Exact Results in D=2 Supersymmetric Gauge Theories And Applications

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1 Exact Results in D=2 Supersymmetric Gauge Theories And Applications Jaume Gomis Miami 2012 Conference arxiv: with Doroud, Le Floch and Lee arxiv: with Lee

2 N = (2, 2) supersymmetry on S 2 Described by the SU(2 1) algebra { Q α, Q β } = γαβj m m 1 2 ε αβr J m : SU(2) isometry generators of S 2 R: vector R-symmetry Comments: subalgebra of N = (2, 2) superconformal algebra preversing S 2 isometries in r limit = N = (2, 2) Super-Poincaré SU(2 1) realized by Killing vectors and conformal Killing spinors i ɛ = γ i ɛ = 1 d γ i / ɛ

3 N = (2, 2) Supersymmetry Transformations on S 2 Find a representation of SU(2 1) on supermultiplets Dimensional reduction of d = 4 N = 1 supersymmetry multiplets vector multiplet: (A i, σ 1, σ 2, λ, λ, D) chiral multiplet: (φ, φ, ψ, ψ, F, F ) Additional multiplets are available in d = 2. In particular twisted chiral multiplet: (Y, Y, χ, χ, G, Ḡ) SU(2 1) transformations obtained by Weyl covariantization of super-poincaré transformations in flat space Obtain an off-shell representation of SU(2 1) on all multiplets

4 N = (2, 2) Supersymmetric Field Theories on S 2 Lagrangian is a deformation of covariantized flat space one L = L r L r 2 L 2 Comments: Field content depends on: G: gauge group for vector multiplet R: representation of G for matter multiplets The Lagrangian depends on two holomorphic functions: W(φ): superpotential W (Y ): twisted superpotential The S 2 partition function can depend on the parameters in L

5 Parameters: g I : superenormalizable gauge coupling for each gauge group factor G I Complexified marginal coupling τ for each U(1) factor τ = θ 2π + iξ where L FI = iξ Tr D L top = i ϑ 2π Tr F These can be recast as twisted superpotential coupling W (Σ) = i τ 2 Σ Σ = D +D V W parameters W parameters Parameters in the Cartan of G F, the flavour symmetry of L where mi: masses qi: R-charges M I = m I + i 2r q I

6 The partition function on S 2 Compute exactly by localization. Path integral localizes: deform by a Q-exact term L L + t Q V partition function is independent of t in the t limit, the semiclassical approximation wrt eff 1/t is exact The domain of integration F is: {Q invariant configurations} {Saddle points of Q V } Exact Partition function: dµ e S F dµ obtained by integrating out fields in Q V at one loop F

7 see also Benini & Cremonesi Consider arbitrary gauge theory with vector and chiral multiplets: Choose Q SU(2 1) obeying Q 2 = J + R 2 Given Q, we have freedom in choosing a deformation term Q V = changes the saddle points We exploit this freedom and find inequivalent representations Z = Z Coul = Z Higgs ZCoul by choice of deformation Q V Coul ZHiggs by choice of deformation Q V Higgs

8 Coulomb Branch Representation F Coulomb has components labeled by quantized flux B on S 2 A = B 2 (±1 cos θ) dϕ σ 1 = B 2r σ 2 = a [a, B] = 0 The partition function is given by Z Coul da e S0 Zone-loop(a, v.m B) Zone-loop(a, c.m B, m) B t S0 = 4πirξ rentr a + iϑ Tr B Z v.m. one-loop(a, B) = Z c.m. one-loop(a, B, m) = w R [ 1 ( α B ) W(G) α ] 2 + 2r + (α a) 2 Γ( ir(w a+m) w B 2 ) 2 ) Γ(1+ir(w a+m) w B

9 Higgs Branch Representation F Higgs are localized vortices and anti-vortices at the poles Sum over all of vortices at the north pole This defines the vortex partition function: Z vortex (v, m, e 2πiτ ) = e 2πiτk k=0 M k vortex Explicitly computable in certain gauge theories e w

10 Putting vortices and anti-vortices together we obtain Z Higgs = e S0 B=0 res a=v [Z one-loop (a, 0, m)] Z vortex (v, m, e 2πiτ ) 2 v Higgs vacua Comments: ZHiggs yields a holomorphically factorized representation of Z Mellin-Barnes representation of ZCoul ZCoul evaluated by residues. Realizes a holomorphically factorized form: Z = Z Coul = Z Higgs For a Landau-Ginzburg model with a twisted superpotential W, the exact two-sphere partition function is: (Y )+iw (Ȳ ) Z = dy dȳ eiw

11 Applications Exact dynamics of nonlinear sigma models (X is a Kähler manifold) ϕ : S 2 X Asymptotically free for R > 0 Non-perturbative corrections due to worldsheet instantons ϕ = 0 Worldsheet instanton generating function F = N β e 2πi β t β H 2 (X) N β : Gromov-Witten invariants X is a Calabi-Yau = Quantum Kähler potential in moduli space K(τ, τ) Computes: Yukawa couplings and kinetic terms for moduli

12 Nonlinear sigma models studied from an UV gauge theory description GLSM NLSM e.g: CP N -model is a U(1) gauge theory with N + 1 chirals of charge +1 The two-sphere partition function of a Calabi-Yau GLSM computes the exact Quantum Kähler potential Z = e K(τ, τ) Conjectured by Jockers et al and proven in J.G & Lee New physics based method for computing worldsheet instantons and Gromov-Witten invariants Other Applications: Z also captures the Seiberg-Witten Kähler potential of 4d N = 2 gauge theories Park & Song Vortex partition functions Topology Change in String Theory.

13 THANK YOU TOM, LUCA AND JOANN!