M-theory and exact results in SUSY gauge theories Kazuo Hosomichi - PowerPoint PPT Presentation

M-theory and exact results in SUSY gauge theories Kazuo Hosomichi (YITP) 7 Feb 2012, Kyoto @YIPQS symposium

16 years after the 2 nd superstring revolution

Superstring Revolution II ('95) Quantum equivalence relations among Dualities : different 10-dim superstring theories Spatially extended solitons Branes : in superstring theories

Duality Web connects 5 superstring theories. Type I Het SO  32  Type IIB TOE Het E 8 × E 8 Type IIA (11-dim) 11 dimensions emerge = M-theory at a corner of the web

p-brane = (p+1)-dim solitonic object p 0 1 2 3 4 5 D-string D3-brane D5-brane IIB string NS5-brane IIA D-particle string D2-brane D4-brane NS5-brane M M2-brane M5-brane Dynamics of Dp-branes = open string theory j = (p+1)-dim SUSY gauge theory i Other branes are harder to understand. 1 2 N

Gauge Theories can describe D-brane dynamics D-branes can engineer SUSY gauge theories in different dimensions, and visualize their strong coupling behaviors The two subjects have been influenced from each other for the last 15 years.

Example: Brane construction of 4D N=2 SUSY YM NS5 NS5 x 4 = Re v Open strings on n D4-branes D4 v = a 1 = SU(n) gauge theory v = a 2 ⋮ ( a 1 , ⋯ ,a n ) v = a n = position of D4-branes L = label of vacua of gauge theory x 6 x 5 = Im v

Example: Brane construction of 4D N=2 SUSY YM NS5 NS5 x 4 = Re v D4 v = a 1 v = a 2 ⋮ v = a n L x 6 x 5 = Im v (strong coupling) IIA M

Example: Brane construction of 4D N=2 SUSY YM NS5 NS5 x 4 = Re v Single smooth M5 wrapping a 2D surface D4 v = a 1 v = a 2 ⋮ v = a n L x 6 x 5 = Im v (strong coupling) IIA M

The shape of the smooth M5-brane (complex curve) : n 2 − t ∏ i = 1 2 n ( v − a i )+Λ SYM = 0 t 2 n = exp (− L / g s ) Λ SYM = Seiberg-Witten curve for N=2 SUSY YM with G=SU(n), which encodes everything about low-energy dynamics

The mesh of the web became very, very fine after 16 years. Type I Het SO  32  IIB Het E 8 × E 8 IIA M Another big discovery: AdS/CFT correspondence (1997)

But still, 10,1 ℝ the (5+1)-dim theory on N flat M5-branes in remains mysterious. “6dim (2,0) theories”

Recent progress in SUSY gauge theories

Localization principle = simplification of (path-)integrals due to (super)symmetry. Non-zero contributions arises only from fixed points . Application to SW theories (=4D N=2 SUSY gauge theories): 4 ℝ ϵ 1 , ϵ 2 -- Partition function on “Omega-background” (Nekrasov '02) 4 -- Partition function & Wilson loop on (Pestun '07) S . . . proved a long-standing conjecture Circular Wilson loop in N=4 SYM = Gaussian matrix integral (Erikson-Semenoff-Zarembo, Drukker-Gross, . . .)

Application to M5-brane physics T ( N , Σ g ,n ) Gaiotto's theory N = SW theory describing M5-branes g = 3 Σ g ,n wrapped on a Riemann surface n = 4 [examples] 2 M5-branes SU(2) SQCD on Σ 0,4 with 4 doublet quarks 2 M5-branes SU(2) SQCD on Σ 1,1 with 1 triplet quark

AGT relation (Alday-Gaiotto-Tachikawa '09) 4 -partition function n-point correlation function S = T ( N , Σ g ,n ) of SU(N) Toda CFT on Σ g of (4dim field theory) (2dim field theory) Comments: * The relation was first found experimentally. * Several ideas for proof have been proposed. * The relation implies 4 S N M5-branes on = SU(N) Toda CFT (not proved yet.)

More recent progress: Localization in 3-dim SUSY theories

SUSY theories on 3-sphere -- Initiated by Kapustin-Willett-Yaakov ('09) on round sphere 3 * cf) Sen('87,'90), Romelsberger('05) : 4D theories on ℝ× S -- Partition function for general N=2 SUSY theories Jafferis('10), Hama-KH-Lee('10) -- Generalization to Non-round spheres Hama-KH-Lee('11), Imamura-Yokoyama('11)

SUSY on 3-sphere (or any curved manifold) . . . in correspondence with Killing Spinor (KS) D μ ε ≡ (∂ μ + 1 ab )ε = Γ μ ̃ 4 Γ ab ω μ ε for some ̃ ε There are 4 KSs on round 3-sphere. 2 = 1 2 + x 1 2 + x 2 2 + x 3 4 in ℝ x 0 2 = μ 1 μ 1 +μ 2 μ 2 +μ 3 μ 3 a ds ( : SU(2) LI 1-forms) μ

Generalization 1 (Hama-KH-Lee '11) 3 : 2 ( x 0 2 + x 1 2 ) + b − 2 ( x 2 2 + x 3 2 ) = 1 Ellipsoid S b b admits no KSs. V μ But with a suitable background field turned on, there are charged KSs. ± ≡ (∂ μ + 1 ab ∓ iV μ )ε ± = Γ μ ̃ ± D μ ε 4 Γ ab ω μ ε

Generalization 2 (Imamura-Yokoyama '11) Squashed 3-sphere: 2 = μ 1 + μ 2 + s 2 μ 1 μ 2 μ 3 μ 3 ( s < 1 ) ds admits no KSs. V μ But with a suitable background field turned on, there are modified KSs satisfying (∂ μ + 1 ± = − is ab )ε ± ± tV ν Γ μ ν ε ± 4 Γ ab ω μ 2 Γ μ ε ( t ≡ √ 1 − s 2 ) b ≡ s + it (∣ b ∣= 1 )

Interesting questions : * Any other 3-manifolds admitting SUSY? * Any other variations of KS equation? Note the relation (Festuccia-Seiberg '11) KS equation Gravitino's transformation law in supergravity

3D N=2 SUSY theories multiplets: “gauge” “matter” G R Rep of G Lie algebra labelled by q U(1) R-charge ϕ σ real scalar complex scalar ψ λ spinor spinor fields A μ vector D F aux. scalar aux. scalar Couplings: YM coupling, Chern-Simons coupling, masses . . . Partition function depends on some of them.

Computation of partition function using localization principle

SUSY localization principle Nonzero contribution to SUSY (path-)integrals localizes to “ fixed points ” satisfying δ SUSY Ψ = 0 for all fields Ψ . For 3D N=2 SUSY theories, ∂ μ σ= A μ =λ=ϕ=ψ= F = 0 at fixed points. (up to gauge choice) ∫ D ( fields ) e ∫ d σ 0 (⋯) − S Path integral Matrix integral

σ 0 For integral over everything except for , Gaussian approximation is exact, since partition function does not change under the shift 2 S → S + t δ SUSY V ( s.t. δ SUSY V = 0 ) * YM action for gauge multiplet, * Kinetic action for matter multiplets are of this class

Result: (Hama-KH-Lee '11) [double-sine function]

3-dim AGT (Dimofte-Gaiotto-Gukov, Cecotti-Cordova-Vafa) another mysterious relation began to be uncovered, between 3 S b -- 3D N=2 SUSY gauge theories on -- SL(n) Chern-Simons path integrals 3 × M 3 n M5-branes on S b (3-manifold with defect curves) SUSY gauge theory SL(n) CS theory = 3 T ( n, M 3 ) S b on M 3 on

Summary M5-branes may provide a new web of duality among different non-gravitating theories (how many more corners?) IIA branes 3D SUSY M5 SW theory Non-compact CS theory Toda CFT Studying this web will lead to a better understanding of M5-branes themselves

Application of the exact 3D result to M2-brane dynamics

“M2-brane mini-revolution” ('08) New 3D Chern-Simons-matter theories with extended SUSY were discovered, and identified with the theory of multiple M2-branes.

ABJM model (Aharony-Bergman-Jafferis-Maldacena '08) U ( N ) k × U ( N ) − k Chern-Simons-matter theory 4 /ℤ k describing N M2-branes in the “ orbifold ” ℂ (bi-fundamental matters) 11d SUGRA on IIA SUGRA on (AdS/CFT)

A puzzle: free energy at large N λ ≡ N / k : 't Hooft coupling of ABJM model 2 F ∼ N ( k ≫ N ) At weak coupling : (since ABJM model is a field theory of NxN matrices) 3 / 2 F ∼ N ( k ≪ N ) At strong coupling : (from SUGRA analysis) How are these two connected?

Solution Drukker-Marino-Putrov ('10) studied the formula for partition function on 3-sphere, using techniques of large-N matrix models. (eigenvalue distribution analysis) reproduced the SUGRA result

Conclusion * Exact analysis of SUSY gauge theories ( localization ) * Rigid SUSY theories on compact curved spaces ( spheres ) -- Application to multiple M2-brane theories * New relations among QFTs in different dimensions ( AGT, DGG ) -- Better understanding of M5-branes