Lecture III: Coherent states, loops and
effective actions in NC field theory
Harold Steinacker
Department of Physics, University of Vienna
H.S., arXiv:1606.00646
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
The (maximally SUSY) IKKT matrix model
Ishibashi, Kawai, Kitazawa, Tsuchiya 1996
0
0
S[X , Ψ] = −Tr [X a , X b ][X a , X b ]ηaa0 ηbb0 + Ψ̄γa [X a , Ψ]
X a = X a † ∈ Mat(N, C) ,
a = 0, ..., 9
(N → ∞)
Ψ ... Majorana-Weyl fermions
1) nonpert. def. of IIB string theory (on R10 )
(IKKT )
2) N = 4 SUSY Yang-Mills gauge thy. on “noncommutative“ R4θ
Symmetries:
gauge symmetry X a → UX a U −1 ,
U ∈ U(H)
SO(10) rotations X a → Λab X a , similarly spinor
translations X a → X a + c a 1l
SUSY
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
pre-geometric; geometry emerges on solutions
solutions = (typically) fuzzy spaces
(=”branes“)
fluctuations around solutions = physical fields
quantization well-behaved because of maximal SUSY
leads to interaction between branes:
(non-local UV/IR mixing) ≡ IIB supergravity in R10
(as predicted in string theory)
conjecture: dynamics of brane geometry (for suitable branes):
→ (”emergent”) 4D gravity
fuzzy SN4 seems to work !
H. Steinacker
H.S. arXiv:1606.00769
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Quantization of matrix models:
Z
Z = dX a dΨ e−S[X ]−S[Ψ]
similarly for correlation functions
...non-perturbative!
includes integration over geometries !
probably ill-defined for generic models
(UV/IR mixing → strongly non-local physics)
well-behaved (only?) for IKKT model, D = 10 due to max. SUSY
non-trivial gauge theory can arise from fuzzy extra dims
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
background solutions: generic fuzzy space
X a ∼ x a : M ,→ R10
fluctuation around background X a → X a + Aa (X a )
expand bare action to O(A2 ):
S[X + A] = S[X ] +
2
Tr
g2
2Aa (2 + µ2 )Xa + Aa (2 + µ2 )δba + 2i[Θab , . ] − [X a , [X b , .]] Ab
with iΘab = [X a , X b ]
e.g. one-loop effective action = Gaussian approx. around background
Z
dAdΨe−S[X +A,Ψ] = e−Γeff [X ]
Z [X ] =
Gauss
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Coherent states & applications
recall fuzzy SN2 :
[X a , X b ] = iεabc X c ,
X a Xa =
1 2
(N − 1) = RN2 .
4
X a = πN (J a ) ... irrep of SU(2) on H = CN
functions on SN2 ... AN = End(H) =
H. Steinacker
N−1
L
(2l + 1)
l=0
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Coherent states & applications
recall fuzzy SN2 :
[X a , X b ] = iεabc X c ,
X a Xa =
1 2
(N − 1) = RN2 .
4
X a = πN (J a ) ... irrep of SU(2) on H = CN
functions on SN2 ... AN = End(H) =
H. Steinacker
N−1
L
(2l + 1)
l=0
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
S 2 as group orbit:
2
coherent states on SN
string states
Trace formulas
One-loop propagator
let p ∈ S 2 ... north pole
SU(2)
→
S2
g
7→
g · p =: x
stabilizer K ⊂ SU(2) ⇒ S 2 ∼
= SU(2)/U(1)
coherent states on SN2 :
|pi ∈ HN ... highest weight vector in HN
def.
|xi = gx · |pi,
xa
gx ∈ SU(2) ... coherent states
= hx|X a |xi ≡ hX a i ∈ S 2 ,
x a xa = 14 (N − 1)2 =: rN2
|xi ... one-to-one correspondence to points x on S 2 (up to phase)
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
coherent states are optimally localized, minimize uncertainty
P
a
a 2
δ 2 :=
a h(X − hX i) i
P
a a
a
a
=
a hp|X X |pi − hp|X |pihp|X |pi
=
RN2 − rN2 =
=: L2NC ≈
2
N
N−1
2
RN2
Exercise 9 (challenge) : show that the highest weight states
minimize the uncertainty δ 2 .
Hint: for given state |ψi consider the vector ~x (ψ) := hψ|X a |ψi and
rotate it such that ~x (ψ) points along ~ez
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
overlap of coherent states:
N−1
|hx|y i|2 = ( 1+x·y
2 )
φ
char. angle φN =
≈ exp(− 14 φ2 (N − 1)) =
δN (x, y)
= ](x, y )
√π
N
char. angular momentum l ∼
H. Steinacker
1
cN
√
N
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
overcompleteness:
Z
1lH = cN
dx|xihx|,
cN =
S2
dim H
VolS2
(by SU(2) invariance)
trace of any operator O ∈ End(H)
dim H
trO =
VolS2
Z
dxhx|O|xi
S2
generalizes to any quantized coadjoint orbit
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
focus at north pole p ∈ S 2
relation with R2θ , Q.M:
rescale R → ∞ s.t. θ =
2R 2
N
= const
[X i , X j ] = iθij
+ O(
1
)
N
coherent state at “origin” = north pole:
|0i ≡ |pi,
a|0i ∼ (X1 + iX2 )|0i = 0
highest weight state
shifted (rotated) coherent states:
|xi = Ux |0i where
Ux = exp(iφi J i ),
localization:
i
i
0j
hx 0 |xi = e− 2θ x εij x e−
x i = R ij φj
|x−x 0 |2
4θ
covers area AN = θ
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
operators and symbols
For an operator O ∈ End(H), define symbol of O as
O(x) = hx|O|xi
... de-quantization of O, “semi-classical limit”
conversely:
Z
O = cN
dx Õ(x)|xihx|
S2
in particular
Ŷml = cN
Z
S2
dxYml (x)|xihx|
however very delicate for large momenta, misleading in UV
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
wavefunctions and UV / IR sectors
IR (“semi-classical”) sector:
non-local matrix elements decay at distances |x − y| ∼ LNC
hx|O|y i ≈ hx|O|xi hx|yi ≈
1
hx|O|xi δ̃N (x, y)
cN
i.e. O(x) varies slowly on uncertainty scale δ = LNC
max. angular momentum l ≤
√
N, uncertainty ∆2 = L2NC
optimally localized semi-classical function
|pihp| =:
H. Steinacker
1
cN
δN (X ; p)
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
UV sector:
most O ∈ End(H) have l >
√
string states
Trace formulas
One-loop propagator
N, not in semi-classical sector.
best described by non-local string states
ψx,y := |xihy|
∈ End(H)
most extreme “function” on SN2 :
N−1
YN−1
= |pih−p|,
has lUV = N, maximally de-localized
most NC “functions” are non-local
⇒ non-local contributions in loops!
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Quasi-coherent states
analogous (quasi-) coherent states exist on generic fuzzy spaces MN
L. Schneiderbauer HS arXiv:1601.08007; cf. Ishiki arXiv:1503.01230
Add a “point probe” brane px at x ∈ RD ,
matrix Laplacian for background MN ∪ px with point brane:
a
X
0
Xa =
∈ End(H ⊕ C),
a = 1, . . . , D
0 xa
string sector = off-diagonal “functions” Φ ∈ End(H ⊕ C) connecting
MN with px , of the form

0
 ..

Φ= .
 0
···
..
.
···
hφ|
0
..
.
0


|φi


0
∈ H ⊕ HT
where |φi ∈ H.
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Consider energy (=Laplacian) of these string states:
P
P a
a
a a
a a
a
a
2X Φ =
a (X X Φ + ΦX X − 2X ΦX )
a [X , [X , Φ]] =


0 ··· 0
P a


.. . .
..
(X − x a )2 |φi

.
.
.

a
= 
 0 ··· 0

 P a

a 2
hφ| (X − x )
0
a
... shifted harmonic oscillator for |φi on M
(recall: MN has structure of phase space in QM,
P 2
cf. P 2 + Q 2 ≡
Xi ... harmonic oscillator at origin)
a
X
X =
0
a
H. Steinacker
0
,
xa
a = 1, . . . , d
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Laplacian for string sector reduces to
2x =
d
X
2
(X a − x a )
a=1
consider quadratic form
1
†
2 tr(Φ 2X Φ)
= hφ|2x |φi =
P
a
2
(∆φ X a ) +
P
a
2
(hφ|X a |φi − x a )
= δ 2 (φ) + |~x(φ) − ~x |2
=: E(~x )
Exercise 10 : Check this ! Here
~ |φi
~x(φ) = hφ|X
and
δ 2 (φ) =
X
hφ|(X a )2 |φi − hφ|X a |φi2 =
a
X
hφ|(X a − xa (φ))2 |φi
a
is the dispersion of the state φ.
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Quasi-coherent states
Let ~x be a point in target space RD . Then the quasi-coherent state(s)
at ~x are defined to be the ground state(s) Ψ of 2x , and their
eigenvalue
2x Ψ = E(~x )Ψ
is the displacement energy.
Quasi-coherent states minimize
δ 2 (φ) + |~x(φ) − ~x |2 = E(~x ) = Dispersion + displacement2
Perelomov coherent states are quasi-coherent states.
denote set of quasi-coherent states by SE (possibly with cutoff
prescription, e.g. upper bound on E)
Then
M := ~x(SE ) := {hψ|X a |ψi;
ψ ∈ SE }
⊂ RD
provides a semi-classical picture of the (generic) fuzzy space.
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
example: fuzzy shere SN2
can show:
The quasi-coherent state at ~x ∈ R3 coincides with the Perelomov
coherent state on SN2 which is closest to ~x ∈ R3 .
and
M = ~x(SE ) = S 2
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
Measuring finite Quantum Geometries via
Quasi-Coherent States
Compute E(x) for all x ∈ RD .
For x ∈ M, expect Hessian Hµν = ∇µ ∇ν E to have 2n = dim M small
eigenvalues, clearly separated from the remaining higher EV’s (which
measure the transversal separation).
The corresponding 2n eigenvectors of Hµν define the tanget space of
M at x.
Can measure the location of the brane M ⊂ RD by scanning target
space RD and looking for (“quasi-”) minima of E(~x ), by following
these tangential directions, possibly with cutoff
Mathematica package BProbe,
https://github.com/lschneiderbauer/BProbe/tree/v0.9.0
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
string states
Iso Kawai Kitazawa hep-th/0001027; H.S., arXiv:1606.00646
x y
x
y
:= ψx,y := |xihy |
:=
†
ψx,y
∈ End(H)
:= |yihx|
momentum operators
Pa O
2O
expectation values
x a x
=
y P y
x x a
=
y P Pa y
:=
[X a , O],
:= P a Pa O
tr ψy,x [X a , ψx,y ] = ~x(x) − ~x(y )
tr ψy,x [X a , [Xa , .]]ψx,y = Exy
Exy = (~x(x) − ~x(y))2 + 2∆2
energy of string state = length 2 + zero point energy
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
general matrix elements
x P a P x 0
a y 0
y
=
hx|X a X a |x 0 ihy 0 |yi + hx|x 0 ihy 0 |X a X a |yi − 2hx|X a |x 0 ihy 0 |Xa |yi
≈
(2∆2 + ~x 2 + ~y 2 − 2~x ~y )) hx|x 0 ihy 0 |yi
≈
Exy hx|x 0 ihy 0 |yi
nearly diagonal
good localization properties in both position and momentum !!
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
propagator
claim:
2 −1
(2̄ + µ )
:=
cN2
Z
dxdy xy
M
1
Exy + µ2
x
y
≈ (2 + µ2 )−1
is excellent approximation to the propagator
because:
(2̃ + µ2 )−1 (2 + µ2 ) xy
=
≈
≈
0
dx 0 dy 0 xy 0
0
R
cN2 dx 0 dy 0 xy 0
x y
cN2
R
1
Ex 0 y 0 +µ2
0
x (2 + µ2 ) x
y
y0 1
(Exy
Ex 0 y 0 +µ2
+ µ2 )hx 0 |xihy|y 0 i
completely regular since Exy ≥ ∆2
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
trace formula for End(H)
TrEnd(H) O =
(dim H)2
(VolM)2
Z
dxdy
x
x
y O y
M×M
proof:
rhs = unique functional on End(End(H)) invariant under GL × GR
= Tr
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
example:
TrEnd(H) [X a , [Xa , .]]
=
N2
(VolS2 )2
=
N2
(VolS2 )2
=
≈
=
N2
VolS2
dxdx 0 tr(|xihx 0 |) |x 0 a − x a |2 + 2∆2 (|x 0 ihx|)
R
S 2 ×S 2
dxdx 0 (|x 0 a − x a |2 + 2∆2 )
R
S 2 ×S 2
R
dx(|x a (e) − x a (x)|2 + 2∆2 )
S2
2
2
1 N (N −1)
4
4π 2
1 2
N (N
2
−
R
S2
1)2
dx(|e3 − x|2 + O( N1 ))
1 + O( N1 )
using rN2 = x a xa = 41 (N − 1)2 and ∆2 ≈
N
2
good agreement with exact result:
TrEnd(H) [X a , [Xa , .]] =
N−1
X
j=0
H. Steinacker
j(j + 1)(2j + 1) =
1 2 2
N (N − 1).
2
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
more generally:
for any smooth function f
TrEnd(H) f (2)
=
N2
(VolS2 )2
=
N2
VolS2
R
S2
R
S2
dx
R
S2
dy f (RN2 |x − y |2 + 2∆2 )
dxf (rN2 |e3 − x|2 + 2∆2 )
Rπ
N2
= 2π VolS
dϑ sin ϑf (rN2 (1 − cos θ)2 + sin2 θ) + 2∆2 )
2
0
2 R1
= N2 −1 duf (2rN2 (1 − u) + 2∆2 )
RN
PN−1
≈ 0 dj 2jf (j 2 + 2∆2 ) ≈ j=0 (2j + 1)f j(j + 1) + 2∆2
= Trjmax f (2g + 2∆2 )
= Trjmax f (2g )
shift by 2∆2 = N − 1 negligible for N 1
UV dominates!
works because spec(2) = spec(2g ) also in UV
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
one-loop propagator for φ4 on SN2
1 1
g tr φ(2 + µ2 )φ + φ4 = S0 [φ] + Sint [φ] .
N
2
4!
1-loop effective action (= Gaussian approx.)
Γeff [φ] = S[φ] + 12 TrEnd(H) log S 00 [φ]
(ψ, S 00 [φ]ψ) = N1 tr ψ(2 + µ2 )ψ + g3 φ2 ψ 2 + g6 ψφψφ
S[φ] =
expanded:
Γ1−loop [φ]
=
=
H. Steinacker
Tr log(.(2 + µ2 ). + g3 .φ2 . + g6 .φ.φ)
g 2
g
1
Tr log(2 + µ2 ) + Tr . 2+µ
+ O(φ4 )
2 ( 3 φ . + 6 φ.φ)
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
2
coherent states on SN
Coherent states
string states
Trace formulas
One-loop propagator
assume φ = φ(X ) slowly varying, IR regime
⇒ φψyx ≈ φ(y)ψyx in string basis
Tr(.φ2 .)
=
=
N2
Vol(M)2
N2
Vol(M)
R
dxdy tr(ψy,x φ2 ψx,y )
M×M
R
dx hx|φ2 |xi .
M
Similarly, “planar” contribution
Tr (.2−1 φ2 .)
H. Steinacker
=
N2
Vol(M)2
≈
N2
Vol(M)2
=
N2
Vol(M2 )
=
µ2N
Vol(M)
R
dxdy tr(ψy,x (2 + µ2 )−1 (φ2 ψx,y ))
M×M
R
dxdy r 2 |x−y|2 1+2∆2 +µ2 tr(ψy,x φ2 ψx,y )
N
M×M
R
dxdy r 2 |x−y1|2 +µ̃2 hx|φ2 |xi
N
M×M
R
2
dx φ (x)
M
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
2
coherent states on SN
Coherent states
string states
Trace formulas
One-loop propagator
assume φ = φ(X ) slowly varying, IR regime
⇒ φψyx ≈ φ(y)ψyx in string basis
Tr(.φ2 .)
=
=
N2
Vol(M)2
N2
Vol(M)
R
dxdy tr(ψy,x φ2 ψx,y )
M×M
R
dx hx|φ2 |xi .
M
Similarly, “planar” contribution
Tr (.2−1 φ2 .)
H. Steinacker
=
N2
Vol(M)2
≈
N2
Vol(M)2
=
N2
Vol(M2 )
=
µ2N
Vol(M)
R
dxdy tr(ψy,x (2 + µ2 )−1 (φ2 ψx,y ))
M×M
R
dxdy r 2 |x−y|2 1+2∆2 +µ2 tr(ψy,x φ2 ψx,y )
N
M×M
R
dxdy r 2 |x−y1|2 +µ̃2 hx|φ2 |xi
N
M×M
R
2
dx φ (x)
M
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
1-loop “planar” mass renormalization
µ2N
=
=
N2
Vol(S 2 )
N2
2rN2
= 2
Rπ
0
R
dy
S2
1
rN2 |e−y|2 +µ̃2
dϑ sin ϑ
r
N
R1
1
du
2
−1
2−2u+ µ̃2
r
≈
N
P
j=0
H. Steinacker
1
2
(1−cos ϑ)2 +sin ϑ2 + µ̃2
N
2j+1
j(j+1)+µ2
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
“nonplanar” contribution
Tr(.(2 + µ2 )−1 φ.φ)
=
=
=
H. Steinacker
N2
Vol(M)2
N2
Vol(M)2
N2
Vol(M)2
R
dxdy tr(ψy,x (2 + µ2 )−1 (φψx,y φ))
M×M
R
dxdyhx|(2 + µ2 )−1 φ|xihy |φ|y i
M×M
R
dxdy r 2 |x−y1|2 +µ̃2 φ(x)φ(y )
M×M
N
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
one-loop quantum effective action:
S1−loop ∼ S0 +
1
g
3 Vol(M)
Z
dxµ2N φ(x)2 +
M
long-range non-locality
g
N2
6 Vol(M)2 r2N
Z
dxdy
M×M
from UV sector
φ(x)φ(y)
|x − y|2 +
µ̃2
rN2
+ O(φ4 )
(UV/IR mixing)
applies to any compact fuzzy space
check for SN2 : agrees with traditional mode expansion
Z
g
1
Φ(µ2N −
h(2̃))Φ + o(1/N)
S1−loop = S0 +
2
12π
Chu Madore HS hep-th/0106205
where
1
h(L) = −
2
Z1
dt
−1
L
X
1
1
(PL (t) − 1) =
1−t
k
k=1
less transparent, requires asymptotics of 6J symbols etc.
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
2
coherent states on SN
Coherent states
R 2 = r 2 RN2 =
Moyal-Weyl plane limit R2θ
ΓNP
string states
≈
gN 2
6Vol(M)2
=
g
6π 2 θ 2
Trace formulas
One-loop propagator
Nθ
4
φ(x)φ(y)
dxdy |x−y|
2 +µ2
R
RM×M φ(x)φ(y)
dxdy |x−y|2 +µ2
M×M
where VolM = 4πR 2 = πNθ
R 2
plane wave basis φ(x) = d2πk φk (eixk + e−ikx ).
ΓNP
≈
=
g
6π 2 θ 2
g
6π 2 θ 2
R
d 2 k φ(k )2
R
d 2 z |z|21+µ2 eiki z
i
g
R
d 2 k φ(k )2
0
R
ij
d 2 p pi pj G1ij +µ2 eiki θ pj .
0
replacing z i = θij pj , and Gij = θii θjj δi 0 j 0
... familiar form in NCFT, IR divergence for k → 0 from UV loop,
UV/IR mixing
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
significance of UV/IR mixing:
UV sector in loops = virtual long strings |xihy|
R
φ(x)φ(y)
lead to long-range non-locality in
dxdy |x−y|
2 +µ2
M×M
interpret NCFT as (non-critical) string theory!
open strings beginning and ending on D-branes
universal, same on any fuzzy space M, any dimension
accumulates at higher loops, inacceptable as fundamental theory
except in SUSY case: cancellations!
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
higher loops
t’Hooft double line formalism, ribbon graphs
lines labeled by positions x, preserved by propagators (!!)
much simpler than in ordinary QFT, directly in position space !
H.S., arXiv:1606.00646
(to be developed)
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory
IKKT model
Coherent states
2
coherent states on SN
string states
Trace formulas
One-loop propagator
summary & outlook
fuzzy spaces = noncommutative spaces embedded in RD
realized by (finite-dim.) matrices X a , a = 1, ..., D
can realize generic geometries
physical models naturally formulated as matrix models
coherent states |xi, “string states” |xihy | useful
UV/IR mixing understood due to long strings mediating
interactions
supersymm. IKKT model → mild non-locality (=IIB supergravity)
4D gravity should (?!!) emerge on suitable branes (SN4 )
→ candidate for theory of fundamental interactions including
gravity
H. Steinacker
Lecture III: Coherent states, loops and effective actions in NC field theory