Appendix A Special Functions
We summarize the special functions used in the manuscript.
A.1 Gamma Functions
We define the Barnes zeta function for (i )i=1,...,k for Re(s)>k and Re(i )>0:
1 ζ (s, z; ,..., ) = . k 1 k s (A.1.1) (z + n11 +···+nk k ) n1,...,nk ≥0
Then, the multiple gamma function is defined as ∂ (z; ,..., ) = ζ (s, z; ,..., ) k 1 k exp ∂ k 1 k s s=0 1 = . (A.1.2) z + n11 +···+nk k n1,...,nk ≥0
Precisely speaking, the infinite product in the second line should be a formal expres- sion since the corresponding series expansion is available only for Re(s)>k.Nev- ertheless, we interpret this as the zeta function regularization of the infinite product. This gamma function is constructed to obey a functional relation
k (z + i ; 1,...,k ) = k−1(z; 1,...,ˇi ,...,k ) (A.1.3) k (z; 1,...,k )
© The Editor(s) (if applicable) and The Author(s), under exclusive license 251 to Springer Nature Switzerland AG 2021 T. Kimura, Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, https://doi.org/10.1007/978-3-030-76190-5 252 Appendix A: Special Functions where ˇi means that the dependence on i on the right hand side is removed. We remark that the gamma function k (z; 1,...,k ) has poles at z + n11 +···+ nk k = 0 and no zero. The degree-one case is related to the standard definition of the gamma function:
/− 1 z 2 1(z; ) = √ (z/) . (A.1.4) 2π
Therefore, the asymptotic behavior is given by Stirling’s formula:
lim log 1(z; ) = z log z − z . (A.1.5) →0
A.1.1 Reflection Formula
Together with the infinite product formula of the sine function
∞ sin πz z2 = 1 − , (A.1.6) πz n2 n=1
∞ and the zeta function regularization 2n2 = 2π/, we obtain the reflection n=1 formula 1 (z; ) ( − z; ) = . (A.1.7) 1 1 2sinπz/
It is also possible to derive this formula form the standard version of the formula through the relation (A.1.4): π (z)(1 − z) = . (A.1.8) sin πz
A.1.2 Multiple Sine Function
One may define multiple sine functions through the reflection formula for the multiple gamma functions [3],
(−1)k k ( − z; 1,...,k ) Sk (z; 1,...,k ) = , (A.1.9) k (z; 1,...,k ) Appendix A: Special Functions 253 where
k = i . (A.1.10) i=1
For example, the double sine function is given as follows:
( + − z; , ) z + n + m S (z; , ) = 2 1 2 1 2 = 1 2 . 2 1 2 (z; , ) + − z + n + m 2 1 2 n,m≥0 1 2 1 2 (A.1.11)
In this case, it is defined as a ratio of the double gamma functions. We instead obtain the Upsilon function with their product,
1 ϒ(z; 1,2) = . (A.1.12) 2(z; 1,2)2(1 + 2 − z; 1,2)
A.2 q-Functions
A.2.1 q-Shifted Factorial
We define the q-shifted factorial (also known as the q-Pochhammer symbol):
n−1 m (z; q)n = (1 − zq ). (A.2.1) m=0
The multivariable analog is similarly defined as
(z1,...,zk ; q)n = (z1; q)n ···(zk ; q)n . (A.2.2)
The q-shifted factorial for n →∞is given for |q| < 1as ∞ ∞ m m z (z; q)∞ = (1 − zq ) = exp − . (A.2.3) m(1 − qm ) m=0 m=1
For |q| > 1, it is given through the analytic continuation:
−1 −1 −1 (z; q)∞ = (zq ; q )∞ . (A.2.4) 254 Appendix A: Special Functions
We remark the relation ∞ m mn (z; q)∞ z 1 − q (z; q)n = = exp − . (A.2.5) (zqn; q)∞ m 1 − qm m=1
A.2.2 Quantum Dilogarithm
Let q = e, and consider the expansion around = 0oftheq-factorial. We remark the expansion
∞ 1 1 1 n =− =− B (A.2.6) 1 − q e − 1 n n! n=0 where (Bn)n≥0 are the Bernoulli numbers. Then the q-shifted factorial is given by ∞ 1 n 1 (z; q)∞ = exp Li − (z) B = exp (Li (z) + O()) (A.2.7) 2 n n n! 2 n=0 where we define the polylogarithm function for |z| < 1as
∞ zm Li (z) = . (A.2.8) p m p m=1
In this sense, the q-shifted factorial is interpreted as a q-deformation of the dilog- arithm, which is called the quantum dilogarithm [2]. We remark that this quantum dilogarithm is related to, but different from Faddeev’s quantum dilogarithm [1].
A.2.3 q-Gamma Functions
The formulas shown above imply that the q-shifted factorial is interpreted as a q- analog of the gamma function1:
1This is slightly different from the standard definition of the q-gamma function, ( ; ) 1−x q q ∞ q (x) = (1 − q) , (A.2.9) (qx ; q)∞ which obeys the relation
1 − qx q (x + 1) = q (x) =[x]q q (x). (A.2.10) 1 − q Appendix A: Special Functions 255
1 q (z; q) := (A.2.11) (z; q)∞ with poles at zqn = 1forn ≥ 0. In this convention, the relation (A.2.5)isgivenby
( n; ) q zq q n−1 = (z; q)n = (1 − z) ···(1 − zq ). (A.2.12) q (z; q)
A q-analog of the multiple gamma function is defined as
(−1)k , (z; q ,...,q ) = (z; q ,...,q ) q k 1 k 1 k ∞ ∞ zm 1 = exp (−1)k+1 (A.2.13) m (1 − qm ) ···(1 − qm ) m=1 1 k with the multiple version of the q-shifted factorial for |q1|,...,|qk | < 1: ( ; ,..., ) = ( − n1 ··· nk ) z q1 qk ∞ 1 zq1 qk ≤ ,..., ≤∞ 0 n1 nk ∞ zm 1 = exp − . (A.2.14) m (1 − qm ) ···(1 − qm ) m=1 1 k
This q-gamma function obeys the functional relation
q,k (zqi ; q1,...,qk ) = q,k−1(z; q1,...,qˇi ,...,qk ). (A.2.15) q,k (z; q1,...,qk )
From this point of view, we may also consider the multiple q-gamma function of negative degree as follows: ∞ m k+1 z m m ,− (z; q ,...,q ) = exp (−1) (1 − q ) ···(1 − q ) . (A.2.16) q k 1 k m 1 k m=1
The case with k = 2 is the (K-theoretic) S-function (2.2.24), and the case with k = 4 is used in the context of A0 quiver. See Sect. 7.5.1.
A.2.4 Partition Sum
Let λ be a (two-dimensional) partition. Then the summation over the partitions is given by Euler’s product formula:
1 q|λ| = . ( ; ) (A.2.17) λ q q ∞ 256 Appendix A: Special Functions
A similar result is available for the sum over the plane partitions (three-dimensional partitions) by the MacMahon function:
∞ 1 q|π| = . (A.2.18) (1 − qn)n π n=1
A.3 Elliptic Functions
A.3.1 Theta Function
The theta function with the elliptic nome p = e2πiτ ∈ C× is given by ⎛ ⎞ n −1 ⎝ z ⎠ θ(z; p) = (z; p)∞(z p; p)∞ = exp − (A.3.1) n(1 − pn) n∈Z=0 where (z; p)∞ is the p-shifted factorial (A.2.1). It obeys the reflection relation
θ(z−1; p) = (−z−1)θ(z; p). (A.3.2)
We remark that, since the q-shifted factorial is identified with the q-gamma func- tion (A.2.11), the relation (A.3.1)isaq-analog of the reflection formula of the gamma function discussed in Sect. A.1.1. In this sense, the theta function is interpreted as a q-analog of the sine function having zeros at
log z = Z + τZ . (A.3.3) 2πi An Identity
We start with Ramanujan’s identity (also known as 1ψ1 formula) for |b/a| < |z| < 1: ( , / , , / ; ) ( ; ) az p az p b z p ∞ a p n n = z = 1ψ1(a; b; z, p) (A.3.4) (z, b/az, b, p/a; p)∞ (b; p) n∈Z n where we denote the bilateral basic hypergeometric series by r ψs (a1,...,r ; b1,...,s ; z, q). We put b = ap, then we obtain
2 (az, p/az, p, p; p)∞ θ(az; p)(p; p) LHS = = ∞ (1 − a), (A.3.5a) (z, p/z, ap, p/a; p)∞ θ(z; p)θ(a; p) 1 − a RHS = zn , (A.3.5b) 1 − apn n∈Z Appendix A: Special Functions 257 which leads to the identity
θ(az; p)(p; p)2 zn ∞ = . (A.3.6) θ(z; p)θ(a; p) 1 − apn n∈Z
A.3.2 Elliptic Gamma Functions
We define the elliptic gamma function for |p|, |q| < 1: ⎛ ⎞ −1 m (z pq; p, q)∞ ⎝ z 1 ⎠ e(z; p, q) = = exp , (z; p, q)∞ m (1 − pm )(1 − qm ) n,m≥0 m∈Z=0 (A.3.7) which obeys the relation
(zp; p, q) (zq; p, q) e = θ(z; q), e = θ(z; p). (A.3.8) e(z; p, q) e(z; p, q)
In this case, the analog of the reflection formula (Sect. A.1.1)isgivenby
−1 e(z; p, q)e(pqz ; p, q) = 1 . (A.3.9)
We remark that the (inverse of) double sine function (A.1.11) is obtained in the β β β scaling limit of the elliptic gamma function with (z, p, q) = (e x , e 1 , e 2 ) and taking β → 0. Elliptic Double Gamma Function The elliptic analog of the double gamma function is given by
−1 , (z; q , q , q ) = (z; q , q , q )∞(z q q q ; q , q , q )∞ e 2 1 2 3 ⎛1 2 3 1 2 3 1 2 3 ⎞ zm 1 = exp ⎝− ⎠ , (A.3.10) ( − m )( − m )( − m ) m 1 q1 1 q2 1 q3 m∈Z=0 which obeys the relation
e,2(zq1; q1, q2, q3) = e(z; q2, q3), etc. . (A.3.11) e,2(z; q1, q2, q3)
We can similarly construct the elliptic analog of the multiple gamma functions, 258 Appendix A: Special Functions ⎛ ⎞ ∞ m ⎝ k+1 z 1 ⎠ , (z; q ,...,q + ) = exp (−1) , e k 1 k 1 ( − m ) ···( − m ) m 1 q1 1 qk+1 m∈Z=0 (A.3.12) which obeys a similar shift relation to (A.2.15). See [4, 5] for details.
A.3.3 Elliptic Analog of Polylogarithm
We define an elliptic analog of polylogarithm function:
n z 1 p→0 Li (z; p) = −−→ Li (z). (A.3.13) k nk 1 − pn k n∈Z=0
The first example is given by
p→0 Li1(z; p) =−log θ(z; p) −−→ Li1(z) =−log(1 − z). (A.3.14)
The elliptic gamma function has the asymptotic expansion in terms of the elliptic polylogarithm functions: ∞ n 1 1 e(z; p, e ) = exp − Li − (z; p) Bn = exp − (Li (z; p) + O()) , 2 n n! 2 n=0 (A.3.15) which is analogous to the expansion (A.2.7). References
1. L.D. Faddeev, Discrete Heisenberg–Weyl group and modular group. Lett. Math. Phys. 34, 249–254 (1995). arXiv:hep-th/9504111 [hep-th] 2. L.D. Faddeev, R.M. Kashaev, Quantum dilogarithm. Mod. Phys. Lett. A9, 427– 434 (1994). arXiv:hep-th/9310070 [hep-th] 3. N. Kurokawa, S.-Y. Koyama, Multiple sine functions. Forum Math. 15, 839– 876 (2003) 4. A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions. Adv. Math. 189(2), 247–267 (2004). arXiv:math/0306164 [math.QA] 5. [Nis01] M. Nishizawa, An elliptic analogue of the multiple gamma function. J. Phys. A 34, 7411–7421 (2001) Appendix B Combinatorial Calculus
B.1 Partition
The partition λ is a sequence of non-increasing non-negative integers:
λ = (λ ≥ λ ≥···≥ ) ∈ Z∞ . 1 2 0 ≥0 (B.1.1)
We denote the transposed partition of λ by λˇ. The size of the partition is defined as
∞ ∞ ˇ ˇ |λ|= λi = λi =|λ| . (B.1.2) i=1 i=1
For the partition λα, we define the arm and leg lengths for s = (s1, s2):
( ) = λ − ,( ) = λˇ − . aα s α,s1 s2 α s α,s2 s1 (B.1.3)
We remark that not necessarily s ∈ λα, so that (aα(s), α(s)) may be negative. Then the relative hook length is defined
( ) = ( ) + ( ) + = λ + λˇ − − + . hαβ s aα s β s 1 α,s1 β,s2 s1 s2 1 (B.1.4)
B.2 Instanton Calculus
We summarize the combinatorics calculus of the partition for the instanton partition function. Summation over the partition is expressed in the following two ways,
© The Editor(s) (if applicable) and The Author(s), under exclusive license 259 to Springer Nature Switzerland AG 2021 T. Kimura, Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, https://doi.org/10.1007/978-3-030-76190-5 260 Appendix B: Combinatorial Calculus
λˇ λ λ λˇ 1 s1 1 s2 = = . (B.2.1)
s∈λ s1=1 s2=1 s2=1 s1=1
B.2.1 U(n) Theory
We consider the instanton contribution to the Chern character of the bifundamental hypermultiplet (see Sect. 1.9):
bf, inst =−μ ∧ ∨ ∨ chT He:i→ j e chT Q chT Ki chT K j + μ ∨ + μ −1 ∨ e chT Ni chT K j eq chT Ki chT N j ni n j ν j,β =: μe [λi,α,λj,β ] (B.2.2) ν ,α α=1 β=1 i where we define − − −s +s −s +s [λ ,λ ]=−( − 1)( − 1) 1 1 2 2 α β 1 q1 1 q2 q1 q2
s∈λα s ∈λβ − − s −1 s −1 + s1 s2 + 1 2 q1 q2 q1 q2 (B.2.3)
s∈λα s ∈λβ
From this expression, we obtain a combinatorial formula (see, for example, [1]) ( ) ( ) [λ ,λ ]= β s −aα (s)−1 + −α (s)−1 aβ s . α β q1 q2 q1 q2 (B.2.4) s∈λα s∈λβ where the arm and leg lengths for each box s = (s1, s2) in the partition are defined in (B.1.3). We remark q [λα,λβ ] = [λβ ,λα] . (B.2.5) , −1, −1 q1 q2 q1 q2
The vector multiplet contribution has a similar expression
inst = ∧ ∨ ∨ − ∨ − −1 ∨ chT Vi chT Q chT Ki chT K j chT Ni chT K j q chT Ki chT N j ni νi,β =− [λi,α,λi,β ] . (B.2.6) ν ,α α,β=1 i Appendix B: Combinatorial Calculus 261
Proof of the formula (B.2.4) We prove the combinatorial formula (B.2.4). We partially perform the summation for the first term in (B.2.3),
λˇ λ α,1 β,1 λˇ −s +s −s +s β,s − −λα, s −1 − ( − −1)( − −1) 1 1 2 2 = ( − 2 ) s1 ( − s1 ) 2 1 q1 1 q2 q1 q2 1 q1 q1 1 q2 q2 s∈λα s ∈λ s =1 = β 1 s2 1 ⎡ ⎤ λˇ λ α,1 β,1 λˇ −s λˇ β,s 1 −λα, +s −1 − s −1 β,s − s −1 − −λα, s −1 = ⎣ 2 s1 2 − s1 2 + ( − 2 ) s1 2 + s1 ( − s1 ) 2 ⎦ . q1 q2 q1 q2 1 q1 q1 q2 q1 1 q2 q2 s =1 = 1 s2 1 (B.2.7)
The third and fourth terms in (B.2.7) are then given by
ˇ λˇ λ λˇ λ λβ, α,1 β,1 α,1 β,1 s2 λˇ β,s − s −1 −s +s −1 s −1 ( − 2 ) s1 2 = ( − ) 1 1 2 1 q1 q1 q2 1 q1 q1 q2 = = = = = s1 1 s2 1 s1 1 s2 1 s1 1 ˇ −λα, s −1 s −1 =− ( − 1 ) 1 2 , 1 q1 q1 q2 (B.2.8a)
s ∈λβ
ˇ ˇ λ λα,1 λβ,1 λα,1 λβ,1 α,s 1 − −λα, s −1 − − −s +s s1 ( − s1 ) 2 = s1 ( − 1) 2 2 q1 1 q2 q2 q1 1 q2 q2
s1=1 s =1 s1=1 s =1 s2=1 2 2 − λ − =− s1 ( − β,1 ) s2 . q1 1 q2 q2 (B.2.8b) s∈λα
Combining them together, (B.2.3) becomes
ˇ λα,1 λβ,1 λˇ − β,s s1 −λα, +s −1 − s −1 [λ ,λ ]= 2 s1 2 − s1 2 α β q1 q2 q1 q2 = = s1 1 s2 1 ˇ − λβ, −s −λα, +s −1 s −1 + s1 1 2 + 1 1 2 . q1 q2 q1 q2 (B.2.9)
s∈λα s ∈λβ
We divide it into the negative and positive parts
[λα,λβ ]= <0 [λα,λβ ]+ ≥0 [λα,λβ ] (B.2.10) q2 q2 where <0 consists of monomials with negative powers of q2, while ≥0 consists q2 q2 of positive ones. Let us focus on <0 with (B.2.9). q2
• For λβ, >λα, , the first term in (B.2.9) may contribute to the negative part <0 . 1 s1 q2 • For λβ, ≤ λα, , the first and third terms may contribute to <0 . 1 s1 q2 262 Appendix B: Combinatorial Calculus
In both cases, the negative part <0 is given by q2 λˇ β, − −λα, + − ( ) s2 s1 s1 s2 1 β s −aα (s)−1 <0 [λα,λβ ]= q q = q q . (B.2.11) q2 1 2 1 2 s∈λα s∈λα
We can similarly obtain the positive part ≥0 by utilizing the formula (B.2.5), q2 −α (s)−1 aβ (s) ≥0 [λα,λβ ]= q q . (B.2.12) q2 1 2 s∈λβ
This proves the formula (B.2.4).
B.2.2 U(n0|n1) Theory
For the supergroup theory, we consider the following contribution to the Chern char- acter (see Sect. 3.4):
bf,inst =−μ ∧ ∨ σ ∨ σ + μ σ ∨ σ chT He:i→ j,σ σ e chT Q chT Ki chT K j e chT Ni chT K j + μ −1 σ ∨ σ eq chT Ki chT N j
ni n j ν j,β σ σ =: μ [λ ,λ ] e σσ i,α j,β (B.2.13) ν ,α α=1 β=1 i where the diagonal factors are written using (B.2.3)as
00[λα,λβ ]=[λα,λβ ] ,11[λα,λβ ]=[λβ ,λα] . (B.2.14)
The vector multiplet contribution (3.4.22a) is given by
ni ni ν j,β σ σ inst =− [λ ,λ ] . chT Vi,σ σ σσ i,α i,β (B.2.15) ν ,α α=1 β=1 i
The off-diagonal factors are − − −s −s +1 −s −s +1 [λ ,λ ]=−( − 1)( − 1) 1 1 2 2 01 α β 1 q1 1 q2 q1 q2
s∈λα s ∈λβ − − −s −s + s1 s2 + 1 2 q1 q2 q1 q2 (B.2.16a)
s∈λα s ∈λβ Appendix B: Combinatorial Calculus 263 − − s +s −1 s +s −1 [λ ,λ ]=−( − 1)( − 1) 1 1 2 2 10 α β 1 q1 1 q2 q1 q2
s∈λα s ∈λβ − − s −1 s −1 + s1 1 s2 1 + 1 2 q1 q2 q1 q2 (B.2.16b)
s∈λα s ∈λβ
We remark that these off-diagonal factors are symmetric under λα ↔ λβ ,
01[λα,λβ ]=01[λβ ,λα] ,10[λα,λβ ]=10[λβ ,λα] , (B.2.17) and q 01[λα,λβ ] = 10[λα,λβ ] (B.2.18) , −1, −1 q1 q2 q1 q2
We apply a similar computation to (B.2.16a) as discussed in Sect. B.2.1. The first term in (B.2.16a) yields − − −s −s +1 −s −s +1 − ( − 1)( − 1) 1 1 2 2 1 q1 1 q2 q1 q2
s∈λα s ∈λβ ˇ λα,1 λβ,1 −λˇ β,s − −λα, −s =− ( − 2 ) s1 ( − s1 ) 2 1 q1 q1 1 q2 q2 = = s1 1 s2 1 ˇ λα,1 λβ,1 −λˇ − −λˇ β,s s1 −λα, −s − −s β,s − −s =− 2 s1 2 − s1 2 + ( − 2 ) s1 2 q1 q2 q1 q2 1 q1 q1 q2
s1=1 s =1 2
− −λα, −s + s1 ( − s1 ) 2 . q1 1 q2 q2 (B.2.19)
The third and fourth terms in (B.2.19)aregivenby
ˇ λα,1 λβ,1 −λˇ ˇ β,s − −s −λα, −s −s − ( − 2 ) s1 2 =− ( − 1 ) 1 2 , 1 q1 q1 q2 1 q1 q1 q2 (B.2.20a)
= = ∈λβ s1 1 s2 1 s ˇ λα,1 λβ,1 − −λα, −s − −λβ, − − s1 ( − s1 ) 2 =− s1 ( − 1 ) s2 . q1 1 q2 q2 q1 1 q2 q2 (B.2.20b)
= = ∈λα s1 1 s2 1 s
Hence we obtain λˇ α, λβ, 1 1 −λˇ − β,s s1 −λα, −s − −s [λ ,λ ]=− 2 s1 2 − s1 2 01 α β q1 q2 q1 q2 s =1 = 1 s2 1 − −λβ, − −λˇ α, −s −s + s1 1 s2 + 1 1 2 , q1 q2 q1 q2 (B.2.21)
s∈λα s ∈λβ 264 Appendix B: Combinatorial Calculus and similarly
ˇ λα,1 λβ,1 λˇ + − β,s s1 1 λα, +s −1 − s −1 [λ ,λ ]=− 2 s1 2 − s1 1 2 10 α β q1 q2 q1 q2 = = s1 1 s2 1 ˇ − λβ, +s −1 λα, +s −1 s −1 + s1 1 1 2 + 1 1 2 . q1 q2 q1 q2 (B.2.22)
s∈λα s ∈λβ
In contrast to the diagonal part 00(11), further simplification does not occur for these off-diagonal ones. Reference
1. H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces (American Mathematical Society, 1999) Appendix C Matrix Model
In this Chapter, we summarize the geometric and algebraic aspects of the matrix model, which exhibit similar perspectives to gauge theory discussed in this manuscript. See also the manuscripts [3, 5] for details on this topic.
C.1 Matrix Integral
The partition function of the matrix model (the path integral of zero-dimensional QFT) is given as an integral over the self-conjugate (real symmetric, complex Her- mitian, quarternion self-dual) matrix: − 1 ( ) Z = dHe tr V H (C.1.1) where ⎧ ⎨⎪H T (real symmetric) rk H = N , H = H † (complex Hermitian) , (C.1.2) ⎩⎪ H D (quaternion self-dual) with the coupling constant . We define the potential function
+ d 1 t V (x) = k xk , (C.1.3) k k=1
© The Editor(s) (if applicable) and The Author(s), under exclusive license 265 to Springer Nature Switzerland AG 2021 T. Kimura, Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, https://doi.org/10.1007/978-3-030-76190-5 266 Appendix C: Matrix Model which is a polynomial function of degree d + 1. 2 The matrix measure dH is given as a product of the Lebesgue measures of all real components of the matrix H, denoted (α) α = ,..., β − by Hij for 0 2 1:
N = (α) , dH dHii dHij (C.1.4) i=1 1≤i< j≤N α=0,...,2β−1 where the symmetry parameter β is given by
1 R : β = , C : β = 1 , H : β = 2 . (C.1.5) 2
C.1.1 Eigenvalue Integral Representation
The matrix measure and the potential term in the matrix integral (C.1.1) are invariant under the similarity transformation:
H −→ UHU† , U ∈ (O(N), U(N), Sp(N)) . (C.1.6)
Utilizing this symmetry, we choose the basis diagonalizing the matrix (gauge fixing),
† N H = UXU , X = diag(x1,...,xN ) ∈ R . (C.1.7)
Then, we can write the partition function (C.1.1) in terms of the eigenvalues:
N 1 dxi − β ( ) β Z = e V xi | (X)|2 N! 2π N i=1 N β 1 dxi − S(X) =: e 2 (C.1.8) N! 2π i=1 where we rescale the potential function V (x), and the Jacobian term is given as the Vandermonde determinant
N N (X) = (x j − xi ). (C.1.9) i< j
2 We should impose a certain condition for the coefficients (tk )k=1,...,d+1 for convergence of the matrix integral, which could be relaxed via complexification of the integration contour. See [3] for details. Appendix C: Matrix Model 267
We also define the effective action S(X) as
N N 2 S(X) = V (xi ) − 2 log |xi − x j | . (C.1.10) i=1 i< j
We discuss the properties of the matrix model mainly based on this eigenvalue integral representation of the partition function in the following.
C.2 Saddle Point Analysis
We are in particular interested in the asymptotic regime of the matrix model, which is called the ’t Hooft limit:
N −→ ∞ , −→ 0 , t : fixed , (C.2.1) where t is the ’t Hooft coupling defined as
t = N = O(1). (C.2.2)
In this limit, we can apply the saddle point analysis to the matrix model: A specific configuration dominates in the integral satisfying the saddle point equation,
∂ S(X) 0 = ∂xi N 1 = V (x ) − 22 =: V (x ) (C.2.3) i x − x eff i i=1 i j (i= j) for i = 1,...,N. We denote the effective potential by Veff(x), which consists of the one-body potential and the interaction with other eigenvalues,
N Veff(xi ) = V (xi ) − 2 log |xi − x j | . (C.2.4) i=1 (i= j)
In order to solve the saddle point equation, we define an auxiliary function, called the resolvent,3
3Precisely speaking, we should distinguish the resolvent and its average, W(x) =W(x) , whereas, in the ’t Hooft limit, the average of the observable is given by its on-shell value in general, O = 1 − 1 tr V (H) dHe O(H) ≈ O(H∗) with the solution to the saddle point equation denoted by H∗. Z 268 Appendix C: Matrix Model
N 1 1 x→∞ t W(x) = tr = −−−→ . (C.2.5) x − H x − x x i=1 i
We see that the saddle point equation (C.2.3) is equivalent to the differential equation for the resolvent,
W(x)2 + W (x) − V (x)W(x) + P(x) = 0 (C.2.6) where we define the polynomial function of degree d − 1,
N V (x) − V (x ) P(x) = i . (C.2.7) x − x i=1 i
Furthermore, in the semiclassical limit → 0, we may omit the derivative term in the Eq. (C.2.6), and thus the saddle point equation is reduced to the algebraic equation
W(x)2 − V (x)W(x) + P(x) = 0 , (C.2.8) which solves the resolvent
V (x) 1 W(x) = − V (x)2 − 4P(x). (C.2.9) 2 2 We remark that the sign of the square root is fixed to be consistent with the asymptotic behavior of the resolvent (C.2.5).
C.2.1 Eigenvalue Density Function
From the saddle point equation (C.2.8), we can construct the eigenvalue density function
1 1 N ρ(x) = tr δ(x − H) = δ(x − x ), (C.2.10) N N i i=1 satisfying the normalization condition dx ρ(x) = 1 . (C.2.11)
Since the delta function is described as Appendix C: Matrix Model 269
1 1 1 1 δ(z) =∓ Im := lim ∓ Im , (C.2.12) π z ± i0 →0 π z ± i the density function is given as the imaginary part of the resolvent,
1 ρ(x) = (W(x − i0) − W(x + i0)) 2πi ⎧ ⎪ n ⎨⎪ 1 |M(x)| − (x − x−)(x − x+)(x ∈ C) = π α α , (C.2.13) ⎪2 α= ⎩⎪ 1 0 (x ∈/ C) where we denote
n > ( ∈ C) 2 2 − + 0 x 4P(x) − V (x) =−M(x) (x − xα )(x − xα ) (C.2.14) < 0 (x ∈/ C) α=1 with
!n − + C = Cα , Cα = (xα , xα ). (C.2.15) a=1
We remark that, since deg V (x) = d,wehave1≤ n ≤ d and the degree of the polynomial function M(x) is deg M(x) = d − n, which is called the n-cut solution. In order to characterize the n-cut solution profile, we define the filling fraction
Nα εα = ∈ R+ ,α= 1,...,n (C.2.16) N where
Nα = dx ρ(x) = #{eigenvalues on the cut Cα} (C.2.17) Cα together with the normalization condition
n εα = 1 . (C.2.18) α=1
In fact, the filling fractions are the parameters characterizing the saddle point of the matrix integral, which are interpreted as proper coordinates of the moduli space associated with the matrix model. 270 Appendix C: Matrix Model
C.2.2 Functional Representation
Using the density function (C.2.10), the potential term in the effective action (C.1.10) is written as N t N V (x ) = dx δ(x − x ) V (x) = t dx ρ(x) V (x). (C.2.19) i N i i=1 i=1
Applying a similar argument to the interaction term, we obtain the functional repre- sentation of the effective action S[ρ(x)]=t dx ρ(x) V (x) − t2 − dxdx ρ(x)ρ(x ) log |x − x | x=x n + α εα − dx ρ(x) , (C.2.20) C α=1 α " − where we denote the principal value integral by dx, and (α)α=1,...,n are the Lagrange multipliers to impose the condition (C.2.17). Let us consider the saddle point analysis in this functional representation. We take the functional derivative of the effective action, δ [ρ( )] 1 S x −1 = V (x) − 2t − dx ρ(x ) log |x − x |−t α t δρ(x) −1 = Veff(x) − t α , x ∈ Cα , (C.2.21) where Veff(x) is the functional version of the effective potential given in (C.2.3). Thus, from the equation of motion for the effective action, the effective potential takes a constant value for each cut, δ [ρ( )] S x −1 = 0 =⇒ V (x) = t α , x ∈ Cα . (C.2.22) δρ(x) eff
Then, the (functional version of) saddle point equation (C.2.3) is obtained from the effective potential, ρ(x ) V (x) = V (x) − 2t − dx = 0 . (C.2.23) eff x − x
In this formalism, the resolvent takes a form of ρ(x ) W(x) = t dx . (C.2.24) x − x Appendix C: Matrix Model 271
Namely, the resolvent is given by the Hilbert transform of the density function ρ(x). Since the integrand of this expression has a singularity at x = x ∈ C, we define the regularized version of the resolvent with the principal value integral, ρ(x ) W reg(x) = t − dx (x ∈ C) x − x 1 = (W(x + i0) + W(x − i0)) = Re W(x ± i0). (C.2.25) 2 Recalling the imaginary part of the resolvent is given by the eigenvalue density function (C.2.13), we obtain the Kramers–Kronig relation,
1 W(x ± i0) = V (x) ∓ πiρ(x)(x ∈ C), (C.2.26) 2 ⇐⇒ ( ) =± π ρ( )(∈ C). Veff x 2 i x x (C.2.27)
C.3 Spectral Curve
The algebraic equation (C.2.8) defines the spectral curve of the matrix model, which describes the saddle point configuration of the matrix integral,
={(x, y) ∈ C × C | H(x, y) = 0} . (C.3.1)
The algebraic function H(x, y) is defined as
H(x, y) = y2 − V (x) y + P(x) (C.3.2) where we identify y = W(x), and we define the one-form and the symplectic two- form on the curve,
λ = ydx,ω= dλ = dy ∧ dx . (C.3.3)
We also use another expression of the curve via the symplectic transform, y → y + V (x)/2,4
4 This algebraic function is given as the characteristic polynomial of the Lax matrix, L(x) ∈ sl2, associated with the orthogonal polynomials,
H(x, y) = det(y − L(x)) (C.3.4) with 1 tr L(x) = 0 , det L(x) = P(x) − V (x)2 . (C.3.5) 4 See [3] for details. 272 Appendix C: Matrix Model
1 H(x, y) = y2 − V (x)2 + P(x). (C.3.6) 4 The degree of the spectral curve is (2, 2d). From this expression, the spectral curve is written in the form of the hyperelliptic curve
1 : y2 = P(x) − V (x)2 . (C.3.7) 4 In particular, the n-cut solution (C.2.13) gives rise to the spectral curve with genus g = n − 1.
C.3.1 Cycle Integrals
For the spectral curve of the matrix model with genus g = n − 1, we define the A-cycle and B-cycle as in Figs. 4.1 and 4.2. Namely, the A-cycle is a contour sur- rounding the cut Cα, so that the filling fraction is given as # 1 1 εα = dx ρ(x) = dx W(x). π (C.3.8) N Cα 2 it Aα
The B-cycle is the contour from the α-th cut to (α + 1)-st cut, then (α + 1)-st to α-th on the other sheet. Recalling (C.2.22), we obtain # α − α+ λ = 1 . (C.3.9) Bα t
Thus, the contour integrals of the one-form λ along the A and B-cycles are given by # 1 λ = t εα , (C.3.10a) 2πi #Aα 1 λ = 1 ( − ) . π π α α+1 (C.3.10b) 2 i Bα 2 it
We remark that, from the effective action (C.2.20), we have the relation
∂ S α = . (C.3.11) ∂εα
This implies the relation between the A and B-cycle integrals through the effective action, Appendix C: Matrix Model 273 # ∂ S ∂ S λ = t−1 − , ∂ε ∂ε (C.3.12) Bα α α+1 which is analogous to the Seiberg–Witten geometry discussed in Sect. 4.2. In order to obtain a closer expression to the Seiberg–Witten geometry, we take a linear combination,
ε¯α = εα − εα+1 ,α= 1,...,n − 1,, (C.3.13) which corresponds to the simple root of the Lie group SU(n). Similarly define the ¯ modified A-cycle, Aα = Aα − Aα+1, then we obtain # # 1 1 1 ∂ S λ = t ε¯α , λ = ,α= 1,...,n − 1 , (C.3.14) π ¯ π π ∂ε¯ 2 i Aα 2 i Bα 2 it α where the modified filling fraction and the effective action correspond to the Coulomb moduli and the prepotential of the Seiberg–Witten theory.
C.4 Quantum Geometry
Let us discuss how to quantize the spectral curve of the matrix model. Recall the saddle point equation at finite (C.2.6) is a Riccati-type differential equation for the resolvent. In order to linearize the differential equation, we write the resolvent as
ψ (x) d W(x) = = log ψ(x), (C.4.1) ψ(x) dx where ψ(x) is the characteristic polynomial, called the wave function in this context,
N ψ(x) = (x − xi ) = det(x − H). (C.4.2) N i=1
Then, we obtain the second order linear ODE for the wave function,
2ψ (x) − V (x)ψ (x) + P(x)ψ(x) = 0 . (C.4.3)
This ODE is also written in the operator form,5
H(xˆ, yˆ)ψ(x) = 0 , (C.4.4)
5Precisely speaking, we should properly impose the ordering of the operators since the operators (xˆ, yˆ) are noncommutative. 274 Appendix C: Matrix Model where H(x, y) is the algebraic function defined in (C.3.2), and the operator pair is given by
d xˆ ψ(x) = x ψ(x), yˆ ψ(x) = ψ(x), (C.4.5) dx obeying the relation $ % yˆ , xˆ = . (C.4.6)
This is the canonical commutation relation with respect to the symplectic two-form defined in (C.3.3). In this sense, this Schrödinger-type ODE is interpreted as quan- tization of the spectral curve (quantum curve), and ψ(x) plays a role of the wave function.
C.4.1 Baker–Akhiezer Function
In order to see the connection to the integrable hierarchy, we consider the expectation value of the characteristic polynomial,
ψ(x) =det(x − H) ∞ x N 1 x−k = dH exp − tr V (H) − tr H k Z(t) k k=1 Z(t +[x]) =: x N , (C.4.7) Z(t) where we explicitly show the t-dependence of the matrix integral, and we denote & ' [x]= x−1, x−2,... . (C.4.8)
From this point of view, we in practice consider the situation with d →∞.We remark that such a shift of the t-variable can be imposed by the vertex operator (Sect. C.5.4). In fact, the matrix integral Z(t) is identified with the N-soliton solution to the τ-function of the KP hierarchy, and the wave function is the corresponding Baker–Akhiezer function. See, for example, [2, 4] for more details.
C.4.2 Quantization of the Cycle
We can also see quantization of the cycle at finite . As seen from the definition of the resolvent (C.2.5), it has poles with the residue , so that the A-cycle integral Appendix C: Matrix Model 275 counts the number of eigenvalues on the cut Cα, # 1 λ = Z. π (C.4.9) 2 i Aα
This is the Bohr–Sommerfeld quantization condition with the quantum parameter .
C.5 Quantum Algebra
In this Section, we explore the algebraic structure of the matrix model. We see that the loop equation, which provides a relation for the correlation functions, is characterized using the infinite dimensional algebra.
C.5.1 Loop Equation
Let O(H) be a non-singular function of the matrix H (also called the observable). Then, as long as its expectation value O(H) is finite, the following identity holds: ( ) ∂ − 1 ( ) dH e tr V H O(H) = 0 , (C.5.1) ∂ H because the boundary value of the integrand is suppressed by the exponential term − 1 ( ) e tr V H . In the eigenvalue representation, we instead obtain the identity:
N ( ) ∂ − β ( ) β dX e tr V X (X)2 O(X) = 0 , (C.5.2) ∂x N k=1 k where we can interpret β as an arbitrary parameter.* O( ) = k = N k =: ( ) ( ) We consider the case with H tr H i=1 xi pk X , where pk X is the k-th power sum polynomial of X = (xi )i=1,...,N . In this case, the identity (C.5.2) gives rise to + ⎛ ⎞ , N k 1 N 2xk 0 = β ⎝ xk−1 − V (x )xk + i ⎠ β i i i x − x i=1 j(=i) i j k−1 β = β H m H k−m−1 +( − β) k H k−1 − H k V (H) . tr tr 1 tr tr (C.5.3) m=0
Then, the identity with 276 Appendix C: Matrix Model
∞ 1 O(H) = tr = x−k−1 tr H k , (C.5.4) x − H k=0 is obtained from the relation (C.5.3) by multiplying the factor x−k−1 and summing over k = 0,...,∞, + , + , - . 1 2 1 2 β V (H) 0 = β tr + (1 − β) tr − tr x − H x − H x − H + , + , 1 2 1 2 = β tr + (1 − β) tr x − H x − H - . - . β 1 β V (x) − V (H) − V (x) tr + tr , (C.5.5) x − H x − H which is called the loop equation. Inserting more generic operator O(H), we obtain various relations among the correlation functions.
C.5.2 Operator Formalism
We then discuss the underlying algebraic structure of the loop equation obtained above. For this purpose, we introduce the operator formalism as follows. Recalling the potential function takes a form of (C.1.3), one may express the matrix moment as 1 − β ( ) ∂ tr H n = dHe tr V H tr H n =−n log Z , (C.5.6) Z β ∂tn which implies the correspondence ∂ tr H n ⇐⇒ − n (C.5.7) β ∂tn
∀ in the operator formalism. In order to apply this correspondence for n ∈ Z>0,we take d →∞for the moment. As in Sect. 6.1, we introduce the Fock space F = C[[tn,∂n]] | 0 with the vac- uum state ∂n | 0 = 0, generated by the Heisenberg algebra H = (tn,∂n)n≥1 with the algebraic relation [∂n, tn ]=δn,n . For the latter purpose, we introduce another set of the oscillators, / 0 ∂ β 2 −1 an = n , a−n = tn (C.5.8) β ∂tn 2 Appendix C: Matrix Model 277 obeying the algebraic relation
[an, am ] = n δn+m,0 . (C.5.9)
With these operators, the loop equation (C.5.3) is written as follows:
Lk−1 Z = 0 , k ≥ 0 , (C.5.10) which is called the Virasoro constraint. The operators (Lk−1)k≥0 are given in terms of the a-modes,
k−1 ∞ ( ) 1 1 − L − = a a − − + a− a + − + √ β − β 1 ka − k 1 2 n k n 1 n n k 1 k 1 n=0 n=1 2 ( ) 1 1 − = : a a − − :+√ β − β 1 ka − , (C.5.11) 2 n k n 1 k 1 n∈Z 2 which obeys the algebraic relation for the Virasoro algebra,
[Ln, Ln ] = (n − n )Ln+n . (C.5.12)
We remark that the full Virasoro algebra (Ln)n∈Z obeys the relation
c 2 [L , L ] = (n − n )L + + n(n − 1), (C.5.13) n n n n 12 where c is called the central charge. The current matrix model construction (C.5.12) does not provide the central term, because it appears from the commutation relation between the positive and negative modes, Ln and L−n; There appear only a half of the generators from the matrix model. Nevertheless, we can formally define the negative generators from (C.5.11) to realize the full Virasoro algebra with the central charge
( ) 2 1 (β = 1) c = 1 − 6 β − β−1 = . (C.5.14) −2 (β = 2±1)
C.5.3 Gauge Theory Parameter
The expression of the central charge in terms of the symmetry parameter β implies the correspondence to the -background parameter [1], & ' −1 1 (1,2) = , −β ⇐⇒ (,β) = 1, − . (C.5.15) 2 278 Appendix C: Matrix Model
Under this identification, exchanging 1 ↔ 2 implies the following symmetry on the matrix model parameters: & ' −1 −1 1 ←→ 2 ⇐⇒ (,β) ←→ −β ,β . (C.5.16)
In fact, from the expression (C.1.8), we obtain the effective action in the asymptotic limit
2 →0 − Z = Z −−→ S(α,α), β log 1 2 log (C.5.17) which is analogous to the asymptotic behavior of the partition function discussed in D Sect. 5.3, and the effective action S(α,α) plays a role of the prepotential F(aα, aα ) in gauge theory. See also Sect. C.3.1.
C.5.4 Vertex Operators
From the a-modes (C.5.8), we define the current operator, which is identified with the (derivative of) effective potential through the identification (C.5.7), −n−1 J(x) = an x n∈Z 0 0 1 β β =− 2β tr + −1 V (x) = −1 V (x). (C.5.18) x − H 2 2 eff
Similarly, we define the free boson operator a φ(x) =− n x−n + a log x +˜a n 0 0 n∈Z= 0 0 0 β β =− 2β tr log(x − H) + −1 V (x) +˜a = −1 V (x) +˜a 2 0 2 eff 0 (C.5.19) with the zero mode
[an, a˜0]=δn,0 . (C.5.20)
The free boson and the current operator have the relation
J(x) = ∂x φ(x). (C.5.21) Appendix C: Matrix Model 279
We also define the vertex operator with the free boson,
αφ(x) Vα(x) =:e : (C.5.22) √ where α is called the momentum parameter. Put α =∓ β/2, then the vertex operator gives rise to the characteristic polynomial in the matrix model ( ) β ±1 √ − ( ) β ( ) = 2 V x ( − ) × V∓ β/2 x e det x H const. (C.5.23)
Generating Current Let us consider the generating current of the Virasoro generators, which is called the energy-momentum tensor, and also the stress tensor,
L T (x) = n . (C.5.24) xn+2 n∈Z
In fact, this generating current has a realization in terms of the current operator,
1 T (x) = : JJ :(x) − ρ J (x) (C.5.25) 2 where 1 ρ = √ ( β − β−1). (C.5.26) 2
We remark ρ = 0forβ = 1. Since the current operator has the expression (C.5.18), the generating current is given by 1 2 β 1 β T (x) = β tr − V (x) tr + V (x)2 x − H x − H 42 1 2 1 − β + (1 − β)tr + V (x). (C.5.27) x − H 2
The algebraic relation (C.5.13) is equivalent to the OPE between the generating currents,
c/2 2 T (x ) T (x ) T (x)T (x ) = + + +··· . (C.5.28) (x − x )4 (x − x )2 x − x 280 Appendix C: Matrix Model
C.5.5 Z-State
The OPE between the generating current and the vertex operator is given by α 1 ∂ T (x)Vα(x ) = + Vα(x ) +··· (C.5.29) (x − x )2 x − x ∂x where the parameter & ' 1 2 2 α = (α + ρ) − ρ (C.5.30) 2 √ = is called the conformal weight. We define the screening current with − 2β 1,
( ) = √ ( ). S x V− 2β x (C.5.31)
Then, the OPE becomes ∂ 1 T (x)S(x ) = S(x ) +··· , (C.5.32) ∂x x − x which is a total derivative, so that the integral of the screening current commutes with the generating current $ % T (x), S = T (x), dx S(x ) = 0 . (C.5.33)
We define the screening charge as the integral of the screening current, S = dx S(x). (C.5.34)
Then, the Z-state, obtained from the operator analog of the partition function through the operator/state correspondence, is constructed from the screening charge as follows6: 1 N N N | Z = SN | 0 = dx (x − x )2β : S(x ) : | 0 N! i i j i i=1 i< j i=1 N 1 − β ( ) β = dx e V xi (X)2 | 0 . (C.5.35) N! i N i=1
6 Precisely speaking, we should take care of the ordering of the integral variables, x1 < ···< xN . Here we relax this condition via the analytic continuation, and multiply the factor 1/N!. Appendix C: Matrix Model 281
In order to obtain the partition function, which takes a value in C, we define the dual state & √ ' −1 d + 1 | β/2 tn (n ∈ (1,...,d + 1)) d + 1 | an = (C.5.36) 0 (n ∈/ (1,...,d + 1))
having the same charge as the Z-state | Z . We remark that the eigenvalue of the oscillator an (namely, the parameter tn) is now a c-number. Then, the partition function is given as a chiral correlator of the screening charges,
N 1 − β ( ) β Z =d + 1 | Z =d + 1 | SN | 0 = dx e V xi (X)2 , N! i N i=1 (C.5.37)
where the potential function is now a polynomial of degree d + 1. Virasoro Constraint Revisited Let us discuss the Virasoro constraint from the Z-state perspective. Applying the Virasoro generating current to the vacuum, we have −n−2 T (x) | 0 = Ln | 0 x . (C.5.38) n∈Z
Regularity at x → 0 implies
Ln | 0 = 0 , n ≥−1 . (C.5.39)
Since the screening charge commutes with the generating current, the same argu- ment is applied to obtain the Virasoro constraint (C.5.10). Actually, compared to the expression of the generating current (C.5.27), the loop equation (C.5.5) implies the negative power modes vanish in the generating current. References 1. L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four- dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219 [hep-th] 2. O. Babelon, D. Bernard, M. Talon, Introduction to Classical Integrable Systems. Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2003) 3. B. Eynard, T. Kimura, S. Ribault, Random matrices. arXiv:1510.04430 [math-ph] 4. M. Jimbo, T. Miwa, Solitons and infinite dimensional lie algebras. Publ. Res. Inst. Math. Sci. Kyoto 19, 943–1001 (1983) 5. M. Mariño, Instantons and Large N (Cambridge University Press, 2015) Index
A dual—, 104 Adams operation, 32, 60, 74 ADHM construction, 8, 63 ALE space, 63 D supergroup, 86 Delta function, 268 A-function, 153, 154 multiplicative—, 44 AGT relation, xi, 197, 241 D-module, 174 ALE space, 63, 121 Doubling trick, 238 A-operator, 203 Dual Dirac operator, 10, 87 —charge, 233 elliptic—, 247 fractional—, 217 E (A)SD YM Effective twisted superpotential, 160, 171 —connection, 5, 10, 87 Equivariant —equation, 5, 8, 50 —action, 19, 26, 42, 51, 68 —cohomology, 15, 23 —form, 16, 23 B —index, 29, 52, 74, 91 Baker–Akhiezer function, 203, 274 —localization, 17, 25, 67, 89 Baker–Campbell–Hausdorff formula, 194 Equivariantly compact, 15, 19 Bethe equation, 178, 229 Bianchi identity, 5 BPS/CFT correspondence, 197 BPS spectrum, 104, 111 F Framing, 9 —bundle, 30, 50, 89 C Fugacity (instanton), 6, 26, 53, 106 Cartan matrix classical—, 61 full—, 60 G half—, 60 Gamma function, 251 inverse of—, 198 q—, 254 symmetrized—, 73 elliptic—, 257 Coulomb moduli, 102, 107, 109, 117, 167, multiple—, 251 172, 181, 273 Gauge origami, 59, 70, 170
© The Editor(s) (if applicable) and The Author(s), under exclusive license 283 to Springer Nature Switzerland AG 2021 T. Kimura, Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, https://doi.org/10.1007/978-3-030-76190-5 284 Index
H Prepotential, 104, 158, 172, 190, 273 Holomorphic deformation, 190
Q I q-character, 160, 163, 173, 175, 176, 227 Instanton bundle, 30, 50, 89 Q-function, 173, 179 iWeyl reflection, 153, 204, 218 qq-character, 154, 171, 234 A1, 161, 211, 248 A2, 165, 212, 249 L A p, 214 Lax matrix, 113–115, 174, 177, 229 B3, 222 LMNS formula, 25, 42 BC2, 220 fractional quiver gauge theory, 75 C3, 224 gauge origami, 170 D4, 214 quiver gauge theory, 55, 232 G2, 226
supergroup, 96 A0, 166, 231 —with flavor, 28 q-shifted factorial, 253 Loop equation, 276 Quantum curve, 174–177, 229, 274 Quantum dilogarithm, 254 Quiver, 49 M fractional—, 70 Mathai–Quillen formalism, 22 fractional—W-algebra, 216 Matter bundle, 35, 50 —variety, 61, 65, 171, 231 McKay correspondence, 63 —W-algebra, 209 Moduli space of vacua Coulomb branch, 102, 104, 123, 131 Higgs branch, 102, 123 R Multiple sine function, 252 Radial ordering, 193, 200, 239 Resolvent, 267, 270, 273 Root of Higgs branch, 40, 123, 179 N Normal ordering, 193 S Saddle point analysis, 158, 178, 229, 267 O Screening current, 193, 280 Operator Product Expansion (OPE), 193 elliptic—, 238 A and A, 204 fractional—, 216 A and S, 205 S-duality, 133, 135 A and V, 204, 247 Seiberg–Witten curve A and Y, 204 A2 quiver, 114 S and S, 195, 238 A3 quiver, 114 S and V, 198, 245 5d theory, 132 S and Y, 202, 217, 246 6d theory, 137 T and T (A1), 212 SQCD, 111 T and T (A2), 213 SU(2), 108 T and T (BC2), 220 SU(n), 110 T and T (elliptic A1), 248 supergroup, 116 T and T (elliptic A2), 249 S-function, 25, 43, 58, 76 Operator-state correspondence, 192, 239 Stability condition, 14, 21, 62, 88
P T Pit condition, 40 Taub–NUT space, 121 Potential term, 190, 195, 265 Theta function, 256 Index 285
’t Hooft limit, 267 Virasoro constraint, 277 Time variable, 190, 240 V-operator, 198 T-operator, see qq-character elliptic—, 244 Topological twist, 7 TQ-relation Toda chain, 173 Y XXX spin chain, 175, 180 Yang–Mills action, 4 T-system, 163 supergroup—, 81 Y-function, 148 Y-operator, 201 U elliptic—, 245 Uhlenbeck compactification, 13 fractional—, 217
V Vacuum expectation value (vev), 102 Z Vandermonde determinant, 266 Z-state, 192, 280