Exact Results in Massive N=2 Theories Konstantin Zarembo (Nordita, Stockholm) Integrability and Chaos in Multicomponent Systems, Vladivostok, 7.10.17

AdS/CFT correspondence z 5D bulk strings gauge fields 4D boundary

Breaking scale invariance “IR cutoff” horizon or domain wall massive field theory asymptotically AdS metric approximate scale invariance at short distances Routinely used in many contexts Very few quantitative tests so far

Example: N=2* theory N=4 SYM hypermultiplets vector multiplet mass =

Holographic dual Domain wall “AdS6” N=2* theory AdS5 Pilch,Warner’00

Weak coupling Strong coupling UV regularization of pure N=2 SYM In this talk: N=∞ λ = gYM2 N UV regularization of pure N=2 SYM Weak coupling Strong coupling Hoyos’10 Dimensional crossover by Eguchi-Kawai mechanism Young,Z.’14

Holographic Wilson loops Area law: Maldacena’98 Rey,Yee’98

Example: Circle Minimal surface: Area: regularized area Drukker,Gross,Ooguri’99 Berenstein,Corrado,Fischler,Maldacena’98 Area: regularized area

Wilson loop in perturbation theory Consider - Gaussian vector field - Gaussian scalar field - circular contour

θ =

Summing rainbow diagrams Random matrix model: Large-N solution: Wigner distribution

Strong-weak coupling interpolation λ SYM perturbation theory String perturbation theory 1 + + + … Circular Wilson loop (exact): Erickson,Semenoff,Zarembo’00 Drukker,Gross’00 Minimal area law in AdS5

Partition function of any N=2 theory on S4: Localization Pestun’07 Partition function of any N=2 theory on S4: Vector multiplet: Fund. hypermultiplet: Adj. hypermultiplet:

Saddle-point equations: Example: pure N=2 Saddle-point equations: Decompactification limit Douglas, Shenker’95

Super-QCD Two phases: Heavy quarks (M>μ) Light quarks (M<μ)

N=2*

Weak Coupling OPE in condensates β-function of N=2 SYM! Douglas,Shenker’95 OPE in condensates β-function of N=2 SYM!

Phase transtion resonance on massless hyper: Weak-coupling solution is valid up to resonance on massless hyper:

Phase diagram R: radius of S4

Strong coupling

Perimeter law (perimeter law) substitute <Φ> Buchel,Russo,Z.’13 Chen-Lin,Gordon,Z.’14 Z.’14

Dual geometry Pilch,Warner’00

Minimal surface for circle: (domain wall region)

agrees with localization! Minimal area renormalized away agrees with localization!

String fluctuations Setup: straight Wilson line

Semiclassical string path integral:

Fluctuation determinants

Density of states and phaseshifts Cancels among bosons and fermions

Divergences cancel among bosons and fermions Area law Lüscher discrepancy? vs. field-theory prediction:

Fradkin-Tseytlin term

Lüscher Fradkin-Tseytlin agree! Matrix model:

Conclusions Localization is a powerful probe of N=2 gauge theories fisrt (?) quantum test of non-conformal holography what are the implications of the phase transitions AdS/CFT?