Looking Beyond Complete Intersection Calabi-Yau Manifolds. Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R.

Transcription

1 Looking Beyond Complete Intersection Calabi-Yau Manifolds Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R. Morrison

2 Who and Why Def: X is Calabi-Yau (CY) if X is a Ricci-flat, Kähler manifold. Ansatz consistent 10D string background with reasonable phenomenology low-scale supersymmetry, gauge groups, chiral matter, Yukawa couplings etc.

3 Calabi-Yau Questions Phenomenology is a function of X. What is the structure of the configuration space? Are there finitely many (Hodge numbers)? Is the space of possibilities connected? Is there a classification? What is the analogue of the beautiful story for K3 surfaces?

4 How do we construct CYs? Specify polynomial equations in an ambient space (like ). Complete intersection: # eqns = codimension Large classes of CICYs known. How complete are these lists?

5 Complexity with Codimension

6 Complexity with Codimension Codim 1: Only hypersurfaces.

7 Complexity with Codimension Codim 1: Only hypersurfaces. Codim 2: Only complete intersections.

8 Complexity with Codimension Codim 1: Only hypersurfaces. Codim 2: Only complete intersections. Codim 3 (Buchsbaum-Eisenbud): Every variety is specified by the Pfaffian minors of a anti-symmetric matrix.

9 Complexity with Codimension Codim 1: Only hypersurfaces. Codim 2: Only complete intersections. Codim 3 (Buchsbaum-Eisenbud): Every variety is specified by the Pfaffian minors of a anti-symmetric matrix. Non CI! equations, but only codim 3. Codim > 3:????

10 We would like to construct, maybe systematically, examples of Calabi-Yau manifolds that are NOT complete intersections.

11 How can physicists help? Gauged Linear Sigma Model (Witten) N=2 NLSM with CY target space X

12 How can physicists help? Gauged Linear Sigma Model (Witten) UV N=2 NLSM with CY target space X Marginal def: Kähler, CS of X N=2 SCFT IR

13 How can physicists help? Gauged Linear Sigma Model (Witten) Kähler: FI+theta CS: Superpotential Abelian Gauge thy with matter (N=2) UV N=2 NLSM with CY target space X Marginal def: Kähler, CS of X N=2 SCFT IR

14 Example: The Quintic Fields +1-5

15 Example: The Quintic Fields +1-5 Superpotential F-I parameter, theta angle Completely general polynomial

16 Example: The Quintic F-terms Fields +1-5 D-term Superpotential F-I parameter, theta angle Completely general polynomial

17 Assume Genericity:, can integrate out gauge fields. and

18 Assume Genericity:, can integrate out gauge fields. and

19 Assume Genericity:, can integrate out gauge fields. and The describe a with Kähler class. The poly specifies a quintic hypersurface.

20 Assume Genericity:, can integrate out gauge fields. and The describe a with Kähler class. The poly specifies a quintic hypersurface. Landau-Ginzburg orbifold. Massless interacting through.

21 Non-anomalous R symmetry Calabi-Yau condition is exactly marginal GLSM flows to IR SCFT SCFT moduli space = Quantum corrected complexified Kähler moduli space of CY

22 Non-anomalous R symmetry Calabi-Yau condition is exactly marginal GLSM flows to IR SCFT SCFT moduli space = Quantum corrected complexified Kähler moduli space of CY Coulomb branch CY LG

23 Non-anomalous R symmetry Calabi-Yau condition is exactly marginal GLSM flows to IR SCFT SCFT moduli space = Quantum corrected complexified Kähler moduli space of CY Coulomb branch CY LG ALL complete intersections in toric varieties can be described.

24 What about non-complete intersections? The naïve generalization fails. When are not a CI, they do not impose independent conditions locally. Lagrange multipliers compact directions. do not vanish non

25 What is the missing box? Abelian GLSM Complete Intersections in Toric Varieties Non-abelian GLSM? Hori and Tong (2006) and Hori (2010) have developed techniques to understand the low-energy dynamics.

26 Determinantal Varieties Variety given by the rank degeneration locus of a matrix. Example (KM): Rank 2 locus of a 4x4 matrix codim 4. Not CI: Sixteen 3x3 minors must vanish. Generalize: rectangular, symmetry, weights, consider CIs.

27 Nonabelian GLSM is natural On the variety, we can decompose BUT, there is a redundancy. Introduce quarks X, Y and a U(2) gauge field. Then, use a superpotential

28 KM example Field R-symmetry anomaly free Two U(1) factors two Kähler parameters Central charge of SCFT is 3

29 F-terms: U(1) D-term: U(2) D-term:

30 F-terms: U(1) D-term: U(2) D-term: KM: Determinantal variety KM Gr KM Gr Gr: small resolution of a nodal hypersurface in the Grassmanian with 56 nodes.

31 The phases: KM and Gr

32 The phases: KM and Gr KM is rank 2. fibered over rank 2 locus, fiber={pt} pt KM

33 The phases: KM and Gr KM Gr is rank 2. fibered over rank 2 locus, fiber={pt} pt KM, is a nodal quartic. P is fibered, resolves all 56 nodes. pt

34 Summary so far We can construct determinantal CYs using nonabelian GLSMs. We constructed an example of a codim 4 non-ci Calabi-Yau (KM). It is connected to a (resolved) nodal hypersurface in G(2,4). Similar to Hori and Tong's analysis of the codim 3 Pfaffian Calabi-Yau.

35 Mirror Symmetry

36 Mirror Symmetry Non-trivial statement about NLSMs on CYs NLSM on X NLSM on X' RG flow N=2 SCFT Automorphism: = N=2 SCFT Kähler(X) CS(X) CS(X') Kähler(X') Compute quantum Kähler moduli space.

37 Mirror Symmetry for KM Cross-check GLSM analysis. Mirror symmetry is not well understood for noncis handful of examples. Tropical geometry provides a candidate mirror family for the KM example.

38 (CS) Moduli Space of Mirror family Singular divisors Blow up the moduli space till the boundary consists of a set of normal crossing divisors.

39 (CS) Moduli Space of Mirror family Singular divisors Large CS points Blow up the moduli space till the boundary consists of a set of normal crossing divisors.

40 Quantum Kähler Moduli Space Two large volume points: smooth Gr KM GLSM 56 rational curves in one class. We need to see this in the instanton expansion! Computation in progress...

41 Aside: Linear PDEs can secretly have a regular singular pt at z=0, even if M has higher order poles. Systematic pole reduction algorithms exist. We have a 6x6 matrix PDE in two vars. Not aware of a systematic algorithm a fun problem to work out.

42 Conclusions, and looking ahead... Non-abelian gauged linear sigma models naturally describe determinantal varieties. Go beyond CIs! Mirror symmetry not understood. Batyrev-Borisov generalization? Maybe tropical geometry? Enumerate these. Do they outnumber CIs? Consider more general gauge theories with a nonanomalous R-symmetry and central charge 3. Could hint at other algberaic structures for codim > 3.