A resurgence analysis of the $SU(2)$ Chern-Simons partition functions on a Brieskorn homology sphere $\Sigma(2,5,7)$

0.39 MB·

by  Sungbong Chun

PreprinttypesetinJHEPstyle-HYPERVERSION
A resurgence analysis of the SU(2) Chern-Simons
partition functions on a Brieskorn homology sphere
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1 Sungbong Chun

] Walter Burke Institute for Theoretical Physics, California Institute of Technology,
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t Pasadena, CA 91125 USA
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[ Abstract: We perform a resurgence analysis of the SU(2) Chern-Simons partition function

on a Brieksorn homology sphere Σ(2,5,7). Starting from an exact Chern-Simons partition
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v function, we study the Borel resummation of its perturbative expansion.
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Contents
1. Introduction 1
2. Setups for the Borel resummation in Chern-Simons theory 2
2.1 Borel resummation basics 2
2.2 Chern-Simons partition function as a trans-series 3
3. Exact partition function Z (Σ(2,5,7)) 4
CS
4. Asymptotics of Z (Σ(2,5,7)) 5
CS
5. Resurgence analysis of Z (Σ(2,5,7)) 6
CS
5.1 Borel transform and resummation of Z (M(2,5,7)) 6
CS
6. Homological block decomposition of Z (Σ(2,5,7)) and the modular trans-
CS
form 10
1. Introduction
We perform a resurgence analysis of the SU(2) Chern-Simons partition function on a Birek-
sorn homology sphere, following [1]. Consider the Chern-Simons action with a gauge group
G on a 3-manifold M :
3
(cid:90)
1 2
CS(A) = A∧dA+ A∧A∧A,
8π2 3
M3
where A is a Lie algebra (adG) valued 1-form on M . Classical solutions of this action are
3
the flat connections, satisfying F = dA+A∧A = 0. The Chern-Simons partition function
A
at level k can be expanded with a perturbation parameter 1/k, around the flat connections:
(cid:88)
Z (M ) = e2πikCS(α)Zpert. (1.1)
CS 3 α
α∈Mflat(M3,G)
Above, M (M ,G) is the moduli space of flat G-connections on M , and we have assumed
flat 3 3
a discrete moduli space. When k is an integer, CS(A) is only defined modulo 1.
The exact partition function Z (M ) = (cid:82) DAe2πikCS(A) can be recovered from its per-
CS 3
turbative expansion by a resurgence analysis of Jean E´calle [2]. We first analytically con-
tinue k to compex values and apply the method of steepest descent. Then, perform a Borel
– 1 –
transformation and resummation of the perturbative partition function, to recover the exact
partition function. Surprisingly, the exact partition function is now written as a linear sum
of the “homological blocks” [1]:
(cid:88)
Z (M ) = e2πikCS(α)Z . (1.2)
CS 3 a
aabelian
Above, Z gets contributions from both the abelian flat connection a and the irreducible
a
flat connections. In [3], it was proposed that the partition function in this form allows a
“categorification,” in a sense that it is a “S-transform” of a vector whose entries are integer-
coefficient Laurent series in q = e2πi/k.
In this paper, we provide a supporting example of [1]. First, we perform a resur-
gence analysis of SU(2) Chern-Simons partition function on a Brieskorn homology sphere,
M = Σ(2,5,7). We start with the exact partition function Z (Σ(2,5,7)), which is written
3 CS
as a linear sum of “mock modular forms” [4]. Then, we consider its perturbative expan-
sion and perform a Borel resummation. The Borel resummation in effect recovers the full
partition function Z (Σ(2,5,7)), and we observe a Stokes phenomenon which encodes the
CS
non-perturbative contributons to the partition function.
2. Setups for the Borel resummation in Chern-Simons theory
In this section, we provide necessary notations and setups for the Borel resummation in
Chern-Simons theory. A complete and concise review can be found in section 2 of [1].
Let us start with the exact Chern-Simons partition function Z (M ) = (cid:82) DAe2πikCS(A),
CS 3
integrated over G = SU(2) connections. Next, analytically continue k to complex values
and apply the method of steepest descent on the Feynman path integral [5–14]. Then, the
integration domain is altered to a middle-dimensional cycle Γ in the moduli space of G =
C
SL(2,C) connections, which is the union of the steepest descent flows from the saddle points.
To elaborate, the moduli space is the universal cover of the space of SL(2,C) connections
modulo “based” gauge transformations, in which the gauge transformations are held to be 1
at the designated points. In sum, the partition function becomes:
(cid:90)
Z (M ) = DAe2πikCS(A), k ∈ C. (2.1)
CS 3
Γ
2.1 Borel resummation basics
Partition function of form Equation 2.1 is interesting, for its perturbative expansion can be
regarded as a trans-series expansion, which can be Borel resummed. Let us provide here the
basics of Borel resummation, following [5]. The simplest example of a trans-series is a formal
power series solution of Euler’s equation:
dϕ A (cid:88) A−nn!
+Aϕ(z) = , ϕ (z) = .
dz z 0 zn+1
n≥0
– 2 –
One may view the above trans-series as a perturbative (in 1/z) solution to the differential
equation,butthesolutionhaszeroradiusofconvergence. BytheBorelresummation,however,
one can recover a convergent solution. When a trans-series is of form ϕ(z) = (cid:80) a /zn
n≥0 n
with a ∼ n!, its Borel transformation is defined as:
n
(cid:88) ζn−1
ϕˆ(ζ) = a .
n
(n−1)!
n≥1
The Borel transformation ϕˆ(ζ) is analytic near the origin of ζ-plane. If we can analytically
continue ϕˆ(ζ) to a neighborhood of the positive real axis, we can perform the Laplace trans-
form:
(cid:90) ∞
S ϕ(z) = a + e−zζϕˆ(ζ)dζ,
0 0
0
where the subscript “0” indicates that the integration contour is along the positive real axis,
{arg(z) = 0}. It can be easily checked that the asymptotics of the above integral coincides
with that of ϕ(z). When S ϕ(z) converges in some region in the z-plane, ϕ(z) is said to be
0
Borel summable, and S ϕ(z) is called the Borel sum of ϕ(z).
0
2.2 Chern-Simons partition function as a trans-series
SaddlepointsoftheChern-SimonsactionformthemodulispaceofflatconnectionsM˜, whose
connected components M˜ are indexed by their “instanton numbers,”
α˜
α˜ = (α,CS(α˜)) ∈ M (M ,SL(2,C))×Z.
flat 3
Here, CS(α˜) denotes the value of Chern-Simons action at α, without moding out by 1.
Following [1], we will call a flat connection abelian (irreducible, resp.), if the stabilizer is
SU(2) or U(1) ({±1}, resp.) action on Hom(π (M ),SU(2)).
1 3
Now, letΓ betheunionofsteepestdescentflowsinM˜, startingfromα˜. Theintegration
α˜
cycle Γ is then given by a linear sum of these “Lefshetz thimbles.”
(cid:88)
Γ = n Γ , (2.2)
α˜,θ α˜,θ
α˜
where θ = arg(k), and n ∈ Z are the trans-series parameters, given by the pairing between
α˜,θ
the submanifolds of steepest descent and ascent. The value of θ is adjusted so that there is no
steepest descent flow between the saddle points. Let I be the contribution from a Lefshetz
α˜,θ
thimble Γ to Z (M ) in Equation 2.1:
α˜,θ CS 3
(cid:90)
I = DAe2πikCS(A),
α˜,θ
Γ
α˜,θ
which can be expanded in 1/k near α˜ as:
∞
(cid:88)
I ∼ e2πikCS(α˜)Zpert, where Zpert = aαk−n+(dα−3)/2, d = dim M˜ .
α˜,θ α α n α C α˜
n=0
– 3 –
In sum, we can write the Chern-Simons partition function in the form:
(cid:88) (cid:88)
Z (M ;k) = n I ∼ n e2πikCS(α˜)Zpert(k), (2.3)
CS 3 α˜,θ α˜,θ α˜,θ α
α˜ α˜
which is a trans-series expansion of the Chern-Simons partition function. From the asymp-
totics given by this trans-series, we can apply Borel resummation and recover the full Chern-
Simons partition function. Note that Equation 2.3 depends on the choice of θ = arg(k). In
fact, as we vary θ, the value of I jumps to keep the whole expression continuous in θ as
α˜,θ
follows:
β˜
I = I +m I . (2.4)
α˜,θα˜β˜+(cid:15) α˜,θα˜β˜−(cid:15) α˜ β˜,θα˜β˜−(cid:15)
This is called the Stokes phenomenon, and it happens near the Stokes rays θ = θ ≡
α˜β˜
1 arg(S −S ). The trans-series parameters n jump accordingly to keep Z (M ;k) con-
i α˜ β˜ α˜,θ CS 3
β˜
tinuous in θ. The coefficients m are called Stokes monodromy coefficients.
α˜
3. Exact partition function Z (Σ(2,5,7))
CS
Before going into the resurgence analysis of Z (Σ(2,5,7)), let us provide here the exact
CS
partition function Z (Σ(2,5,7)). We first compute the Witten-Reshetikhin-Turaev (WRT)
CS
invariant τ (Σ(p ,p ,p )) and then write the exact SU(2) Chern-Simons partition function
k 1 2 3
in terms of WRT invariants as follows:
τ (Σ(p ,p ,p ))
k 1 2 3
Z (Σ(p ,p ,p )) = . (3.1)
CS 1 2 3 τ (S2×S1)
k
Here, k is the level of Chern-Simons theory.1
WRTinvariantsforSeiferthomologyspherescanbecomputedfromtheirsurgerypresen-
tations[15]. Inthispaper,wefocusonaspecifictypeofSeiferthomologyspheres,theso-called
Bireskorn homology spheres. A Brieskorn manifold Σ(p ,p ,p ) is defined as an intersection
1 2 3
of a complex unit sphere |z |2+|z |2+|z |2 = 1 and a hypersurface zp1+zp2+zp3 = 0. When
1 2 3 1 2 3
p ,p ,p are coprime integers, Σ(p ,p ,p ) is a homology sphere with three singular fibers.
1 2 3 1 2 3
From the surgery presentation of Σ(p ,p ,p ), we can write its WRT invariant, which can be
1 2 3
writtenalinearsumofmockmodularforms[4,16]. Inparticular,when1/p +1/p +1/p < 1,
1 2 3
we can write:
e2kπi(φ(p1,4p2,p3)−21)(e2kπi −1)τk(Σ(p1,p2,p3)) = 21Ψ˜p(11,p12,1p)3(1/k). (3.2)
Let us decode Equation 3.2. First of all, τ (Σ(p ,p ,p )) is the desired WRT invariant,
k 1 2 3
(cid:113)
normalizedsuchthatτ (S3) = 1andτ (S2×S1) = k 1 .Next,thenumberφ(p ,p ,p )
k k 2sin(π/k) 1 2 3
1Tobemoreprecise,k mustbereplacedbyk+2. However,ourinterestinthispaperistorecoverthefull
partitionfunction fromaperturbative expansion in1/k. Therefore, we willassumek tobelarge, and replace
k+2 with k here.
– 4 –
is defined as:
1
φ(p ,p ,p ) = 3− +12(s(p p ,p )+s(p p ,p )+s(p p ,p )),
1 2 3 1 2 3 2 3 1 3 1 2
p p p
1 2 3
b−1
1 (cid:88) nπ naπ
where s(a,b) = cot( )cot( ).
4b b b
n=1
Finally, Ψ˜(1,1,1) is a linear sum of mock modular forms Ψ˜a , namely:
p1p2p3 p1p2p3
(cid:40)
Ψ˜a(1/k) = (cid:88)ψa (n)qn2/4P, where ψa (n) = ±1 n ≡ ±a mod 2P (3.3)
P 2P 2P
0 otherwise
n≥0
Ψ˜(1,1,1)(1/k) = −1 (cid:88) (cid:15) (cid:15) (cid:15) Ψ˜p1p2p3(1+(cid:80)j(cid:15)j/pj)(1/k), (3.4)
p1p2p3 2 1 2 3 p1p2p3
(cid:15)1,(cid:15)2,(cid:15)3=±1
where q in Equation 3.3 is given by e2πi/k.
Now, let us restrict ourselves to (p ,p ,p ) = (2,5,7). First of all, p = 2,p = 5,p = 7
1 2 3 1 2 3
are relatively prime, so Σ(2,5,7) is a homology sphere. Next, 1/p +1/p +1/p < 1, so we
1 2 3
can write the WRT invariant as a linear sum of mock modular forms:
e2kπi(φ(2,45,7)−12)(e2kπi −1)τk(Σ(2,5,7)) = 12Ψ˜7(10,1,1)(1/k)
1
= (Ψ˜11−Ψ˜31−Ψ˜39+Ψ˜59)(1/k), (3.5)
70 70 70 70
2
where φ(2,5,7) = −19. From Equation 3.1 and 3.5, we can explicitly write the exact Chern-
70
Simons partition function Z (Σ(2,5,7)) as follows:
CS
1
Z (Σ(2,5,7)) = √ (Ψ˜11−Ψ˜31−Ψ˜39+Ψ˜59)(1/k). (3.6)
CS 70 70 70 70
iqφ(2,5,7)/4 8k
4. Asymptotics of Z (Σ(2,5,7))
CS
Before proceeding to the Borel transform and resummation of the exact partition function,
let us briefly consider the its asymptotics in the large k limit. This can be most easily done
by considering the “mock modular” property of mock modular forms:
Ψ˜a(q) = −(cid:114)k (cid:88)p−1(cid:114)2 sin πabΨ˜b(e−2πik)+(cid:88) L(−2n,ψ2ap)(cid:18) πi (cid:19)n, (4.1)
p p
i p p n! 2pk
b=1 n≥0
(2p)n (cid:88)2p (cid:18)m(cid:19)
where L(−n,ψa ) = − ψa (m)B , (4.2)
2p 2p n+1
n+1 2p
m=1
and B stands for the (n+1)-th Bernoulli polynomial. For integer values of k,
n+1
Ψ˜b(e−2πik) = (1− b)e−2π2ikpb2,
p p
– 5 –
and in large k limit, we may consider the second summation in Equation 4.1 as “perturba-
tive” contributions, while the first summation standing for “non-perturbative” contributions.
Therefore, the asymptotics of Z (Σ(2,5,7)) can be written as (p = 70, below):
CS
√
iq−19/280 8kZ (Σ(2,5,7)) =
CS
(cid:114) 70−1(cid:114) (cid:18) (cid:19)
− k (cid:88) 2 sin 11bπ −sin 31bπ −sin 39bπ +sin 59bπ (1− b)e−2π4ikpb2
i 70 70 70 70 70 p
b=1
√
+iq−19/280 8kZ (1/k), (4.3)
pert
√
where the perturbative contributions iq−19/280 8kZ (1/k) can be explicitly written as:
pert
Z (1/k) = Z11 (1/k)−Z31 (1/k)−Z39 (1/k)+Z59 (1/k),
pert pert pert pert pert
√ (cid:88) ba
where i 8q−19/280Za (1/k) = n for a = 11,31,39,59
pert kn+1/2
n≥0
L(−2n,ψa )(cid:18)πi(cid:19)n
and ba = 2p . (4.4)
n
n! 2p
Onecaneasilyseethatthesum(cid:0)sin 11bπ−sin 31bπ−sin 39bπ+sin 59bπ(cid:1)inEquation4.3is
70 70 70 70
nonzero if and only if b is not divisible by 2,5 or 7. We will later see that these b’s correspond
to the positions of the poles in the Borel plane.
5. Resurgence analysis of Z (Σ(2,5,7))
CS
In this section, we perform a resurgence analysis of the partition function and decompose
Z (M(2,5,7)) into the homological blocks:
CS
(cid:88)
Z (Σ(2,5,7)) = n e2πikCS(α)Z ,
CS α α
α
where α runs over the abelian/reducible flat connections. Since Z gets contributions from
α
boththeabelian/reducbleflatconnectionαandtheirreducibleflatconnections,itisnecessary
to study how the contributions from the irreducible flat connections regroup themselves into
the homological blocks. We accomplish the goal in three steps. First, we study the Borel
transform and resummation of the partition function and identify the contributions from the
irreducible flat connections. Then, the contributions from the irreducible flat connections are
shown to enter in the homological blocks via Stokes monodromy coefficients.
5.1 Borel transform and resummation of Z (M(2,5,7))
CS
Recall that the perturbative contributions Za (1/k) have the following asymptotics:
pert
√ (cid:88) ba
i 8qφ(2,5,7)/4Za (1/k) = n . (5.1)
pert k(n+1/2)
n≥0
– 6 –
Now, consider its Borel transform:
(cid:88) ba
BZa (ζ) = n ζn−1/2 (5.2)
pert
Γ(n+1/2)
n≥1
√
1 (cid:88) 4n n! (cid:18) π(2n)!(cid:19)
= √ ba√ ζn ∵ Γ(n+1/2) = (5.3)
ζ n π(2n)! 4n n!
n≥0
(cid:115)
1 (cid:88) n! 2πi
= √ ca z2n, where z = ζ. (5.4)
n
πζ (2n)! p
n≥0
In the last equality, we have simply changed the variable from ζ to z and absorbed all other
factors into the coefficients ca.
n
Although the coefficients ca only appear in the perturbative piece of the partition func-
n
tion, we can recover the exact partition function from them. Let us first consider generating
functions which package the coefficients ca:
n
sinh((p−a)z) (cid:88) n! (cid:88)
= ca z2n = ψa e−nz.
n 2p
sinh(pz) (2n)!
n≥0 n≥0
Now we can write the mock modular forms in an integral from, using these generating func-
tions:
sinh(p−a)η (cid:88)
= ψa (n)e−nη (5.5)
2p
sinhpη
n≥0
(cid:90) (cid:90)
⇒ dηsinh(p−a)ζe−k2pπηi2 = dη(cid:88)ψ2ap(n)e−nηe−k2pπηi2 (5.6)
sinhpη
iR+(cid:15) iR+(cid:15)
n≥0
(cid:115)
⇒ (cid:90) dηsinh(p−a)ηe−k2pπηi2 = 2π2i√1 Ψ˜ap(q). (5.7)
iR+(cid:15) sinhpη p k
In the second line, the integral is taken along a line Re[η] = (cid:15) > 0, where the integral
converges, and the third line is simply a Gaussian integral. The change of variables
pη2
ζ =
2πi
alters the integration contour from a single line to the union of two rays from the origin,
ieiδR and ie−iδR . In sum,
+ +
(cid:18) (cid:113) (cid:19)
sinh (p−a) 2πiζ
1 1(cid:18)(cid:90) (cid:90) (cid:19) dζ p
√ Ψ˜a(q) = + √ e−kζ. (5.8)
k p 2 ieiδR+ ie−iδR+ πζ sinh(cid:18)p(cid:113)2πiζ(cid:19)
p
Thus we have recovered the entire mock modular form from its perturbative expansion.
Sincethepartitionfunctionisalinearsumofmockmodularforms, thisimpliesthattheBorel
– 7 –
resummation of BZ will return the exact partition function. Furthermore, the poles of
pert
generating functions sinh((p−a)z)/sinh(pz) encodes the information of the non-perturbative
contributions, as we exhibit below.
First of all, since Z (Σ(2,5,7)) ∼ (Ψ˜11 −Ψ˜31 −Ψ˜39 +Ψ˜59)(q), the Borel transform of
CS 70 70 70 70
Z is given by:
pert
sinh(59η)−sinh(39η)−sinh(31η)+sinh(11η) 4sinh(35η)sinh(14η)sinh(10η)
= , (5.9)
sinh(70η) sinh(70η)
Note that the RHS of Equation 5.9 has only simple poles at η = nπi/70 for n non-divisible
by 2, 5, or 7. In particular, the poles are aligned on the imaginary axis, so we choose the
same integration contours as in Equation 5.6 - 5.8. The Borel resummation of Equation 5.9
is then the average of Borel sums along the two rays depicted in Figure 1(a):
(cid:20) (cid:21)
1
ZCS(Σ(2,5,7)) = Sπ−δZpert(1/k)+Sπ+δZpert(1/k) . (5.10)
2 2 2
Im Im
ie−iδR+ ieiδR+
Re Re
(a) (b)
Figure 1: (a)Anintegrationcontourintheζ-plane,madeoftworaysfromtheorigin. Dotsrepresent
thepoles. (b)Anequivalentintegrationcontour. Thecontributionfromtheintegrationalongthereal
axis must be doubled.
ToevaluatetheRHSofEquation5.10, weintegratealonganequivalentcontourinFigure
1(b). Note that as we change to the contour in Figure 1(b), a Stokes ray ie−iδR has crossed
+
thepolesontheimaginaryaxis,towardsthepositiverealaxis. Asareult,thepolescontribute
to the Borel sums with residues, which is precisely a Stokes phenomenon. Since each pole is
located at η = nπi/70, its residue includes a factor of e−kζ = e−k720πηi2 = e2πik(−2n820). Shortly,
we will exhibit that these factors precisely correspond to the Chern-Simons instanton actions,
so let us regroup the poles (n modulo 140) by their instanton actions:
• n = 9,19,51,61,79,89,121,131,forwhichCS = − 92 andresidues{1,1,1,1,−1,−1,−1,−1}
280
with overall factor i (cos 3π −sin π ).
35 35 70
• n = 3,17,53,67,73,87,123,137,forwhichCS = − 32 andresidues{−1,−1,−1,−1,1,1,1,1}
280
with overall factor i (cos π +cos 6π).
35 35 35
– 8 –
• n = 23,33,37,47,93,103,107,117,forwhichCS = −232 andresidues{1,1,1,1,−1,−1,−1,−1}
280
with overall factor i (cos 4π +sin 13π).
35 35 70
• n = 13,27,43,57,83,97,113,127,forwhichCS = −132 andresidues{−1,−1,−1,−1,1,1,1,1}
280
with overall factor i (sin 3π +sin 17π).
35 70 70
• n = 11,31,39,59,81,101,109,129,forwhichCS = −112 andresidues{1,−1,−1,1,−1,1,1,−1}
280
with overall factor i (cos 8π +sin 9π).
35 35 70
• n = 1,29,41,69,71,99,111,139,forwhichCS = − 12 andresidues{1,−1,−1,1,−1,1,1,−1}
280
with overall factor i (cos 2π −sin 11π).
35 35 70
The top four groups of poles correspond to the four irreducible SU(2) flat connections,
whiletheremainingtwocorrespondtothecomplexflatconnections. Toseethis, firstconsider
the moduli space of flat connections M (Σ(2,5,7),SL(2,C)). Since Σ(2,5,7) is a homol-
flat
ogy 3-sphere, it has only one abelian flat connection α , which is trivial. Next, there are
0
(2−1)(5−1)(7−1)
total = 6 irreducible SL(2,C) flat connections, four of which are conjugate to
4
SU(2) and the remaining two are “complex” (conjugate to SL(2,R)) [17–19]. To compute
their Chern-Simons instanton actions, we characterize all six flat connections by their “rota-
tion angles,” which we will briefly explain here. Consider the following presentation of the
fundamental group of Σ(2,5,7).
π (Σ(2,5,7)) = (cid:104)x ,x ,x ,h|hcentral,x2 = h−1,x5 = h−9,x7 = h−5,x x x = h−3(cid:105).
1 1 2 3 1 2 3 1 2 3
(5.11)
When a representation α : π (Σ(2,5,7)) → SL(2,C) is conjugate in SU(2), α(h) is equal
1
to ±1, and the conjugacy classes of α(x ) can be represented in the form (cid:0)λj 0 (cid:1) for some
j 0 λ−1
j
|λ | = 1. There are four triples (λ ,λ ,λ ) satisfying the relations in Equation 5.11:
j 1 2 3
(l ,l ,l ) = (1,1,3), (1,3,1), (1,3,3), (1,3,5) where λ = eπilj/pj. (5.12)
1 2 3 j
Each triple corresponds to one of the four irreducible SU(2) flat connections, which we will
call α ,α ,α and α . From the rotation angles of an irreducible flat connection A, we can
1 2 3 4
read off its Chern-Simons instanton action:
CS(A) = −p1p2p3(1+(cid:88)l /p )2
j j
4
i
92 32 232 132
⇒ CS(α ) = − , CS(α ) = − , CS(α ) = − , CS(α ) = − , (5.13)
1 2 3 4
280 280 280 280
which is in agreement with the instanton actions of the poles in the Borel plane. Likewise,
one can compute the Chern-Simons instanton actions of the two complex flat connections α
5
and α ,
6
112 12
CS(α ) = − , CS(α ) = − .
5 6
280 280
– 9 –