Exact Geometric Phases and K hler Cohomology Julian Sonner - - PowerPoint PPT Presentation

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Exact Geometric Phases and K hler Cohomology Julian Sonner - - PowerPoint PPT Presentation

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Exact Geometric Phases and K hler Cohomology Julian Sonner University of Cambridge Great Lakes Strings Conference Madison, April 2008 Whats the point? Compute non - abelian Berrys Phase in strongly interacting QM systems

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Exact Geometric Phases and Kӓhler Cohomology

Julian Sonner University of Cambridge

Great Lakes Strings Conference Madison, April 2008

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Geometric Phases in String Theory Julian Sonner, DAMTP

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

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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

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Geometric Phases in String Theory Julian Sonner, DAMTP

Berry Philosophy [M. Berry, B.Simon]

  • Set system up in a particular energy eigenstate
  • Change parameters slowly: Adiabatic theorem

means that system clings on to eigenstate

Parameters

Hamiltonian

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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
  • slowly change magnetic field
  • Adiabatic Theorem: Cling on to eigenstate
  • Quantum Evolution gives law of parallel

transport

H = B · σ

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Geometric Phases in String Theory Julian Sonner, DAMTP

Review of Berry Phase II

(the canonical example)

  • Physics near generic two-level crossing is

described by this term

  • Quantization:
  • Now ask what happens as is varied

H1/2 = B · σ

H1/2|B± = ±B|B±

  • B

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Geometric Phases in String Theory Julian Sonner, DAMTP

Review of Berry Phase III

(the general picture)

  • Induce gauge connection on

A = iB+(t)|d|B+(t)

|B+(t) = exp

  • −i

t E+(t′)dt′

  • exp
  • i
  • C

B+|d|B+

  • |B+(0)

Dynamical phase Geometric phase

θ ∈ U(1)

R3 R3

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For our spin-1/2 Hamiltonian:

A = ADirac

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Geometric Phases in String Theory Julian Sonner, DAMTP

Non-Abelian Berry Phase

[F. Wilzcek, A. Zee]

  • If there is degeneracy: Instantaneous basis has

natural U(n) action. Pick basis

Aab = ia(t)|d|b(t)

  • Really: Berry holonomy

|a → P exp

  • −i
  • (Aµ)ab dXµ
  • |b

{|a}n

a=1

RN

  • θ ∈ U(n)

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Example: D0-D4 [C.Pedder, JS, D.Tong]

ASU(2) = AYang

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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?)

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Geometric Phases in String Theory Julian Sonner, DAMTP

(2,2) Geometric Phase

  • (2,2) SUSY Quantum Mechanics: R-Symmetry
  • 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

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SU(2) × U(1)

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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]

  • More complicated interacting theories: Non-renormalization dictates

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A =

N

  • i=1

±ADirac

i

Lm = | ˙ φ|2 + i ¯ ψ ˙ ψ − m2|φ|2 − ¯ ψ(m · σ)ψ

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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

  • Simplest example: with potential
  • Potential has two minima
  • There are two ground states for all

values of , even at:

[E. Witten]

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P1

V

H0 ∼ = H∗

¯ ∂(P1) = C ⊕ C

P1 K¨ ahler class R m m = 0

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Geometric Phases in String Theory Julian Sonner, DAMTP

Building the connection

  • In each vacuum , but the full

connection takes the form

  • Off-diagonal elements are given by BPS

instantons

  • Leading order result:

[C. Pedder, JS, D.Tong, 2007]

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A = ±ADirac f = 4mRe−2mR A(P1) = 1|d|1 1|d|2 2|d|1 2|d|2

  • =

ADirac f inst (f inst)∗ −ADirac

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Geometric Phases in String Theory Julian Sonner

Smoothness Property

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  • Via gauge transformation:
  • At origin , no change in degeneracy

happens for topological reasons

➡ Connection must be smooth there ➡ Further corrections to should smooth out

Question: Can we get exactly?

Aµ = εµνρ mνσρ 2m2 (1 − f(mR)) Aµ f f m = 0

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Geometric Phases in String Theory Julian Sonner

NLSM: Exact result for f(mR)

  • Change of perspective: NLSM with target

space and potential given by holomorphic isometry

  • W
  • rk in manifestly geometric language
  • Supercharge:
  • This defines an SU(2)-twisted version of

equivariant cohomology

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P1 Q = D + (m · σ)J (k) D = ∗∂∗ ∂

  • J (k) =
  • ı¯

k

∗ı¯

k∗

  • k = kz(z)∂z
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Geometric Phases in String Theory Julian Sonner

Twisted Equivariant Cohomlogy

  • Find such that
  • Can do this explicitly for . W

e find

  • Normalized cohomology classes are identified

with ground states

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ω ∈ Ω∞,∗(P1) Qω = Q∗ω = 0 H∞,∗

SU(2)(P1) ∼

= H∗

¯ ∂(P1) = C ⊕ C

{ω1, ω2} → {|1 , |2} P1

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Geometric Phases in String Theory Julian Sonner

The exact answer

  • Overlaps of states are given via Hodge inner

product extended to

  • This leads to the full connection
  • W

e recognize an old friend: BPS ‘t Hooft Polyakov monopole!

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Ω∞,∗(P1) Aµ = εµνρ mνσρ 2m2

  • 1 −

2mR sinh (2mR)

  • 1|d|2 =
  • P1 ω1 ∧ ∗dω2 =

2mR sinh (2mR) ∼ 4mRe−2mR

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Geometric Phases in String Theory Julian Sonner

Conclusions and Outlook

  • Non-abelian Berry phase: Multiple vacua protected via Witten index
  • BPS instantons fill in off-diagonal components
  • Gauging isometries in NLSM: Ground states define new(?) twisted

equivariant cohomology for Kӓhler manifolds

  • Allows exact all-instanton computation of Berry connection
  • Relation to SU(2) Lefschetz actions
  • General theory for Kӓhler manifolds: dynamical equation for A?
  • SO(5) action for hyper-Kӓhler manifolds [V

erbitsky and others]

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