t Hooft loop path integral in N = 2 gauge theories Jaume Gomis (based on work with Takuya Okuda and Vasily Pestun) Perimeter Institute December 17, 2010 Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 1 / 23
Introduction Solving exactly four dimensional gauge theories is out of reach The constraints imposed by supersymmetry have resulted in novel insights into the dynamics of gauge theories: non-perturbative effects strong coupling dynamics (Seiberg-Witten) dualities (S-duality, Seiberg dualities, mirror symmetry...) holography mathematical connections Ubiquity of non-local operators in gauge theories Wilson loops t Hooft loops surface operators domain walls Play a central role in dualities Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 2 / 23
Introduction Loop operators are important gauge theory observables: Gauge Theories can be formulated in terms of them Basic Order Parameters for the phases of gauge theories They are two basic types of loop operators in gauge theories: Wilson Loops A Wilson loop inserts an electrically charged probe particle W R (C) = tr R Pexp A dx C - a contour R - an electric weight (a representation of gauge group G) Their physical properties well understood and is textbook material Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 3 / 23 C
Introduction t Hooft Loops Introduced by t Hooft in 1978 to probe confinement in QCD A t Hooft loop inserts an magnetically charged probe particle C - a contour B - a magnetic weight (a representation of the dual gauge group G ) It is a disorder operator specified by a singularity/boundary condition F = B 4 ɛ ijk x i dx j dx k, x R 3 We compute the exact expectation value of circular t Hooft operators in N = 2 gauge theories by explicitly evaluating the path integral = Localization of path integral [Witten] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 4 / 23
Introduction Localization Idea Use of a fermionic symmetry Q to reduce the original path integral to one which is one-loop exact with respect to an auxiliary parameter = 1/t It however captures results to all orders with respect to the original gauge coupling! Consider path integral (enriched with operators invariant under Q) Z 0 = [Dφ] e S 0[φ] S 0 - the physical action Q - global fermionic symmetry, QS 0 = 0 Q 2 = global bosonic symmetry Now deform the original action with Q 2 V = 0 S t [φ] = S 0 [φ] + tqv Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 5 / 23
Introduction And consider instead the path integral Z t = [Dφ] e S 0[φ] tqv [φ] Integrating by parts we get d dt Z t = 0 = Z 0 = Z Evaluate the path integral at t =, where the semiclassical evaluation of Z t with respect to = 1/t is exact!! Calculation for Wilson loops was carried out by Pestun [Pestun 07] We apply localization to compute the exact expectation value of supersymmetric t Hooft loop operators in N = 2 gauge theories on S 4 [Gomis, Okuda, Pestun] Our results reproduce the formulas obtained from Liouville/Toda CFT! [Drukker,Gomis,Okuda,Teschner 09], [Alday, Gaiotto, Gukov, Tachikawa, Verlinde 09], [Gomis, Floch 10] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 6 / 23
Outline of Computation Start with Lagrangian of N = 2 gauge theory on S 4 [Pestun 07] : vectormultiplet: (A µ, Φ 0, Φ 9, Ψ) (adjoint of gauge group G) hypermultiplet: (q, q, χ) (representation R of G) Allow masses for the hypermultiplet (flavour symmetries) Gauge theories invariant under OSp(2 4) 8 fermionic generators Sp(4) SO(5) bosonic subgroup - isometry of S 4 X1 2 +... + X5 2 = 1 SO(2) R symmetry Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 7 / 23
Outline of Computation Evaluate t Hooft loop path integral by deforming the action with V = (Ψ, QΨ) S t = S 0 + tqv Evaluate the path integral by integrating over the saddle points: QΨ = 0 in the presence of a supersymmetric t Hooft loop operator at the equator of S 4 F B 4 ɛ ijk x i dx j dx k Φ 9 B 1 2 x B is a magnetic weight, defining a homomorphism: U(1) G. Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 8 / 23
Outline of Computation In short, we have to carry the following steps: 1 Find the most general solution to the saddle point equations: QΨ = 0 = Deformed monopole equations 2 Evaluate the gauge theory action on the saddle points 3 Calculate the one loop determinants of all fields in the saddle point background 4 Identify nonperturbative contributions to the path integral: Instantons: from North Pole of S 4 [Nekrasov 02] Anti-instantons: from South Pole of S 4 Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 9 / 23
Saddle point equations We must solve saddle point equations: QΨ = 1 2 F mnf mn ε 1 2 φ aγ aµ µ ε + ik i Γ 8i+4 ε = 0 where ε is a conformal Killing spinor on S 4 parametrizing transformation generated by Q (SU(1 1) OSp(2 4)) Q 2 = J + R + gauge transformation R = SO(2) R symmetry in OSp(2 4) and J = U(1) SO(5) acts X 1 + ix 2 e iα (X 1 + ix 2 ) X 3 + ix 4 e iα (X 3 + ix 4 ) Represent S 4 as S 1 fibration over 3d solid ball B 3 (x i x i < 1, i = 1... 3): ds 2 = dx 2 i (1 + x 2 ) 2 + (1 x2 ) 2 (1 + x 2 ) 2 dτ 2 The t Hooft loop runs along the S 1 fiber τ 0... 2π at x i = 0. Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 10 / 23
Saddle point equations In the B 3 S 1 coordinates (x i, τ) the equations QΨ = 0 are [ F 1τ + D 1, i ( 1 + x 2 ) ] Φ 0 x 3 Φ 9 + x 1 F 12 = 0 2 [ F 2τ + D 2, i ( 1 + x 2 ) ] Φ 0 x 3 Φ 9 x 2 F 21 = 0 2 [ F 3τ + [ D 3, i 2 ( 1 + x 2 ) Φ 0 x 3 Φ 9 ] + x 1 F 32 x 2 F 31 = 0 ( 1 + x 2 ) Φ 0 x 3 Φ 9 ] + x 1 F τ2 x 2 F τ1 = 0 D τ, i 2 [ [Φ 9, D τ ] + Φ 9, i ( 1 + x 2 ) ] Φ 0 + x 1 [Φ 9, D 2 ] x 2 [Φ 9, D 1 ] = 0 2 Interpretation: Saddle point field configurations are Q 2 -invariant = Reduce to D = 3 equations Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 11 / 23
Saddle point equations The interesting remaining equations are the D = 3 equations on B 3. Let s define: a i = 1 2 (D iφ 9 + 1 2 ɛ ijkf jk ), b i = 1 2 (D iφ 9 1 2 ɛ ijkf jk ). where a i, b i are the Bogomolny equations Then the remaining localization equations can be written as where b i δ i x i Φ 9 + δ i x 2 T ij a j = 0 δ 1 = δ 2 = δ 3 = 1 T ij = δ ij 2x ix j x 2 = Deformed Monopole Equations Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 12 / 23
Classical The smooth saddle solutions are the classical field configurations x i F jk = B 2 ɛ ijk x 3, F iˆ4 = ig2 θ B x i 16π 2 x 3, Φ 9 = B 2 x, Φ 0 = g 2 θ B 1 16π 2 x + a a 1 + x 2 K 3 = (1 + x 2 ) 2. Path integral localizes to integration over Φ 0 zeromode a At N and S poles get extra contributions, solutions to F + = 0 and F = 0 (singular instantons and anti-instantons) = Nekrasov instanton partition function [Nekrasov 02] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 13 / 23
Classical Evaluating classical gauge theory action on the saddle points we get ( S cl [a] = 8π2 2π 2 g 2 Tr a2 + g 2 + g2 θ 2 ) 32π 2 Tr B 2 In terms of we can rewrite where τ = θ 2π + 4πi g 2 S cl [a] = πiτ Tr â(n) 2 + πi τ Tr â(s) 2 â(n) = iφ 0 (N) Φ 9 (N) â(s) = iφ 0 (S) + Φ 9 (S) is the parameter of the gauge transformation generated by Q 2 Q 2 = J + R + [â(n), ] Q 2 = J + R + [â(s), ] at fixed points of J, the North and South poles of S 4 Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 14 / 23
Classical The classical answer can be written in terms of the classical part of the Ω-background partition function (regularized R 4 partition function) [Nekrasov 02] Z cl Ω (â) = exp ( πiτ Tr â 2) Classically, we get for t Hooft loop on S 4 T B = [da]z cl Ω (ia B, τ)z cl Ω(ia, τ) Wilson loop of weight W character exp(2πiwa) t Hooft loop of coweight B shift operator exp(b a ) Agrees with dual computation in Liouville/Toda theory [Drukker,Gomis,Okuda,Teschner 09] [Alday, Gaiotto, Gukov, Tachikawa, Verlinde 09] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 15 / 23
Corrections Beyond the classical contribution. Remaining ingredients: Atiyah-Singer computation of the one-loop determinant Point instanton corrections at the North and South poles Monopole screening for t Hooft loops of higher weights Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 16 / 23
One-loop One loop determinants: We can apply Atiyah-Singer theorem if hypermultiplets are in real representation of the gauge group and the flavour symmetry group. Then we can split ind(d) = ind(d)(n) + ind(d)(s) and factorize the one loop determinant We use Z 1-loop (S 4 ) = Z 1-loop Ω (N)Z 1-loop Ω (S) â(n) = iφ 0 (N) Φ 9 (N) â(s) = iφ 0 (S) + Φ 9 (S) Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 17 / 23
Vectormultiplet (α - roots of g): [ Z 1-loop-vector,N = (α,b)>0 Z 1-loop-vector,S = (α,b)<0 One-loop (α,b)<0 G(α(ia 1 2 B))G(2 + α(ia 1 2 B) G( α(ia 1 2 B))G(2 α(ia 1 2 B)) ] 1/2 [ (α,b)>0 G(z) - Barnes double Gamma function G(1 + z) = (2π) z/2 e ((1+γz2 )+z)/2 G(α(ia + 1 2 B))G(2 + α(ia + 1 2 B)) G( α(ia + 1 2 B))G(2 α(ia + 1 2 B)) ] 1/2 n=1 ( 1 + z ) n e z+ z2 2n. n Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 18 / 23
Hypermultiplet: One-loop Z 1-loop-hyper,N,f = [ G(1 + w(ia 1 2 B) + im f )G(1 + w(ia 1 2 B) im f ) (w,b)<0 (w,b)>0 G(1 w(ia 1 2 B) + im f )G(1 w(ia 1 2 B)) im f ] 1/2 Z 1-loop-hyper,S,f = [ G(1 + w(ia + 1 2 B) + im f )G(1 + w(ia + 1 2 B) im f ) (w,b)>0 (w,b)<0 G(1 w(ia + 1 2 B) + im f )G(1 w(ia + 1 2 B im f )) ] 1/2 Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 19 / 23
Instantons Including the singular contributions at North and South Poles we get North pole: F + = 0 point instantons South pole: F = 0 point anti-instantons These contributions are captured by ZΩ inst, but the t Hooft loop background shifts the argument of Nekrasov s partition function. Combining classical, one-loop and instanton contributions together we get the t Hooft loop answer T (B) S 4 = [da]z 1-loop (a, B) Zcl Ω ( ia B ) ( ZΩ inst ia B ) 2 2 2 Agrees with AGT-dual computation in Liouville and Toda theories [Drukker,Gomis,Okuda,Teschner 09], [Alday, Gaiotto, Gukov, Tachikawa, Verlinde 09], [Gomis, Floch 10] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 20 / 23
Monopole screening At x 0 our 3d equations are Bogomolny equations F A = D A Φ 9 Smooth non-abelian monopoles can screen the magnetic charge B of the singular t Hooft monopole by a coroot amount, such that at large distances another smaller charge v is observed Φ = w 1 2 x, x 0 Φ = v 1 2 x, x Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 21 / 23
Monopole screening Smooth screening monopoles can be arbitrarily small. From analysis of supersymmetric equation we conclude that the path integral is contributed by the U(1) fixed points on the moduli space of screening monopoles T (B) = [da] v Z 1-loop (a, B, v) ZΩ cl ( ia v ) 2 ( ZΩ inst ia v ) 2 2 Z 1 loop is computed using Atiyah-Singer index theorem applied to the fixed point on the monopole moduli space M(v, B) Checked agreement with dual computation of loop operators in Liouville/Toda theory as in [Drukker,Gomis,Okuda,Teschner 09], [Alday, Gaiotto, Gukov, Tachikawa, Verlinde 09], [Gomis, Floch 10] Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 22 / 23
Conclusion Conclusion We performed exact localization computation for the expectation value T of supersymmetric t Hooft operator in N = 2 gauge theories on S 4. The T receives two types of non-perturbative corrections point instantons on the North and South poles screening monopoles near the t Hooft loop These results confirm that under S-duality: Wilson Loops t Hooft Loops Jaume Gomis (Perimeter Institute) t Hooft loop path integral in N = 2 gauge theories December 17, 2010 23 / 23