫A↵

◆

SCS(A) �! SCS(Ag) = SCS(A) + 2⇡k!(g)

Under Aµ ! Agµ = gAµg

�1 � (@µg)g�1

!(g) =1

24⇡2

Zd

3x ✏

µ⌫↵Tr(g�1

@µgg�1

@⌫gg�1

@↵g)

is an integer, called the winding number.

k has to be an integereiSCS(A) = eiSCS(Ag)

Field equations:

We choose the temporal gauge and , z = x� iy

z = x+ iy

Chern-Simons Theory

is the Gauss’ law of CS theory

Ga =ik

2⇡F azz

is the generator of infinitesimal gauge transformations

SCS = � k

4⇡

Z

⌃⇥[ti,tf ]

d

3x ✏

µ⌫↵Tr

✓Aµ@⌫A↵ +

2

3AµA⌫A↵

◆

The conjugate momenta are

and ⇧az = � ik

4⇡Aa

z⇧az =ik

4⇡Aa

z

Chern-Simons Theory

Then the inner product is

h1|2i =Z

d�(M)�⇤1�2 !

Zd�(M)e�K ⇤

1 2

⌦ =ik

2⇡

Z

⌃

�Aaz�A

az

K =k

2⇡

Z

⌃Aa

zAaz

The phase space is Kähler with

and Kähler potential

We choose the Kähler polarization

�[Az, Az] = e�12K [Az]

The Wave Functional for CS[3,4]

Aaz [A

az ] =

2⇡

k

�

�Aaz [Aa

z ]

[3] M. Bos and V.P. Nair, "Coherent State Quantization of Chern-Simons Theory", Int. J. Mod. Phys. A5, 959 (1990).[4] V.P.Nair, "Quantum Field Theory - A Modern Perspective", Springer, (2005).

The quantum wave-functional must satisfy the Gauss’ law constraint F a

zz [Aaz ] = 0

If Σ is simply connected we can parametrize the gauge fields as

Az = �@zUU�1 Az = (U †�1)@zU† U 2 SL(N,C)

U(x, 0, C) = Pexp

0

@�Z

x

0C

(Az

dz +Az

dz)

1

A

@zAz � @zAz + [Az, Az] = 0

where

and

U ! gU

An infinitesimal gauge transformation on the wave functional

=

Zd2z✏a

✓@z

�

�Aaz+ fabcAb

z�

�Acz

◆

�✏ [Az] = � k

2⇡

Zd2z✏a(F a

zz � @zAaz)

=k

2⇡

Zd2z✏a(@zA

az)

�✏ [Az] =

Zd2z �✏A

az

�

�Aaz

then using , we getAaz [A

az ] =

2⇡

k

�

�Aaz [Aa

z ]

The Wave Functional for CS

�✏Aaz = Dz✏

a

�✏ =k

2⇡

Zd2z✏a(@zA

az)

[Az] = exp(kSWZW (U))

This is a well known condition and it is solved by

Az = �@zUU�1

The Wave Functional for CS

= ��

Generally the wave-functional is in the form

satisfies the Gauss’ law

(gauge invariant) required to satisfy the Schrödinger’s equation

� = 1H = 0

we take

is where the scale dependence would be hidden�( )

The Measure (CS)The metric of the space of gauge potentials

ds2SL(N,C) = 8

ZTr[(�UU�1)(U †�1�U†)]

The metric of SL(N,C)

Then the measure is

dµ(A ) = det(DzDz)dµ(U,U†)

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A<latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit><latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit>

ds

2A =

Zd

2x �A

ai �A

ai = �8

ZTr(�Az�Az)

=8

ZTr[Dz(�UU

�1)Dz(U†�1

�U

†)]<latexit sha1_base64="ap6Vlt+VXvOTe+JnMZs+QXH1DAM=">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</latexit><latexit sha1_base64="ap6Vlt+VXvOTe+JnMZs+QXH1DAM=">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</latexit>

where and

det(DzDz) = constant⇥ e

2cASWZW (H)

H = U†U H 2 SL(N,C)/SU(N)

The Inner Product for CS TheoryThe inner product is given by

h1|2i =Z

d�(M)�⇤1�2 !

Zd�(M)e�K ⇤

1 2

h | iCS =

Zdµ(H)e(2ca+k)SWZW (H)

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CS inner product

Quantization of Topologically Massive Yang-Mills Theory

Topologically Massive Yang-Mills TheoryThe action is given by

Here m is called the topological mass. The field equations of this theory are,

✏µ↵�F↵� +1

mD⌫F

µ⌫ = 0

STMYM =SCS + SYM

=� k

4⇡

Z

⌃⇥[ti,tf ]

d

3x ✏

µ⌫↵Tr

✓Aµ@⌫A↵ +

2

3AµA⌫A↵

◆

� k

4⇡

1

4m

Z

⌃⇥[ti,tf ]

d

3x Tr Fµ⌫F

µ⌫

Topologically Massive Yang-Mills TheoryTo simplify the notation, we define,

whereAz = Az + Ez Az = Az + Ez

Ez =i

2mF 0z

Ez = � i

2mF 0z

then the momenta are

⇧az =ik

4⇡Aa

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⇧az = � ik

4⇡Aa

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(transform like gauge fields)

The Kähler potential is K =k

4⇡

Z

⌃(Aa

zAaz +Aa

zAaz)

⌦ =ik

4⇡

Z

⌃

(�Aaz�A

az + �Aa

z�Aaz)The symplectic two-form is

Topologically Massive Yang-Mills Theory

Aµ = Aµ +1

2m✏µ↵�F

↵�

Bz =1

2(A1 + iA2)

Bz =1

2(A1 � iA2)

Cz =1

2(A1 + iA2)

Cz =1

2(A1 � iA2)

Using the mixed gauge fields

⌦ =ik

4⇡

Z

⌃

(�Baz �B

az + �Ca

z �Caz )

TMYM phase space consists of two Chern-Simons phase spaces with levels k/2

We choose the Kähler polarization

�[Az, Az, Az, Az] = e�12K [Az, Az]


An infinitesimal gauge transformation on the wave-functional

�✏ [Az, Az] =

Zd2z

✓�

�Aaz�✏A

az +

�

�Aaz

�✏Aaz

◆

=k

4⇡

Zd2z✏a

⇣@zAz + @zAz � 2Fzz �DzEz +DzEz

⌘a

The Gauss law [2Fzz +DzEz �DzEz] = 0

then the infinitesimal gauge transformation becomes

�✏ =k

4⇡

Zd2z✏a(@zA

az + @zA

az)


same solution, using Az = �@zU U�1

[Az, Az] = exp

k

2(SWZW (U) + SWZW (U))

��

Here is a gauge invariant functional. It is required to satisfy the Schrödinger’s equation.

�

The Hamiltonian

[Eaz (x), E

bz(y)] = �8⇡

k

�

ab�

(2)(x� y)Ez

Ez is the creation andis the annihilationoperator

H =m

2↵(Ea

zEaz + Ea

zEaz )

| {z }+

↵

mBaBa

| {z }T V

To get rid of the infinite energy term, Hamiltonian needs to be normal ordered as

H =m

↵Ea

zEaz +

↵

mBaBa

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The vacuum wave-functional is given by H = 0<latexit sha1_base64="NIE2dL+DdcZa3wGnsRsnufnQwms=">AAAB+XicbVBNS8NAFHypX7V+pXr0slgETyUVUS9C0UuPFYwtNKFstpt26WYTdjdKif0pXjyoePWfePPfuGlz0NaBhWHmPd7sBAlnSjvOt1VaWV1b3yhvVra2d3b37Or+vYpTSahLYh7LboAV5UxQVzPNaTeRFEcBp51gfJP7nQcqFYvFnZ4k1I/wULCQEayN1LerXoT1iGCetaZeW7Erp2/XnLozA1omjYLUoEC7b395g5ikERWacKxUr+Ek2s+w1IxwOq14qaIJJmM8pD1DBY6o8rNZ9Ck6NsoAhbE0T2g0U39vZDhSahIFZjIPqha9XPzP66U6vPQzJpJUU0Hmh8KUIx2jvAc0YJISzSeGYCKZyYrICEtMtGmrYkpoLH55mbin9fO6c3tWa14XbZThEI7gBBpwAU1oQRtcIPAIz/AKb9aT9WK9Wx/z0ZJV7BzAH1ifP0+Qk5A=</latexit><latexit sha1_base64="NIE2dL+DdcZa3wGnsRsnufnQwms=">AAAB+XicbVBNS8NAFHypX7V+pXr0slgETyUVUS9C0UuPFYwtNKFstpt26WYTdjdKif0pXjyoePWfePPfuGlz0NaBhWHmPd7sBAlnSjvOt1VaWV1b3yhvVra2d3b37Or+vYpTSahLYh7LboAV5UxQVzPNaTeRFEcBp51gfJP7nQcqFYvFnZ4k1I/wULCQEayN1LerXoT1iGCetaZeW7Erp2/XnLozA1omjYLUoEC7b395g5ikERWacKxUr+Ek2s+w1IxwOq14qaIJJmM8pD1DBY6o8rNZ9Ck6NsoAhbE0T2g0U39vZDhSahIFZjIPqha9XPzP66U6vPQzJpJUU0Hmh8KUIx2jvAc0YJISzSeGYCKZyYrICEtMtGmrYkpoLH55mbin9fO6c3tWa14XbZThEI7gBBpwAU1oQRtcIPAIz/AKb9aT9WK9Wx/z0ZJV7BzAH1ifP0+Qk5A=</latexit>

↵ =4⇡

k

The Hamiltonian

In the strong coupling limit(large m), we can ignore the potential energy term

To find the vacuum wave-functional

Ez 0 = 0

� = exp

✓� k

8⇡

ZEa

z Eaz

◆= 1 +O(1/m2)

Eaz = �Ea

z � 8⇡

k

�ln�

�Eaz Az = Az + Ezwhere

The Measure(TMYM)The metric of the space of gauge potentials

ds2A =� 4

ZTr(�Az�Az + �Az�Az)

=4

ZTr[Dz(�U U�1)Dz(U

†�1�U†) +Dz(�UU�1)Dz(U†�1�U †)]

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The measure

dµ(A ) = det(DzDz)det(DzDz)dµ(U†U)dµ(U†U)

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A<latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit><latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit>

|{z} |{z}N N†= =

where

det(DzDz)det(DzDz) = constant⇥ e

2cA�SWZW (N)+SWZW (N†)

�

TMYM and CS

�⇤0�0 = e�

k8⇡

R(Ea

z Eaz +Ea

z Eaz ) = 1 +O(1/m2)

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h | iCS =



CS inner product

Two CS parts!

h 0| 0iTMYMk⇡Z

dµ(N)dµ(N†)e(2cA+ k2 )�SWZW (N)+SWZW (N†)

�= h | i2CSk/2

TMYM inner product

h 0| 0i =Z

dµ(N)dµ(N†)e(2cA+ k2 )�SWZW (N)+SWZW (N†)

�e�

k8⇡

R(Ea

z Eaz +Ea

z Eaz )

CS Splitting and Gauge Invariance

h 0| 0iTMYMk = h | iCSk/2h | iCSk/2

+O(1/m2)

1

2SCS(B) +

1

2SCS(C) ! 1

2SCS(B) +

1

2SCS(C) + 2⇡k!

Gauge invariance:

Quantization of Pure Yang-Mills Theory

Pure Yang-Mills TheoryThe action is given by

SYM = � k

4⇡

1

4m

Z

⌃⇥[ti,tf ]

d

3x Tr (Fµ⌫F

µ⌫)

The symplectic two-form is

⌦ =

Z

⌃

(�Eaz �A

az + �Aa

z�Eaz )

Gauss’ law is

DzEaz �DzE

az = 0

Phase Space Geometry of YM

⌦ =

Z

⌃

(�Aaz�A

az � �Aa

z�Aaz)

Symplectic two-form can be written as

Az = Az + Ez

Az = Az � Ez

where

⌦ =ik

4⇡

Z

⌃

(�Baz �B

az � �Ca

z �Caz )

Bz =1

2(A1 + iA2)

Using the mixed gauge fields

Cz =1

2(A1 + iA2)

YM phase space consists of two CS phase spaces with levels k/2 and -k/2

YM Wave-functionalOnce again, we choose the holomorphic polarization�[Az, Az, Az, Az] = e�

12K [Az, Az]

�✏ =k

4⇡

Zd2z✏a (@zE

az �DzE

az +DzE

az )

Infinitesimal gauge transformation on wave-functional

�✏ =k

4⇡

Zd2z ✏a (@zE

az )

=k

4⇡

Zd2z ✏a

⇣@zA

az � @zA

az

⌘

=k

4⇡

Zd2z ✏a

⇣@zA

az � @zA

az

⌘

After forcing Gauss’ law

YM Wave-functionalSolution is

�[Az, Az] = exp

k

2

�SWZW (U)� SWZW (U)

��

�[Az, Az] = exp

k

2

�SWZW (U)� SWZW (U)

��or equally

In temporal gauge TMYM and YM Hamiltonians are the same.Similarly, Schrödinger’s equation leads to

� = 1 +O(1/m2)

Measure

ds2A =� 4

ZTr(�Az�Az � �Az�Az)

= 4

ZTr[Dz(�U U�1)Dz(U

†�1�U†)�Dz(�UU�1)Dz(U†�1�U†)]

The metric of the space of gauge potentials A<latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit><latexit sha1_base64="ogCDyQ1nIIIz/VJofZm/sC+SkCU=">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</latexit>

dµ(A ) = det(DzDz)det(DzDz)dµ(U†U)dµ(U†U)|{z} |{z}

H2H1

dµ(A ) = e2cA�SWZW (H1)+SWZW (H2)

�dµ(H1)dµ(H2)

Then the gauge invariant measure is

YM and CS

h | iCS =



CS inner product

YM inner product

h 0| 0i =Z

dµ(H1)dµ(H2)e(2cA+ k

2 )SWZW (H1)+(2cA� k2 )SWZW (H2) +O(1/m2)

h 0| 0iYMk = h | iCSk/2h | iCS�k/2

+O(1/m2)

Gauge invariance:

1

2SCS(B)� 1

2SCS(C) ! 1

2SCS(B)� 1

2SCS(C) + ⇡k! � ⇡k!

Wilson Loops and Chern-Simons Splitting

Wilson Loops

Let us define TR

(C) = TrR

P e�HC Aµdx

µ

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WR

(C) = TrR

P e�HC Aµdx

µ

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for TMYM theory

T (C1)W (C2) = e2⇡ik l(C1,C2)W (C2)T (C1)

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T(C) is like a ’t Hooft loop for TMYM theory

Wilson Loops

At large finite distances, TMYM and pure YM theories act analogous to topologically massive AdS gravity (at corresponding limits) and their observables are link invariants.

hWR1(C1)TR2(C2)iTMYM2k =

✓hWR1(C1)iCSk

◆✓hWR2(C2)iCSk

◆+O(1/m2)

For TMYM theory with even level number

hWR1(C1)TR2(C2)iYM2k =

✓hWR1(C1)iCSk

◆✓hWR2(C2)iCS�k

◆+O(1/m2)

For YM theory with even level number

Thank You

TMYM - ASU seminar 03:27:2015

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