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Deformed Supersymmetric Gauge Theories from String and M-Theory

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Deformed Supersymmetric Gauge Theories from String and M-Theory Deformed supersymmetric gauge theories from String and M-Theory Susanne Reffert based on work with with D. Orlando, S. Hellerman, N. Lambert arXiv:1106.2097, 1108.0644, 1111.4811, 1204.4192, 1210.7805, 1304.3488, 1309.7350, work in progress Introduction In recent years, N=2 supersymmetric gauge theories and their deformations have played an important role in theoretical physics - very active research topic. Examples: 2d gauge/Bethe correspondence (Nekrasov/Shatashvili): relates 2d gauge theories with twisted masses to integrable spin chains. 4d gauge/Bethe correspondence (Nekrasov/Shatashvili): relates Omega-deformed 4d gauge theories to quantum integrable systems. AGT correspondence (Alday, Gaiotto, Tachikawa): relates Omega-deformed super-Yang-Mills theory to Liouville theory. Introduction All these examples have two things in common: 1. A deformed supersymmetric gauge theory is linked to an integrable system. Relation between two very constrained and well- behaved systems that can be studied separately with different methods. Transfer insights from one side to the other, cross- fertilization between subjects! 2. The deformed gauge theories in question can be realized in string theory via the fluxtrap background! The string theory construction provides a unifying framework and a different point of view on the gauge theory problems. Introduction Aim: Realize deformed supersymmetric gauge theories via string theory. Gauge theories encode fluctuations on the world-volume of D-branes. Many parameters can be tuned by varying brane geometry. Here: Deform the string theory background (“fluxtrap”) into which the branes are placed (Hellerman, Orlando, S.R.) ⇒ different brane set-ups give rise to different gauge theories with seemingly unrelated deformations! Use the fluxtrap construction to unify and meaningfully relate and reinterpret a large variety of existing results. Introduction Our string theoretic approach enables us moreover to generate new deformed gauge theories in a simple and algorithmic way. Today: panoramic overview over the many applications of the fluxtrap background: - 2d effective gauge theories with deformations - 4d effective gauge theories with deformations Fluxtrap background as toolbox to generate deformed gauge theories and analyze them via string theoretic methods. Introduction M-theory 3 T MFT MFB M5 M5 Lift Reduce string theory FT Reduce +T Reduce topological Reduce string theory D1 D2 D3 D4 NS5 NS5 NS5 gauge theory N=4 w. N=2* їGHI їGHI6: їGHI real masses 1 6<0 1 6<0 N=(8,8) w. 6GXDORI tw. masses їGHI reciproc. їGHI6: N=(2,2) w. 1 6<0 gauge th. tw. masses їGHI 1 6<0 2d 3d 4d 5d The same string theory background can give rise to many different deformations depending on how we place branes in it! Outline • Introduction, Motivation • The Fluxtrap Background • Deformed gauge theories • 2d Gauge Theories with twisted masses • N=2* theory • Polchinski/Strassler type gravity dual • Ω–deformed N=2 SYM • Ω–deformed N=1 SYM • Ω–deformed SW • Summary The Fluxtrap Background The Fluxtrap Background M-theory 3 T MFT MFB M5 M5 Lift Reduce string theory FT Reduce +T Reduce topological Reduce string theory D1 D2 D3 D4 NS5 NS5 NS5 gauge theory N=4 w. N=2* їGHI їGHI6: їGHI real masses 1 6<0 1 6<0 N=(8,8) w. 6GXDORI tw. masses їGHI reciproc. їGHI6: N=(2,2) w. 1 6<0 gauge th. tw. masses їGHI 1 6<0 2d 3d 4d 5d The Fluxtrap Background Geometrical realization of Nekrasov's construction of the equivariant gauge theory. Start with metric with 2 periodic directions and at least a U(1)xU(1) symmetry, no B-field, constant dilaton. Fluxbrane background with 3 independent deformation parameters: T 2 x 0123456789 8 8 x x +2πR8 (ρ1, θ1)(ρ2, θ2)(ρ3, θ3)(ρ4, θ4) v 9 9 x x +2πR9 fluxbrane 1 2 3 4 ◦◦ Impose identifications: fluxbrane parameters x8 x8 +2πR n x9 x9 +2πR n 8 8 9 9 θ θ +2πRR n θ θ +2πI R n k k k 8 8 k k k 9 9 This corresponds to the well-known Melvin or fluxbrane background. The Fluxtrap Background Introduce new angular variables with disentangled φ = θ Rx8 I x9 = θ Re( v¯) periodicities: k k − k − k k − k R I 8 9 k = +i v = x +ix k k 2 8 Fluxbrane metric ( T -fibration over Ω-deformed R ): V RV R dxi dxj V RV R dxi dxj ds2 =dx2 i j i j 0...7 − 1+V R V R − 1+V R V R · · 2 V R dxi + 1+V R V R dx8 i · − 1+V R V R · 2 V I dxi + 1+V I V I dx9 i +2V R V I dx8 dx9 · − 1+V I V I · · Generator of rotations: V = V R +iV I = x1 ∂ x0 ∂ + x3 ∂ x2 ∂ 1 0− 1 2 2− 3 + x5 ∂ x4 ∂ + x7 ∂ x6 ∂ 3 4− 5 4 6− 7 The Fluxtrap Background The general case breaks all supersymmetries. Impose condition N k =0 k=1 Find preserved Killing spinor γ K = exp φ ρkθk Πfluxη k 2 k k with projector Πflux = 1 (1 γ ) k 2 − ρkθkρN θN Each projector breaks half of the supersymmetries: 6 N 2 − susys are preserved The Fluxtrap Background T-dualize along torus directions and take decompactification limit to discard torus momenta: Fluxtrap background Before T-duality, locally, the metric was still flat, but some of the rotation symmetries were broken globally. R I Bulk fields after T-duality (case V V =0 , 1 R , 2 i R , 3 = 4 =0 ): · ∈ ∈ not anymore flat 2 2 2 2 2 2 7 2 2 ρ1 dφ1 +dx8 2 ρ2 dφ2 +dx9 k 2 ds =dρ1 + 2 2 +dρ2 + 2 2 + (dx ) , 1+1ρ1 1+2ρ2 k=4 2 2 B-field has appeared ρ1 ρ2 B = 1 2 2 dφ1 dx8 + 2 2 2 dφ2 dx9 , 1+1ρ1 ∧ 1+2ρ2 ∧ √ Φ0 creates a potential that Φ α e− 2 2 2 2 e− = (1 + ρ )(1+ ρ ) R 1 1 2 2 localizes instantons 1 φ1 R2 1 r R S1 × Figure 1: Cartoon of the geometry of the base of the manifold M3(1): a cigar with asymptotic radius 1/1. The Fluxtrap Background The space splits into a product M = M ( ) M ( ) R3 S1 ,(2.14) Study10 resulting3 1 × 3 geometry.2 × × 3 1 where R is generatedSpace by (ρ3, ψsplits, x7), the intoS is generated by (x6, x7), and M3 is a three- dimensional manifold which is obtained as a R foliation (generated by x8 or x9) over 4 M10 = M3(1) M3(2) R the cigar with asymptotic radius 1/i described by (ρ1, φ1) or (ρ2, φ2) ×(see the cartoon× in Figure 1): R x M ( ) 8 3 1 R-foliation over the cigar cigar ρ , φ 1 1 (2.15) This shows that the effect of the Ω–deformation is to regularize the rotations generated by ∂φ1 and ∂φ2 in the sense that the operators become bounded: φ 1 2 2 1 1 2 ρ1 1 R2 2 ρ2 1 ∂φ = < , ∂φ = < .(2.16) 1 1 + 2ρ2 2 2 1 + 2ρ2 2 1 1 1 2 2 2 r R S1 In a different frame this will translate into a bound on the asymptotic coupling of the × effective gauge theory for the motion of a D–brane. As a final remark we observe that even though the background in Equation (2.13) where the contributions of the two i are decoupled was obtained as a limit, it is by The generator of rotations is bounded (by asymptotic( ) radius). itself a solution of theFigure ten-dimensional1: Cartoon supergravity of the geometry equations of motion the base for anyof the value manifold M3 1 : a cigar with asymptotic radius 1/1. of ρi. What we have obtained is the starting point of the chain of dualities leading The space splits into a product eventually to the reciprocal background, as detailed in Table 1. 3 1 M10 = M3(1) M3(2) R S ,(2.14) M–theory. As a first step we dualize in x6 to type iia and then× lift to M–theory.× × A remarkable feature of the M–theory background is the fact that it is symmetric R3 1 under the exchangewhereρ , φ , x , is generatedρ , φ , x by, (.ρ This3, ψ, isx7 the), the originS ofis the generated S–duality by (x6, x7), and M3 is a three- { 1 1 8 1} ↔ { 2 2 9 2} covariance of the finaldimensional type iib background. manifold This which has is to obtained be contrasted as a withR foliation the fact (generated by x8 or x9) over that the directions xthe6 and cigar the M–circle with asymptoticx10 appear radiusin a non-symmetric1/i described fashion. by This(ρ1, isφ1) or (ρ2, φ2) (see the cartoon in Figure 1): 5 R x M ( ) 8 3 1 cigar ρ , φ 1 1 (2.15) This shows that the effect of the Ω–deformation is to regularize the rotations generated by ∂φ1 and ∂φ2 in the sense that the operators become bounded: 2 2 2 ρ1 1 2 ρ2 1 ∂φ1 = 2 2 < 2 , ∂φ2 = 2 2 < 2 .(2.16) 1 + 1ρ1 1 1 + 2ρ2 2 In a different frame this will translate into a bound on the asymptotic coupling of the effective gauge theory for the motion of a D–brane. As a final remark we observe that even though the background in Equation (2.13) where the contributions of the two i are decoupled was obtained as a limit, it is by itself a solution of the ten-dimensional supergravity equations of motion for any value of ρi. What we have obtained is the starting point of the chain of dualities leading eventually to the reciprocal background, as detailed in Table 1.
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