Albert Einstein Center for Fundamental Physics University of Bern 5 September 2017 | 京都 work in collaboration with: S. Reffert, Y. Sekiguchi (AEC Bern); S. Hellerman (IPMU); N. Lambert (King’s College). Domenico Orlando Deformations, defects and a noncommutative spectral curve Deformations,defects and anoncommutativespectralcurve Domenico Orlando
Outline Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity from geometry Wilson lines and surfaces Domenico Orlando Deformations, defects and a noncommutative spectral curve
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects Outline Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity from geometry Wilson lines and surfaces Domenico Orlando Deformations, defects and a noncommutative spectral curve
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects The Ω deformation The Ω background was introduced by Nekrasov as a way of regularizing the and Witten. The path integrals localize on a discrete set of points. Domenico Orlando Deformations, defects and a noncommutative spectral curve four-dimensional instanton partition function and reproducing the results of Seiberg One introduces an appropriate deformation of the four-dimensional theory , with parameters ε 1 and ε 2 , breaking rotational invariance of R 4 . The k -instanton contribution to the prepotential for the original (undeformed) theory is found in the limit ε i → 0.
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects Philosophy If a problem is hard, make it harder (and new structures will appear). Domenico Orlando Deformations, defects and a noncommutative spectral curve
Introduction and motivation In fact this turned out to be a much richer subject. Domenico Orlando six-dimensional theory on the Ω background. The fmuxtrap topological strings on a CY related to the spectral curve; Finite ε Defects Noncommutativity The Ω deformation Deformations, defects and a noncommutative spectral curve The partition function in the Ω background has a meaning also for fjnite values of ε . ▶ In the limit ε 1 = − ε 2 ∝ g s the partition function is the same as the one for ▶ In the limit ε 1 = 0 the gauge theory is closely related to quantum integrable h = ε 2 ; models with ¯ ▶ In the general case ε 1 ̸ = ε 2 , we have the refjnement of topological strings ; ▶ The AGT construction can be understood in terms of compactifjcations of a
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects The fmuxtrap But there is more. The string theory background that realizes this deformation has many interesting properties: dimensions Domenico Orlando Deformations, defects and a noncommutative spectral curve ▶ it’s an exact CFT ▶ it’s directly related to noncommutativity ▶ can be used to realize explicitly fjeld theories in presence of defects of different ▶ it’s the common origin of different gauge theories that seem unrelated.
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects Outline Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity from geometry Wilson lines and surfaces Domenico Orlando Deformations, defects and a noncommutative spectral curve
Introduction and motivation monodromy Domenico Orlando T-duality is the string theory version of a reduction on S 1 . The fmuxtrap M We want to write the String Theoretical analog to a compactifjcation with a Wilson line. Defects The Ω deformation Noncommutativity Deformations, defects and a noncommutative spectral curve Melvin construction in fjeld theory In the Melvin construction one starts with an S 1 fjbration over R 4 , with a non-trivial S 1 ( ˜ u ) { u ∼ ˜ u + 2 π n u , ˜ n u ∈ Z θ k ∼ θ k + 2 πε k ˜ Rn u , R 4 ( ρ k , θ k )
Introduction and motivation The fmuxtrap Domenico Orlando to R This modifjes the boundary conditions from Deformations, defects and a noncommutative spectral curve Defects The String Theory version The Ω deformation Noncommutativity Start from a Ricci-fmat metric d s 2 = g ij d x i d x j + d ( ˜ x 9 = � x 9 ) 2 , where ˜ u , where g has N ≤ 4 R ˜ (non-bounded) rotational isometries generated by ∂ θ k . Pass to a set of disentangled variables φ k = θ k − ε k � R ˜ u , ( ) + 2 π n k ( 0 , 1 ) 1 , ε k ˜ u , θ k ) ∼ ( ˜ ( ˜ u , θ k ) + 2 π n u u , φ k ) ∼ ( ˜ ( ˜ u , φ k ) + 2 π n u ( 1 , 0 ) + 2 π n k ( 0 , 1 ) . The price to pay is the appearance of a graviphoton ε U i d x i .
Introduction and motivation The fmuxtrap Domenico Orlando All the local degrees of freedom are physical . u ) have been removed (they turn into infjnitely heavy winding modes). R Deformations, defects and a noncommutative spectral curve u . We get a B -fjeld and a non-trivial dilaton: the fmuxtrap The generic fmuxtrap The Ω deformation Noncommutativity Defects Now T-dualize in ˜ ( d x 9 ) 2 d s 2 = g ij d x i d x j − ε 2 U i U j d x i d x j + 1 + ε 2 U i U i , 1 + ε 2 U i U i B = ε U i d x i ∧ d x 9 1 + ε 2 U i U i , √ √ α ′ e − Φ 0 e − Φ = 1 + ε 2 U i U i , We have taken the limit ˜ R → 0: in this picture the irrelevant degrees of freedom (rotations around ˜
Introduction and motivation R Domenico Orlando U i U i N duality isometries before and after the The fmuxtrap background Deformations, defects and a noncommutative spectral curve The generic fmuxtrap The Ω deformation Noncommutativity Defects ▶ For ε = 0 this is the initial Ricci-fmat d s 2 = g ij d x i d x j + ( d x 9 ) 2 − ε 2 U i U j d x i d x j ▶ U is the generator of the rotational , 1 + ε 2 U i U i B = ε U i d x i ∧ d x 9 1 + ε 2 U i U i , √ √ α ′ e − Φ 0 ε U i ∂ i = ∑ e − Φ = ε k ∂ φ k 1 + ε 2 U i U i , k = 1 ▶ branes will be trapped in U = 0 by the terms in the denominators ▶ ε regularizes the rotation , which is always bounded if ε ̸ = 0 ∥ U ∥ 2 trap = 1 + ε 2 U i U i < 1 ε 2 . ▶ the dilaton has a maximum when U = 0.
Introduction and motivation v ). Domenico Orlando φ ρ After two T-dualities, the space takes the form of a product The fmuxtrap Deformations, defects and a noncommutative spectral curve To get an intuitive picture of the deformation, start with fmat space and twist in two The Ω deformation Noncommutativity Defects Fluxtrap around fmat space directions ( ˜ u and ˜ { { u ∼ ˜ v ∼ ˜ ˜ u + 2 π n u , ˜ v + 2 π n v , θ 1 ∼ θ 1 + 2 πε 1 ˜ θ 2 ∼ θ 2 + 2 πε 2 ˜ R u n u , R v n v , M 10 = M 3 ( ε 1 ) × M 3 ( ε 2 ) × R 4 where M 3 ( ε ) is a R fjbration (the dual direction) over a cigar with asymptotic radius 1 / ε . The NS three-form is the volume of M 3 . 1 / ε R 2
Introduction and motivation with Domenico Orlando φ k N η R φ k The fmuxtrap η L N iia Supersymmetry in type IIA The Ω deformation Noncommutativity Defects Deformations, defects and a noncommutative spectral curve T–duality maps the Killing spinors η iib into local type iia Killing spinors η iia . Using an appropriate vielbein for the T–dual metric they take the form η iia = η L iia + η R [ ] ∏ iia = ( 1 + Γ 11 ) exp P fmux η w , 2 Γ ρ k θ k k = 1 [ ] ∏ iia = ( 1 − Γ 11 ) Γ u exp P fmux η w , 2 Γ ρ k θ k k = 1 where Γ u is the gamma matrix in the u direction normalized to unity.
Introduction and motivation The fmuxtrap Domenico Orlando independent ε breaks 1/2 of the supersymmetry ; Examples: Supersymmetry Defects Noncommutativity The Ω deformation Deformations, defects and a noncommutative spectral curve Depending on η w , the projector P fmux can either break all supersymmetries or preserve some of them . In the latter case, at least 1 / 2 N − 1 of the original ones are preserved. ▶ In fmat space η w is a constant spinor with 32 independent components. Each ▶ There are special confjgurations with 12 supercharges ▶ In the Taub–nut case, the orientation is fjxed by the triholomorphic U ( 1 ) isometry: ▶ The choice ε 1 = − ε 2 preserves all supersymmetries. ▶ The choice ε 1 = ε 2 breaks all supersymmetries.
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects The point Domenico Orlando Deformations, defects and a noncommutative spectral curve ▶ We look at a String Theory realization of the Melvin construction ▶ T-duality removes the non-physical degrees of freedom ▶ We fjnd a background where all local degrees of freedom are physical ▶ We can study this background using String Theory ▶ Supersymmetry in terms of Killing spinors in the bulk
Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity Defects Outline Introduction and motivation The fmuxtrap The Ω deformation Noncommutativity from geometry Wilson lines and surfaces Domenico Orlando Deformations, defects and a noncommutative spectral curve
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