Stringy Instantons, Backreaction and Dimers. Eduardo García-Valdecasas Tenreiro Instituto de Física Teórica UAM/CSIC, Madrid Based on 1605.08092 and 1704.05888 by E.G. & A. Uranga and ongoing work String Phenomenology 2017, Virginia Tech, 6 th July 2017
Outline Motivation and Introduction 1 Motivation and Introduction. Instantons and Induced Operators. Quiver gauge theories and dimer diagrams. 2. 3. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 2 / 19
Motivation and Introduction Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 3 / 19
Motivation Motivation and Introduction Non-perturbative effects are essential to understand in string theory, both formally and phenomenologically. D-brane instantons are understood from the open string perspective. A closed string (geometric) dual description should exist, in a gauge/gravity duality spirit. (Klebanov & Strassler, 2000) Indeed, such a dual description exists. Taking instantons into account amounts to modifying the geometry. (Koerber & Martucci, 2007) (Garca-Valdecasas & Uranga, 2016) Toric geometries are well understood in terms of quiver gauge theories and dimer diagrams. Studying non-perturbative effects in these set-ups is easier. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 4 / 19
Operators induced by D-brane Instantons. Open String: Euclidean Dp-branes (D-brane instantons) induce non-perturbative field theory operators. (Aldazabal et al., 2001) Consider N a, N b D6-branes on intersecting cycles Π a, Π b, I ab = [Π a] [Π b ]. Introducing a D2-brane instanton on Π 3 produces fields: D2-D2 sector: Universal goldstone zero modes x µ and two goldstinos θ α. Upon integration of these, 4d effective action: d 4 x d 2 θ A e Z (...) neutral O charged (1) (*) This sector may actually give rise to additional fermion zero modes if too much SUSY is originally present. Not relevant for our purposes. (**)For simplicity orientifolds are not considered, but the results extend to that case as well. D2-D6 a sector: I aπ3 fermion zero modes in the fundamental of U(N a), λ a. O arises upon integration over these additional fermion zero modes. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 5 / 19
Operators induced by D-brane Instantons: Example. Consider two stacks of N D6-branes in cycles Π a, Π b with a D2-brane instanton on Π 3 such that I aπ3 = 1, I bπ3 = 1, I ab = 1. This configuration has D2-D6 fermion zero modes λ a, λ b and a D6 a-d6 b bifundamental multiplet Φ ab. These couple through a WS instanton as S z.m. λ aφ ab λb. So, the induced charged operator is, O charged dλ a d λ b e Sz.m. = det(φ ab ) which breaks the symmetry group: λ a D2 D6 A Φ ab U(N) a U(N) b SU(N) a SU(N) b U(1) λ b D6 B Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 6 / 19
WS instantons and D-branes. Eduardo García-Valdecasas Tenreiro Stringy Instantons, BackreactionFigure and Dimers. 7 / 19 Usual view: Worldsheet Instanton contributions need to be summed over in the path integral. Including the no-instanton contribution. Brane configurations may restrict the allowed patterns of WS instantons as follows. Consider D6-brane intersecting a D2-brane. Flux induced by the D6 inside the D2 (and vice-versa) Freed-Witten Anomaly String worldsheet needed for consistency.
Toric Gauge Theories and Dimers. D3-branes at a toric CY singularity have toric quiver gauge theory inside them, naturally encoded in a bipartite graph representing a tiling of the torus called dimer diagram. (Franco et al., 2006) T-dual picture where the geometry is encoded in a web-diagram of NS5 branes. There is yet a mirror description where physics is encoded in the tiling of a dual Riemann surface Σ, physically realised in the mirror geometry. Zig-Zag paths represent D6-branes. (Feng et al., 2008) 1 2 C F A 3 B F A F E C B D F A D 1 2 3 4 F C 3 4 E D 4 C 1 E B 2 Gauge Theory Quiver Dimer Mirror Σ SU(N) gauge factor Node Face Zig-Zag Path Bifundamental Matter Arrow Edge Zig-Zag Path Intersections Superpotential Terms Additional Info B/W node WS Instantons (nodes) Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 8 / 19
Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 9 / 19
Backreacting the geometry. Consider type IIA on a CY 3. A D2-brane instanton wrapping Π 3 can be given a dual description in terms of deformed geometry. The induced deformation is encoded in: dt 2 = W npδ 3 (Π 3 ) Rename: dα 2 = β 3 (2) (Where T e ij for a CY.) Change in the topology! β 3 becomes exact Π 3 becomes a boundary of a 4-chain. In turn the hodge dual Π 3 becomes a 3-chain with a 2-boundary. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 10 / 19
Brane Recombination. Consider now adding two D6-branes on 3-cycles Π + and Π such that [Π 3 ] [Π ± ] = ±1, so that: [Π ± ] = ±[ Π 3 ] +... (3) In the dual picture (after backreaction): [Π 3 ] = 0, Π 3 = Σ 2 Π ± = ±Σ 2 D6 s become inconsistent! But their recombination, [Π +] + [Π ] is consistent! Thus, the D-brane instanton triggers the recombination of the D6 s. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 11 / 19
Backreaction in the dimer. Consider a single instanton and abelian D6 s on dimer diagram. Mirror Riemann surface Σ has zig-zag paths corresponding to D6 s and a D2-instanton. Backreacting the instanton amounts to: 1 Step 1: Cut. Remove the instanton 1-cycle. This cuts Σ, each of its boundaries is a point. 1-cycles intersecting the removed cycle are split and become 1-chains. 2 Step 2: Reccombine. The 1-chains at each side of the cut are recombined without crossing edges of the tiling of Σ. This is unambiguous. 3 Step 3: Field Theory Operator. The 4d operator induced by the D2-instanton is now created by a WS instanton bounded by the recombined D6-branes. As we will see there are, broadly, two kinds of backreacted geometry: 1 The first, which correspond to complex deformations of the toric singularity, will split Σ, corresponding to the splitting of the web diagram. 2 The remaining produce non-cy geometries. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 12 / 19
Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 13 / 19
An example: The Conifold. Introduce an instanton in one of the nodes of a conifold quiver: Backreaction: The zig-zag path is removed, the surface cut and the remaining brane, recombined: 1 2 WS instantons are shown in light shaded colors. The surface splits like in a complex deformation. Note: the field theory operator is trivial in both pictures. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 14 / 19
Further examples: Double Conifold Introduce one D2-brane instanton in the double conifold, in blue. 3 3 1 2 F F A D A D 4 C 1 4 C B E B E 3 4 2 2 The backreacted geometry is the conifold, agreeing with the complex deformation. Again, Σ splits. In the original geometry fermion zero modes couplings are λ 13 X 34 X 43 λ31 + λ 12 X 24 X 42 λ21 (4) Saturating zero modes gives operator: O charged (X 34 X 43 )(X 24 X 42 ) In backreacted geometry, the remaining WS instanton induces the same operator, as predicted. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 15 / 19
Further examples: dp 0 dp 0 has no complex deformations. Let s see what happens. 1 1 1 3 2 In the backreacted geometry Σ is not split, but its genus has decreased by 1. The geometry is non-toric. In the original geometry fermion zero modes couplings are ɛ ijk λ i 12 X j λ k 23 31, i, j, k = 1, 2, 3 (5) Saturating zero modes gives the operator: O charged = (X 1 23 )(X 2 23 )(X 3 23 ) Again, in backreacted geometry, the remaining WS instanton induces the same operator, as predicted. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 16 / 19
Some generalisations. This mechanism can be generalized as follows, Multi-Instantons: Instantons wrapping a combination of cycles in homology can also be considered and the results are consistent. The only subtlety are D2 D2 zero modes, but these can be nicely understood. Non-compact instantons: These appear as links between faces in Σ. The recipe can be applied as: Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 17 / 19
Conclusions & Outlook There is a dual description to the open string D2-brane instanton in terms of a modified geometry. The cycle wrapped by the instanton develops a boundary and disappears from homology. Branes intersecting it are cut and recombine. This process can be understood in dimer language. Generalisations to multiple-instanton processes and non-compact instantons are straight forward. Further work: Understand better the non-toric geometries. Bipartite field theories? Find setups where the gauge/gravity duality is better understood. Find concrete examples in global compactifications and applications to phenomenology. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 18 / 19
Thank You! Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 19 / 19
The Eq. in more detail. F-flatness equation Applying to T one finds, φ W 3( φ logn )W = 0 (6) δ T W 3(δ T logn )W = 0 d H Z = 2iW 0 e 2A Imt (7) While for Z one finds, δz, d H Ret if W M 2N e 4A Z = 0 (8) Which is more complicated. Adding a non-perturbative contribution and neglecting the first term one finds, d H Z 2iW npj np = 2iW npδ (2) (Π) (9) Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 19 / 19
References Aldazabal, G., Franco, S., Ibanez, Luis E., Rabadan, R., & Uranga, A. M. 2001. Intersecting brane worlds. JHEP, 02, 047. Feng, Bo, He, Yang-Hui, Kennaway, Kristian D., & Vafa, Cumrun. 2008. Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys., 12(3), 489 545. Franco, Sebastian, Hanany, Amihay, Kennaway, Kristian D., Vegh, David, & Wecht, Brian. 2006. Brane dimers and quiver gauge theories. JHEP, 0601, 096. Garca-Valdecasas, Eduardo, & Uranga, Angel. 2016. On the 3-form formulation of axion potentials from D-brane instantons. Klebanov, Igor R., & Strassler, Matthew J. 2000. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 19 / 19
References Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities. JHEP, 0008, 052. Koerber, Paul, & Martucci, Luca. 2007. From ten to four and back again: How to generalize the geometry. JHEP, 08, 059. Eduardo García-Valdecasas Tenreiro Stringy Instantons, Backreaction and Dimers. 19 / 19