Integrability and the Conformal Field Theory of the Higgs branch - - PowerPoint PPT Presentation
Integrability and the Conformal Field Theory of the Higgs branch Bogdan Stefaski, jr. City University London 17 November 2015 Based on 1411.3676 , JHEP 1506 (2014) 103 with O. Ohlsson Sax, A. Sfondrini AdS 3 with 8 + 8 susys and integrability
Based on 1411.3676, JHEP 1506 (2014) 103 with O. Ohlsson Sax, A. Sfondrini
String theory on AdS3 × S3 × M4 where M4 =
String theory on AdS3 × S3 × M4 where M4 =
String theory on AdS3 × S3 × M4 where M4 =
All-loop integrable wsheet 2-body S matrix known
[Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli]
String theory on AdS3 × S3 × M4 where M4 =
All-loop integrable wsheet 2-body S matrix known
[Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli]
expectation: integrability solves spectral problem
[Abbott, Aniceto, Babichenko, Bianchi, Beccaria, David, Dekel, Engelund, Hernandez, Hoare, Levkovich-Maslyuk, Macorini, McKeown, Nieto, Pittelli, Prinsloo, Regelskis, Roiban, Sahoo, Stepanchuk, Sundin, Tseytlin, Wolf, Wulff, Zarembo]
String theory on AdS3 × S3 × M4 where M4 =
All-loop integrable wsheet 2-body S matrix known
[Borsato, Ohlsson Sax, Lloyd, Sfondrini, BS, Torrielli]
expectation: integrability solves spectral problem
[Abbott, Aniceto, Babichenko, Bianchi, Beccaria, David, Dekel, Engelund, Hernandez, Hoare, Levkovich-Maslyuk, Macorini, McKeown, Nieto, Pittelli, Prinsloo, Regelskis, Roiban, Sahoo, Stepanchuk, Sundin, Tseytlin, Wolf, Wulff, Zarembo]
This talk focuses on pure R-R, M4 = T 4 case
λ −
[Ohlsson Sax, BS, Torrielli ’11]
λ −
[Ohlsson Sax, BS, Torrielli ’11]
Sites of spin-chain
λ −
[Ohlsson Sax, BS, Torrielli ’11]
Sites of spin-chain
1 2-BPS irrep of psu(1, 1|2) with bos/ferm h.w. state
λ −
[Ohlsson Sax, BS, Torrielli ’11]
Sites of spin-chain
1 2-BPS irrep of psu(1, 1|2) with bos/ferm h.w. state
R sites same as L sites
1 UV gauge theory
2 CFTH
3 ∆ from Leff and spin-chains 4 Outlook and Conclusions
D1-D5 branes
D1-D5 branes
D1-D5 branes
D1-D5 branes
D5: break susy to (4, 4)
YM
YM
YM
YM
No H − T couplings
YM
No H − T couplings LUV has two branches of susy vacua:
YM
No H − T couplings LUV has two branches of susy vacua:
gYM dimensionful: theory flows to CFT in IR
YM
No H − T couplings LUV has two branches of susy vacua:
gYM dimensionful: theory flows to CFT in IR IR CFT = CFTC ⊕ CFTH
[Witten ’95, ’97]
YM
No H − T couplings LUV has two branches of susy vacua:
gYM dimensionful: theory flows to CFT in IR IR CFT = CFTC ⊕ CFTH
[Witten ’95, ’97]
CFTH dual to AdS3
[Maldacena ’97]
In IR gYM → ∞ so LΦ irrelevant and can be dropped [Witten ’97]
In IR gYM → ∞ so LΦ irrelevant and can be dropped [Witten ’97]
LIR marginal if Φ has geometric dimensions
In IR gYM → ∞ so LΦ irrelevant and can be dropped [Witten ’97]
LIR marginal if Φ has geometric dimensions
Φ enter quadratically as auxiliary fields in LIR
Φ auxiliary: eliminate using eoms
[Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99]
Φ auxiliary: eliminate using eoms
[Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99]
LADHM is (4, 4) σ-model with target space
Φ auxiliary: eliminate using eoms
[Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99]
LADHM is (4, 4) σ-model with target space
LADHM gives conventional picture of Higgs branch:
Φ auxiliary: eliminate using eoms
[Witten 97] [Berkooz, Verlinde ’99] [Aharony, Berkooz ’99]
LADHM is (4, 4) σ-model with target space
LADHM gives conventional picture of Higgs branch:
LADHM has small instanton singularity:
CFTH states localised near origin of Higgs branch (near small
CFTH states localised near origin of Higgs branch (near small
For such states integrate out H
[Witten 97] [Aharony, Berkooz 99]
CFTH states localised near origin of Higgs branch (near small
For such states integrate out H
[Witten 97] [Aharony, Berkooz 99]
This "re-animates" Φ
CFTH states localised near origin of Higgs branch (near small
For such states integrate out H
[Witten 97] [Aharony, Berkooz 99]
This "re-animates" Φ Nf factor comes from Nf copies of H
CFTH states localised near origin of Higgs branch (near small
For such states integrate out H
[Witten 97] [Aharony, Berkooz 99]
This "re-animates" Φ Nf factor comes from Nf copies of H LT unaffected since no T − H couplings
Φ becomes dynamical. 2pt fn fixed by conformal invariance
Φ becomes dynamical. 2pt fn fixed by conformal invariance
Leff interactions follow from LH interactions and integrating out
Φ becomes dynamical. 2pt fn fixed by conformal invariance
Leff interactions follow from LH interactions and integrating out
Φ becomes dynamical. 2pt fn fixed by conformal invariance
Leff interactions follow from LH interactions and integrating out Rescaling
1 Nf as coupling constant.
Gauge invariant states built from adjoint fields Φ and T
Gauge invariant states built from adjoint fields Φ and T Nc → ∞: perturbation series becomes ’t Hooft expansion in
c
Gauge invariant states built from adjoint fields Φ and T Nc → ∞: perturbation series becomes ’t Hooft expansion in
c
In this limit single-trace ops dominate
Gauge invariant states built from adjoint fields Φ and T Nc → ∞: perturbation series becomes ’t Hooft expansion in
c
In this limit single-trace ops dominate
Such operators correspond to spin-chains with sites
Gauge invariant states built from adjoint fields Φ and T Nc → ∞: perturbation series becomes ’t Hooft expansion in
c
In this limit single-trace ops dominate
Such operators correspond to spin-chains with sites
[Ohlsson Sax, BS, Torrielli ’11]
Gauge invariant states built from adjoint fields Φ and T Nc → ∞: perturbation series becomes ’t Hooft expansion in
c
In this limit single-trace ops dominate
Such operators correspond to spin-chains with sites
[Ohlsson Sax, BS, Torrielli ’11]
Local spin-chain appears very naturally
We restrict to O(λ) in so(4) subsector
We restrict to O(λ) in so(4) subsector
Ground state is 1/2 BPS: ∆ = J
We restrict to O(λ) in so(4) subsector
Ground state is 1/2 BPS: ∆ = J
Planar gauge theory
L
We restrict to O(λ) in so(4) subsector
Ground state is 1/2 BPS: ∆ = J
Planar gauge theory
L
Ground state protected so
L
We restrict to O(λ) in so(4) subsector
Ground state is 1/2 BPS: ∆ = J
Planar gauge theory
L
Ground state protected so
L
2 N−2
Power-counting divergent leading order diagrams
Power-counting divergent leading order diagrams
Second and fourth diagrams give trivial so(4) structure.
Power-counting divergent leading order diagrams Expanding interactions in diagrams with non-trivial so(4) structure
Power-counting divergent leading order diagrams Expanding interactions in diagrams with non-trivial so(4) structure Diagrams involving a gluon exchange vanish due to symmetry
Power-counting divergent leading order diagrams Expanding interactions in diagrams with non-trivial so(4) structure Diagrams involving a gluon exchange vanish due to symmetry Only "φ4" diagram is divergent and has non-trivial so(4) structure
Expanding the "φ4" diagram
Expanding the "φ4" diagram
First diagram has trivial so(4) structure.
Expanding the "φ4" diagram
First diagram has trivial so(4) structure. Second diagram is UV finite
Expanding the "φ4" diagram
First diagram has trivial so(4) structure. Second diagram is UV finite Only third diagram UV divergent and so(4) non-trivial
Compute diagram, find c2 = 1 and hence so(4) dilatation operator
Nf L
Compute diagram, find c2 = 1 and hence so(4) dilatation operator
Nf L
Perturbative calculation in larger sector?
Perturbative calculation in larger sector? Dilatation operator from symmetries?
AdS3/CFT2 likely to be an example of holographic integrability.
AdS3/CFT2 likely to be an example of holographic integrability.
I focused on D1-D5 pure R-R flux.
AdS3/CFT2 likely to be an example of holographic integrability.
I focused on D1-D5 pure R-R flux.
What is the connection to other points in the moduli space, such
AdS3/CFT2 likely to be an example of holographic integrability.
I focused on D1-D5 pure R-R flux.
What is the connection to other points in the moduli space, such
What about D1-D5-D5’ and its CFT2 [Tong]
AdS3/CFT2 likely to be an example of holographic integrability.
I focused on D1-D5 pure R-R flux.
What is the connection to other points in the moduli space, such
What about D1-D5-D5’ and its CFT2 [Tong] Integrability in AdS3/CFT2 has a rich structure that needs to be