Holographic Yang-MillsChern-Simons Defects

Why (YM)-CS Holography
with/without Defects?
SA =124g3dRk3µ⌫d x Tr(F Fµ⌫ ) + 4⇡

Confinement/TopologicalT r (A ^ F )FQHE / Edge States

Level-Rank
Dualityxy

SU (N )k $ U (k) N, NR⌫[Aharony 1512.00161]⌫e2= ⌫h1=2k + 1
1=
2U (1)2k+1
SU (2)2[Fradkin-Nayak-Tsvelik-Wilczek ‘98]

Chern-Simons Level-Changing Defects
Two regions w/different level:

Z

k1
SCS =4⇡k2!3 (A) +4⇡

y<0!3 (A)y>0Gauge transformation:
Zk1 k2SCS =T r( F )4⇡
y=0Z

2T r(A ^ dA + A ^ A ^ A)
3

not gauge invariant

level-changing defectFor k fundamental chiralfermions trapped on defect:
ZSF = d2 x i† (i@+ + A+ ) i
Gauge Anomaly

Fermion effective action:
ZkSF =d2 xT r( F )4⇡
For k = k2 - k1:[Laughlin ’81, Callan-Harvey ‘85]

total action is gauge invariantHolographic YMCS & Defect D7 Branes
124g3dRk4⇡

Rd3 xT rSU (N ) F 2

d3 xT rSU (N ) A ^ F

2g3d= gs MKK

Fujita+Li+Ryu+Takayanagi 0901.0924

Fujita, Melby-Thompson, Meyer, Sugimoto 1601.00525

S =R2dxy=0†(i@+ + A+ )

Holography of Defect D7 Branes

D3D7¯
D70xxx

1xxx

2x

3x

4

5

6

7

8

9xxxxxxxxxxxx

Holography of Defect D7 Branes

D3D7¯
D7txxxxxxxyxy(r)
y(r)

⌧x
⌧ (r)
⌧ (r)

r

S5S5S5S5S5xxxxxxxxxxxx

Holography of Defect D7 Branes

Fractional (?) QH SampleU (1)⇢U (N )N e2
= k hxy[Fujita+Li+Ryu+Takayanagi 0901.0924]

Holography of Defect D7 Branes

y5, extended in xM = (t, x1 , u) ,Single D7 wrapped on S
Bosonic fields: y(xM ) , ⌧ (xM ) , aM (xM )S 5 singlet states onlyFields at right-hand defect (left-hand defect has opposite chirality):operator†J ⇠†Oy ⇠F+y†2O+ ⇠F+y1
35

sourcea+yavevaya+[Harvey+Royston 0804.2854](F statistical SU(N) gauge field strength)

Field
yaa+

Asymptotic form (w = u/u0 )Rw⇤34w4 + . . .
(0)

y0(4)
a w4 + (. . . ) + a + . . .
(0)( 4)
a+ + a+ w 4 + . . .

[Fujita, Melby-Thompson, Meyer, Sugimoto]Topology vs. Confinement
Topological Wilson Loop
Level-Rank Duality
SU (N )k $ U (k) N, N

[Fujita+Li+Ryu+Takayanagi 0901.0924]

L<<RkkL>>RkkD7

One Point Functions & Anomaly(1) Scalar condensate:@ShOy i ⇠⇠ h † F+y@Li⇠Known?
⇠+N (0)a4⇡(3) Chiral edge bosons:(0)a± ⇠ @ ±hJ i =(2) Current:N

23d MKK64⇡ 2N f03
13d⇡ 2 MKK L3

N@+ hJ i =@ A+2⇡
Chiral anomaly
A+ = 0@+ = 0hO O+ i 'N u80 e2c0 (2MKK )2Le
164⇡ R(4) Dim. 5 operator:o.s.
StotalhO+ i ==C

N c+ u4⇤ ⇠e084⇡ 8⇡↵ R)

Known?

Edge Chiral Symmetry Breaking
hJ+ i ⇠ @+0hJ i ⇠ @Chiral currents: Flat part of the connection a(0) = d
Shift in
Shift inRRR
R+ 0 doesn’t change a(0) = d =0L

does0U(1)diagLNG boson (“Pion”, as in [Sakai-Sugimoto])(lifted by quantum effects)

BH Horizon(restored at T>>MKK)Edge Chiral Symmetry Breaking
D7F1hEdge mode=Nambu-Goldstoneboson of broken
chiral symmetry
Known?

†iPe
L

RLA
R

Ri = e

F1
Sren[Aharony+Kutasov 0803.3547]

Correlations and Phase Transitionslc ~ LKKCorrelations and Phase Transitions

Dimension 5 Correlations
Large l > 1/Gap: adjacent correlations hO
Small l < 1/Gap: distant correlations

(1)

(2)

O± ⇠

(2)

O+ i =
6 0 hO

hO+ O

(3)

i=
6 0 hO†2F⌥ ±y⌥(3)

O+ i =60(4)(1)

O+ i =
60(4)Known?
(F statistical gauge field)(vanish at T > gap)

Correspondence between FQHE and 2D QCD(1) Basically 2+1 dimensional holographic domain wall fermions= 2D holographic QCD with all its features (chiral symmetry
breaking, Goldstone modes, anomaly)(2) By compactifying and T-dualizing perpendicular to the defect, thesystem becomes the holographic dual to 2D QCD of [Yee-Zahed 1103.6286](3) LEEA of (abelian) FQH States: U(1) CS theory
LEEA of single flavored 2D QCD:
U(1) DBI-CS Theory on a D7 brane [Yee-Zahed 1103.6286]•Anyonic quasiparticle (couples with unit charge to the statisticalgauge field) ~ Fundamental quark (baryon number charge 1/N)
~ Fundamental string ending on D7 brane•

Electron ~ Baryon vertex (bound state of N quarks has baryon
number charge (electric charge) 1)

Conclusions/Outlook•

YM-CS theory with level-changing defects in Holography•U (1)⇢U (N )N e2= k hxy

Similarities to FQH samples[Fujita+Li+Ryu+Takayanagi 0901.0924]•Chiral Symmetry Breaking between Edges•

VEVs involving edge fermions & statistical gauge field,Transitions between samples, Dim. 5 correlations•Edge transport, Impurity scattering, Relation to 2D QCD,
Backreaction, Tunneling of anyons/electrons, …

•Non-abelian (YM)CS in strongly coupled systems