The sound of Knots and 44-manifolds
-manifolds
Sergei Gukov
Mikhail Khovanov
Piotr Sulkowski
Knots
?
=
Knots
?
=
4-manifolds
4_1
FIGURE EIGHT
ALEXANDER
JONES
1
SL(3)
GENUS
AMPHICHEIRAL
HOMFLY-PT
COLORED
Why integer coefficients?
1
i
[M.Khovanov]
2
1
0
-1
-2
-5
-3
-1
1
3
5
j
Paul Dirac (1939):
“Pure mathematics and physics are
becoming ever more closely connected,
though their methods remain different …
It is difficult to predict what the result of
all this will be. Possibly, the two subjects
will ultimately unify … At present we
are, of course, very far from this stage,
even with regard to some of the most
elementary questions. For example,
only four-dimensional space is of
importance in physics, while spaces
with other numbers of dimensions are of
about equal interest in mathematics.”
Extra Dimensions
Kaluza-Klein compactification
10d string theory
on
10-n
x Mn
“effective” theory
T[Mn ]
in 10-n dimensions
depends on
topology and
geometry of
Mn
10 = 4 + 6
E.Calabi
S.-T.Yau
Geometry of extra dimensions
determines the spectrum of 4d particles
and their interactions
0
104
108
1012
Energy Scale (GeV)
1016
R.Dijkgraaf ‘03
•
= spectrum of BPS particles in
supersymmetric theories
Strings in 6d
6d fivebrane theory
on
6-n
x Mn
“effective” theory
T[Mn ]
in 6-n dimensions
depends on
topology and
geometry of
Mn
6=3+3
3-manifold M3
complex flat connections
3d N = 2 theory
T[M 3]
supersymmetric vacua
6=3+3
3d N = 2 theory
3-manifold M3
T[M 3]
supersymmetric vacua
complex flat connections
spectrum of BPS
states
Knot homology
=
BPS
Classifying Phases of Matter
Quiver Chern-Simons theory
vertex
a
U(1) Chern-Simons at level a
a
ai
edge
aj
cf. [D.Belov, G.Moore]
[A.Kapustin, N.Saulina]
[J.Fuchs, C.Schweigert, A.Valentino]
:
Quiver Chern-Simons theory
integrate out A
3d Kirby moves
3d Kirby moves
A is Lagrange multiplier
Integrating out A makes B pure gauge
and removes all its Chern-Simons couplings
Plumbing graphs
Intersection form on
:
6=2+4
4-manifold M4
2d N = (0,2) theory
T[M 4]
6d fivebrane theory
on
2
x M4
depends on
topology and
geometry of
M4
6=2+4
4-manifold M4
very rich math
2d N = (0,2) theory
T[M 4]
very rich physics
6=2+4
4-manifold M4
0-handle
adding 2-handles: ADE
more general plumbing
graphs, Kirby moves, …
Vertex Operator
Algebra
Heisenberg algebra
affine Kac-Moody
VOAs associated with
even positive lattices …
line
26
/7
BMY
c = 12c
(abelian
)
Unitarity
constraints
rm
o
f
e
d
E Het
at
ns
o
i
at
ge
r
la
he
t
e
No
N
ine
l
r
c = 0 (large N)
Phases
Phases
[M.Stephanov]
Can we quantitatively understand
confinement and the mass gap?
• Extensively tested in computer
simulations
• Paper-and-pencil computation?
$1,000,000 Prize
The answer may involve gravity!
Confinement in 2d
‘t Hooft
Solvable Gauge Theories
•
•
•
•
2d Yang-Mills: almost “topological”
QED: confinement / screening
2d N = 0 QCD: one Regge trajectory
QCD with a massive adjoint: higher Regge
Solvable Gauge Theories
•
•
•
•
•
•
•
2d Yang-Mills: almost “topological”
QED: confinement / screening
2d N = 0 QCD: one Regge trajectory
QCD with a massive adjoint: higher Regge
2d N = (1,1) SQCD: very similar
2d N = (0,2) SQCD: ?
2d N = (2,2) SQCD: ?
:
2d N = (0,2) SQCD
• New (0,2) SCFTs
• Exactly solvable
string phenomenology
• Holographic dual of
the Veneziano limit
higher spin theory
Building walls …
… vs building bridges
Interview with Sir Michael Atiyah on math, physics and fun
What makes a mathematics problem fun for you?
The main thing that interests me in
mathematics always is the
interconnection between different
parts of mathematics, the fact
that one problem may have half a
dozen different ways of being
looked at in different subjects, a
bit of algebra, a bit of geometry, a
bit of topology. It's this
interaction and bridges that
interest me.