Candu,Mitev,Quella,VS,Saleur

Harmonics of
Gauge Theory
Darmstadt, July 2009
Volker Schomerus

Intro: QFT & Statistical Mechanics
QFT in path integral formulation
System of statistical mechanics
lattice
i j
critical exponent

Intro: ST & Harmonic Analysis
Particle in S 1 with radius R:
q = e -β
String on S 1: momentum winding
T-duality: R 2/R ; w ↔ n
oscillations

Intro: Gauge/String duality
(SUSY) U(N c) Quantum gauge theories in 4D
↔ String theories in 5D Anti-deSitter geometry
[Polyakov] [Maldacena]
Observables ψ Stringy Harmonics
crit. exponents Eigenvalues of Δ S
Plan: 1 – AdS/CFT Correspondence
Gauge Theory, N=4 SYM, CFT; Strings in AdS, supercosets
2 – Stringy Harmonic Analysis
Compact supercosets; e.g. stringy HA on Calabi-Yau CP N-1|N

Quantum Gauge Theory
Observables are
gauge invariants
Lattice:
lattice
holonomy
U ij
N c x N c
matrices
Weight of plaquette:

N=4 Super Yang-Mills Theory
same on all
length scales
6 matrix valued scalars
→ Conformal Quantum Field Theory
Symmetries: U(4) ~ SO(6) `R-symmetry’ and
4D conformal group SO(2,4)
Poincare, Dilations,
Special Conformal
combine with 4 x 4 fermionic symmetries into
Lie Supergroup PSU(2,2|4)

Conformal Quantum Field Theory
There is unitary representation R of SO(2,4)
Dilations:
Eigenvalues of Δ = conformal weights Δ ψ
0
Δψ depends on gYM & Nc: Δψ = Δψ + δψ(λ,Nc) 2
λ = gYM Nc ‘t Hooft coupling
classical dim anomalous dim

Maldacena’s AdS/CFT duality
Conjecture: [Maldacena] N=4 SYM is dual to
String theory on AdS 5 x S 5
x = (x 0 ,..x 3 )
Parameters are related by
line element on S 5
string coupling string length

ST in AdS 5 x S 5 & Supercosets
Symmetries of AdS 5 x S 5 : SO(2,4) x SO(6)
bosonic
PSU(2,2|4)
SO(1,4) x SO(5)
same as in Gauge Theory!
Construction of superstring on AdS 5 x S 5
involves coset superspace
admits action of PSU(2,2|4)
bosonic
base is
[Metsaev,Tseytlin]
[Berkovits]
AdS 5 x S 5
+32 fermionic
coordinates

Compact Symmetric Superspaces
with Ricci-flat metric:
OSP(2S+2|2S)
OSP(2S+1|2S)
U(N|N)
U(N-1|N)xU(1)
→ S2S+1|2S
→ CPN-1|N
.
includes S 5
Calabi-Yau
including string corrections
G/G Z2
OSP(2S+2|2S)
OSP(2S+2-n|2S) x SO(n)
U(N|N)
U(N-n|N) x U(n)
• note: c v (GL(N|N)) = 0 = c v (OSP(2S+2|2S))

Calabi-Yau Calabi Yau Superspaces CP N-1|N 1|N
Twistorial CY CP 3|4 ↔ N=4 Super Yang Mills [Witten]
complex bosonic fermionic
CP N-1|N possess single Kaehler parameter t
complexified size
Complex line bundles with
Perform Harmonic Analysis for as function of parameter t

Harmonic Analysis for CP 1|2
Recall: Harmonic Analysis on CP 1 ~ S 2 gives
u(2) Casimir / R 2 Cartan generator
monopole charge k
Harmonic Analysis for CP 1|2
pu(2|2) character

ST Harmonic Analysis for CP 1|2
open string has
two ends
[Candu,Mitev,Quella,VS,Saleur]
• Obtained by summing all order perturbative expansion
possible because of target space SUSY
• Tested through extensive numerical lattice simulations

ST Harmonic Analysis for CP 1|2
[Candu,Mitev,Quella,VS,Saleur]
λ t is universal (depends only on t, k 1,k 2)

ST Harmonic Analysis for CP 1|2
[Candu,Mitev,Quella,VS,Saleur]
Branching fcts at R = ∞ from decomposition of
explicitly known

ST Harmonic Analysis for CP 1|2
[Candu,Mitev,Quella,VS,Saleur]
Value of Quadratic Casimir
in representation of pu(2|2)

ST Harmonic Analysis for CP 1|2
[Candu,Mitev,Quella,VS,Saleur]
Character χ Λ of
representation Λ

f 00
Lattice Models and Numerics
boundary term
level 1
level 2
level 3
w=w(R)
acts on states space:
w=w(R)=0 : non-inter-
secting loop model
[Candu,Read,Saleur]
[Candu,Mitev,Quella,
VS, Saleur]

Conclusions & Open Problems
Similar results exist for supersphere S 2S+1|2S
[Mitev, Quella VS]
At R=1: Z(S 3|2 ) decomposes into characters
of affine osp(4|2) at level k = -1/2
[Candu,
Saleur]
→ there exists description as Gross-Neveu model
~ Landau Ginsburg description of Calabi-Yau
• Affine symmetries for Calabi-Yau CP N-1|N ?
• Ext. to non-compact Supercosets (AdS 5) ?