An Uncertainty Principle for Topological Sectors

Zurich, July 2, 2007

J Äurg FrÄohlich Fest

Based on: hep-th/0605198, hep-th/0605200 with D. Freed and G. Segal

\Generalized abelian gauge theories:" { the space of gauge-invariant ¯eld con-¯gurations \A=G" is a generalized di®erential cohomology group in thesenseofHopkins & Singer.

In particular \ A=G" is an abelian group.

IntroductionToday I'll be talking about some interesting subtleties in the quantization of Maxwell's theory and some of its generalizations.

There are many different generalizations of Maxwell's theory…

Abelian gauge theories: ¯eldstrengths aredi®erential forms F 2 ­ `(M ).

These kinds of theories arise naturally in supergravity and superstring theories, and indeed play a key role in the theory of D-branes and in claims of moduli stabilization in string theory made in the past few years.

Summary of the Results

2. Group theoretic approach Manifestly electric-magnetic dual formulation of the Hilbert space.

This is surprising and nontrivial!

1. The Hilbert space can be decomposed into electric andmagnetic flux sectors, but the operators that measure electric andmagnetic topological sectors don't commute and cannot be simultaneously diagonalized.

3. Group theoretic approach to the theory of a self-dual field.

4. In particular: the K-theory class of a RR field cannot be measured!

Generalized Maxwell Theory

more sophisticated…

dF = 0 ) [F ]2 H `DR (X )

If M = X £ R wehavea Hilbert spaceH.

Spacetime = M, with dim(M) =n

F 2 ­ `(M )Gauge invariant fieldstrength is a differential form:

S =¼R2RM F ¤F

H =©mHm m2 H `(X ;Z):

Free field action:

[F] is conserved Grade the Hilbert space by ``Flux sectors’’

Example: Maxwell Theory on a Lens Space

Degree­l =2

M =S3=Zk £ R

c1(L ) 2 H 2(X ;Z) »= Zk

Motivating Question:

Canwesimultaneously decomposeH intoelectric andmagnetic °ux sectors?

H ?= ©e;m He;m:

FD 2 ­ n¡ `(M ) RRD = ¹h

Electric-Magnetic duality: For dim(M)=n, there is an equivalent formulation with

H =©eHe e2 H n¡ `(X ;Z);

There must also be a grading by the (topological class of) electric flux:

``Isn’t it obviously true?’’

Measuremagnetic °ux:R§ 1F where § 1 2 Z`(X )

Measureelectric °ux:R§ 2¤F where§ 2 2 Zn¡ `(X )

Canonical momentum conjugate to A is ¦ » (¤F )X .

[Z

§ 1

F;Z

§ 2

¤F ] » [Z

§ 1

F;Z

§ 2

¦ ] (1)

= [Z

X! 1F;

Z

X! 2¦ ]

= i¹hRX ! 1d! 2 =0 (2)

where ! i areclosed forms Poincar¶edual to § i .

For the periodic 1+1 scalar this is just the decomposition into momentum + winding sectors!

Fallacy in the argument

Measuring flux sectors by periods misses our uncertainty principle

The topological sectors e and m are elements of integral cohomology groups.

These abelian groups in general have nontrivial torsion subgroups.

But! the above period integrals only measure the flux modulo torsion.

Recall: A ``torsion subgroup’’ of an abelian group is the subgroupof elements of finite order:

0! H `T (X ;Z) ! H`(X ;Z) ! H `D R (X )

What it is not:

1. We are not saying [E i (x);B j (y)] 6= 0

2. We are also not saying [Z

§ 1F;Z

§ 2¤F ]= L(C1;C2)

@§ 1 =C1 @§ 2 =C2where

3. Our uncertainty principle concerns torsion classes of topological sectors.

Differential Cohomology

De¯nition: By \generalizedMaxwell theory" wemeana¯eld theory such thatthespaceof gauge inequivalent ¯elds is ·H `(M ) for some .

Correct Mathematical Framework: Deligne-Cheeger-Simons Cohomology

·H `(M ).To a manifold M and degree l we associate

·H 1(S1) =M ap(S1;U(1)) = LU(1)

Simplest example: Periodic Scalar in 1+1 dimensions: F = d

Next we want to get a picture of the space ·H `(M ) in general

Structure of the Differential Cohomology Group - I

Fieldstrength exact sequence:

Characteristic class exact sequence:

F

A

0!

°atz }| {H `¡ 1(M ;R=Z) ! ·H `(M )

¯eldstrength¡ ! ­ `Z(M ) ! 0

Connected!

0! ­ `¡ 1(M )=­ `¡ 1Z (M )| {z }Topologically trivial

! ·H `(M ) char:class¡ ! H `(M ;Z)| {z }

Topological sector

! 0

Structure of the Differential Cohomology Group - II

V: In¯nite-dimensional vector spaceof \ oscillator modes." V »= Imdy.

H `(M ;Z)

T: Connected torus of topologically trivial °at ¯elds:

¡ : Discrete (possibly in¯nite) abelian group of topological sectors: H `(M ;Z).

W`¡ 1(M ) = H `¡ 1(M ;Z) ­ R=Z

·H ` = T £ ¡ £ VThe space of differential characters has the form:

W`¡ 1

V

Example: Loop Group of U(1)

·H 1(S1) =M ap(S1;U(1)) = LU(1)

w2 H 1(S1;Z) »= Z

Configuration space of a periodic scalar field on a circle:

Topological class = Winding number:

H 0(S1;R=Z) »= R=ZFlat fields = Torus of constant maps: T

V = ­ 0=R·H 1(S1) = T £ Z £ V

Vector Space:

' (¾) = exp·2¼iÁ0+2¼iw¾+

X

n6=0

Ánne2¼in¾

¸

Loops admitting a logarithm.

This corresponds to the explicit decomposition:

The space of flat fields

… it is not necessarily connected!

Example:

Discrete Wilson lines, defining topologically nontrivial flat gauge fields. Entirely determined by holonomy:

»= [F inite group]£ [Torus]

`= 2, M = S3=Zk

H 1(M ;R=Z) »= Zk

H1(M ;Z) »= H 2(M ;Z) »= Zk

H `¡ 1(X ;R=Z) Is a compact abelian group

W`¡ 1(M ) = H `¡ 1(M ;Z) ­ R=ZConnected component of identity:

=H `T (M ;Z)Group of components:

Âr (°) = e2¼ir=k = ! rk r 2 Zk

Poincare-Pontryagin Duality

M is compact, oriented, dim(M) = n

There is a very subtle PERFECT PAIRING on differential cohomology:

·H `(M ) £ ·H n+1¡ `(M ) ! R=Z

h[ ·A1];[ ·A2]i :=Z ·H

M[ ·A1]¤[ ·A2]

On topologically trivial fields:

h[A1];[A2]i =Z

MA1dA2 mod Z

Example:Cocycle of the Loop Group Recall ·H 1(S1) = LU(1):

' = exp(2¼iÁ) Á : R ! R

Á(s+1) = Á(s) +w w2 Z is thewinding number.

h' 1; ' 2i =Z 1

0Á1dÁ2

dsds ¡ w1Á2(0) mod Z

Note! This is (twice!) the cocyclewhich de nes thebasic central extensionof LU(1).

h' 1; ' 2i =??

Hamiltonian Formulation of Generalized Maxwell Theory

But for the moment we use this formulation.

Hilbert space: H = L2( ·H `(X ))

S =¼R2Z

MF ¤F

Spacetime: M = X £ R.Generalized Maxwell ¯elds: [ ·A]2 ·H `(M ).

This breaks manifest electric-magnetic duality.

There is a better way to characterize the Hilbert space.

Defining Magnetic Flux Sectors

Return to our question:

Can we simultaneously decompose into electric andmagnetic flux sectors?

H ?= ©e;m He;m:

Since

Definition: A state of definite topological class of magneticflux has its wavefunction supported on the component labelled by

m2 H `(X ;Z):

Electric flux can be defined similarly in the dual frame, but…

H

H »= L2( ·H n¡ `)

H »= L2( ·H `(X )) and since ·H `(X ) has components,

We need to understand more deeply the grading by electric fluxin the polarization

Definition: A state of definite topological class of electricflux is an eigenstate under translation by flat fields

8Áf 2 H `¡ 1(X ;R=Z); Ã( ·A + ·Áf ) = expµ2¼iZ H

XeÁf

¶Ã( ·A)

The topological classes of electric flux are labelled by e2 H n¡ `(X ;Z):

Diagonalizing (¤F )X means diagonalizing ¦ ,

¦ is thegenerator of translations.

Defining Electric Flux Sectors

H »= L2( ·H `(X )):

Noncommuting Fluxes

So: the decomposition wrt electric flux is diagonalizing the group

of translations by flat fields:

Such a state cannot be in an eigenstate of translations by flat characters, if there are any flat, but topologically nontrivial fields.

Translation by such a nontrivial flat field must translate the support of to a different magnetic flux sector.

Therefore, one cannot in general measure both electric and magnetic flux.

H `¡ 1(X ;R=Z)

Supposeà is in a stateof de nitemagnetic °ux m2 H `(X ;Z):

the support of thewavefunction à is in the topological sector ·H `(X )m.

m

Group Theoretic Approach

1! U(1) ! Heis(S £ S) ! S £ S ! 1

H = L2(S) is a representation of S: 8s0 2 S

(Ts0Ã)(s) := Ã(s+s0):

H = L2(S) is also a representation of S: 8Â 2 S

(MÂÃ)(s) := Â(s)Ã(s)

Ts0MÂ =Â(s0)MÂTs0 :But!

Let S be any (locally compact) abelian group (with a measure)

H = L2(S):

SLet be the Pontryagin dual group of characters of S

So is a representation of the Heisenberg group central extension:

Heisenberg GroupsT heorem A Let G bea topological abelian group. Central extensions, ~G, ofG by U(1) arein one-onecorrespondencewith continuousbimultiplicativemapss : G £ G ! U(1) which arealternating (and henceskew).

T heorem B: (Stone-von Neuman theorem). If ~G is a Heisenberg group thentheunitary irrep of ~G whereU(1) actscanonically is uniqueup to isomorphism.

De¯nition: If s is nondegenerate then ~G is a Heisenberg group.

s(x1+x2;y) = s(x1;y)s(x2;y) & s(x;y1+y2) = s(x;y1)s(x;y2)

• s is alternating: s(x,x) = 1

•s is skew: s(x,y) = s(y,x) -1

If x 2 G lifts to ~x 2 ~G then s(x;y) = ~x~y~x¡ 1~y¡ 1.

• s is bimultiplicative:

Heisenberg group for generalized Maxwell theory

~G := Heis¡ ·H `(X ) £ ·H n¡ `(X )

¢

via the group commutator:

s¡([ ·A1];[ ·AD1 ]);([ ·A2];[ ·A

D2 ])¢= exp

·2¼i¡h[ ·A2];[ ·AD1 ]i ¡ h[ ·A1];[ ·A

D2 ]i¢¸:

The Hilbert space of the generalized Maxwell theoryis the unique irrep of the Heisenberg group

N.B! This formulation of the Hilbert space is manifestly electric-magnetic dual.

~G

If S = ·H `(X ), then PP duality ) S = ·H n¡ `(X ):

Flux Sectors from Group Theory

Electric flux sectors diagonalize the flat fields

Magnetic flux sectors diagonalize dual flat fields

These groups separately lift to commutative subgroups of

However they do not commute with each other!

UE (´e) := translation operator by ´e 2 H `¡ 1(X ;R=Z)

UM (´m) := translation operator by ´m 2 H n¡ `¡ 1(X ;R=Z)

[Ue(´e);Um(´m)]= T(´e;´m) = expµ2¼iZ

X´e¯´m

¶

T: torsion pairing, ¯= Bockstein: ¯(´m) 2 H n¡ `T (X ;Z).

H `¡ 1(X ;R=Z)

H n¡ `¡ 1(X ;R=Z)

~G := Heis( ·H ` £ ·H n¡ `).

The New Uncertainty Relation[Ue(´e);Um(´m)]= T(´e;´m) = exp

µ2¼iZ

X´e¯´m

¶

T torsion pairing, Bockstein.

H =©¹e; ¹mH ¹e; ¹m:

However: The pairing does not commute on the subgroups of all flat fields.

It descends to the ``torsion pairing'' or ``link pairing'':

H `T (X ) £ Hn¡ `T (X ) ! R=Z

This is a perfect pairing, so it is maximally noncommutative on torsion.

Translations by W`¡ 1(X ) and by Wn¡ `¡ 1(X ) commute

Maxwell theory on a Lens space

Acting on the Hilbert space the flat fields generate a Heisenberg group extension

0! Zk ! Heis(Zk £ Zk) ! Zk £ Zk ! 0

This has unique irrep P = clock operator, Q = shift operator

PQ = e2¼i=kQP

States of definite electric and magnetic flux jei =1pk

X

m

e2¼iem=k jmi

This example already appeared in string theory in Gukov, Rangamani, and Witten, hep-th/9811048. They studied AdS5xS5/Z3 and in order to match nonperturbative states concluded that in the presence of a D3 brane one cannot simultaneously measure D1 and F1 number.

S3=Zk £ R H 1(L k;R=Z) »= H 2(Lk;Z) = Zk

Remarks- Principles of Quantum Mechanics

The pairing of topologically nontrivial flat fields has no tunableparameter. Therefore, this is a quantum effect which does not become small in the semiclassical or large volume limit.

In general, there are no clear rules for quantization of disconnectedphase spaces. We are extending the standard rules of quantum mechanics by demanding electric-magnetic duality.

This becomes even more striking in the theory of the self-dual field.

An Experimental TestSince our remark applies to Maxwell theory: Can we test it experimentally?

Discouraging fact: No region in has torsion in its cohomology

With A. Kitaev and K. Walker we noted that using arrays of Josephson Junctions, in particular a device called a ``superconducting mirror,’’we can ``trick’’ the Maxwell field into behaving as if it were in a space with torsion.

To exponentially good accuracy the groundstates of the electromagneticfield are an irreducible representation of

R3

Heis(Zn £ Zn)

See arXiv:0706.3410 for more details.

Self-dual fields

We can impose a self-duality condition F = *F.

For the non-self-dual field we represent Heis( ·H `(X ) £ ·H `(X ))

Proposal: For the self-dual field we represent: Heis( ·H `(X ))

Attempt to define this Heisenberg group via

strial([ ·A1];[ ·A2]) = exp2¼ih[ ·A1];[ ·A2]i :

It is skew and and nondegenerate, but not alternating!

strial([ ·A];[ ·A]) = (¡ 1)RXº2k m Gomi 2005

Now supposedimM = 4k+2, and `= 2k+1.

This is a very subtle quantum theory: Differential cohomology is the ideal tool.

-graded Heisenberg groups

Theorem A’: Skew bimultiplicative maps classify -graded Heisenberg groups.

grading in our case:²([ ·A]) = 0 if

Zº2k m=0mod2

²([ ·A]) = 1 ifZº2k m=1mod2

Theorem B’: A -graded Heisenberg group has a unqiue -graded irreducible representation.

Z2

Z2

Z2Z2

This defines the Hilbert space of the self-dual field

Z2

Example: Self-dual scalar: k = 0. By bosonization à = eiÁ.TheZ2-grading is just fermion number!

a torsor for 2-torsion points in H 2k+1(X ;Z) ­ R=Z.

Thus, self-dual theory generalizes VOA theory

.Remark: One can show that the nonself-dual field at a special radius, decomposes into

Hnsd »=©®Hsd;® ­ Hasd;®

Relation to Nonselfdual Field

For the self dual scalar ®labels R and NS sectors.

Thesubscript ®is a sumover \ generalized spin structures" -

R2 =2¹h,

String Theory Applications

1. RR field of type II string theory is valued in differential K-theory ·K (X )

RR field is self dual, hence Hilbert space = unique irrep of Heis( ·K (X ))

Hence – the full K theory class is not measurable!

2. B-fields

3. AdS/CFT Correspondence

Open Problems1. What happens when X is noncompact?

2. Complete the quantum theory of the free self-dual field.

3. If one cannot measure the complete K-theory class ofRR flux what about the D-brane charge?

a.) If no, we need to make an important conceptual revision ofthe standard picture of a D-brane

b.) If yes, then flux-sectors and D-brane charges areclassified by different groups tension with AdS/CFT andgeometric transitions.

4. What is the physical meaning of the fermionic sectors inthe RR Hilbert space?

5. How is this compatible with noncommutativity of 7-formPage charges in M-theory?