Gauge invariance and topologyGauge fields ↔ by 1-forms in Minkowski spaceGauge fields ↔ gauge connections in a generic manifoldPath-integral computation: a nontrivial example

E. Guadagnini and F. Thuillier, ArXiv:1301.6407

M = 3-manifold; a good covering of M is given by the atlas U = {Ua} inwhich each chart Ua (with a = 1, 2, 3, ...) is a contractible open set homeomor-phic with R3, and each intersection Ua1 ∩ Ua2 ∩ · · · ∩ Uam is either empty orcontractible.

A U(1) gauge connection A on M is defined by a triplet of local variables

A = {va,λab, nabc} ,

va = 1-forms in the open sets Ua,λab = 0-forms in the intersections Ua ∩ Ubnabc’s are integers in the intersections Ua ∩ Ub ∩ Uc.

Wednesday, June 5, 2013

Inside Ua ∩ Ub one has: vb − va = dλab .

In the intersections Ua ∩ Ub ∩ Uc: λbc − λac + λab = nabc .

U(1) principal bundle over M ;

transition functions gab : Ua ∩ Ub → U(1) are : gab = e2πiλab

.Cocycle consistency condition ; gab gbc gca = 1

WA(C[q]) = exp

�2πiq

�

CA�

= exp

�2πiq

��

Ca

�

Ca

va −�

Ca∩Cb

λab��

Connection 2

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If ω ∈ Ω1(M)

A+ ω = {va + ωa,λab, nabc} = {va,λab, nabc}+ {ωa, 0, 0}

Ω1Z(M) = space of closed 1-forms with integral periods. If ωI ∈ Ω1Z(M), theintegral of ωI along C is an integer,

�C ωI = n ∈ Z.

For any oriented colored link L, the holonomy of A along L satisfies

exp

�2πi

�

L(A+ ωI )

�= exp

�2πi

�

LA�

exp

�2πi

�

LωI

�= exp

�2πi

�

LA�

Gauge transformation: A → A+ ωI , (1)

The set of the equivalence classes of U(1) gauge connections modulo the trans-formations (1) is called the Deligne-Beilinson (DB) cohomology group of degreeone = H1D(M).

3gauge invariance

A ∈ H1D(M) , A ↔ {va,λab, nabc}

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A ∈ H1D(M) is represented by A ↔ {va,λab, nabc},the abelian Chern-Simons action is given by

S[A] = 2πk

�

MA ∗A =

= 2πk

��

Ma

�

Ma

va ∧ dva +

�

Sba

�

Sba

λabdva +�

lcba

nabc

�

lcba

va +

�

xdcba

nabcλab

�

Abelian Chern-Simons theory 4

Configuration space of gauge orbits, or Deligne-Beilinson classes,

0 → Ω1(M)/Ω1Z(M) → H1D(M) → H2(M) → 0

By Poincaré duality: H2(M) � H1(M)

Wednesday, June 5, 2013

5space of gauge orbits

H (M)10

A0

γ

Aγ^

^A = �Aγ + ω

H1(M) = T (M) = Zp1 ⊕ Zp2 ⊕ · · ·⊕ Zpwtorsion numbers {p1, p2, ..., pw} are fixed by the convention that pi divides pi+1

Wednesday, June 5, 2013

6normalized partition function

Zk(M) =

�γ∈H1(M)

�Dω eiS[

�Aγ+ω]�Dω eiS[ω]

Select �Aγ gauge orbit A0γ of a flat connection,

Zk(M) =

�γ∈H1(M)

�Dω eiS[A

0γ+ω]

�Dω eiS[ω]

=�

γ∈H1(M)

eiS[A0γ ]

Indeed, S[A0γ + ω] = S[A0γ ] + S[ω] + 2πk

�A0γ ∗ ω = S[A0γ ] + S[ω]

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7computation

Generators {h1, h2, ..., hw} for H1(M); the element hi is a generator for Zpi ,with pihi = 0. A generic element γ ∈ H1(M) can be described by means of thesum γ =

�wi=1 nihi with integers {ni}.

eiS[A0γ ] = e

2πik�

ij ninjQij

where the matrix Qij determines a Q/Z. -valued quadratic form Q on the torsiongroup T (M). Path-integral invariant :

Zk(M) =

p1−1�

n1=0

p2−1�

n2=0

· · ·pw−1�

nw=0

e2πik

�ij ninjQij

One has

Zk(M) = (p1p2 · · · pw)1/2 Ik(M)

where Ik(M) = Reshetikhin-Turaev U(1) surgery invariant of M .

Wednesday, June 5, 2013

8examples

Lens spaces Lp/r; in this case H1(Lp/r) = Zp

Zk(Lp/r) =p−1�

n=0

exp

�2πikr

pn2

�

− 3

− 1

+ 3

Manifold M2,6; in this case H1(M2,6) = T (M2,6) = Z2 ⊕ Z6

Zk(M2,6) =1�

n1=0

5�

n2=0

e2πik(−3n21+n

22)/6

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