Exact Geometric Phases and K ӓ hler Cohomology Julian Sonner University of Cambridge Great Lakes Strings Conference Madison, April 2008
What’s the point? • Compute non - abelian Berry’s Phase in strongly interacting QM systems • “The Geometric Phase in Supersymmetric Quantum Mechanics”; arXiv 0709.0731 and Phys. Rev. D77, 2008. [ C. Pedder, JS, D. Tong ] • “The Geometric Phase and Gravitational Precession of D - branes”; arXiv 0709.2136 and Phys. Rev. D76, 2007. [ C. Pedder, JS, D. Tong ] • “The Berry Phase of D0 - Branes”; JHEP 0803:065, 2008. [ C. Pedder, JS, D. Tong ] • Recent work on exact ( all - instanton ) results via algebraic geometry: SU ( 2 )- twisted equivariant cohomology. [ JS, D.Tong ] • Find applications to condensed - matter systems and/or topological quantum computation 2 Geometric Phases in String Theory Julian Sonner, DAMTP
Contents of the Talk • Review of ( non - abelian ) geometric phase • Summary of results • ( 2,2 )- models: - Explicit GLSM computation - New exact results from twisted equivariant cohomology • Conclusions, future directions 3 Geometric Phases in String Theory Julian Sonner, DAMTP
Berry Philosophy [ M. Berry, B.Simon ] Parameters Hamiltonian • Set system up in a particular energy eigenstate • Change parameters slowly: Adiabatic theorem means that system clings on to eigenstate 4 Geometric Phases in String Theory Julian Sonner, DAMTP
Review of Berry Phase I • Canonical Example of Abelian Berry Phase: • Spin 1/2 in external magnetic field H = � B · � σ • slowly change magnetic field • Adiabatic Theorem: Cling on to eigenstate • Quantum Evolution gives law of parallel transport 5 Geometric Phases in String Theory Julian Sonner, DAMTP
Review of Berry Phase II ( the canonical example ) H 1 / 2 = � B · σ • Physics near generic two - level crossing is described by this term • Quantization: H 1 / 2 | B ± � = ± B | B ± � � • Now ask what happens as is varied B 6 Geometric Phases in String Theory Julian Sonner, DAMTP
Review of Berry Phase III ( the general picture ) R 3 θ ∈ U (1) • Induce gauge connection on R 3 A = i � B + ( t ) | d | B + ( t ) � For our spin - 1/2 Hamiltonian: A = A Dirac � t � � � � � E + ( t ′ ) dt ′ | B + ( t ) � = exp − i exp � B + | d | B + � | B + (0) � i C Dynamical phase Geometric phase 7 Geometric Phases in String Theory Julian Sonner, DAMTP
Non - Abelian Berry Phase [ F. Wilzcek, A. Zee ] • If there is degeneracy: Instantaneous basis has {| a � } n natural U ( n ) action. Pick basis a =1 A ab = i � a ( t ) | d | b ( t ) � R N � θ ∈ U ( n ) Example: D0 - D4 [ C.Pedder, JS, D.Tong ] A SU (2) = A Yang • Really: Berry holonomy � � � ( A µ ) ab dX µ | a � → P exp − i | b � 8 Geometric Phases in String Theory Julian Sonner, DAMTP
Summary of Results • Exact results for abelian ( 2,2 ) Berry phases in interacting theories • Non - abelian Berry phase with ( 4,4 ) SUSY : application to D0 - D4 system - W eak coupling computation agrees with result for SO ( 5 )- invariant high - TC superconductors - AdS/CFT interpretation of Berry phase as gravitational precession • D0 - branes: - Relation of Berry phase to division algebras - D0 branes confined to d=2 are anyons ( M - theory interpretation? ) 9 Geometric Phases in String Theory Julian Sonner, DAMTP
( 2,2 ) Geometric Phase • ( 2,2 ) SUSY Quantum Mechanics: R - Symmetry SU (2) × U (1) • Same SUSY as N=1 in four dimensions • Parameters: 1 ) complex superpotential [ Cecotti V afa: tt* ] 2 ) triplet of real scalars from vector multiplet • Focus on the latter and compute associated geometric phases • Supersymmetry helps in many ways • Non - renormalisation of abelian phase [ F. Denef ] • Exact results for non - abelian phases 10 Geometric Phases in String Theory Julian Sonner, DAMTP
Non - Renormalization • Single ground state: - Free ( massive ) chiral exhibits U ( 1 )- Dirac Berry phase over complete range of parameters [ C.Pedder, JS, D.Tong ] φ | 2 + i ¯ L m = | ˙ ψ ˙ ψ − m 2 | φ | 2 − ¯ ψ ( m · σ ) ψ - More complicated interacting theories: Non - renormalization dictates N � ± A Dirac A = i i =1 11 Geometric Phases in String Theory Julian Sonner, DAMTP
Non - abelian case • Witten index protects number of ground states - Can use multiple ground states to investigate non - abelian phases P 1 • Simplest example: with potential - Potential has two minima V - There are two ground states for all m = 0 values of , even at: m P 1 ∂ ( P 1 ) = C ⊕ C H 0 ∼ = H ∗ ¯ K¨ ahler class R [ E. Witten ] 12 Geometric Phases in String Theory Julian Sonner, DAMTP
Building the connection A = ± A Dirac • In each vacuum , but the full connection takes the form � � 1 | d | 1 � � A Dirac � f inst � � 1 | d | 2 � A ( P 1 ) = = ( f inst ) ∗ − A Dirac � 2 | d | 1 � � 2 | d | 2 � • O ff- diagonal elements are given by BPS instantons f = 4 mRe − 2 mR • Leading order result: [ C. Pedder, JS, D.Tong, 2007 ] 13 Geometric Phases in String Theory Julian Sonner, DAMTP
Smoothness Property • Via gauge transformation: m ν σ ρ A µ = ε µ νρ 2 m 2 (1 − f ( mR )) • At origin , no change in degeneracy m = 0 happens for topological reasons ➡ Connection must be smooth there ➡ Further corrections to should smooth out A µ f Question: Can we get exactly? f 14 Geometric Phases in String Theory Julian Sonner
NLSM: Exact result for f ( mR ) • Change of perspective: NLSM with target P 1 space and potential given by holomorphic k = k z ( z ) ∂ z isometry • W ork in manifestly geometric language • Supercharge: Q = D + ( m · σ ) J ( k ) � ∗ ∂ ∗ � � � ı ¯ k J ( k ) = D = ∂ ∗ ı ¯ k ∗ • This defines an SU ( 2 )- twisted version of equivariant cohomology 15 Geometric Phases in String Theory Julian Sonner
Twisted Equivariant Cohomlogy ω ∈ Ω ∞ , ∗ ( P 1 ) • Find such that Q ω = Q ∗ ω = 0 P 1 • Can do this explicitly for . W e find H ∞ , ∗ SU (2) ( P 1 ) ∼ ∂ ( P 1 ) = C ⊕ C = H ∗ ¯ • Normalized cohomology classes are identified with ground states { ω 1 , ω 2 } → {| 1 � , | 2 � } 16 Geometric Phases in String Theory Julian Sonner
The exact answer • Overlaps of states are given via Hodge inner Ω ∞ , ∗ ( P 1 ) product extended to 2 mR � sinh (2 mR ) ∼ 4 mRe − 2 mR � 1 | d | 2 � = P 1 ω 1 ∧ ∗ d ω 2 = • This leads to the full connection � � 2 mR m ν σ ρ A µ = ε µ νρ 1 − 2 m 2 sinh (2 mR ) • W e recognize an old friend: BPS ‘t Hooft Polyakov monopole! 17 Geometric Phases in String Theory Julian Sonner
Conclusions and Outlook • Non - abelian Berry phase: Multiple vacua protected via Witten index • BPS instantons fill in o ff- diagonal components • Gauging isometries in NLSM: Ground states define new ( ? ) twisted equivariant cohomology for K ӓ hler manifolds - Allows exact all - instanton computation of Berry connection - Relation to SU ( 2 ) Lefschetz actions - General theory for K ӓ hler manifolds: dynamical equation for A? - SO ( 5 ) action for hyper - K ӓ hler manifolds [ V erbitsky and others ] 18 Geometric Phases in String Theory Julian Sonner
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