-connections on circle bundles over space-time

1. -connections on circle bundles over space-time Distortion of gauge fields and order parameters Michael Freedman Roman Lutchyn September, 2014

2. Outline I will discuss the possibility of slightly generalizing -principal bundles familiar in • Electromagnetism • Superfluids • Superconductors

3. What we propose is looking at things “halfway” between • ​-principal bundle with -connection and • ​-principal bundle with -connection.1 1. Going this far is problematic: the definite Killing directions yield, in quantum theories, excitations of negative energy.

4. Namely • ​-principal bundle with local gauge symmetry of the Lagrangianbut with an -connection (i.e., not “left invariant”) • Or an associated bundle to built from the quotient action: • This bundle has an connection. • There is no canonical-symmetry. But imposing one leads back to case 1. repelling attracting

5. Energy Penalty • Possible to measure distortion of the Mobius structure on and charge some cost: , “nematic distortion energy” or “Beltrami energy”,where are the Killing direction of . • Idea: Consider models which tolerate a little (elliptical) distortion (at a price). Study the limit: distortion . • This new flexibility has some surprising consequences. • At the end of this talk I’ll consider and sl(2,R) G-L theory:

6. Preview: • One may write a Ginzburg-Landau Lagrangian in the context Beltrami energy ​ is (local) -gauge invariant: ,

7. Literature • Witten: – gravity(Nucl. Phys. B311 (1988), 46–78, 0712.0155, 1001.2933) • Haldane: anisotropic model for FQHE (1201.1983, 1202.5586) Effective mass tensor compared to Coulomb • ​ (kinetic energy of free electrons in crystal with -field) • Coulomb interaction energy (inside lattice)

8. Mathematical Starting Point A surprising flexibility of circle bundles over surfaces with flat connection • , real, , • Lie algebra, real, , • via polarization ,

9. Mathematical Starting Point commutes with commutes with hyperbolic elliptic parabolic

10. Fact For a closed surface of genus has a flat connection (acting projectively between fibers). Proof Unwrap to , . Geodesic flow canonically identifies all unit tangentcircles to withthe circle at infinity . This integratesthe connection .

11. Fact • This gives a (actually, many) irreps . (Such geometric reps and their Galois conjugates are the chief source of examples.) • “Chern number”: for • Although , defines a flat -bundle with structure group : ,

12. Fact • For ,cannot extend as rep over any bounding -manifold , but by Thurston’s orbifold theorem extensions over • For , “tripus”

13. Fact • This can be used to make pairs of -monopoles if one allows not to act near a point. • Recall: In EM, over a spatial surface , . • If topology is standard and (nomonopoles).

14. Fact But expanding the nominal fiber from to and letting the connection (potential) of EM take values near a point creates a -monopole (charge). fibers Tripus time flux appears from projection of to . This projection creates curvature. No flux, flat

15. Chern-Weil Theory • What is going on? How can you have a characteristic class without curvature? • char. class func. (curvature) for rational classes has an exceptions when structure group is noncompact: • Exceptionfor Euler Class, group non-compact. • For connections , where • However, for , there is no such formula even though and have equivalent bundle theories. “Pfaffian”

16. Chern-Weil Theory • Fact: There exist -bundles with which admit flat connections. • Milnor (1958) proved a sharp threshold for surfaces : Given , a flat linearconnection, and Wood (1970) showed a flat projective connection.

17. Infinitesimal Milnor For genus g , has an -flat bundle with and holonomies, , radian rotation (Commutating boots yield a rotation.)

18. Infinitesimal Milnor To second order: But this is not exact. However, topology implies an exact solution. is to. Since this is non-contractable, perturbations remain surjective. This defines a representation near boost on meridian and longitude and pure rotation around puncture. Band summing copies “infinitesimal Milnor”

19. There is a converse A principal bundle with flat satisfies const. Proof: Use flat to trivialize over top cell . Comparing this trivialization with “round” structure on each fiber gives a map . , where is the th component of , , . But . □ “Beltrami energy”

20. Application of hyperbolic geometry in condensed matter physics • Quantum Hall effect: Haldane et al., 2011–12, Maciejko et al., 2013) • Superfluids and superconductors: Freedman and Lutchyn, 2014 (this talk ☺) Properties of two dimensional models on a space with negative curvature are very different! • No long range interaction between vortices in XY model: Callan and Wilczek, Nucl. Phys. B340 (1990), 366–386 • In contrast, the last bit of this talk is about curvature the target space • Connections may boost as well as rotate • Exotic quantization condition

21. Synthetic Gauge Fields in Cold Atoms • The cold atoms community is now proficient at simulating and gauge fields. • We suspect that a similar technique would permit simulation of gauge field-gravity in the lab. Specifically, a rotationally symmetric, pure boost might be imposed on a ring of cold atoms.

22. Cold Atoms • Recall the Poincaré disk model: Metric: Stereographic projection: On the next slide we see that the boost connection is precisely the angular component of the Levi-Civita connection. also has the interpretation of the tangents to the circle of radius in the hyperbolic plane .

23. Cold Atoms Hyperbolic geometry arises as when has the bi-invariant Killing metric. loop of all ellipses of eccentricity

24. Cold Atoms • An order parameter coupled to (not!) will have energy • All zero-energy solutions: () • The quantization condition is integral.

25. The uninitiated would have an experimental surprise • A pure boost integratesto purely rotational holonomy • Hyperbolic quantization conditions • Energy vs. , not: but

26. superconductivity? • Let us assume EM is pure . Is there a role for in an effective theory of superconductivity? • The simplest opportunity is a spin polarized, two-dimensional, superconductor. • Consider the GL Lagrangian density:

27. Understanding the Order Parameter • First, in what complex line should a spin-polarized (fixed -vector)-wave superconducting order parameter take its values? • It is a section of , where τ is the tangent bundle to the superconducting space-time, and is the bundle of electromagnetism, EM. • . The comes from :, , , etc… . GL-Hamiltonian has symmetry • -wave:ground state symmetry , since and if implements and implements spatial , then,

28. Since the tangent bundle to a sample is only an abstraction coarsely connected to the experimental reality, it does not seem essential to postulate . • A small amount of “slop” in metric transport is modeled at lowest order by elliptic distortion (). Considering the effect of , -shifts, one calculates that is a section of . Thus lies in .

29. One may write a Ginzburg-Landau Lagrangian in the context • Notation: ​is, as in EM, a -connection. is an enhancement, an -connection. Using orthogonal basis: , project: Beltrami energy -part, boost -part rotation

30. All bundles have structure group , but possess an connection, . • Dynamic variables: , • ​ is (local) -gauge invariant: , • Under a local -gauge transformation , is invariant. • Since conjugation by is an isometry of and the contribution is parallel to and thus projected out.

31. In such a Lagrangian there are new ways to trade energy around, specifically between , , and the Beltrami term . • We expect some modification to Meissner physics, and vortex geometry. • Also new Josephson equations, if the Beltrami energy were coupled to (as is charge density).

32. U(1) Meissner effect– geometry of vortices picture Energy balance winding h/2e h/2e SC SC Large Small|^2 Small Large |^2

33. Vortices • Small SC stiffness Type II, for usual connection Moderate SC stiffness and small Beltrami coefficient“genus transition”, • In this transition, a planar 2DEG becomes high genus • Either by spontaneously adding handles, or • In a bilayer system via “genons”* • High genus makes independent of , enabling both and to be reduced at the expense of increasing the energy of nematic distortion, • “Infinitesimal Milnor” allows “designer connections” *[Barkeshli, Jian, Qi] arXiv:1208.4834

34. Since a polarized chiral -wave 2DEG supports non-abelian Ising excitations, genus ground state degeneracy (for a fixed order parameter configuration). • This degeneracy could have observable consequences, e.g., entropy as it affects specific heat. genus transition vortex core 2DE6

35. Summary • connections on -principal bundles allow surprising flexibilities and may be useful in model building: • High energy (We discussed the -sector of the standard model only) • Low energy, superfluids, superconductors, and possibly QHE • Non-compact forms of other Lie algebras can be considered in a similar vein.