1. CHERN-SIMONS DECOMPOSITION
OF 3D GAUGE THEORIES AT LARGE
DISTANCES
Tuna Yıldırım
(UIOWA, ASU)
Arizona State University
March 27, 2015
• Int.J.Mod.Phys.A, 30(7):1550034, 2015, arXiv:1311.1853
• arXiv:1410.8593 (preprint)

2. 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

3. Wilson Loops and
Knot Theory

4. Wilson Loops
Area 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)
. . .

5. 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 the
integral of a three-form on a three-manifold.
Gauge invariance and general covariance are the real reasons for the properties
of the expectation value (1.17) that we have observed. Gauge invariance forced us
to choose the external source to be expressed in terms of closed paths (conserved
external currents), since only gauge-invariant quantities have an intrinsic mean-
ing in gauge theories. Because of general covariance, the final result (1.17) only
depends on the topological structure of the closed contours. This is why there is
invariance 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.
A link is a union of
non-intersecting
knots.

6. A 3D Knot

7. 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)

8. Jones Polynomial of the Trefoil Knot
We 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

9. 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

10. The Wilson loop integral is
WR(C) = TrR
✓
Pexp i
I
c
Aµdxµ
◆
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]

11. SL+
1
SL = 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 1
hWL i = z( )hWL0
i
k: level number of CS

12. Topologically Massive
AdS Gravity

13. Topologically Massive AdS Gravity[3,4]
The action is
S =
Z
d3
x

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±
µ
a
b[e] = !µ
a
b[e] ± ✏a
bceµ
c
SCS[A] =
1
2
Z
✏µ⌫⇢
✓
Aµ
a
b@⌫A⇢
b
a +
2
3
Aµ
a
cA⌫
c
bA⇢
b
a
◆
where
and
[3] S. Deser, R. Jackiw, and S. Templeton, 1982.
[4] A. Achúcarro and P.K. Townsend, 1986.

14. Topologically Massive AdS Gravity
For small values of μ (near CS limit)
S[e] ⇡
1
2µ
SCS
⇥
A+
[e]
⇤
+
1
2µ
SCS
⇥
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
2
SCS
⇥
A [e]
⇤ 1
2
SCS
⇥
A+
[e]
⇤

15. Geometric Quantization of
Chern-Simons Theory

16. Chern-Simons Theory
SCS =
k
4⇡
Z
⌃⇥[ti,tf ]
d3
x ✏µ⌫↵
Tr
✓
Aµ@⌫A↵ +
2
3
AµA⌫A↵
◆
SCS(A) ! SCS(Ag
) = SCS(A) + 2⇡k!(g)
Under Aµ ! Ag
µ = gAµg 1
(@µg)g 1
!(g) =
1
24⇡2
Z
d3
x ✏µ⌫↵
Tr(g 1
@µgg 1
@⌫gg 1
@↵g)
is an integer, called the winding number.
k has to be an integer
eiSCS (A)
= eiSCS (Ag
)

17. 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⇡
Fa
z¯z
is the generator of infinitesimal
gauge transformations
SCS =
k
4⇡
Z
⌃⇥[ti,tf ]
d3
x ✏µ⌫↵
Tr
✓
Aµ@⌫A↵ +
2
3
AµA⌫A↵
◆

18. The conjugate momenta are
and ⇧a¯z
=
ik
4⇡
Aa
z⇧az
=
ik
4⇡
Aa
¯z
Chern-Simons Theory
Then the inner product is
h1|2i =
Z
d (M) ⇤
1 2 !
Z
d (M)e K ⇤
1 2
⌦ =
ik
2⇡
Z
⌃
Aa
¯z Aa
z
K =
k
2⇡
Z
⌃
Aa
¯zAa
z
The phase space is Kähler with
and Kähler potential
We choose the Kähler polarization
[Az, A¯z] = e
1
2 K
[A¯z]

19. The Wave Functional for CS[3,4]
Aa
z [Aa
¯z] =
2⇡
k Aa
¯z
[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
Fa
z¯z [Aa
¯z] = 0
If Σ is simply connected we can parametrize the gauge fields as
A¯z = @¯zUU 1 Az = (U† 1
)@zU† U 2 SL(N, C)
U(x, 0, C) = Pexp
0
@
Z x
0
C
(A¯zd¯z + Azdz)
1
A
@zA¯z @¯zAz + [Az, A¯z] = 0
where
and
U ! gU

20. An infinitesimal gauge transformation on the wave functional
=
Z
d2
z✏a
✓
@¯z
Aa
¯z
+ fabc
Ab
¯z
Ac
¯z
◆
✏ [A¯z] =
k
2⇡
Z
d2
z✏a
(Fa
z¯z @zAa
¯z)
=
k
2⇡
Z
d2
z✏a
(@zAa
¯z)
✏ [A¯z] =
Z
d2
z ✏Aa
¯z
Aa
¯z
then using , we getAa
z [Aa
¯z] =
2⇡
k Aa
¯z
[Aa
¯z]
The Wave Functional for CS
✏Aa
¯z = D¯z✏a

21. ✏ =
k
2⇡
Z
d2
z✏a
(@zAa
¯z)
[A¯z] = exp(kSW ZW (U))
This is a well known condition and it is solved by
A¯z = @¯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
= 1
H = 0
we take
is where the scale
dependence would be hidden( )

22. The Measure (CS)
The metric of the space of gauge potentials
ds2
SL(N,C) = 8
Z
Tr[( UU 1
)(U† 1
U†
)]
The metric of SL(N,C)
Then the measure is
dµ(A ) = det(D¯zDz)dµ(U, U†
)<latexit sha1_base64="KiYftcdnHd6FKoetbaYQhBMFEZg=">AAACr3icbVHLbtQwFPWEVymvKSzZRIyQZiQ0SipUuqnUli5YFonQkZI03Dg3Gau2E9kOdGrlT/o1bOEH+Bs86SCYKVeydHTOub6vvOFMmyD4NfDu3L13/8HWw+1Hj588fTbcef5Z162iGNGa12qWg0bOJEaGGY6zRiGInONZfvF+qZ99RaVZLT+ZRYOpgEqyklEwjsqGe0Ui2nEiwMw1VfaomxwUaMYnmU1yUPaq606yq0lvit5E50kBVYVqkg1HwTTow78NwhUYkVWcZjuDPClq2gqUhnLQOg6DxqQWlGGUY7edtBoboBdQYeygBIE6tf2Anf/aMYVf1so9afye/TfDgtB6IXLn7CfZ1Jbk/7S4NeV+aplsWoOS3hQqW+6b2l9uyy+YQmr4wgGgirlefToHBdS4na5VWf6tdKnXJrEcDF66zhwr8RuthQBZ2ETQzvY7p8A3pAa6OExtwrE041GYKFbNzWTTlP81xX9MaeeuEm7e4DaIdqd70+Dj29Hh8eo8W+QleUXGJCTvyCH5QE5JRCi5Jt/JD/LT2/Vm3rn35cbqDVY5L8haeOw3ApPYcg==</latexit><latexit sha1_base64="KiYftcdnHd6FKoetbaYQhBMFEZg=">AAACr3icbVHLbtQwFPWEVymvKSzZRIyQZiQ0SipUuqnUli5YFonQkZI03Dg3Gau2E9kOdGrlT/o1bOEH+Bs86SCYKVeydHTOub6vvOFMmyD4NfDu3L13/8HWw+1Hj588fTbcef5Z162iGNGa12qWg0bOJEaGGY6zRiGInONZfvF+qZ99RaVZLT+ZRYOpgEqyklEwjsqGe0Ui2nEiwMw1VfaomxwUaMYnmU1yUPaq606yq0lvit5E50kBVYVqkg1HwTTow78NwhUYkVWcZjuDPClq2gqUhnLQOg6DxqQWlGGUY7edtBoboBdQYeygBIE6tf2Anf/aMYVf1so9afye/TfDgtB6IXLn7CfZ1Jbk/7S4NeV+aplsWoOS3hQqW+6b2l9uyy+YQmr4wgGgirlefToHBdS4na5VWf6tdKnXJrEcDF66zhwr8RuthQBZ2ETQzvY7p8A3pAa6OExtwrE041GYKFbNzWTTlP81xX9MaeeuEm7e4DaIdqd70+Dj29Hh8eo8W+QleUXGJCTvyCH5QE5JRCi5Jt/JD/LT2/Vm3rn35cbqDVY5L8haeOw3ApPYcg==</latexit>
dµ(A ) = det(D¯zDz)dµ(H)<latexit sha1_base64="HD3wGYWslVZIGU02gp2WbVZ8HbY=">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</latexit><latexit sha1_base64="HD3wGYWslVZIGU02gp2WbVZ8HbY=">AAACpXicbVFda9RAFJ2NX7V+bfXRl+Ai7IIsSRH1Rajahz5JhW5bSEK4mdzsDp2ZhJkbdTvkT/hrfNV/4b9xNl3R3Xph4HDOuXO/ikYKS1H0axDcuHnr9p2du7v37j94+Gi49/jU1q3hOOO1rM15ARal0DgjQRLPG4OgColnxcWHlX72GY0VtT6hZYOZgrkWleBAnsqHL8pUteNUAS0sN+5dN3lbIo0Pc5cWYNxl1x3ml5PedDTJh6NoGvURXgfxGozYOo7zvUGRljVvFWriEqxN4qihzIEhwSV2u2lrsQF+AXNMPNSg0GauH6sLn3umDKva+Kcp7Nl/Mxwoa5eq8M6+/21tRf5PS1qq3mRO6KYl1PyqUNXKkOpwtaOwFAY5yaUHwI3wvYZ8AQY4+U1uVFn9bWxlNyZxEgi/+s48q/ELr5UCXbpU8c71m+Ygt6QGuiTOXCqxovEoTo2YL2iybSr+mpI/pqzzV4m3b3AdzPanr6bRp5ejg/fr8+ywp+wZG7OYvWYH7Igdsxnj7Bv7zn6wn8E4+BicBKdX1mCwznnCNiLIfwMk69Rc</latexit>
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>
ds2
A =
Z
d2
x Aa
i Aa
i = 8
Z
Tr( A¯z Az)
=8
Z
Tr[D¯z( 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(D¯zDz) = constant ⇥ e2cASW ZW (H)
H = U†
U H 2 SL(N, C)/SU(N)

23. The Inner Product for CS Theory
The inner product is given by
h1|2i =
Z
d (M) ⇤
1 2 !
Z
d (M)e K ⇤
1 2
h | iCS =
Z
dµ(H)e(2ca+k)SW ZW (H)
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CS inner product

24. Quantization of Topologically
Massive Yang-Mills Theory

25. Topologically Massive Yang-Mills Theory
The action is given by
Here m is called the topological mass.
The field equations of this theory are,
✏µ↵
F↵ +
1
m
D⌫Fµ⌫
= 0
ST MY M =SCS + SY M
=
k
4⇡
Z
⌃⇥[ti,tf ]
d3
x ✏µ⌫↵
Tr
✓
Aµ@⌫A↵ +
2
3
AµA⌫A↵
◆
k
4⇡
1
4m
Z
⌃⇥[ti,tf ]
d3
x Tr Fµ⌫Fµ⌫

26. Topologically Massive Yang-Mills Theory
To simplify the notation, we define,
where˜Az = Az + Ez
˜A¯z = A¯z + E¯z
Ez =
i
2m
F0¯z
E¯z =
i
2m
F0z
then the momenta are
⇧az
=
ik
4⇡
˜Aa
¯z
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⇧a¯z
=
ik
4⇡
˜Aa
z
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(transform like
gauge fields)
The Kähler potential is K =
k
4⇡
Z
⌃
( ˜Aa
¯zAa
z + Aa
¯z
˜Aa
z)
⌦ =
ik
4⇡
Z
⌃
( ˜Aa
¯z Aa
z + Aa
¯z
˜Aa
z)The symplectic two-form is

27. Topologically Massive Yang-Mills Theory
˜Aµ = Aµ +
1
2m
✏µ↵ F↵
Bz =
1
2
( ˜A1 + iA2)
B¯z =
1
2
( ˜A1 iA2)
Cz =
1
2
(A1 + i ˜A2)
C¯z =
1
2
(A1 i ˜A2)
Using the mixed gauge fields
⌦ =
ik
4⇡
Z
⌃
( Ba
¯z Ba
z + Ca
¯z Ca
z )
TMYM phase space
consists of two
Chern-Simons phase
spaces with levels k/2

28. We choose the Kähler polarization
[Az, A¯z, ˜Az, ˜A¯z] = e
1
2 K
[A¯z, ˜A¯z]
Topologically Massive Yang-Mills Theory
An infinitesimal gauge transformation on the wave-functional
✏ [A¯z, ˜A¯z] =
Z
d2
z
✓
Aa
¯z
✏Aa
¯z +
˜Aa
¯z
✏
˜Aa
¯z
◆
=
k
4⇡
Z
d2
z✏a
⇣
@z
˜A¯z + @zA¯z 2Fz¯z DzE¯z + D¯zEz
⌘a
The Gauss law [2Fz¯z + DzE¯z D¯zEz] = 0

29. then the infinitesimal gauge transformation becomes
✏ =
k
4⇡
Z
d2
z✏a
(@¯z
˜Aa
z + @¯zAa
z)
Topologically Massive Yang-Mills Theory
same solution, using ˜A¯z = @¯z
˜U ˜U 1
[A¯z, ˜A¯z] = exp

k
2
(SW ZW ( ˜U) + SW ZW (U))
Here is a gauge invariant functional. It is
required to satisfy the Schrödinger’s equation.

30. The Hamiltonian
[Ea
z (x), Eb
¯z(y)] =
8⇡
k
ab (2)
(x y)
Ez
E¯z is the creation and
is the annihilation
operator
H =
m
2↵
(Ea
¯z Ea
z + Ea
z Ea
¯z )
| {z }
+
↵
m
Ba
Ba
| {z }
T V
To get rid of the infinite energy term,
Hamiltonian needs to be normal ordered as
H =
m
↵
Ea
¯z Ea
z +
↵
m
Ba
Ba
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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

31. 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⇡
Z
Ea
¯z Ea
z
◆
= 1 + O(1/m2
)
Ea
z = Ea
z
8⇡
k
ln
Ea
¯z
˜Az = Az + Ezwhere

32. The Measure(TMYM)
The metric of the space of gauge potentials
ds2
A = 4
Z
Tr( ˜A¯z Az + A¯z
˜Az)
=4
Z
Tr[ ˜D¯z( ˜U ˜U 1
)Dz(U† 1
U†
) + D¯z( UU 1
) ˜Dz( ˜U† 1 ˜U†
)]
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The measure
dµ(A ) = det( ˜D¯zDz)det(D¯z
˜Dz)dµ( ˜U†
U)dµ(U† ˜U)<latexit sha1_base64="kGffsJZLXqNjyYN9nSQj9Y0Pfvs=">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</latexit><latexit sha1_base64="kGffsJZLXqNjyYN9nSQj9Y0Pfvs=">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</latexit>
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( ˜D¯zDz)det(D¯z
˜Dz) = constant ⇥ e2cA SW ZW (N)+SW ZW (N†
)

33. TMYM and CS
⇤
0 0 = e
k
8⇡
R
(Ea
z Ea
¯z +Ea
¯z Ea
z )
= 1 + O(1/m2
)<latexit sha1_base64="bJkbdw5nQFpUnET3dE94NRMeQCc=">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</latexit><latexit sha1_base64="bJkbdw5nQFpUnET3dE94NRMeQCc=">AAAC7XicbVHdbtMwGHXC31b+OrjkxqJCapnokgnBbiZNoErcMSTKJsVJ5LhOa9V2ItsBWsuvwRXilmeCp8HpOlg7PsnyyTnH8efzFTVn2kTRryC8cfPW7Ts7u5279+4/eNjde/RJV40idEwqXqnzAmvKmaRjwwyn57WiWBScnhXzt61+9pkqzSr50Sxqmgo8laxkBBtP5V2DyIzlUfa83TMLXR4d08y+QKXCxM6dPUI1c4hJA/ujDOdLiASxI+ehRQVWdunc/ujq1199CQfuON5HApsZwdy+d/34QGSHg07e7UXDaFXwOojXoAfWdZrvBQWaVKQRVBrCsdZJHNUmtVgZRjh1HdRoWmMyx1OaeCixoDq1q3gcfOaZCSwr5Zd/x4q9esJiofVCFN7Z9qq3tZb8n5Y0pjxKLZN1Y6gkFxeVDYemgm3WcMIUJYYvPMBEMd8rJDPsgzV+Ihu3tP9WutQbL7EcG/rVd+ZZSb+QSggsJ9YH7OxlqltSjV0SpxZxWpp+L0aKTWdmsG0q/pmSS1Pq/FTi7RlcB+PD4ath9OFl7+TNejw74Al4CvogBq/BCXgHTsEYEPA7AMFu0Anr8Fv4PfxxYQ2D9ZnHYKPCn38Ak0vuJA==</latexit>
h | iCS =
Z
dµ(H)e(2ca+k)SW ZW (H)
<latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit><latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit>
CS inner product
Two CS parts!
h 0| 0iT MY Mk
⇡
Z
dµ(N)dµ(N†
)e(2cA+ k
2 ) SW ZW (N)+SW ZW (N†
)
= h | i2
CSk/2
TMYM inner product
h 0| 0i =
Z
dµ(N)dµ(N†
)e(2cA+ k
2 ) SW ZW (N)+SW ZW (N†
)
e
k
8⇡
R
(Ea
z Ea
¯z +Ea
¯z Ea
z )

34. CS Splitting and Gauge Invariance
h 0| 0iT MY Mk
= h | iCSk/2
h | iCSk/2
+ O(1/m2
)
1
2
SCS(B) +
1
2
SCS(C) !
1
2
SCS(B) +
1
2
SCS(C) + 2⇡k!
Gauge invariance:

35. Quantization of Pure
Yang-Mills Theory

36. Pure Yang-Mills Theory
The action is given by
SY M =
k
4⇡
1
4m
Z
⌃⇥[ti,tf ]
d3
x Tr (Fµ⌫Fµ⌫
)
The symplectic two-form is
⌦ =
Z
⌃
( Ea
¯z Aa
z + Aa
¯z Ea
z )
Gauss’ law is
DzEa
¯z D¯zEa
z = 0

37. Phase Space Geometry of YM
⌦ =
Z
⌃
( ˜Aa
¯z Aa
z Aa
¯z
ˆAa
z)
Symplectic two-form can be written as
˜A¯z = A¯z + E¯z
ˆAz = Az Ez
where
⌦ =
ik
4⇡
Z
⌃
( Ba
¯z Ba
z Ca
¯z Ca
z )
Bz =
1
2
( ˜A1 + iA2)
Using the mixed gauge fields
Cz =
1
2
(A1 + i ˆA2)
YM phase space consists
of two CS phase spaces
with levels k/2 and -k/2

38. YM Wave-functional
Once again, we choose the holomorphic polarization
[Az, A¯z, ˆAz, ˜A¯z] = e
1
2 K
[A¯z, ˜A¯z]
✏ =
k
4⇡
Z
d2
z✏a
(@zEa
¯z DzEa
¯z + D¯zEa
z )
Infinitesimal gauge transformation on wave-functional
✏ =
k
4⇡
Z
d2
z ✏a
(@zEa
¯z )
=
k
4⇡
Z
d2
z ✏a
⇣
@z
˜Aa
¯z @zAa
¯z
⌘
=
k
4⇡
Z
d2
z ✏a
⇣
@zAa
¯z @z
ˆAa
¯z
⌘
After forcing Gauss’ law

39. YM Wave-functional
Solution is
[A¯z, ˜A¯z] = exp

k
2
SW ZW ( ˜U) SW ZW (U)
[A¯z, ˆA¯z] = exp

k
2
SW ZW (U) SW ZW ( ˆU)
or equally
In temporal gauge TMYM and YM Hamiltonians are the same.
Similarly, Schrödinger’s equation leads to
= 1 + O(1/m2
)

40. Measure
ds2
A = 4
Z
Tr( ˜A¯z Az A¯z
ˆAz)
= 4
Z
Tr[ ˜D¯z( ˜U ˜U 1
)Dz(U† 1
U†
) D¯z( UU 1
) ˆDz( ˆU† 1 ˆU†
)]
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( ˜D¯zDz)det(D¯z
ˆDz)dµ( ˆU†
U)dµ(U† ˜U)|{z} |{z}
H2H1
dµ(A ) = e2cA SW ZW (H1)+SW ZW (H2)
dµ(H1)dµ(H2)
Then the gauge invariant measure is

41. YM and CS
h | iCS =
Z
dµ(H)e(2ca+k)SW ZW (H)
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CS inner product
YM inner product
h 0| 0i =
Z
dµ(H1)dµ(H2)e(2cA+ k
2 )SW ZW (H1)+(2cA
k
2 )SW ZW (H2)
+ O(1/m2
)
h 0| 0iY Mk
= h | iCSk/2
h | iCS k/2
+ O(1/m2
)
Gauge invariance:
1
2
SCS(B)
1
2
SCS(C) !
1
2
SCS(B)
1
2
SCS(C) + ⇡k! ⇡k!

42. Wilson Loops and
Chern-Simons Splitting

43. Wilson Loops
Let us define TR(C) = TrR P e
H
C
˜Aµdxµ
<latexit sha1_base64="OX+KCxgBjNK7iVaJwJwJW5JjNBU=">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</latexit><latexit sha1_base64="OX+KCxgBjNK7iVaJwJwJW5JjNBU=">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</latexit>
WR(C) = TrR P e
H
C
Aµdxµ
<latexit sha1_base64="d7Xpib2RqZ0P9Dxh53mJGODTQQg=">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</latexit><latexit sha1_base64="d7Xpib2RqZ0P9Dxh53mJGODTQQg=">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</latexit>
for TMYM theory
T(C1)W(C2) = e
2⇡i
k l(C1,C2)
W(C2)T(C1)<latexit sha1_base64="5sz6/GQ1LP7S9v4i44db96Bsov0=">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</latexit><latexit sha1_base64="5sz6/GQ1LP7S9v4i44db96Bsov0=">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</latexit>
T(C) is like a ’t Hooft loop for TMYM theory

44. 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)iT MY M2k
=
✓
hWR1
(C1)iCSk
◆✓
hWR2
(C2)iCSk
◆
+ O(1/m2
)
For TMYM theory with even level number
hWR1
(C1)TR2
(C2)iY M2k
=
✓
hWR1
(C1)iCSk
◆✓
hWR2
(C2)iCS k
◆
+ O(1/m2
)
For YM theory with even level number

45. Thank You