Heterotic String Compactication with Gauged Linear Sigma Models
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1 Heterotic String Compactication with Gauged Linear Sigma Models Fabian Rühle Bethe Center for Theoretical Physics Bonn University LMU Fields & Strings 04/25/2013 Based on: [Lüdeling,FR,Wieck: ], [Blaszczyk,Groot Nibbelink,FR: ], [Blaszczyk,Groot Nibbelink,FR: ]
2 Motivation Start with heterotic string on M 1,3 X 4D physics is a function of compactication manifold X Want N = 1 SUSY in 4D X is CY threefold [Candelas,Horowitz,Strominger,Witten] Want G SM = SU(3) SU(2) U(1) gauge group Choose holomorphic, stable vector bundle V to break E 8 E 8 G SM ( G hidden ) Study resulting string theory Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 1 / 34
3 Motivation Start with heterotic string on M 1,3 X 4D physics is a function of compactication manifold X Want N = 1 SUSY in 4D X is CY threefold [Candelas,Horowitz,Strominger,Witten] Want G SM = SU(3) SU(2) U(1) gauge group Choose holomorphic, stable vector bundle V to break E 8 E 8 G SM ( G hidden ) Study resulting string theory Problem Resulting string theory described by complicated NLSM Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 1 / 34
4 Motivation Solution ideas Study special CYs that allow for a free CFT description (Orbifolds) [Dixon,Harvey,Vafa,Witten],[Blaszczyk,Buchmüller,Groot Nibbelink,Nilles, Raby,Ramos Sanchez,Ratz,FR,Vaudrevange,... ] Study generic CYs in supergravity approximation [Anderson,Bouchard,Braun,Donagi,Gray,He,Lukas,Ovrut,Palti,Pantev,Waldram,... ] Study easy non-conformal GLSM that ows in the IR to conformal NLSM [Witten],[Adams,Blaszczyk,Blumenhagen,Carlevahro,Distler,Groot Nibbelink,Israel,Rahn,FR,... ] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 2 / 34
5 Motivation Our procedure Start with orbifold where string theory under control Find consistency requirements from exact free CFT calculations [Blaszczyk,Groot Nibbelink,Ratz,FR,Trapletti,Vaudrevange] Unknown how to get them in other phases like SUGRA approximation Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 3 / 34
6 Motivation Our procedure Start with orbifold where string theory under control Find consistency requirements from exact free CFT calculations [Blaszczyk,Groot Nibbelink,Ratz,FR,Trapletti,Vaudrevange] Unknown how to get them in other phases like SUGRA approximation Depart from orbifold geometry Study more general points in moduli space Phenomenology (SUSY+exotics+GG) requires departure from orbifold point Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 3 / 34
7 Motivation Our procedure Start with orbifold where string theory under control Find consistency requirements from exact free CFT calculations [Blaszczyk,Groot Nibbelink,Ratz,FR,Trapletti,Vaudrevange] Unknown how to get them in other phases like SUGRA approximation Depart from orbifold geometry Study more general points in moduli space Phenomenology (SUSY+exotics+GG) requires departure from orbifold point Go to GLSM description of the orbifold [Blaszczyk,Groot Nibbelink,FR] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 3 / 34
8 Motivation Our procedure Start with orbifold where string theory under control Find consistency requirements from exact free CFT calculations [Blaszczyk,Groot Nibbelink,Ratz,FR,Trapletti,Vaudrevange] Unknown how to get them in other phases like SUGRA approximation Depart from orbifold geometry Study more general points in moduli space Phenomenology (SUSY+exotics+GG) requires departure from orbifold point Go to GLSM description of the orbifold [Blaszczyk,Groot Nibbelink,FR] Study other GLSM phases like smooth CY Phenomenological models require torsion and often NS5 branes Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 3 / 34
9 Outline 1 N = (2, 2) GLSMs Introduction GLSM description of compactication spaces Study of moduli space 2 N = (0, 2) GLSMs & Anomalies Introduction Anomalies + Cancelation mechanism Examples 3 N = (0, 2) GLSMs & Discrete Symmetries Introduction Study of discrete symmetries 4 Conclusion Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 4 / 34
10 Part I N = (2, 2) GLSMs
11 Field content Think of 2D N = (2, 2) superspace as a dim. reduction of 4D N = 1 superspace with Lorentz group SO(1, 1) [Witten] Superspace coordinates Two WS coordinates σ 1, σ 2 Two complex Grassmann variables θ +, θ Multiplets Chiral multiplet (D ± Z = 0): Z = (z, ψ +, ψ, F ) Vector multiplet (V = V ): V = (A, A σ, A σ, λ +, λ, D) Twisted chiral multiplet (D + Σ = D Σ = 0) Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 5 / 34
12 GLSM Action GLSM Action S GLSM = S kin + S FI + S W S kin = d 2 σd 4 θze 2qV Z + 1 e 2 Σ Σ S FI = d 2 σd θ dθ + ρσ + h.c. with ρ = b + iβ S W = d 2 σd 2 θcp(z) + h.c. with q(c) < 0,q(Z) > 0, q(p(z)) = q(c) = i q(z i) Algebraic EOMs D term: i q i z i 2 + q c c 2 = b F term: P(z) = 0, c P(z) z i = 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 6 / 34
13 Relation to target space (2, 2) GLSMs describe heterotic string in standard embedding Relations GLSM elds Z i Toric ambient space coordinates GLSM gaugings U(1) N Weights of toric spaces F terms hypersurface constraints q(c) + i q(p(z i)) = 0 c 1 (TX ) = 0 = c 1 (V ) FI terms b Kähler parameters D terms Kähler cone, SR ideal X is Kähler and Ricci-at X has SU(N) holonomy Spin connection is identied with gauge connection Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 7 / 34
14 Example: T 2 GLSM Fields Z 1 Z 2 Z 3 C U(1) charges Superpotential W = C(Z Z3 2 + Z3 3 + κz 1Z 2 Z 3 ) κ encodes CS of T 2 relate to τ via Weierstrass function Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 8 / 34
15 Example: T 2 GLSM Fields Z 1 Z 2 Z 3 C U(1) charges Superpotential W = C(Z Z3 2 + Z3 3 + κz 1Z 2 Z 3 ) κ encodes CS of T 2 relate to τ via Weierstrass function D Term z z z c 2 = b b > 0: SR={z 1 z 2 z 3 }, c = 0 (from F term) smooth T 2 b < 0: SR={c}, z 1 = z 2 = z 3 = 0 (from F term) Z 3 LG orbifold Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 8 / 34
16 Example: T 2 GLSM Fields Z 1 Z 2 Z 3 C U(1) charges Superpotential W = C(Z Z3 2 + Z3 3 + κz 1Z 2 Z 3 ) κ encodes CS of T 2 relate to τ via Weierstrass function D Term z z z c 2 = b b > 0: SR={z 1 z 2 z 3 }, c = 0 (from F term) smooth T 2 b < 0: SR={c}, z 1 = z 2 = z 3 = 0 (from F term) Z 3 LG orbifold Note: We are not after the (non-geometric) LG orbifold but after the (geometric) singular CY orbifold Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 8 / 34
17 Example: (T 2 ) 3 /Z 3 orbifold e 2πi/3 e 2πi/3 e 2πi/3 Start with compactication space (T 2 ) 3 : Choose complex coordinates z 1, z 2, z 3 and lattice Λ T Remove parallelizable spinors by introducing non-trivial holonomy: Choose CS such that Λ T has Z 3 symmetry Λ T is root lattice A 2 A 2 A 2 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 9 / 34
18 Example: (T 2 ) 3 /Z 3 orbifold Start with compactication space (T 2 ) 3 : Choose complex coordinates z 1, z 2, z 3 and lattice Λ T Remove parallelizable spinors by introducing non-trivial holonomy: Choose CS such that Λ T has Z 3 symmetry Λ T is root lattice A 2 A 2 A 2 Mod out Z 3 : Action described by orbifold twist v = (v 1, v 2, v 3 ): θ : (z 1, z 2, z 3 ) (e 2πiv 1 z 1, e 2πiv 2 z 2, e 2πiv 3 z 3 ) 27 xed points CY constraint: Ω = dz 1 dz 2 dz 3 invariant v 1 +v 2 +v 3 =0 mod 1 Modular invariance: Augment orbifold twist by shift V in gauge DOFs: θ : Λ E8 E 8 Λ E8 E 8 + V with V 2 v 2 = 0 mod 2 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 9 / 34
19 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 R R R Fterms for c i and z iρ : z i1 3 + z i2 3 + z i3 3 + κ i z i1 z i2 z i3 = 0, i = 1, 2, 3, c i (z iρ 2 + κ i z iµ z iν ) = 0, Dterms: z i1 2 + z i2 2 + z i3 2 3 c i 2 = a i, i = 1, 2, 3 Geometry: z i1, z i2, z i3 : coordinate of i th torus Fix CS at τ i = e 2πi/3 κ i = 0 from Weierstrass map i, ρ = 1, 2, 3, ρ µ ν ρ Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 10 / 34
20 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 R R R Fterms for c i and z iρ : z i1 3 + z i2 3 + z i3 3 = 0, i = 1, 2, 3, c i z iρ 2 = 0, i, ρ = 1, 2, 3 Dterms: z i1 2 + z i2 2 + z i3 2 3 c i 2 = a i, i = 1, 2, 3 Geometry: a i : sizes of tori a i > 0 at least 3 z i,ρ 0 c i = 0 (assume this case for now) Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 11 / 34
21 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 x 1 R R R E Fterms for c i : zi1 3 x 1 + zi2 3 + z3 i3 = 0, i = 1, 2, 3 Dterms: z i1 2 + z i2 2 + z i3 2 = a i, i = 1, 2, 3 z z z x 1 2 = b 1 Geometry: [Aspinwall, Plesser] b 1 : sizes of exceptional cycles b 1 < 0 x 1 0 x 1 breaks E 1 to Z 3 with 27 FP z 11 =z 21 =z 31 =0 b 1 > 0 FP z 11 =z 21 =z 31 =0 forbidden smooth Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 11 / 34
22 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 x 1 x 2 R R R E E Fterms for c i : zi1 3 x 1 + zi2 3 x 2 + zi3 3 = 0, i = 1, 2, 3 Dterms: z i1 2 + z i2 2 + z i3 2 = a i, i = 1, 2, 3 z z z x 1 2 = b 1 z z z x 2 2 = b 2 Geometry: b 1 < 0 x 1 0 x 1 breaks E 1 to Z 3 with 9 FP z 11 =z 21 =z 31 =0 b 2 < 0 x 2 0 x 2 breaks E 2 to Z 3 with 9 FP z 12 =z 22 =z 32 =0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 11 / 34
23 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 x 1 x 2 R R R E E E z i1 2 + z i2 2 + z i3 2 = a i, i = 1, 2, 3 z z z x 1 2 = b 1 z z z x 2 2 = b 2 Where are last 9 xed points? Combine U(1)'s or D terms: z z z x x 2 2 = i a i + b 1 + b 2 Geometry: x FP z 11 =z 21 =z 31 =0 x FP z 12 =z 22 =z 32 =0 Last 9 FP z 13 =z 23 =z 33 =0. Note: Not smooth for b 1,2 > 0! Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 11 / 34
24 Example: (T 2 ) 3 /Z 3 orbifold U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 x 1 x 2 x 3 R R R E E E Fterms for c i : z 3 i1 x 1 + z 3 i2 x 2 + z 3 i3 x 3 = 0, i = 1, 2, 3 Dterms: z i1 2 + z i2 2 + z i3 2 = a i, i = 1, 2, 3 z 1ρ 2 + z 2ρ 2 + z 3ρ 2 3 x ρ 2 = b ρ, ρ = 1, 2, 3 Geometry: x 1 =0 generates Z 3 with FP z 11 =z 21 =z 31 =0 x 2 =0 generates Z 3 with FP z 12 =z 22 =z 32 =0 x 3 =0 generates Z 3 with FP z 13 =z 23 =z 33 =0 3 9 xed points Smooth for b ρ > 0! Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 11 / 34
25 General procedure: To build an orbifold/resolution GLSM model 1 Choose toric description appropriate for orbifold action 2 Introduce exceptional divisors to smoothen the singularities 3 Set FI parameters a 0 > b to study orbifold or a b > 0 to study blowup 4 Construct inherited divisors and linear equivalences 5 Read o intersection numbers In this way, the resolution phase can be studied using a GLSM which can be smoothly connected to the orbifold. The procedure yields GLSM description of previous approaches which proceeded via polytopes and gluings. [Lust,Reert,Scheidegger,Stieberger] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 12 / 34
26 Study of Moduli Space But we can do more! Having a GLSM realization, we can probe whole moduli space by simply varying FI parameter. U(1)'s z 11 z 12 z 13 z 21 z 22 z 23 z 31 z 32 z 33 c 1 c 2 c 3 x 1 x 2 x 3 R R R E E E Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 = a i, i = 1, 2, 3 3 i=1 z iρ 2 3 x ρ 2 = b ρ, ρ = 1, 2, 3 For simplicity: Set a i = a and b ρ = b. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 13 / 34
27 Phase I: Non-Geometric Regime b Superpotential: W = i,ρ c iz 3 iρ x ρ a Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric 3 i=1 z iρ 2 3 x ρ 2 =b a < 0, b < 0: c i = a 3, x ρ = b 3, z iρ = 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
28 Phase I: Non-Geometric Regime b Superpotential: W = i,ρ c iz 3 iρ x ρ a Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric 3 i=1 z iρ 2 3 x ρ 2 =b a < 0, b < 0: c i = a 3, x ρ = b 3, z iρ = 0 Target space is a point. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
29 Phase II: Orbifold b Superpotential: W = i,ρ c iz 3 iρ x ρ a Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 0, b < 0: c i = 0, x ρ > 0, z iρ 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
30 Phase II: Orbifold b Superpotential: W = i,ρ c iz 3 iρ x ρ a Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 0, b < 0: c i = 0, x ρ > 0, z iρ 0 Target space is the T 6 /Z 3 orbifold. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
31 Phase III: Blowup b blowup a Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 3b > 0: c i = 0, x ρ 0, z iρ 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
32 Phase III: Blowup b blowup a Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 3b > 0: c i = 0, x ρ 0, z iρ 0 Target space is the resolution CY of the T 6 /Z 3 orbifold. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
33 Phase IV: Singular Over-Blowup b orbifold (singular over-blowup) non-geometric blowup orbifold a Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a 3 i=1 z iρ 2 3 x ρ 2 =b a < 0, b > 0: Note complete symmetry of the model under x ρ c i, z iρ z ρi, a b c i > 0, x ρ = 0, z iρ 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
34 Phase IV: Singular Over-Blowup b orbifold (singular over-blowup) non-geometric blowup orbifold a Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a 3 i=1 z iρ 2 3 x ρ 2 =b a < 0, b > 0: Note complete symmetry of the model under x ρ c i, z iρ z ρi, a b c i > 0, x ρ = 0, z iρ 0 Target space again T 6 /Z 3 orbifold, with x and c exchanged. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
35 Phase V: Over-Blowup b orbifold (singular over-blowup) blowup a over-blowup Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b b > 3a > 0: c i 0, x ρ = 0, z iρ 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
36 Phase V: Over-Blowup b orbifold (singular over-blowup) blowup a over-blowup Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b b > 3a > 0: c i 0, x ρ = 0, z iρ 0 Target space is the resolution CY of the other T 6 /Z 3 orbifold. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
37 Phase VI: Critical Blowup b orbifold (singular critical over-blowup) blowup blowup a over-blowup Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 0, b [ a 3, 3a] : c i ρ 0, x ρ 0, z iρ 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
38 Phase VI: Critical Blowup b orbifold (singular critical over-blowup) blowup blowup a over-blowup Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a non-geometric orbifold 3 i=1 z iρ 2 3 x ρ 2 =b a > 0, b [ a 3, 3a] : c i ρ 0, x ρ 0, z iρ 0 Target space is hybrid phase with blowup limits for b a 3 & b 3a. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
39 Summary b orbifold (singular over-blowup) non-geometric over-blowup critical blowup blowup orbifold a Superpotential: W = i,ρ c iz 3 iρ x ρ Dterms: 3 ρ=1 z iρ 2 3 c i 2 =a 3 i=1 z iρ 2 3 x ρ 2 =b Note that in general The dimension of the TS can jump between the phases There can be op-transitions also outside the CY There can be several distinct singular phases [Aspinwall, Greene, Morrison, Plesser,... ] There is a vast (moduli) space to be explored! Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 14 / 34
40 Part II N = (0, 2) Models & Anomalies
41 Denition of GLSM supereld type notation charge bosonic DOF fermionic DOF chiral Ψ a (q I ) a z a ψ a chiralfermi Λ α (Q I ) α F α λ α gauge (V, A) I 0 aσ, I aσ I, D I Φ I Fermigauge σ i 0 s i ϕ i chiral Φ m (q I ) m x m ψ m chiralfermi Γ µ (Q I ) µ F µ γ µ Think of 2D N = (0, 2) superspace as N = (2, 2) superspace and dispense of θ +, θ + [Dine, Seiberg] Multiplets (2, 2) Chiral multiplet Z (2,2) (Z; χ) =(chiral; chiral-fermi) (2, 2) Vector multiplet V 2,2 (V, A; Σ) =(gauge; Fermi-gauge) Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 15 / 34
42 Denition of GLSM supereld type notation charge bosonic DOF fermionic DOF chiral Z a (q I ) a z a ψ a chiralfermi Λ α (Q I ) α F α λ α gauge (V, A) I 0 aσ I, ai σ, D I Φ I Fermigauge Σ i 0 s i ϕ i chiral Φ m (q I ) m x m ψ m chiralfermi Γ µ (Q I ) µ F µ γ µ Geometry: D Term: (q I ) a z a 2 + (q I ) m x m 2 b I = 0, b I : FIparameter F Term: W geom = Γ µ P µ (Z) P µ (Z) = 0 Geometry D Terms give SR ideal Kähler cone, GLSM phase F Terms give restriction to hypersurface Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 15 / 34
43 Denition of GLSM supereld type notation charge bosonic DOF fermionic DOF chiral Z a (q I ) a z a ψ a chiralfermi Λ α (Q I ) α F α λ α gauge (V, A) I 0 aσ, I aσ I, D I Φ I Fermigauge σ i 0 s i ϕ i chiral Φ m (q I ) m x m ψ m chiralfermi Γ µ (Q I ) µ F µ γ µ Gauge group: Fermionic Transformation: δ Θ Λ α = M α i(z)θ i Superpotential: W bundle = Φ m N mα (Ψ)Λ α Φ m N mα (Z) = 0 Gauge group Gauge group and particle content given by (naturally arising) monad construction via ker(n)/im(m). Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 15 / 34
44 Anomalies A IJ := 1 2 (Q I Q J q I q J ), Q I Q J := α (Q I ) α (Q J ) α + µ (Q I ) µ (Q J ) µ, q I q J := a (q I ) a (q J ) a + m (q I ) m (q J ) m. Problem In general many U(1) gauge groups Huge amount of stringent anomaly conditions. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 16 / 34
45 Anomalies A IJ := 1 2 (Q I Q J q I q J ) T IJ, Q I Q J := α (Q I ) α (Q J ) α + µ (Q I ) µ (Q J ) µ, q I q J := a (q I ) a (q J ) a + m (q I ) m (q J ) m. Idea Introduce new elds to obtain GreenSchwarz mechanism on the worldsheet to cancel gauge anomalies. [Adams,Ernebjerg,Lapan] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 16 / 34
46 GreenSchwarz mechanism GreenSchwarz mechanism needs elds that transform with shifts. Our approach Use logarithm of coordinate elds Ψ W FI = [ ] ρ 0 I + T XI ln R X (Z) f I T IJ = ri X T XI A IJ = 1 2 (Q I Q J q I q J ) T IJ with ρ 0 I : constant FI parameter R X (Z): homogeneous polynomials w/ charges ri X T XI : (quantized) coecients: T XI F I Z Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 17 / 34
47 GreenSchwarz mechanism GreenSchwarz mechanism needs elds that transform with shifts. Our approach Use logarithm of coordinate elds Ψ W FI = [ ] ρ 0 I + T XI ln R X (Z) f I T IJ = ri X T XI A IJ = 1 2 (Q I Q J q I q J ) T IJ WS Anom: TS BI's: I 4 = A IJ F I F J dh = ch 2 (V ) ch 2 (TX ) NS5 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 17 / 34
48 GreenSchwarz mechanism GreenSchwarz mechanism needs elds that transform with shifts. Our approach Use logarithm of coordinate elds Ψ W FI = [ ] ρ 0 I + T XI ln R X (Z) f I T IJ = ri X T XI A IJ = 1 2 (Q I Q J q I q J ) T IJ WS Anom: TS BI's: Anom: I 4 = A IJ F I F J dh = ch 2 (V ) ch 2 (TX ) NS5 I 4 = [QI α Qα J qm I qj m ]F I F J [qi a }{{} qa J Qµ I Qµ J ]F I F J T IJ F I F }{{}}{{ J } Z (ch 2 (V )) Z (ch 2 (TX )) Z (NS5) Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 17 / 34
49 1.) No anomalies supereld Z a=1,...,8 Γ µ=1,...,4 Λ α=1,...,8 Φ m=1,...,4 gauge charge P 7 [2, 2, 2, 2] with SU(3) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln Z 1 r 1 1 = 1, T 11 = T, T 2Z V D = 1 [ z a 2 2 x m 2 b T ln z 1 ] 2! = 0 2 a m Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 18 / 34
50 1.) No anomalies supereld Z a=1,...,8 Γ µ=1,...,4 Λ α=1,...,8 Φ m=1,...,4 gauge charge P 7 [2, 2, 2, 2] with SU(3) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln Z 1 r 1 1 = 1, T 11 = T, T 2Z V D = 1 [ z a 2 2 x m 2 b T ln z 1 ] 2! = 0 2 A 11 = 1 2 a m [ 8 (1) ( 2) 2 8 (1) 2 4 ( 2) 2] + T 11! = 0 T 11 = T = 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 18 / 34
51 1.) No anomalies b = 1 b > 0 x a = 0, V D! = 0 8 z a 2 = b a=1 Geometry compact, no anomalies. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 19 / 34
52 2.) T > 0, compact geometry, eective curve supereld Ψ a=1,...,8 Γ µ=1,...,4 Λ α=1,...,4 Φ m=1,2 gauge charge P 7 [2, 2, 2, 2] with SU(2) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln[z 1 ] r 1 1 = 1, T 11 = T, T 2Z V D = 1 [ z a 2 2 x m 2 b T ln z 1 ] 2! = 0 2 a m Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 20 / 34
53 2.) T > 0, compact geometry, eective curve supereld Ψ a=1,...,8 Γ µ=1,...,4 Λ α=1,...,4 Φ m=1,2 gauge charge P 7 [2, 2, 2, 2] with SU(2) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln[z 1 ] r 1 1 = 1, T 11 = T, T 2Z V D = 1 [ z a 2 2 x m 2 b T ln z 1 ] 2! = 0 2 A 11 = 1 2 a m [ 4 (1) ( 2) 2 8 (1) 2 2 ( 2) 2] T 11! = 0 T 11 = T = 2 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 20 / 34
54 2.) T > 0, compact geometry, eective curve b = 1 b = 10 b > 0 x a = 0, V D! = 0 8 z a 2 = b + 2 ln z 1 z 1 2 a=2 Geometry still compact, anomalies canceled by NS5 branes. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 21 / 34
55 3.) T < 0, decompactied geometry, noneective curve supereld Z a=1,...,8 Γ µ=1,...,4 Λ α=1,...,8 Φ m=1,2 gauge charge P 7 [2, 2, 2, 2] with SU(6) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln[z 1 ] r 1 1 = 1, T 11 = T, T 4Z V D = 1 [ z a 2 4 x m 2 b T ln z 1 ] 2! = 0 2 a m Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 22 / 34
56 3.) T < 0, decompactied geometry, noneective curve supereld Z a=1,...,8 Γ µ=1,...,4 Λ α=1,...,8 Φ m=1,2 gauge charge P 7 [2, 2, 2, 2] with SU(6) bundle A 11 = 1 2 [ Q 2 1 q1 2 ] T11 W FI = ρ 0 + T ln[z 1 ] r 1 1 = 1, T 11 = T, T 4Z V D = 1 [ z a 2 4 x m 2 b T ln z 1 ] 2! = 0 2 A 11 = 1 2 a m [ 8 (1) ( 2) 2 8 (1) 2 2 ( 4) 2] T 11! = 0 T 11 = T = 8 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 22 / 34
57 3.) T < 0, decompactied geometry, noneective curve b = 1 b = 10 b > 0 x a = 0, V D! = 0 8 z a 2 = b 8 ln z 1 z 1 2 a=2 Geometry decompactied, anomalies canceled with antins5 branes. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 23 / 34
58 Part III N = (0, 2) Models & Discrete Symmetries
59 Introduction Discrete Symmtries important for Absence of FCNCs Absence of µterm [Dreiner,Lee,Luhn,Raby,Ratz,Ross,Schieren,Schmidt- Hoberg,Thormaier,Vaudrevange,... ] Creation of hierarchies Proton stability Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 24 / 34
60 Introduction Discrete Symmtries important for Absence of FCNCs Absence of µterm [Dreiner,Lee,Luhn,Raby,Ratz,Ross,Schieren,Schmidt- Hoberg,Thormaier,Vaudrevange,... ] Creation of hierarchies Proton stability Discrete symmetries on orbifold Non-R Symmetries from xed point degeneracies R Symmetries from remnants of internal Lorentz group Lee at.al. found nice Z 4 R-symmetry Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 24 / 34
61 Introduction Discrete Symmtries important for Absence of FCNCs Absence of µterm [Dreiner,Lee,Luhn,Raby,Ratz,Ross,Schieren,Schmidt- Hoberg,Thormaier,Vaudrevange,... ] Creation of hierarchies Proton stability Discrete symmetries on orbifold Non-R Symmetries from xed point degeneracies R Symmetries from remnants of internal Lorentz group Lee at.al. found nice Z 4 R-symmetry Discrete symmetries on CY Symmetries aries as isometries in GLSM Charges from graded cohomology Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 24 / 34
62 Z 3 Orbifold θ : (z 1, z 2, z 3 ) (e 2πi/3 z 1, e 2πi/3 z 2, e 2πi 2/3 z 3 ) Orbifold action given by twist vector v = 1 3 (1, 1, 2) Gauge sector given by shift V in the Λ E8 E 8 Choose Standard embedding V = 1 3 (1, 1, 2, 05 )(0 8 ) = v Gauge group: [E 6 SU(3)] vis [E 8 ] hidden Matter: 3(27, 3; 1) + 27[(27, 1; 1) + 3(1, 3; 1)] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 25 / 34
63 Consistency requirements Want to construct smooth CY with line bundles from orbifold using toric (algebraic) geometry. Impose Bianchi identity (ensures anomaly cancelation): H = db + ω YM ω L D dh = D trr 2 trf 2! = 0 DonaldsonUhlenbeckYau (ensures 4d N = 1 SUSY) J J F = 0 X Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 26 / 34
64 Resolution of T 6 /Z 3 with line bundles Procedure: [Blaszczyk,Groot Nibbelink,Ha,Klevers,FR,Trapletti,Vaudrevange,Walter,... ] Introduce exceptional divisors E αβγ at x αβγ = 0 Introduce gauge ux F = E αβγ V I αβγ H I The H I are the 16 Cartan generators of E 8 E 8 The matrix Vαβγ I describes the gauge line bundle at the 27 xed points Note that in the orbifold limit the E αβγ are shrunk to a point ux is located at xed points Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 27 / 34
65 Resolution of T 6 /Z 3 with line bundles Procedure: [Blaszczyk,Groot Nibbelink,Ha,Klevers,FR,Trapletti,Vaudrevange,Walter,... ] Introduce exceptional divisors E αβγ at x αβγ = 0 Introduce gauge ux F = E αβγ V I αβγ H I The H I are the 16 Cartan generators of E 8 E 8 The matrix Vαβγ I describes the gauge line bundle at the 27 xed points Note that in the orbifold limit the E αβγ are shrunk to a point ux is located at xed points To make contact with the orbifold description: Choose the V αβγ to coincide with the internal E 8 E 8 momentum of some twisted orbifold state located at (α, β, γ) Vev of orbifold state generates the blowup of the E αβγ Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 27 / 34
66 Resolution of T 6 /Z 3 with line bundles Choose 3 dierent bundle vectors from (27, 1) of E 6 SU(3) V 1 = 1 3 (2, 2, 2, 05 )(0 8 ) at 9 xed points V 2 = 1 3 ( 1, 1, 1, 3, 04 )(0 8 ) at 9 xed points V 3 = (V 1 + V 2 ) at 9 xed points F = 9 i=1 E iv1 I H I + 18 j=10 E jv2 I H I E nv3 I H I Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 28 / 34
67 Resolution of T 6 /Z 3 with line bundles Choose 3 dierent bundle vectors from (27, 1) of E 6 SU(3) V 1 = 1 3 (2, 2, 2, 05 )(0 8 ) at 9 xed points V 2 = 1 3 ( 1, 1, 1, 3, 04 )(0 8 ) at 9 xed points V 3 = (V 1 + V 2 ) at 9 xed points F = 9 i=1 E iv1 I H I + 18 j=10 E jv2 I H I E nv3 I H I Gauge group [E 6 SU(3)] [E 8 ] [SO(8) (U(1) A U(1) B ) SU(3)] [E 8 ], U(1) A,B anomalous Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 28 / 34
68 Resolution of T 6 /Z 3 with line bundles Choose 3 dierent bundle vectors from (27, 1) of E 6 SU(3) V 1 = 1 3 (2, 2, 2, 05 )(0 8 ) at 9 xed points V 2 = 1 3 ( 1, 1, 1, 3, 04 )(0 8 ) at 9 xed points V 3 = (V 1 + V 2 ) at 9 xed points F = 9 i=1 E iv1 I H I + 18 j=10 E jv2 I H I E nv3 I H I Gauge group [E 6 SU(3)] [E 8 ] [SO(8) (U(1) A U(1) B ) SU(3)] [E 8 ], U(1) A,B anomalous Bianchi identities E αβγ trf 2 = E αβγ trr 2 V 2 1 = V 2 2 = V 2 3 = 4 3 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 28 / 34
69 Resolution of T 6 /Z 3 with line bundles Choose 3 dierent bundle vectors from (27, 1) of E 6 SU(3) V 1 = 1 3 (2, 2, 2, 05 )(0 8 ) at 9 xed points V 2 = 1 3 ( 1, 1, 1, 3, 04 )(0 8 ) at 9 xed points V 3 = (V 1 + V 2 ) at 9 xed points F = 9 i=1 E iv1 I H I + 18 j=10 E jv2 I H I E nv3 I H I Gauge group [E 6 SU(3)] [E 8 ] [SO(8) (U(1) A U(1) B ) SU(3)] [E 8 ], U(1) A,B anomalous Bianchi identities E αβγ trf 2 = E αβγ trr 2 V 2 1 = V 2 2 = V 2 3 = 4 3 DUY equations J J F = 0 9 i=1 V1 I vol(e i)+ 18 j=10 V2 I vol(e j)+ 27 k=19 V I 3 vol(e k) = 0 Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 28 / 34
70 Remnant non-r symmetries Non-R symmetries arise as discrete subgroups of U(1) A and U(1) B which leave vevs of blowup modes invariant 27 8 s(1, 1) + 8 c(1,1) + 8 v( 2,0) + 1 ( 2, 2) + 1 ( 2,2) + 1 (4,0) Blowup modes: 1 ( 2, 2), 1 ( 2,2), 1 (4,0) corresponding to V 1, V 2, V 3 Leave discrete Z 2 Z 2 symmetry generated by T ± : φ (qa,q b ) e 2πi 2 (q A±q B ) φ (qa,q B ) Both symmetries are non-anomalous Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 28 / 34
71 Remnant R symmetries Properties of R symmetries R symmetries do not commute with SUSY Grassmann coordinate θ transforms under R symmetries R symmetries only dened up to mixing with non-r symmetries Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 29 / 34
72 Remnant R symmetries Properties of R symmetries R symmetries do not commute with SUSY Grassmann coordinate θ transforms under R symmetries R symmetries only dened up to mixing with non-r symmetries R symmetries on orbifolds R charge on orbifold dened via a combination of right-moving momenta q and oscillator numbers N: R = q N with q = 1 3 (1, 1, 1) [Kobayashi,Raby,Zhang] Remnant symmetry of internal space: Sublattice rotations by 2π/3 in each torus: T R k : φ e2πi/3r k φ, k=1,2,3 labels tori Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 29 / 34
73 Remnant R symmetries Orbifold Our orbifold blowup modes have R = q N = 1 (1, 1, 1) 3 To identify remnant R-symmetries, search for invariant combinations of T R k with T U(1) A and T U(1)B : 1 ( 2, 2) (T R 1 ) a (T R 2 ) b (T R 3 ) c T U(1)A T U(1)B 1 ( 2, 2)! = 1 ( 2, 2) 1 ( 2,2) (T R 1 ) a (T R 2 ) b (T R 3 ) c T U(1)A T U(1)B 1 ( 2,2)! = 1 ( 2,2) 1 (4,0) (T R 1 ) a (T R 2 ) b (T R 3 ) c T U(1)A T U(1)B 1 (4,0)! = 1 (4,0) Result One nds that a + b + c = 3 only a (trivial) Z 2 R-symmetry remains in blowup. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 30 / 34
74 Remnant R symmetries GLSM Look at simplied model with 3 exceptional divisors: Symmetries: 0 = z 3 11x 1 + z 3 12x 2 + z 3 13x 3 0 = z 3 21x 1 x 2 x 3 + z z = z 3 31x 1 x 2 x 3 + z z 3 33 a i = z i1 2 + z i2 2 + z i3 2 b α = z 1α 2 + z z x α 2 z iα e 2πi/3 z iα (x 1, x 2, x 3 ) e 2πi/3 (x 1, x 2, x 3 )... Origin of Symmetries Note that the symmetries are inherited from the special choice of complex structure on the orbifold (absence of κ z i1 z i2 z i3 term) Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 31 / 34
75 Remnant R symmetries GLSM How to check which of these symmetries are R-symmetries? R-symmetries will transform the holomorphic (3, 0) form Ω: Ω ηγηdz i dz j dz k Q R (Ω) = Q R (W ) [Witten] Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 32 / 34
76 Remnant R symmetries GLSM How to check which of these symmetries are R-symmetries? R-symmetries will transform the holomorphic (3, 0) form Ω: Ω ηγηdz i dz j dz k Q R (Ω) = Q R (W ) [Witten] How are the R-symmetries broken in blowup? (Presumably) via marginal deformations in Kähler potential under the presence of the gauge bundle: d 2 θ + φ 4D (x µ )N(z, x)λλ φ 4D : 4D modes N(z, x): Polynomial in the geometry elds z iα, x α Λ: WS fermions describing the gauge bundle N(z, x) might not be compatible with rotational symmetries R symmetry broken Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 32 / 34
77 Remnant R symmetries GLSM To check transformation of bundle under discrete symmetries: Tools Find discrete transformations of coordinate elds z, x under symmetry in question Write down gauge bundle in ambient space Restrict bundle to toric hypersurface via Koszul sequence Find contributing monomials Check transformation of monomials under discrete symmetry The last three steps should be automatized using cohomcalg and the Koszul extension developed by Blumenhagen et.al. Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 33 / 34
78 Conclusion N = (2, 2) GLSM GLSM Resolution of Orbifolds worked out Rich moduli space with interesting topology changes Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 34 / 34
79 Conclusion N = (2, 2) GLSM GLSM Resolution of Orbifolds worked out Rich moduli space with interesting topology changes N = (0, 2) GLSM & Anomalies GLSM anomalies can be canceled via log FI Term Description of torsion and NS5 branes Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 34 / 34
80 Conclusion N = (2, 2) GLSM GLSM Resolution of Orbifolds worked out Rich moduli space with interesting topology changes N = (0, 2) GLSM & Anomalies GLSM anomalies can be canceled via log FI Term Description of torsion and NS5 branes N = (0, 2) GLSM & Discrete Symmetries Find phenomenologically interesting discrete symmetries in GLSM Calculate matter charges from kinetic deformations Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 34 / 34
81 Conclusion N = (2, 2) GLSM GLSM Resolution of Orbifolds worked out Rich moduli space with interesting topology changes N = (0, 2) GLSM & Anomalies GLSM anomalies can be canceled via log FI Term Description of torsion and NS5 branes N = (0, 2) GLSM & Discrete Symmetries Find phenomenologically interesting discrete symmetries in GLSM Calculate matter charges from kinetic deformations Thank you for your attention! Fabian Ruehle (BCTP Bonn) Heterotic string with GLSMs LMU (04/25/2013) 34 / 34
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