Topological Strings, Instantons and Asymptotic Forms of Gopakumar-Vafa Invariants

0.28 MB·English

by  Yukiko Konishi

December, 2003
hep-th/0312090
Topological Strings, Instantons and
Asymptotic Forms of Gopakumar–Vafa Invariants
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n Yukiko Konishi
a
J

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2 Research Institute for Mathematical Sciences,
v
0 Kyoto University,
9
0 Kyoto 606-8502, Japan
2
1 [email protected]
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h
t
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p
e
h
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v Abstract
i
X
r We calculate the topological string amplitudes of Calabi–Yau toric threefolds cor-
a
responding to 4D, = 2 SU(2) gauge theory with N = 0,1,2,3,4 fundamental
f
N
hypermultiplets by using the method of the geometric transition and show that they
reproduce Nekrasov’s formulas for instanton counting. We also determine the asymp-
totic forms of the Gopakumar–Vafa invariants of the Calabi–Yau threefolds including
those at higher genera from instanton amplitudes of the gauge theory.
1 Introduction
Recently, remarkable developments occurred in the theory of the topological strings.
We can now compute the Gromov–Witten invariants or Gopakumar–Vafa invariants of a
Calabi–Yau toric threefold at all genera by using the Feynman-like rules [1, 2] which has
been developed from the geometric transition and the Chern–Simons theory [3]. Although
this method is most powerful compared to other methods such as localization and local B-
model calculation, we still cannot obtain the exact form of the topological string amplitude
in general cases because we have to sum over several partitions. However, it is found
that we can perform the summation for some special types of tree graphs by using the
identities on the skew-Schur functions [4, 5, 6]. A simplest example is the resolved conifold
( 1) ( 1) P1.
O − ⊕O − →
Another interesting application is the geometric engineering of the gauge theories [7,
8, 4, 5]. In this article we study the cases with the gauge group SU(2) and with N =
f
0,1,2,3,4 fundamental hypermultiplets. It has been known that the corresponding Calabi–
Yau threefolds are the canonical bundles of the Hirzebruch surfaces blown up at N -points.
f
We calculate the topological string amplitudes of these Calabi–Yau threefolds and show
that they reproduce Nekrasov’s formulas for instanton counting [9] in a certain limit. The
SU(n+1) cases without hypermultiplets and the SU(2) case with one hypermultiplet have
been studied in [7, 8, 4] and the calculations in this article are essentially the same. We
also determine the asymptotic forms of the Gopakumar–Vafa invariants of these Calabi–
Yau threefolds from the relation between the topological string amplitudes and Nekrasov’s
formula. This result is the generalization of the genus zero results [10, 11] to higher genus
cases.
The organization of the paper is as follows. In section 2, we calculate the topological
string amplitudes of Calabi–Yau toric threefolds that correspond to 4D, = 2 SU(2)
N
gauge theories with N = 0,1,2,3,4 fundamental hypermultiplets. In section 3, we show
f
that the topological amplitude reproduces Nekrasov’s formula. In section 4, we derive
the asymptotic form of the Gopakumar–Vafa invariants of the Calabi–Yau toric threefolds.
Appendices contain formulas and the calculation of the framing.
1
2 Topological String Amplitude
In this section, we calculate the topological string amplitudes of Calabi–Yau toric three-
folds that reproduce four-dimensional = 2 supersymmetric gauge theories with gauge
N
group SU(2) and with N = 0,1,2,3,4 fundamental hypermultiplets.
f
First we briefly review the calculation of the topological string amplitudes of Calabi–
Yau threefolds X following [2] when X is the canonical bundle of a smooth toric surface
classified in [12]. Recall that a two-dimensional integral polytope (the section of the fan at
the height 1) of X has only one interior integral point (0,0) and this point and each integral
point v (1 i k) on the boundary span an interior edge (here we define k to be the
i
≤ ≤
number of the interior edges and take v ,v ,... in the clockwise direction). Therefore the
1 2
corresponding web diagram consists of a polygon with k-edges and external lines attached
to it. We take the orientation of edges on the polygon in the clockwise direction, and that
of the external lines in the outgoing direction. Then the integer m of the framing for an
i
interior edge dual to v is given by
i
m = γ 1. (1)
i i
− −
Here γ is the self-intersection number of the P1 associated to v and computed from the
i i
equation
γ v = v +v . (2)
i i i 1 i 1
− − −
The derivation of m is included in appendix. Then we assign a partition Q to each interior
i i
edge and the partition of zero to each external edge. Finallywe obtain the topological string
amplitude by multiplying all quantities associated to vertices and edges and by summing
over all partitions. The brief summary of the rule is as follows: to a trivalent vertex we
associate the three-point amplitude C if the orientation of all the edges are outgoing,
R1,R2,R3
(−1)l(R1)CR1t,R2,R3 if the orientation of one edge with a partition R1 is incoming, etc, where
R ,R ,R are partitions assigned to three edges attached to the vertex; to an interior edge
1 2 3
we associate ( 1)ml(R)q mκ(R)e l(R)t where m is the integer coming from the framing and R
− 2 −
−
is the partition assigned to the edge and t is the K¨ahler parameter of the corresponding P1.
Here l(R) := µ for a partition R = (µ ,µ ,...) and κ(R) := µ (µ 2i+1). Thus
Pi i 1 2 Pi i i −
2
the topological string amplitude for X is written as
k
Z = X YCQti,∅,Qi+1(−1)γil(Qi)qγi2+1κ(Qi). (3)
Q1,···,Qk i=1
Here we have defined γ := γ ,Q := Q . This result was derived in [13, 14, 4]. The
k+1 1 k+1 1
relation between the topological string amplitude and the Gromov–Witten invariants is
Z
that log is the generating function of the Gromov–Witten invariants where g is
Z|q=e√−1gs s
the genus expansion parameter. This statement has been proved for the canonical bundle
of fano toric surfaces [14] 1.
Next we derive more compact formulas for the toric Calabi–Yau threefolds that corre-
sponding to 4D, = 2 SU(2) gauge theories with N = 0,1,2,3,4 fundamental hyper-
f
N
multiplets. The Calabi–Yau threefolds are the canonical bundles of the Hirzebruch surfaces
F ,F , or F blown up at N points. There exist 3,2,3,3,2 such Calabi–Yau threefolds for
0 1 2 f
N = 0,1,2,3,4.2 The fans and the web diagrams for these threefolds are shown in figures
f
1 and 3. We take [C ],[C ],[C ] (1 i N ) as a basis of the second homology where C
B F Ei ≤ ≤ f B
(resp. C ) is the base P1 (resp. fiber P1) and C is an exceptional curve. The intersections
F Ei
are
C .C = b, C .C = 1, C .C = 0,
B B − B F F Ei (4)
C .C = 0, C .C = 0, C .C = δ
F F F Ei Ei Ej − i,j
for 1 i,j N . The values of b are 1 or 2 and will be listed later. Here t ,t ,t (1
≤ ≤ f B F Ei ≤
i 3) denote the K¨ahler parameters of the base P1, the fiber P1, the i-th ( 1)-curve and
≤ −
qB := e−tB, qF := e−tF, qi := e−tEi.
Now wecompute thetopologicalstringamplitudes byusing thesame strategyas[7,8,4].
Let us take N = 2 cases shown in figure 1 as examples. We first cut the polygon in the
f
web diagram into two upper and lower parts (as shown by dotted line) and compute the
amplitudes separately. Then we glue the two amplitudes together along the vertical edges
1P2 and (2)(3)(5)(7) in figures 1, 3.
2Although there are 16 smooth toric surfaces classified in [12], three among them (1,10,16in figure 1 in
[12]) do not correspond to the four-dimensional gauge theories.
3
(7) (8) (9)
tE1-tE2 tE1-tE2
tF-tE1
0 tE1,0 tF-tE1 1 tE2,0 tF-tE1 1 tE2,0
tB,0 0 0
tF-tE2,0 t0E2-ttBE+1-t0tFE2 tB,1 t-tBE+1-2ttEF2 tB,0 tF,-1 -ttBE+10-ttFE2
tF,-1
-1
Figure 1: The fans and the web diagrams for the Calabi–Yau toric 3-folds that correspond
to four-dimensional = 2 SU(2)gauge theory with N = 2 fundamental hypermultiplets.
f
N
For N = 0,1,2,4 see figure 3.
f
x
Q,x Qx2 xQ1 x3 Q22 x1 xQ3 Q ,x
-1 20 0 1 Q3 1 Q1 x4 13 1 2 2
0 0 Q4 0 0Q1,x
R R R R R R R R
2 1 2 1 2 1 2 1
(k)
Figure 2: The web diagrams corresponding to H (x ,...,x ) (1 k 4).
R1,R2 1 k ≤ ≤
to obtain the whole topological string amplitude. They are written as follows:
(7): = H(2) (q ,q q 1)H(2) (q q 1,q ) (5)
Z X R1,R2 1 F 1− R2,R1 F 2− 2
R1,R2
q l(R1)+l(R2)q l(R1)(q q ) l(R1),
B F 1 2 −
×
(8): = H(3) (q ,q q 1,q q 1)H(1) (q ) (6)
Z X R1,R2 2 1 2− F 1− R2,R1 F
R1,R2
( 1)l(R1)+l(R2)qκ(R21)−κ(R22)qBl(R1)+l(R2)qF2l(R1)(q1q2)−l(R1),
× −
(9): = H(3) (q ,q q 1,q q 1)H(1) (q ) (7)
Z X R1,R2 2 1 2− F 1− R2,R1 F
R1,R2
q l(R1)+l(R2)q l(R1)(q q ) l(R1).
B F 1 2 −
×
(k)
HereR (resp. R )isapartitionassignedtotheright(resp. left)verticaledge. H (x ,...,x )
1 2 R1,R2 1 k
4
(1 k 4) are amplitudes corresponding to the web diagrams in figure 2 and defined by
≤ ≤
HR(11),R2(x) := Xqκ(2Q)xl(Q)(−1)l(R2)C∅,R1,QtCQ,R2t,∅, (8)
Q
k
HR(k1),R2(x1,...,xk) := X (−1)l(R2)+l(Q1)+l(Qk)q−κ(Q2)+···2+κ(Qk−1) Yxil(Qi) (9)
Q1,···,Qk i=1
C C C C (2 k 4).
× ∅,R1,Q1t Q2t,∅,Q1··· Qkt,∅,Qk−1 Qk,R2t,∅ ≤ ≤
Using the expression for C written in terms of the skew-Schur functions and the
R1,R2,R3
(k)
identities (see appendix), H (x ,...,x ) becomes
R1,R2 1 k
H(k) (x ,...,x ) = ( 1)l(R2)W W K(x x )g (x x )
R1,R2 1 k − R1 R2t 1··· k R1,R2 1··· k
k 1 k
− 1
K(x x )g (x x ) K(x x )g (x x ) −
×hY 1··· j R1,∅ 1··· j Y j··· k R2,∅ j··· k i
j=1 j=2
K(x x ).
× Y i··· j
2 i j k 1
≤≤ ≤ −
(10)
The second (resp. third ) line should be set to 1 for k = 1 (resp. for k 2).
≤
∞ ∞ qkxk
K(x) := (1 x i j+1) 1 = exp (11)
X − −− − hX k(qk 1)i
i,j=1 k=1 −
[µ µ +j i] d(R) µi 1
κR i j
WR := q 4 Y −[j i] − Y Y [v u+d(R)], (12)
1 i<j d(R) − i=1 v=1 −
≤ ≤
∞ (1 xqµ1,i−i+µ2,j−j+1)
g (x) := − (13)
R1,R2 Y h (1 xq i j+1) i
−−
i,j=1 −
1 1
= . (14)
Y (1 xqµ1,i−j+µ2,j−i+1) Y (1 xq−µ∨1,j+i−µ∨2,i+j−1)
(i,j) R1 − (j,i) R2 −
∈ ∈
k k
Here[k] := q q ,l(R) := µ forapartitionR = (µ ,µ ,...),κ(R) := µ (µ 2i+1)
2− −2 Pi i 1 2 Pi i i−
and d(R) is the length of R. (µ ) (resp. (µ ) ) is the conjugate partition of R (resp.
∨1,i i 1 ∨2,i i 1 1
≥ ≥
R ) and (i,j) R means that there is a box in the place of i-th row and j-th column in the
2
∈
partition R regarded as a Young diagram. We have used an identity in appendix to obtain
the expression (14). The final form of the topological string amplitudes for the N = 2 cases
f
5
become
= Z Z
0 1
Z ≥
Nf 1 (7)
Z0 = K(qF)2YK(qj)−1K(qFqj−1)−1 ×
j=1 K(q1q2−1) (8)(9)
(15)


Nf
Z = g (1)g (1)g (q )2 g (q ) 1g (q q 1) 1
≥1 X R1,R1t R2,R2t R1,R2 F Y R1,∅ j − R2,∅ F j− −
R1,R2 j=1
×qBl(R1)+l(R2)qFbl(R1)(q1···qNf)−l(R1)(−1)m1l(R1)+m2l(R2)q−m1κ(R1)+2m2κ(R2).
The numbers b are 1,2,1 and (m ,m ) are (0,0),( 1,1),(0,0)for (7)(8)(9). In Z we have
1 2 1
− ≥
used the identity
W W = ( 1)l(R)g (1).
R Rt − R1,R1t
The generating function of the Gromov–Witten invariants is obtained from the topolog-
ical string amplitude by taking the logarithm and substituting e√ 1gs into q:
−
∞
logZ|q=e√−1gs = Xgs2g−2 X Ng,dB,dF,d1,d2qBdBqFdFq1d1q2d2
g=0 dB,dF,d1,d2 (16)
= ∞ ∞ ngdB,dF,d1,d2 2sin kgs 2g−2 q dBq d1q dFq d1q d2 k.
X X X k (cid:16) 2 (cid:17) (cid:0) B 1 F 1 2 (cid:1)
g=0 dB,dF,d1,d2 k=1
N denotes the genus zero, 0-pointed Gromov–Witten invariant for an integral
g,dB,dF,d1,d2
homology class d [C ]+d [C ]+d [C ]+d [C ] and ng denotes the Gopakumar–
B B F F 1 E1 2 E2 dB,dF,d1,d2
Vafa invariant.
One can read off the Gromov–Witten invariants with d = 0 from Z , because only
B 0
logZ gives the terms with degree zero in q :
0 B
∞ qkqFk Nf ∞ qkqik ∞ qk(qFqi−1)k
logZ = 2 +
0 X k(qk 1)2 −XhX k(qk 1)2 X k(qk 1)2 i
k=1 − i=1 k=1 − k=1 −
∞ qk(q1q2−1)k
+ for (8)(9), 0 for (7)
X k(qk 1)2
k=1 −
(17)
q=e=√−1gs ∞ 1 2sin kgs −2 2q k + Nf q k +(q q 1)k
X k(cid:16) 2 (cid:17) h− F X(cid:0) F F i− (cid:1)i
k=1 i=1
+ ∞ −1 2sin kgs −2 q q 1 k for (8)(9), 0 for (7).
X k (cid:16) 2 (cid:17) (cid:0) 1 2− (cid:1)
k=1
6
HencethenonzeroGopakumar–Vafainvariantsforasecondhomologyclassd [C ]+d [C ]+
B B F F
d [C ]+d [C ] with d = 0 are as follows (we slightly change the notation and write the
1 E1 2 E2 B
Gopakumar–Vafa invariant as ng for a second homology class α):
α
ng=0 = 2, ng=0 = ng=0 = +1 (1 i N ), (18)
[CF] − [CEi] [CF]−[CEi] ≤ ≤ f
and
ng=0 = 1 only for (8)(9). (19)
[CE1]−[CE2] −
The invariants at [C ],[C ] and [C ] [C ] (1 i N ) are the same in all of the three
F Ei F − Ei ≤ ≤ f
cases, while the invariant at [C ] [C ] are different. We will interpret these results from
E1 − E2
the viewpoint of relation to the Seiberg–Witten prepotential of SU(2) gauge theory in the
next section.
The Gopakumar–Vafa invariants with d 1 are given by logZ . We remark two
B 1
≥ ≥
properties. The one is that the Gopakumar–Vafa invariant is nonzero only when
0 d d (1 i N ). (20)
i B ∀ f
≤ − ≤ ≤ ≤
Onecouldreadthisfactfromtheexpression (15)asfollows. Thesummandisthepolynomial
in q 1,q 1 of degree at most l(R )+l(R ) given that g (x) 1 is the polynomial in x of
1− 2− 1 2 R, −
∅
degree l(R). Therefore the degree in q 1 is always equal or smaller than the degree in q ,
i− B
and this fact persists when we take the logarithm. Thus the Gromov–Witten invariants are
zero unless the condition (20) is satisfied, and so are the Gopakumar–Vafa invariants. The
other property is that the Gopakumar–Vafa invariants are symmetric with respect to q ,q .
1 2
This follows from the invariance of Z under the exchange of q and q .
1 1 2
≥
Finally, we summarize the topological string amplitude for all the cases corresponding
to SU(2) gauge theory with N = 0,1,2,3,4 hypermultiplets shown in figures 1, 3.
f
= Z Z
0 1
Z ≥
Nf
Z = K(q )2 K(q ) 1K(q q 1) 1 (a)
0 F Y j − F j− − ×
j=1
(21)
Nf
Z = g (1)g (1)g (q )2 g (q ) 1g (q q 1) 1
≥1 X R1,R1t R2,R2t R1,R2 F Y R1,∅ j − R2,∅ F j− −
R1,R2 j=1
×qBl(R1)+l(R2)qFbl(R1)(q1···qNf)−l(R1)(−1)m1l(R1)+m2l(R2)q−m1κ(R1)+2m2κ(R2).
7
N Nonzero Gopakumar–Vafa invariants
f
0 (2)(3)(4) ng=0
[CF]
1 (5)(6) ng=0 = 2,ng=0 = ng=0 = 1
[CF] − [CE1] [CF]−[CE1]
2 (7) ng=0 = 2,ng=0 = ng=0 = 1(1 i 2)
[CF] − [CEi] [CF]−[CEi] ≤ ≤
(8)(9) ng=0 = 2,ng=0 = ng=0 = 1(1 i 2),n = 1
[CF] − [CEi] [CF]−[CEi] ≤ ≤ [CE1]−[CE2] −
3 (11)(13) ng=0 = 2,ng=0 = ng=0 = 1(1 i 3),
[CF] − [CEi] [CF]−[CEi] ≤ ≤
n = n = n = 1
[CE1]−[CE2] [CE1]−[CE3] [CE2]−[CE3] −
(12) ng=0 = 2,ng=0 = ng=0 = 1(1 i 3),n = 1
[CF] − [CEi] [CF]−[CEi] ≤ ≤ [CE1]−[CE2] −
4 (14) ng=0 = 2,ng=0 = ng=0 = 1(1 i 4),
[CF] − [CEi] [CF]−[CEi] ≤ ≤
n = n = n = 1
[CE1]−[CE2] [CE1]−[CE3] [CE2]−[CE3] −
(15) ng=0 = 2,ng=0 = ng=0 = 1(1 i 4),n = n = 1
[CF] − [CEi] [CF]−[CEi] ≤ ≤ [CE1]−[CE2] [CE3]−[CE4] −
Table 1: The nonzero Gopakumar–Vafa invariants for a second homology class d [C ] +
B B
d [C ]+d [C ]+d [C ] with d = 0.
F F 1 E1 Nf ENf B
where
1 (2)(3)(4)(5)(6)(7)



K(q q 1) (8)(9)(12)
 1 2−
(a) = 

K(q q 1)K(q q 1)K(q q 1) (11)(13)(14)
1 2− 1 3− 2 3−



K(q1q2−1)K(q3q4−1) (15).


b is the self-intersection of the base P1 and m (m ) is the integer of the framing of the right
1 2
(left) vertical edge:
(2) (3) (4) (5) (6) (7) (8) (9) (11) (12) (13) (14) (15)
N 0 1 2 3 4
f
b 0 1 2 1 2 1 2 1 2 2 1 1 1
m 1 2 3 1 2 0 1 0 1 0 0 1 1
1
− − − − − −
m 1 0 1 0 1 0 1 0 0 1 1 1 1
2
−
Note that the properties of the Gopakumar–Vafa invariants mentioned in N = 2 cases
f
hold in all N cases: the nonzero Gopakumar–Vafa invariants for a second homology class
f
d [C ]+d [C ]+d [C ]+d [C ] with d = 0 are summarized in table 1 and one can
B B F F 1 E1 Nf ENf B
8
easily see that in all cases n0 = 2, n0 = n0 = 1 (1 i N ); for d 1,
[CF] − [CEi] [CF]−[CEi] ≤ ≤ f B ≥
the Gopakumar–Vafa invariants is nonzero only if 0 d d (1 i N ) and they are
i B ∀ f
≤ − ≤ ≤ ≤
symmetric with respect to d ,...,d . Note also that Z ’s of (7) and (9) (resp. (14) and
1 Nf ≥1
(15)) are the same, which means that the Gopakumar–Vafa invariants in cases (7) and (9)
(resp. (14) and (15)) with d 1 are completely the same.
B
≥
3 Nekrasov’s Formula
In this section we show that the topological string amplitude gives the one-loop
Z
corrections in the prepotential and Nekrasov’s formula [9] for the 4D, = 2 SU(2) gauge
N
theorywithN fundamentalhypermultiplets atacertainlimit. Theargument inthissection
f
closely follows that in [7, 8, 4].
First let us identify the parameters in the two sides:
t = 2βa, t = β(a+m ) (1 i N ), q = β~. (22)
F − Ei − i ≤ ≤ f −
Then the limit we should take is the limit β 0. Here a = a = a is the vacuum
1 2
→ −
expectation value of the complex scalar field in a gauge multiplets, m (1 i N )’s are
i f
≤ ≤
mass parameters of the fundamental hypermultiplets. g has been introduced as the genus
s
expansion parameter.
Next, let us show that the t independent part Z in the topological string amplitude
B 0
gives the perturbative one-loop correction terms. We again take N = 2 cases (7)(8)(9) as
f
examples. Note that
qk 1
lim = .
~=0 (qk 1)2|q=e−β~ β2~2
−
Then
∞ 1 Nf
lim~2logZ = 2 e 2kβa e β(a+mi)k +e β(a mi)k
~=0 ≥1 X k3h − −X(cid:0) − − − (cid:1)i
k=1 i=1 (23)
∞ 1
+ e kβ(m1 m2) for (8)(9), 0 for (7).
− −
X k3
k=1
Each trilogarithm corresponds to one logarithmic term in the Seiberg–Witten prepotential.
We can see this correspondence in the following way. If we take the third derivative in a,
9