This document discusses Chern-Simons decomposition of 3D gauge theories at large distances. It outlines topics including Wilson loops and knot theory, geometric quantization of Chern-Simons theory, quantization of topologically massive Yang-Mills theory using Chern-Simons splitting, and quantization of pure Yang-Mills theory using Chern-Simons splitting. The document also discusses Wilson loops and their relation to Chern-Simons splitting.
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This lecture notes were written as part of the course "Pattern Recognition and Machine Learning" taught by Prof. Dinesh Garg at IIT Gandhinagar. This lecture notes deals with Linear Regression.
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International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
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The Wow! signal was a strong narrowband radio signal received on August 15, 1977, by Ohio State University's Big Ear radio telescope in the United States, then used to support the search for extraterrestrial intelligence. The signal appeared to come from the constellation Sagittarius and bore the expected hallmarks of extraterrestrial origin.
Let M be a semiprime -ring satisfying a certain assumption. Then we prove that every Jordan left
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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The constructed spl
ine
curves have local shape control that make them useful in such geometric applications a
s
real
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ISSN 2350-1022
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Paper Publications
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
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I will appreciate your comments.
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Let M be a semiprime -ring satisfying a certain assumption. Then we prove that every Jordan left
higher k-centralizer on M is a left higher k-centralizer on M. We also prove that every Jordan higher kcentralizer
of a 2-torsion free semiprime -ring M satisfying a certain assumption is a higher k-centralizer
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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The paper presents a technique for construction
of
C
n
interpolating
rational Bézier
spline curves by means
of blending
rational
quadric Bézier curves. A class of polynomials which satisfy special boundary
conditions is used for blending. Properties of the polynomials
are considered
.
The constructed spl
ine
curves have local shape control that make them useful in such geometric applications a
s
real
-
time
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Маніфэст Беларускага Вызвольнага Руху - вайскова-палітычнай арганізацыі беларускіх эмігрантаў, якая існавала ў 1940-я - 1960-я гг. Адзін з праграмных дакумэнтаў пасьляваеннай беларускай палітычнай эміграцыі
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Plee send comments and suggestions to improvements to solo.hermelin@gmail.com. Thanks/
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Brian Covello: Review on Cycloidal Pathways Using Differential EquationsBrian Covello
Brian Covello uses differential equations to provide an avenue for bridging the tautochrone and brachistochrone. The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659. He proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve was a cycloid.
On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other...[1]
Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle which generates the cycloid, multiplied by π⁄2. In modern terms, this means that the time of descent is , where r is the radius of the circle which generates the cycloid and g is the gravity of Earth.
This solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper (Acta Eruditorum, 1690) that saw the first published use of the term integral.[2]
Schematic of a cycloidal pendulum.
The tautochrone problem was studied more closely when it was realized that a pendulum, which follows a circular path, was not isochronous and thus his pendulum clock would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved to not be useful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock escapements could greatly reduce this source of inaccuracy.
Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem.
A brachistochrone curve (Gr. βράχιστος, brachistos - the shortest, χρόνος, chronos - time) or curve of fastest descent, is the path that will carry a point-like body from one place to another in the least amount of time. The body is released at rest from the starting point and is constrained to move without friction along the curve to the end point, while under the action of constant gravity. The brachistochrone curve is the same as the tautochrone curve for a given starting point.
Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve.
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In order to account for large variance and fat tail of damage by natural disaster, we study a simple model by combining distributions of disaster and population/property with their spatial correlation. We assume fat-tailed or power-law distributions for disaster and population/property exposed to the disaster, and a constant vulnerability for exposed population/property. Our model suggests that the fat tail property of damage can be determined either by that of disaster or by those of population/property depending on which tail is fatter. It is also found that the spatial correlations of population/property can enhance or reduce the variance of damage depending on how fat the tails of population/property are. In case of tornadoes in the United States, we show that the damage does have fat tail property. Our results support that the standard cost-benefit analysis would not be reliable for social investment in vulnerability reduction and disaster prevention.
http://ascelibrary.org/doi/abs/10.1061/9780784413609.277
http://arxiv.org/abs/1407.6209
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...
TMYM - ASU seminar 03:27:2015
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
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.
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
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]
⇤
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=">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</latexit><latexit sha1_base64="KiYftcdnHd6FKoetbaYQhBMFEZg=">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</latexit>
dµ(A ) = det(D¯zDz)dµ(H)<latexit sha1_base64="HD3wGYWslVZIGU02gp2WbVZ8HbY=">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</latexit><latexit sha1_base64="HD3wGYWslVZIGU02gp2WbVZ8HbY=">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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>
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)
<latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit><latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit>
CS inner product
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
<latexit sha1_base64="TesU9nwYU3Zk+nzxUBkamcnVdso=">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</latexit><latexit sha1_base64="TesU9nwYU3Zk+nzxUBkamcnVdso=">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</latexit>
⇧a¯z
=
ik
4⇡
˜Aa
z
<latexit sha1_base64="D9et4XQC+Z1dn+ac3xsyaup2yho=">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</latexit><latexit sha1_base64="D9et4XQC+Z1dn+ac3xsyaup2yho=">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</latexit>
(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=">AAAC7XicbVHdbtMwGHXC31b+OrjkxqJCapnokgnBbiZNoErcMSTKJsVJ5LhOa9V2ItsBWsuvwRXilmeCp8HpOlg7PsnyyTnH8efzFTVn2kTRryC8cfPW7Ts7u5279+4/eNjde/RJV40idEwqXqnzAmvKmaRjwwyn57WiWBScnhXzt61+9pkqzSr50Sxqmgo8laxkBBtP5V2DyIzlUfa83TMLXR4d08y+QKXCxM6dPUI1c4hJA/ujDOdLiASxI+ehRQVWdunc/ujq1199CQfuON5HApsZwdy+d/34QGSHg07e7UXDaFXwOojXoAfWdZrvBQWaVKQRVBrCsdZJHNUmtVgZRjh1HdRoWmMyx1OaeCixoDq1q3gcfOaZCSwr5Zd/x4q9esJiofVCFN7Z9qq3tZb8n5Y0pjxKLZN1Y6gkFxeVDYemgm3WcMIUJYYvPMBEMd8rJDPsgzV+Ihu3tP9WutQbL7EcG/rVd+ZZSb+QSggsJ9YH7OxlqltSjV0SpxZxWpp+L0aKTWdmsG0q/pmSS1Pq/FTi7RlcB+PD4ath9OFl7+TNejw74Al4CvogBq/BCXgHTsEYEPA7AMFu0Anr8Fv4PfxxYQ2D9ZnHYKPCn38Ak0vuJA==</latexit><latexit sha1_base64="bJkbdw5nQFpUnET3dE94NRMeQCc=">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</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:
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)
<latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit><latexit sha1_base64="P8z68TnDjPOmAGjqM42fHT3vpfE=">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</latexit>
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!
43. Wilson Loops
Let us define TR(C) = TrR P e
H
C
˜Aµdxµ
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WR(C) = TrR P e
H
C
Aµdxµ
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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