Syed Azharuddin - Dr. Babasaheb Ambedkar Marathwada

22 Pages·2012·0.3 MB·English

Matrix Models & Knot Theory | Moscow 10/03 (1)
05/03
Matrix Models and Knot Theory
P. Zinn-Justin
References:
P. Zinn-Justin, J.-B. Zuber, math-ph/9904019, math-ph/0002020,
ƒ
math-ph/0303049
P. Zinn-Justin, math-ph/9910010, math-ph/0106005.
ƒ
J. Jacobsen, P. Zinn-Justin, math-ph/0102015, math-ph/0104009.
ƒ
G. Schaefier, P. Zinn-Justin, math-ph/0304034.
ƒ
Classiflcation and Enumeration of Knots, Links, Tangles.
†
Feynman diagrams. O(n) matrix model and renormalization.
†
Universality and conjectures on asymptotic counting.
†
Phase diagram of O(n)-symmetric 2D statistical models.
†
Numerical check: Monte Carlo.
†
Virtual link diagrams and Links on thickened surfaces.
†
Renormalization and the generalized (cid:176)yping conjecture for vir-
†
tual alternating links.
Matrix Models & Knot Theory | Moscow 10/03 (2)
A bit of History
:::
The Gordian knot: (Piotr Pieranski, 2001)
Knots represented by their projection: diagrams (Tait, 1876):
†
6
1
Two diagrams represent the knot/link/tangle ifi they are related
†
by a sequence of Reidemeister moves: (Reidemeister, 1932)
; ;
All knots are connected sums of prime knots (Schubert, 1949):
†
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:1)(cid:2)(cid:1)(cid:0)(cid:2) (cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:3)(cid:4)
Matrix Models & Knot Theory | Moscow 10/03 (3)
Knots, links and tangles
Links are collections of knots: tangles have strings coming out:
Alternating vs Non-Alternating:
†
Tait’s (cid:176)yping conjecture: (Tait, 1898)
Two reduced alternating diagrams represent the same object ifi
they are related by a sequence of (cid:176)ypes:
Proved by Menasco and Thisthlethwaite (’91).
Matrix Models & Knot Theory | Moscow 10/03 (4)
Matrix Models & Knot Theory | Moscow 10/03 (5)
What is the problem?
We want to enumerate (prime) alternating tangles with a given
number of connected components and crossings:
1
p k
¡(n;g) = a g n
k;p
X
k;p=1
Example: tangles with four external legs. Two types:
= + +
3 3
¡ (n;g) = g + g + g +
1
¢¢¢
= + +
2 3 4
¡ (n;g) = g + g + ng +
2
¢¢¢
n = \number of colors": a diagram with k closed loops can be
drawn in nk ways.
Matrix Models & Knot Theory | Moscow 10/03 (6)
Feynman diagrams
Gaussian integral over real variables x , A = AT > 0 def. matrix:
i
(2…)n=2
d 1 x A x
d xe¡2 i ij j =
1
Z
P det2A
1 x A x + b x (2…)n=2 1 b A 1b
dnxe¡2 i ij j i i = e2 i i¡j j
1
Z P P det2A P
Difierentiate w.r.t. b
i
dnx x x x e 1x:A:x
k k k ¡2
x x x = 1 2 ¢¢¢ ‘
k k k
h 1 2 ¢¢¢ ‘i R dnxe 1x:A:x
¡2
@ R @ 1b:A 1:b
= e2 ¡
@b ¢¢¢ @b
k k flb=0
1 ‘
fl
= A 1 fl A 1
¡ ¡
k k k k
P1 P2 ¢¢¢ P‘ 1 P‘
all Xdistinct ¡
pairings P of the k
Non-Gaussian integral: power series (\perturbative") expansion.
Example: (d = 1)
1Ax2 + g x4 2… 21 1 gp x4 p 1Ax2
Z = dxe¡2 4! = dx e¡2
Z A p! Z 4!
‡ · X ‡ ·
p=0
2… 1 1 gp
2 2p
= A
¡
A AutG
‡ · X X
p=0 graphs G with 2p lines j j
and p 4 valent vertices
¡
log Z = connected Feynman diagrams
g g2 1 1
= + + +
8A2 A4 2:4! 24 ¢¢¢
‡ ·
=
2
x =
h i
Matrix Models & Knot Theory | Moscow 10/03 (7)
Matrix Integrals: Feynman Rules
N N Hermitean matrices M,
£
dM = dM d eM d mM
i ii i<j ij ij
< =
Q Q
N[ 1 trM2 + g trM4]
Z = dMe ¡2 4
Z
Feynman rules:
i l
propagator = 1 – –
j k N i‘ jk
p n
m
q
4-valent vertex = gN– – – –
jk ‘m np qi
i l
j k
Count powers of N in a connected diagram:
each vertex N;
† !
each double line N 1;
¡
† !
each loop N.
† !
#vert: #lines + #loops = ´ (§)
Euler
¡
’t Hooft (1974):
g#vert:(§)
2 2genus(§)
log Z = N
¡
symm. factor
X
conn: surf:§
Matrix Models & Knot Theory | Moscow 10/03 (8)
A Matrix Model for Alternating Link Diagrams
n
N tr 1M2 + g(M M )2
Z(N)(n;g) = dMa e ¡2 a 4 a b
Z ¡ ¢
Y
a=1
g
The large N free energy F(n;g) and correlation functions are dou-
ble generating series in n, g (number of connected components,
number of crossings).
F(n;g) counts link diagrams (weighted by their symmetry fac-
tors):
log Z(N)(n;g) 1
p k
F(n;g) = lim = f g n
k;p
N2
N
X
!1 k;p=1
The correlation functions count tangle diagrams:
1
lim tr(M M M M M M ) =
1 2 3 2 1 3
N ¿N (cid:192)
!1 c
Matrix Models & Knot Theory | Moscow 10/03 (9)
From tangle diagrams to tangles:
Renormalization
General idea: removal of the redundancy associated to multiple
equivalent diagrams acts like a \flnite renormalization" on the
model.
Reduced diagrams renormalization of the quadratic term in
† )
the action.
Taking into account the (cid:176)yping equivalence renormalizes the
†
quartic term. However, there are two four-vertex interactions
compatible with the O(n)-symmetry more general O(n) model:
!
n
(N)
Z (n;t;g ;g ) = dM
1 2 a
Z
Y
a=1
N tr tM2 + g1M M M M + g2M M M M
e ¡2 a 4 a b a b 2 a a b b
£ ¡ ¢⁄
t−1 g g
1 2
t, g and g are functions of the renormalized coupling constant
1 2
g, chosen such that the correlation functions are the appropriate
generating series in g of the number of alternating links.
Matrix Models & Knot Theory | Moscow 10/03 (10)
Exactly solved cases
n = 1: the counting of alternating tangles, and more
†
Usual one-matrix model:
N tr tM2 + g0M4
(N)
Z (t;g0) = dM e ¡2 4
Z ¡ ¢
with g = g + 2g .
0 1 2
Renormalization equations recombine into a flfth degree equation:
1 + 2g g2
2 3 4 5
32 64A + 32A 4 ¡ A + 6gA gA = 0
¡ ¡ 1 g ¡
¡
Correlation functions are given in terms of its solution. In partic-
ular, if 1 trM2‘ = a gp is the generating function of
N c 1p=0 p
› fi P
prime alternating tangles with 2‘ legs, then
p
p 5=2
a !1cstg p
p c¡ ¡
»
with g = p21001 101 (g 1 6:1479).
c 270¡ c¡ …
(‘ = 2: Sundberg & Thistlethwaite ’98)
The number f of prime alternating links grows like
p
)
p 7=2
f cstg p
p c¡ ¡
»
(Schaefier & Kunz-Jacques, ’01)
n = 2 :::
† ¡