Gauge invariance and topologytheory.fi.infn.it/cappelli/sft2013/Guadagnini.pdfGauge invariance and...
Transcript of Gauge invariance and topologytheory.fi.infn.it/cappelli/sft2013/Guadagnini.pdfGauge invariance and...
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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.
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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)
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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
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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 .
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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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