On the physical interpretation of the Riemann zeta function, the Rigid Surface Operators in Gauge Theory, the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula, p-adic strings, zeta strings and p-adic cosmology and mathematical connections with some sectors of String Theory and Number Theory.
Michele Nardelli1,2
1Dipartimento di Scienze della Terra
Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy
2Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”
Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy
Abstract
This paper is a review of some interesting results that has been obtained in the study of the physical interpretation of the Riemann zeta function as a FZZT Brane Partition Function associated with a matrix/gravity correspondence and some aspects of the Rigid Surface Operators in Gauge Theory. Furthermore, we describe the mathematical connections with some sectors of String Theory (p-adic and adelic strings, p-adic cosmology) and Number Theory.
In the Section 1 we have described various mathematical aspects of the Riemann Hypothesis, matrix/gravity correspondence and master matrix for FZZT brane partition functions. In the Section 2, we have described some mathematical aspects of the rigid surface operators in gauge theory and some mathematical connections with various sectors of Number Theory, principally with the Ramanujan’s modular equations (thence, prime numbers, prime natural numbers, Fibonacci’s numbers, partitions of numbers, Euler’s functions, etc…) and various numbers and equations related to the Lie Groups. In the Section 3, we have described some very recent mathematical results concerning the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula (connected with the Selberg zeta function), hence, we have obtained some new connections applying these results to the adelic strings and zeta strings. In the Section 4 we have described some equations concerning p-adic strings, p-adic and adelic zeta functions, zeta strings and p-adic cosmology (with regard the p-adic cosmology, some equations concerning a general class of cosmological models driven by a nonlocal scalar field inspired by string field theories). In conclusion, in the Section 5, we have showed various and interesting mathematical connections between some equations concerning the Section 1, 3 and 4.
1. On some equations concerning the physical interpretation of the Riemann zeta function as a FZZT Brane Partition Function associated with a matrix/gravity correspondence and the master matrix of the (2,1) minimal and (3,1) minimal matrix model. [1] [2] [3]
If one can find a special infinite Hermitian matrix M such that: 0
Ξ
( )
z =det(
M0−zI)
, (1.1) where( )
− − + Γ + = Ξ − − 8 1 2 4 1 2 2 1 z 1/4 /2 z2 iz zζ
π
π
iz , (1.2)then the Riemann hypothesis would be true. This is because this function can be written in product form as:
( )
∏
+ − = Ξ n n iz zρ
2 / 1 1 2 1 . (1.3)Thence, we can rewritten the eq. (1.2) also in the following form:
∏
+ − n n izρ
2 / 1 1 2 1 − − + Γ + = − − 8 1 2 4 1 2 2 1 2 2 / 4 / 1 z z izπ
π
izζ
, (1.3b)The eigenvalues of the Hermitian matrix M are denoted by 0
λ
n and are related to the Riemann zeros viaρ
n =iλ
n+1/2. Then the product becomes:
( )
∏
∏
− − = + + − = Ξ n n n n n i z i iz z 2 / 2 1 2 / 1 2 / 1 1 2 1λ
λ
λ
. (1.4)This vanishes at the values
λ
n just as the formal determinant expression. Theλ
n are real if the matrix M is Hermitian and thus the Riemann Hypothesis would be true. 0For a general matrix model with potential V
( )
M the master matrix can be written: M S TS S a t a S n n n + = =∑
∞ = + − − 0 1 1 0 (1.5)where the similarity transformation S is defined so that M0 is Hermitian and the operators a,a+
obey
[ ]
a,a+ =I. One can expand the master matrix as a function of the Hermitian operator+ + =a a xˆ as:
( )
ˆ ˆ ˆ2 ... 2 1 0 x =gx+g x + M (1.6)One can also define an associated complex function:
( )
∑
∞ = + = 0 0 1 n n ny t y y M , (1.7)as well as a conjugate matrix P that satisfies: 0
[
P0,M0]
=I. (1.8)The Master matrix can be determined from the equation:
(
V'(
M0( )
xˆ)
+2P0)
0 =0. (1.9)Here 0 is the vacuum state annihilated by a. The master matrix is closely connected with the resolvent R
( )
z and eigenvalue densityρ
( )
x through:
( )
∫
( )
=−∫
(
−( )
)
− = − = C w M z i dw x z x dx M z Tr z R 0 0 log 2 1π
ρ
. (1.10)The associated function M0
( )
y obeys the relation:R
(
M0( )
y)
=M0(
R( )
y)
= y. (1.11)The function yM0
( )
y is the generating functional of connected Green functions for the generalizedmatrix model.
One observable of matrix models is the exponentiated macroscopic loop or FZZT brane partition function. This is given by:
B
( )
z =det(
M −zI)
. (1.12)This is the characteristic polynomial associated with the matrix M . It’s argument z can be complex. In the context of the Riemann zeta function
ζ
( )
s the variable is related to the usual argument of the zeta function by2 1 + =iz
s . Another observable is the macroscopic loop which is the transform of the Wheeler-DeWitt wave function defined on the gravity side of the correspondence
( )
(
)
(
( ))
+ = − − = ∞∫
− + → ε ε logε
lim log 0 M zI e Tr d zI M Tr z W l l l , (1.13)
( )
( )
− = ∂ ∂ = zI M Tr z z W z R 1 . (1.14)Thence, from the eqs. (1.10) and (1.13), we obtain:
( )
∫
(
( ))
∫
(
( )
)
− = − − = + ∂ ∂ = ∞ − + → C M zI zI M Tr w M z dw i e Tr d z z R log 1 2 1 log lim 0 0ε
π
ε ε l l l . (1.14b)If a special master matrix M can be found then expectation values such as 0
B
( )
z = det(
M −zI)
=∫
DMdet(
M −zI)
e−V( )M =det(
M0−zI)
(1.15)reduce to evaluating the observable at M . In the context of the 0 Ξ
( )
z function the desired relationis of the form:
Ξ
( )
z =det(
M0−zI)
= B( )
z = det(
M −zI)
=∫
DMdet(
M −zI)
e−V( )M . (1.16) Some matrix potentials that have been considered areV
( )
M =Tr( )
M2 (1.17)which describes 2d topological gravity or the (2,1) minimal string theory. A quartic potential: V
( )
M =Tr(
−M2 +gM4)
, (1.18)is used to describe minimal superstring theory. A more complicated matrix potentials is
( )
(
(
)
)
∑
( )
∞ = = − + − = 2 1 log m m M Tr m M I M Tr M V , (1.19)which defines the Penner matrix model and is used to compute the Euler characteristic of the moduli space of Riemann surfaces. Another matrix model that has been introduced is the Liouville matrix model with potential given by:
V
( )
M =Tr(
α
M +µ
eM)
, (1.20) with cosmological constant µ so that:e−V( )M =e−αTrMe−µTreM. (1.21) In this section we will encounter the matrix potential determined by:
( )
∑
( )
∞ = − − − = 1 2 2 2 4 2 2 3 q e Tr q TrM TrM M U M e e q e q eπ
π
π . (1.22)The partition function for this matrix model can be seen as a superposition of partition functions of Liouville matrix models with cosmological constants of the form:
µ 2π
q
= , (1.23) for integer q .
Now we describe the origin of this particular matrix model and it’s relation to the zeta function. To see how the matrix potential (1.22) arises it is helpful consider how the coefficients of the characteristic polynomial observable B
( )
z can be determined by expanding as a series in z . If the function Ξ( )
z is interpreted as a characteristic polynomial then one can obtain these coefficients from the expansion:
( )
( )
( )
∑
∞ = − = Ξ 0 2 2 ! 2 1 n n n n z n a z , (1.24) where∫
( )
∞ − = 1 2 4 / 1 2 log 2 1 4 n n d f a l l l l , (1.25) and( )
∑
∞ = − − = 1 2 / 1 2 2 4 2 2 3 q q e q q f lπ
lπ
l πl. (1.26)Inserting the coefficients a2n into Ξ
( )
z and summing over n we can represent Ξ( )
z as an integral transform:( )
∫
( )∑
∫
( )( )
∞ ∞ = ∞ + − + = − = Ξ 1 1 1 2 / 1 2 / 2 / 1 2 2 2 4 2 / 2 / 1 4 2 3 4 2 q iz q iz f d e q q d z l l l l l l l l l lπ
π
πl . (1.27)Defining the variable φ by l=eφ we have:
[ ]
∫
∑
∞ = − − = Ξ 1 2 2 4 2 2 2 3 k e k iz e e k e k e d z φ φ π φ π φ π φ , (1.28)which is a well known integral expression for the function Ξ
( )
z . For the simple potential( )
( )
2M Tr M
V = the exponentiated macroscopic loop observable (FZZT brane) can be computed. It is given by the Airy function:
( )
=∫
(
−)
−( )
=∫
+ 3 1 3 2 det M zI e TrM dφeizφ iφ DM z Ai . (1.29)Because this function is associated with an Hermitian matrix model it’s zeros are real. This is the analog of the Riemann hypothesis for V
( )
M =Tr( )
M2 . The similarity between the integral representations of (1.28) and (1.29) suggest an analogy between the Airy and zeta functions.The integral representation of the Airy function has a matrix integral generalization. The matrix potential is defined from:
( )
( )
3 3 1 Φ Φ − = i Tr U e e . (1.30)The matrix generalized Airy function is given by:
( )
=∫
Φ iTr( )ZΦ −U( )Φ e e d Z Ai . (1.31)Thence, from the eqs. (1.30) and (1.31), we can write also:
( )
=∫
Φ iTr( )ZΦ e d Z Ai( )
3 3 1 Φ Tr i e . (1.31b)In the above Φ and Z are n×n matrices. The interpretation of this matrix integral is that it describes n FZZT brines. The matrix Φ in the Kontsevich integrand is an effective degree of freedom describing open strings stretched between n FZZT branes. One can try to interpret the integrand of the Ξ
( )
z function in a similar manner. In that case the analog of the potential defined by: ( )∑
∞ = − Φ Φ Φ − Φ − = 1 2 2 4 2 2 2 3 k Tre k Tr Tr U e e k e k e π π π , (1.32)and the analog of the matrix integral describing n FZZT branes is:
[ ]
∫
( )∑
∞ = − Φ Φ Φ Φ − Φ = Ξ 1 2 2 4 2 2 2 3 k Tre k Tr Tr Z iTr e e k e k e D Z π π π . (1.33)The Airy function is the FZZT partition function for the (2,1) minimal matrix model. The FZZT partition function for the generalized
( )
p,1 minimal matrix model with parameters sk is given by:( )
=∫
( ) ∑ ( ) − = + + + + + − 2 1 1 1 1 1 1 1 2 1 pk k k p i k s i p iz e d z B φ φ φφ
π
. (1.34)Unlike the (2,1) matrix model the definition of the generalized
( )
p,1 matrix model requires a two matrix integral of the form:( )
( )
=∫
− ( ( + )− ) AM I M V g p g DMDAe Z 1 1 , . (1.35)Comparison with the integral representation of the Ξ
( )
z function shows that a generalized matrix model for large p can be constructed as an approximation. One writes:
∑
( )
+∑
=−( )
+ ∞ = − + + + − = − 2 1 1 1 1 2 2 4 2 1 1 1 1 2 3 log 2 kp k k p k e k i k s i p e e k e k π φ φ π φ φ π φ . (1.36)In the above formula the function on the left is expanded to order p+1 in the variable φ. We denote this terminated expansion by Ξp
( )
z . Another way to compute the coefficients sk is to differentiate the left hand side and set:( ) 0 1 2 2 4 2 1 2 2 3 log ! = ∞ = − + − − ∂ = φ
∑
φ φ π φ φ π π k e k k k k k e k e e k i s . (1.37)From the integral representation one has:
QΞp
( )
z =zΞp( )
z , PΞp( )
z =−∂zΞp( )
z , (1.38) where: + =∑
− = 1 0 p k k k p P s P Q . (1.39)Inserting this operator into the above equation one has the generalization of the Airy equation given by: P s P p
( )
z z p( )
z p k k k p Ξ = Ξ +∑
− = 1 0 . (1.40)To recover the equation for the full Ξ
( )
z function one has to take p to infinity which agrees with the fact that the zeta function does not obey a finite order differential equation. Note that z and φ are in some sense canonically conjugate. Denote the Fourier transform of the Ξ( )
z function as( )
pΞ~ then:
Ξ
( )
z =∫
dφeiφzΞ~( )
φ . (1.41) The generalized Airy equation then becomes in Fourier space:
φ
φ
p( )
φ
p( )
φ
p k k k p Q s Ξ = Ξ +∑
− = ~ ~ 1 0 . (1.42)This can be written:
(
U'( )
φ −Q) ( )
Ξ~φ =0, (1.43) where: ( )∑
∞ = − − − = 1 2 2 4 2 2 2 3 k e k U e e k e k e φ π φ π φ π φ . (1.44)Equation (1.43) is very similar to the equation for the master matrix. Indeed if we set: φ =M0
( )
y , z=P0( )
y , (1.45)we see that y can be thought of as coordinates of a parametrization of the Riemann surface Mp,1
which is determined from the φ and z constraint U'
( )
φ
−z=0. If we make these variables into operators through:this classical surface is turned into a quantum Riemann surface similar to those studied using noncommutative geometry. Once one has obtained the coefficients sk one can define matrix potential associated with a finite N theory as:
( )
( )
( )
+ =∑
− = ∞ → 2 1 lim p k k k p p Tr V M sV M M V , (1.47) where:( )
∑
(
)
= − = p j j k M I j M V 1 1 . (1.48)Thence, we can write also:
( )
( )
(
)
− + =∑ ∑
− = = ∞ → 2 1 1 1 lim p k p j j k p p Tr V M s j M I M V . (1.48b)A set of orthogonal polynomials with this matrix potential through the integral equation:
( )
( ) ( ) dy y e i n z B n zy y V s y V n p k k k p p 1 2 1 1 lim 1 2 ! 2 1 + + + + + −∫
∑ = − = ∞ → π . (1.49)Or equivalently though the generating function definition:
( ) ( )
( )
∑
∞ = + + + + − = ∑− = ∞ → 0 2 1 1 lim ! 2 1 n n n zy y V s y V n y z B e p k k k p p . (1.50)These are the generalizations of the integral and generating function definitions of the Hermite polynomials associated with the (2,1) minimal model.
Most of this analysis has centred on the matrix side of the matrix/gravity correspondence. The gravity side is related through an integral transform. For example the macroscopic loop observable associated with the Riemann zeta function is given by:
(
)
∫
( )
∞ − − = + 0 2 / 1 2 / 1 logζ
iz l iz W ldl. (1.51)In terms of the λn this observable takes the form:
( )
∑
(
)
( )
− − − = n n W l l l l l l l l log 1 1 log log cos 2 log 1 2 2 / 1 λ . (1.52)Thence, the eq. (1.51) can be written also:
(
)
∫
∞ − − = + 0 2 / 1 2 / 1 logζ
iz l iz(
)
( )
l l l l l l l l n d n∑
− − − log 1 1 log log cos 2 log 1 2 2 / 1 λ . (1.52b)The indefinite integral of this Wheeler-DeWitt wave function is connected to the prime numbers p through:
∫
( )
∑
∑
+ = < ≤ l l l l 2 1 1 2 1 ' ' x pn n pn n d W . (1.53)The FZZT brane partition function can also be represented by prime numbers as:
(
+)
=∑∑
− ( + ) p n iz n p n iz 1/2 1 1/2 logζ . (1.54)Thence, the eq. (1.51) can be written also:
∑∑
− ( + ) p n iz n p n 2 / 1 1∫
∞ − − = 0 2 / 1 iz l(
)
( )
l l l l l l l l n d n∑
− − − log 1 1 log log cos 2 log 1 2 2 / 1 λ . (1.54b)Both of the above formulas follow from the Euler product formula of the zeta function. Much of the physical intuition about the meaning of the FZZT brane and the Wheeler-DeWitt wave function occurs on the gravity side of the correspondence. Thus the connection of Number Theory and Gravity in this context is quite intriguing.
The (2,1) minimal model is defined by the partition function:
∫
dMdPe−V( ) (M +TrPM), (1.55) with:( )
1Tr( )
M2 g M V = , (1.56)and g is the coupling constant. We define a master matrix associated with the model as a matrix whose characteristic polynomial is equal to the matrix integral:
∫
dMdPdet(
M −zI)
e−V( ) (M +TrPM), (1.57) which is the FZZT partition function.The master matrix for the (2,1) minimal model is given by:
− − − = 0 1 0 0 1 0 2 0 0 0 2 0 1 0 0 1 0 2 N N N g M L M O O O M O L . (1.58)
0 7 0 0 0 0 0 0 7 0 6 0 0 0 0 0 0 6 0 5 0 0 0 0 0 0 5 0 4 0 0 0 0 0 0 4 0 3 0 0 0 0 0 0 3 0 2 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 1 0 2 / g . (1.59)
The FZZT partition function for the (2,1) minimal model is:
g HN
(
z g)
N / 4 2 / . (1.60)This coincides with the characteristic polynomial of the master matrix. For the case N =8 this is:
4 3 2 2 4 14 6 8 2 105 2 105 19 105 z gz z g z g g − + − + . (1.61)
The master matrix (1.58) agrees with the master matrix of the Gaussian matrix model which is has the same partition function as the (2,1) minimal model after integration over P .
Because the master matrix is manifestly Hermitian it’s eigenvalues are real. The large N limit of FZZT partition function corresponds to:
N
g→ 1 , z
N
z→−1+ 11/3 , (1.62)
and leads to the Airy function Ai
( )
z . This function is given by the contour integral: Φ( )
=∫
− 0 3/3 2 C z e i d z ϕ ϕ πϕ , (1.63)with contour C starting at infinity with argument 0 −π/3 and ending at infinity with argument
3 /
π . It has the series expansion:
( )
∑
(
(
)
)
(
(
)
)
( )
∞ = + + Γ = 0 3 / 1 3 / 2 sin 2 1 /3 3 ! 3 / 1 3 1 n n z n n n z Ai π π . (1.64)The Airy function obeys the differential equation:
Ai''
( )
z −zAi( )
z =0. (1.65)The Airy function has all it’s zeros on the real axis and this is a manifestation of the Hermitian nature of the master matrix in (1.58).
( )
( )
+ = 2 3 3 1 2 3 1 M Tr M Tr g M V . (1.66)The master matrix of the (3,1) minimal model is the matrix M with nonzero components: Mi,j =
( )(
i−1 i−2)
δi,j+2 +3( )
i−1δi,j+1+gδi+1,j, (1.67)which is of the form:
(
)(
)
(
)
(
)(
) (
)
− − − − − − = 0 1 3 2 1 0 0 2 3 3 2 0 2 0 3 0 0 0 N N N g N N N g g M K M O O O M O K . (1.68)For N =8 this is given by:
0 21 42 0 0 0 0 0 0 18 30 0 0 0 0 0 0 15 20 0 0 0 0 0 0 12 12 0 0 0 0 0 0 9 6 0 0 0 0 0 0 6 2 0 0 0 0 0 0 3 0 0 0 0 0 0 0 g g g g g g g . (1.69)
The characteristic polynomial of this master matrix for g =1/N is given by:
8 6 5 4 3 2 2 21 4 7 32 945 16 105 8 175 256 945 4096 8085 z z z z z z z − + + − − + − , (1.70)
and this correspond to the FZZT partition function of the (3,1) minimal model.
( )
( )
2 0 3 3 1 1 3 2 = − + − ∂ = x xz x x g N x N N z g e Q , (1.71)for N =8. The expression for QN
( )
z can be written using the residue theorem as:N
( ) ( )
= − N∫
N+ e−V( )− z d i N g z Q ϕ ϕ ϕϕ π 1 2 1 ! . (1.72)
N
g→ 1 , z
N
z→−1+ 11/4 , (1.73)
one obtains a generalized Airy function Φ
( )
z defined by the integral: Φ( )
=∫
− 0 4/4 2 C z e i d z ϕ ϕ πϕ . (1.74)The generalized Airy function obeys the differential equation: Φ' ''
( )
z +zΦ( )
x =0, (1.75) with solutions:( )
{ }
{ }
{ }
− + − + − = Φ 64 , 4 5 , 4 3 , 64 , 2 3 , 4 5 , 64 , 4 3 , 2 1 , 4 2 0 4 2 0 2 4 2 0 z F z C z F z B z F A z , (1.76) for constants ,A B and C where pFq is a generalized hypergeometric function. We note, in this , expression, that 64=82 and that 8 is a Fibonacci’s number.Modifying the contour to be along the imaginary axis we can define a modified generalized Airy function Ψ
( )
z by:( )
∫
∞ ∞ − + − = Ψ z e φ iφzdφ 4 4 1 , (1.77) with a series expansion given by:
( )
( )
( )
∑
∞ = + Γ − = Ψ 0 2 2 4 1 ! 2 2 2 1 k k k z k k z . (1.78)This modified generalized Airy function obeys the differential equation: Ψ' ''
( )
z −zΨ( )
x =0, (1.79) with solution:( )
{ }
{ }
Γ − Γ = Ψ 64 , 2 3 , 4 5 , 4 3 64 , 4 3 , 2 1 , 4 1 2 1 4 2 0 2 4 2 0 z F z z F z . (1.80)We note that also in this expression 64=82 and that 8 is a Fibonacci’s number. The Riemann Ξ function is defined by:
( )
− − + Γ + = Ξ − − 8 1 2 4 1 2 2 1 z 1/4 /2 z2 i iz zζ
π
π
iz . (1.81)Also in this expression, we note easily that we have the Fibonacci’s number 8. It is even and can be expressed as an integral along the imaginary axis as:
( )
∫
∞ ( ) ∞ − + − = Ξ φ φφ
d e z U iz , (1.82) where:( )
− − =∑
∞ = − 1 2 2 4 2 2 2 3 log k e k e e k e k U φ π φ π φ π φ . (1.83)This function plays the same role for the Ξ function as the Konsevich potential φ3/3 plays for the Airy function and φ4/4 for the Φ function. For small φ one can develop an expansion:
U
( )
φ =9.36345φ2 +5.95896φ4−2.15104φ6 +O( )
φ8 , (1.84)which is probably why the (3,1) minimal model modified FZZT partition function shares some of the characteristics of the Ξ function. The Ξ function itself can be expanded as:
( )
( )
( )
n n n n z n a z 2 0 2 ! 2 1 − = Ξ∑
∞ = , (1.85) where∫
( )
∞ − = 1 2 4 / 1 2 log 2 1 4 n n d f a l l l l (1.86) and( )
∑
∞ = − − = 1 2 / 1 2 2 4 2 2 3 q q e q q f lπ
lπ
l πl. (1.87)Thence, we can write the eq. (1.85) also:
( )
( )
( )
n n n q n q z n e q q d z 2 0 1 1 2 2 / 1 2 2 4 4 / 1 ! 2 1 log 2 1 2 3 4 2 − − = Ξ∑ ∫
∞∑
= ∞ ∞ = − − l l l l l π π πl . (1.87b)Thus like the Ψ function one can think of the Ξ
( )
z function as an infinite order polynomial expanded in even powers of z .Now we take the pure numbers of the expression (1.84). We obtain an interesting mathematical connection with the aurea section and the aurea ratio. Indeed, we have that:
9.36345+5.95896 – 2.15104 = 13.17137 ≅13.17; = ≅ − − + + + 0901667 . 13 2 1 5 2 1 5 2 1 5 5 2 13.09
Nevertheless keeping the first two terms in the expansion for U
( )
φ
one can derive the following approximate equation for small z :4
(
5.95896) ( ) (
Ξ z '''−29.36345) ( )
Ξ z '−zΞ( )
z ≈0. (1.88) Rescaling the argument of Ξ( )
z we define:Ξ∗
( )
z =Ξ(
2(
5.95896)
1/4)
z. (1.89) So that one has the following approximate equation for small z : Ξ∗( )
z ' ''−s1Ξ∗( )
z '−zΞ∗( )
z ≈0, (1.90) where 3.835753241 95896 . 5 36345 . 9 1 = = s . (1.91)This appear related to the deformed (3,1) minimal model with deformation parameter s1. The solution to the equation for Ξ∗ is denoted by Ψ
( )
z, s1 and is:
( )
∫
∞ ∞ − + − − = Ψ z s e φ sφ iφzdφ 2 1 4 2 1 4 1 1 , . (1.92)We note that the value of s1 is related with the following expressions:
3.82404468 3.82 2 1 5 3 1 2 1 5 2 1 5 3 ≅ = − + − − + 3.8281 3.83 2 1 5 2 1 5 7 / 29 7 / 19 ≅ = − + +
One can improve the approximate equation (1.88) by including higher order terms in the φ expansion of U
( )
φ
. Keeping terms up to φ6 in (1.84) one obtains the approximate differential equation:6
(
2.15104)
Ξ' ''''( ) (
z +45.95896) ( ) (
Ξ z ' ''−29.36345) ( )
Ξ z '−zΞ( )
z ≈0. (1.93) Now rescaling can put the equation in the form:Ξ∗∗
( )
z ''' ''+s3Ξ∗∗( )
z ' ''−s1Ξ∗∗( )
z '−zΞ∗∗( )
z ≈0, (1.94)with deformation parameters s1 and s . Finally we can define a function 3 Φ
( )
z, s1 as the solution to: Φ( )
z,s1 ' ''−s1Φ( )
z,s1 '+zΦ( )
z,s1 =0, (1.95)which is real on the real axis and decays non-oscillatory for large positive y . We have that Φ
( )
z, s1 is the FZZT partition function associated with the matrix potential:
( )
− + + − + + − + + − + + = 1 1/4 3 4 / 1 2 4 / 1 4 / 1 1 1 1 3 1 1 2 3 1 3 1 N x N s N x N x N x N s N x V . (1.96)After rescaling and shifting the point of origin of the potential one can define polynomials for the matrix model deformed by the parameter s1 through:
( )
2 3 1 0 1 1 1 3 1 2 3 3 1 exp 1 , = + + + − + − ∂ + = x N x N N x xz N s x x x N s N N N s s z Q . (1.97)The master matrix which has this as characteristic polynomial is a simple rescaling of the coupling constant of the master matrix of the (3,1) minimal model and is given by:
i j
( )(
)
i j( )
i j i j N s N i i i M 1, 1 1 , 2 , , 1 1 1 3 2 1 + + + + + − + − − = δ δ δ . (1.98)This master matrix can develop complex eigenvalues for large enough N and s1. In particular for
34 ≥
N (note that 34 is a Fibonacci’s number) and s1 given by (1.91) the eigenvalues are complex. However the function Ψ
( )
z, s1 obtained from changing the sign of z in the third term in (1.95) isvery different from Φ
( )
z, s1 in this respect. It would be of interest to determine the master matrix associated with Ψ( )
z, s1 and it’s corrections for terms involving s and higher, which should in 3principle converge to the Riemann Ξ function.
With regard the eq. (1.93), we note that the pure number 2.15104 is related to the following expressions: 2.13014 2.13 2.15104 2 1 5 11/7 ≅ ≅ = + ; 2.1458 2.15 2.15104 2 1 5 2 1 5 2 1 5 4 2 ≅ ≅ = + + − + −
Furthermore, from the eq. (1.93) we have also that:
5.97 5.95896 2 1 5 26/7 ≅ ≅ + ;
(
) (
) (
)
17.94 2 1 5 01518 . 18 7269 . 18 83584 . 23 90624 . 12 36345 . 9 2 95896 . 5 4 15104 . 2 6 7 / 42 ≅ + ≅ = − + = − + 12.72 12.90 2 1 5 37/7 ≅ = + ; 23.62 23.83 2 1 5 46/7 ≅ = + ; 17.94 18.72 2 1 5 42/7 ≅ = +The Ξfunction can be expressed also as Meixner-Pollaczek polynomials. Thence, we have that:
( )
(
)
( )
− + + = ;2 2 3 ; 2 1 4 3 , ! ! 2 ! ! 1 2 ! i 2F1 n i z n n n z pn n . (1.99)These polynomials are the characteristic polynomial of a matrix with nonzero components:
Mi,j i i i,j 1 i 1,j 2 1 + + + + = δ δ . (1.100)
For N =8 this matrix is given by:
0 2 / 105 0 0 0 0 0 0 1 0 36 0 0 0 0 0 0 1 0 2 / 55 0 0 0 0 0 0 1 0 18 0 0 0 0 0 0 1 0 2 / 21 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 1 0 2 / 3 0 0 0 0 0 0 1 0 . (1.100b)
The characteristic polynomial of this matrix is:
2 4 154 6 8 2 10493 2 74247 16 363825 z z z z + − + − , (1.101)
which agrees with (1.99) for N =8. The expansion of the Ξ function with an exponential factor can be expanded in terms of the Meixner-Pollaczek polynomials as:
( )
∑
( )
∞ = − = Ξ 0 4 / n n n z z p b e z π . (1.102)Terminating this series at N one can write this expansion as the characteristic polynomial of a
N
N× matrix. For N =8 this is given by:
+ 6 8 7 8 8 5 8 4 8 3 8 2 8 1 8 0/ / / / / / 105/2 / / 1 0 36 0 0 0 0 0 0 1 0 2 / 55 0 0 0 0 0 0 1 0 18 0 0 0 0 0 0 1 0 2 / 21 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 1 0 2 / 3 0 0 0 0 0 0 1 0 b b b b b b b b b b b b b b b b . (1.103)
When b are taken to zero this reproduces the matrix (1.100). The coefficients n b are linearly n related to the integrals:
∫
∞ + + + = 0 1 2 2 2 1 8 2 sin dy y e y In πy nπ
. (1.104)With regard the pure numbers of the matrix (1.100) and (1.103) we note that:
3, 5, 21, 55 are Fibonacci’s numbers, while 18 = 13 + 5; 36 = 2 + 34 and 105 = 3 + 13 + 89, thence are sum of Fibonacci’s numbers.
1.1 On some equations concerning the partition functions of the rigid string and membrane at any temperature.
The first two terms in the loop expansion
Seff =S0 +S1+... (1.105) of the effective action corresponding to the rigid string
=
∫
σ
[
ρ
−∂ ∂ +λ
(
∂ ∂ −ρδ
)
]
+µ
∫
σρ
α
µ µ µ µ 0 2 2 2 1 2 0 2 1 d X X X X d S a b ab ab , (1.106)where
α
0 is the dimensionless, asymptotically free coupling constant,ρ
the intrinsic metric,µ
0 the explicit string tension (important at low energy) andλ
ab,, 2 , 1 ,b=
a the usual Lagrange multipliers, are given – in the world sheet 0≤
σ
1≤L and ≤σ
2 ≤β
t0 – by S L t
[
λ
λ
tρ
(
α
µ
λ
aa)
]
α
β
+ + − = − 0 0 2 22 11 0 0 2 2 (1.107)at tree level, and by
∑ ∫
∞ −∞ = ∞ + ∞ − + + + − = n t n k t n k dk L d S 2 2 2 22 2 2 11 2 2 2 2 2 2 1 4 4 ln 2 1 2 2β
λ
π
λ
ρ
β
π
π
(1.108)at one-loop order, respectively. Of course, to make sense, this last expression needs to be regularized and its calculation is highly non-trivial. We shall make use of the zeta function procedure and thence, one can write the expression for S1 also:
1=−
(
−2)
A( )
s/2 s=0 ds d L d Sζ
,( )
∑ ∫
(
) (
)
∞ −∞ = ∞ + ∞ − − − − + + + = n s s A s dk k y k y 2 / 2 2 2 / 2 2 2 1 2 /π
ζ
, (1.109) where
(
)
2 / 1 2 / 1 2 11 2 2 22 11 2 11 2 2 4 2 2 + − ± + = ± a t n a t a t n t a y ρ λ ρ λ λ ρ λ , βπ 2 ≡ a . (1.110)We may consider two basic approximations of overlapping validity: one for low temperature,
0
2 µ
β− <<
, and the other for high temperature, 0 0 2 α µ
β− >>
. Both these approximations can be obtained from the expression above, which on its turn can be written in the form
( )
(
)
( )
∑
(
) (
)
∞ −∞ = + − − − Γ − Γ = n s A F s s y y y s s sη
π
ζ
1/2 /2,1/2; ;1 2 1 2 / , 2 2 + − ≡ y y η . (1.111)This is an exact formula.
With regard the low temperature case, the term n=0 in (1.111) is non-vanishing and must be treated separately from the rest. It gives
( )
[
( )
]
(
)
( )
( )
s n A s s s s − = Γ − Γ − Γ = 11 1/2 0 2 / 2 / 1 2 / 1 2 1 2 / λ ρ π ζ ,( )
1 2 1< < s R . (1.112)This is again an exact expression, that yields
0
( )
0 11 2 1 2 / ρλ ζ = = =− s n A s ds d (1.113) and 1( 0) 11 2 2 ρλ − = = d S n . (1.114)For high temperature, the ordinary expansion of the confluent hypergeometric function F of eq.
(1.111) is in order
(
)
( ) ( )
( )
(
)
∑
∞ = − = − 0 1 ! 2 / 1 2 / 1 ; ; 2 / 1 , 2 / k k k k k s k s s s Fη
η
, (1.115)( )
s k =s( ) (
s+1...s+k−1)
being Pochhamer’s symbol (the rising factorial).Now we shall consider the case of the pure bosonic membrane and corresponding p -brane. The
tree level action similar to (1.107) is
( )
(
) (
)
+ − + + = 1 0 0 1 1 2 / 1 0 2 0 2 1 1 1 σ σ λσ λσ β κL t S m (1.116)where λ0 and
λ
1 are Lagrange multipliers and σ0 andσ
1 are composite fields. The one-loop contribution to the action can be written formally as follows( )
(
)
( )
(
)
∑ ∫
∞ −∞ = ∞ + ∞ − + + − = n bm t n k k dk dk L d S 2 2 2 0 2 2 2 2 1 1 2 2 1 2 1 4 2 2 3β
λ
π
λ
π
. (1.117)As in the string case (eq. 1.109), we choose the zeta function method. Calling
ζ
2 the corresponding zeta function, we have( )
(
)
2( )
0 2 1 2 3 = − − = s bm s ds d L d S ζ ,( )
( )
(
)
s n t n k k dk dk s − ∞ −∞ = ∞ + ∞ −∑ ∫
+ + = 2 2 2 0 2 2 2 2 1 1 2 2 1 2 4 ln 2 β λ π λ π ζ . (1.118)After some calculations, we get
( )
( )
4(
2 2)
1 4 1 1 2 2 0 2 1 2 − − = − s t s s R s ζ βπ λ λ π ζ , (1.119)where
ζ
R is Riemann’s zeta function. We thus obtain( ) 2
(
3)
2 2 '( )
2 1 2 0 1 − − − = R bm t L d S ζ β λ πλ . (1.120)In the case of the bosonic p -brane, the corresponding expressions are
( )
(
) (
)
(
)
+ − + + = 0 0 1 1 2 / 1 2 / 1 0 0 2 1 1 1 σ σ λσ λσ β κL t p S p p p (1.121) and 1( )(
)
( )
0 2 1 = − − − = p s p bp s ds d L p d S ζ ,( )
∑ ∫
∞( )
(
)
( )
(
) (
( )
)
(
)
−∞ = − − ∞ + ∞ − − Γ − Γ Γ = + + + = n R s p p p p s p p p p s p t s p s p V t n k k dk dk s 4 2 2 2 / 2 / 4 ... ln 2 ... /2 2 2 0 2 2 / 1 2 2 2 0 2 2 2 1 1 1 ζ βπ λ λ π βλ π λ π ζ (1.122) where V is the “volume” of the p p−1-dimensional unit sphere.Now we shall consider the case of the bosonic membrane with rigid term and the corresponding p
-brane. The tree level action is the same as before, eq. (1.116). The one-loop order contribution for the bosonic membrane is
( )
(
)
2( )
0 2 1 2 3 = − − = s r rm s ds d L d S ζ ,( )
( )
(
)
s n r t n k k t n k k dk dk s − ∞ −∞ = ∞ + ∞ −∑ ∫
+ + + + + = 202 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 1 2 2 1 2 2 4 4 1 2 1β
λ
π
λ
κ
β
π
ρ
π
ζ
, (1.123)where the label r means rigid. We can write
( )
∑ ∫
(
) (
)
∞ −∞ = ∞ − − − + + + = n s s s r c k c k dk s 0 2 2 4 1ρ
π
ζ
, (1.124) with
(
)
2 / 1 2 1 2 2 2 2 2 0 1 1 2 2 2 2 2 4 4 2 4 + − ± + = ±β
π
κρ
λ
κ
ρ
λ
λ
β
π
κρ
λ
t n t n c . (1.125)Here, in analogy with the rigid string case, the term corresponding to n=0 must be treated separately. It yields a beta function. Also as in the rigid string case, the remaining series can be written in terms of a confluent hypergeometric function. The complete result is:
( )
( )
( ) (
)
( )
(
( )
)
∑
(
)
∞ = + − − − − − − − Γ − Γ + Γ − Γ − Γ = 1 2 1 2 2 1 1 2 2 2 ,2 1;2 ;1 / 2 2 1 2 4 1 2 1 n s s s s r c c s s s F c s s s s s sπ
ρ
π
κλ
ρ
ζ
. (1.126)For the general case of the p -brane with rigid term, the one-loop contribution to the action is
1( )
(
)
( )
0 2 1 = − − − = s r p p rp s ds d L p d Sζ
,( )
( )
(
)
∑ ∫
∞ −∞ = − ∞ + ∞ − = + + + + + + + = n s p p p p r p t n k k t n k k dk dk s 2 2 2 0 2 2 2 1 1 2 2 2 2 2 2 2 1 2 1 4 ... 4 ... 1 ... 2 1β
λ
π
λ
κ
β
π
ρ
π
ζ
( )
∑ ∫
(
) (
)
∞ −∞ = ∞ − − − + − + + = n s s p p s p c k c k dkk V 0 1 2 / 2 2 2π
ρ
, (1.127)where V is again the “volume” of the p – 1-dimensional unit sphere and the p c± are again given by (1.125). Considering the n=0 term separately, we obtain the following generalization of the formula corresponding to the rigid membrane:
( )
( )
( )
(
Γ) (
( )
)
+( )
× − Γ − Γ = − − p s p p s p s p p r p V s p s s p V sπ
ρ
π
κλ
ρ
ζ
2 2 / 2 2 / 2 2 2 2 2 / 1 2(
) (
)
( )
∑
(
)
∞ = + − − − − − Γ Γ − Γ × 1 2 2 / / 1 ; 2 ; 2 / 2 , 2 2 / 2 / 2 n s p c c s p s s F c s p p s . (1.128)In the case of the rigid membrane we get the rather simpler result
( )
[
(
( )
)
]
( )
( )
(
1 0)
2 2 2 1 2 1 2 0 2 4 1 2 ' 8 1 ln 4 1κρ
λ
λ
π
β
πζ
β
λ
κρ
λ
κρ
π
ζ
= = − − − − t t s ds d s r . (1.129)The one loop action for the rigid membrane is readily obtained from (1.129)
( )
(
)
[
(
( )
)
]
(
)
( )
( ) (
)
(
1 0)
2 2 2 2 2 1 2 1 2 2 1 8 3 2 ' 3 4 ln 1 8 3λ
λ
κρ
π
ζ
β
π
β
λ
κρ
λ
κρ
π
− − + − − + − − = d L t L d t L d Srm . (1.130)Here, terms up to k=2 in the expansion (1.115) of the hypergeometric function of (1.128) have been taken into account. We note that all higher-order terms would be easy to obtain from (1.128) and that a consistent loop expansion to any desired order can in fact performed. The conditions for extremum of S( )rm =S0( )m +S1( )rm , eqs. (1.116) and (1.130), are obtained by taking the derivatives
t d
β
ρ
π
σ
0 2 4 3 − − = ,(
)
[
(
2 1( )
2)
]
2 1 1 ln 8 3 t t dκρ
λ
β
πβ
ρ
σ
= − − , 0(
1 1)(
1 0)
1/2 − + + =σ
σ
λ
,λ
1=(
1+σ
0)
1/2. (1.131)We can easily identify here the transition that also takes place for the rigid string: for values of the temperature higher than the one coming from the expression
(
)
2 1 3 4ρ
π
β
− = − d t c , (1.132)the values of the parameters, and hence of the action and of the winding soliton mass squared, acquire an imaginary part. Guided by the fact that in the rigid string case this temperature lies above the Hagedorn temperature, we conclude that in order that the whole scheme of the string case can be translated to the membrane situation we must demand that
ρ
2µ
be small.For the Hagedorn temperature, defined as the value for which the winding soliton mass
( ) ( ) 2 1 L S M rm eff rm ≡ (1.133) vanishes, we find
( )
( )
6 1 ln 2 ' 4κρ
κ
ρ
ζ
π
β
− − − ≅ − t H . (1.134)The values of the constants which determine the leading behaviour of the effective action at high temperature, namely the derivative of the zeta function at the point – 2 (in general, −p, respectively), have been calculated. In particular, we have that
ζ
'( )
−1 =−0,16542115,ζ
'( )
−2 =−0,03049103. (1.135) Also here, we note the mathematical connection with the aurea section, i.e.2 1 5− =
φ
. Indeed, we have that:( )
0,16740 2 1 5 7 / 26 7 / 26 =− − − = −φ
;( )
0,030017 2 1 5 7 / 51 7 / 51 =− − − = −φ
. (1.136)Now, we take the pure numbers of the eqs. (1.61), (1.70) and (1.101). We have the following sequence:
We note that: 2 2= , 4=22, 7=7×1=7, 8=23, 14=2×7, 16=24 =2×8, 21=3×7=21, 8 4 2 32= 5 = × , 105=3×5×7=21×5, 154=2×7×11, 175=52×7, 256=28 =4×8×8, 21 5 3 7 5 3 945= 3× × = 2× × , 4096=212 =82×82, 8085=3×5×72×11=5×7×11×21, 1499 7 10493= × , 74247=3×24749, 363825=33×52×72×11=3×52×11×212.
Here, 5, 7 and 11 are prime natural numbers and 2, 3, 5, 8 and 21 are Fibonacci’s numbers. The number 8 is also connected with the “modes” that correspond to the physical vibrations of a superstring by the following Ramanujan function:
( )
+ + + ⋅ = − ∞ −∫
4 2 7 10 4 2 11 10 log ' 142 ' cosh ' cos log 4 3 1 8 2 ' ' 4 0 ' 2 2 w t itw e dx e x txw anti w w t w xφ
π
π
π π . (1.137)With regard the numbers 2, 3, 5, 7 and 11, these are also factors of the numbers of dimension of the Lie’s Groups, connected to the string theory. Indeed, we observe that:
L1 = 2×7; L2 = 22×3; L3 = 2×3×13; L4 = 7×19; L5 = 23×31; L6 = 24×32×5×11; L7 = 23×3×5×7×11×19; L8 = 210×33×5×7×11×23; L9 = 24×36×52×8392, with
1 6 140
839= × − and 140=144−3−1 (that are Fibonacci’s numbers); L10 = 246×320×59×76×112×133×17×19×23×29×31×41×47×59×71.
We have also that:
(
34 55)
/246≅ + ; 20=21−1; 9=8+1; 6=5+1; 2 = 2; 3 = 3; with 2, 3, 5, 8, 21, 34 and 55 that are Fibonacci’s numbers.
With regard the prime natural numbers, we have that: 5, 7, 11, 13, 17, 19, 29, 31 and 47 are of the form 6f ±1 with f =1,2,3,5 and 8 that are Fibonacci’s numbers. With regard the numbers 59 and 71, we have that: 59=6×10−1, with 10 = 8 + 2 (8 and 2 are Fibonacci’s numbers), while
1 12 6
71= × − , with 12 = 13 – 1 (1 and 13 are Fibonacci’s numbers).
We note also that for the Lie’s Groups G2,F4,E6,E7and E , that have dimensions 14, 52, 78, 133 8
and 248, we have that: 7
2
14= × , with 7=6×1+1; 52=4×13, with 13=6×2+1; 78=6×13, with 13=6×2+1; 19
7
133= × , with 19=6×3+1; 248=8×31, with 31=6×5+1. (1, 2, 3 and 5 are Fibonacci’s numbers, while 7, 13, 19 and 31 are prime natural numbers. Furthermore, also here there is the numbers 8 that is related to the physical vibrations of a superstring by eq. (1.137)).
1499 7
10493= × , with 1499 = (1499 + 1)/6 = 250 = 233 + 17 = 233 + [(21 + 13)/2]. Note that 13, 21 and 233 are Fibonacci’s numbers and 17 = 34/2 with 34 = Fibonacci’s number.
24749 3
74247= × , with 24749 = 6×4125−1, with 4125=3×53×11=4181−56, with 56 = 55 + 1. Note that 1, 55 and 4181 are Fibonacci’s numbers. We have also that
25087 – 24749 = 338; 338=6×55+8, with 8 and 55 that are Fibonacci’s numbers. Furthermore, 25087 is a prime natural number. Indeed, 25087=6×4181+1.
2. On some mathematical aspects concerning the rigid surface operators in gauge theory [4]
Now, in this chapter, we describe some interesting aspects of the rigid surface operators in gauge theory for the connections with the Geometric Langlands Program.
The familiar examples of non-local operators in four-dimensional gauge theory include line operators, such as Wilson and ‘t Hooft operators, supported on a one-dimensional curve L in the
space-time manifold M . While a Wilson operator labelled by a representation R of the gauge
group G can be defined by modifying the measure in the path integral, namely by inserting a factor
WR
( )
L =TrRHol( )
=(
∫
)
L RL A Tr Pexp A , (2.1)
an ‘t Hooft operator is defined by modifying the space of fields over which one performs the path integral. Similarly, a surface operator in four-dimensional gauge theory is an operator supported on a two-dimensional submanifold D⊂M in the space-time manifold M .
Four-dimensional gauge theories admit surface operators, and in the supersymmetric case, they often admit supersymmetric surface operators, that is, surface operators that preserve some of the supersymmetry. Now, we consider some mathematical aspects of N =4 super Yang-Mills theory in four dimensions, the maximally supersymmetric case. This theory has many remarkable properties, including electric-magnetic duality, and has been extensively studied in the context of string dualities, in particular in the AdS/CFT correspondence.
Hitchin’s equations are equations in the x2−x3 plane that can be written as follows: FA−
φ
∧φ
=0, dAφ
=0, dA∗φ
=0. (2.2)To define a supersymmetric surface operator, one picks a solution of Hitchin’s equations with a singularity along D , and one requires that quantization of N =4 super Yang-Mills theory should be carried out for fields with precisely this kind of singularity. It is natural to look for surface operators that are invariant under rotations of the x2−x3 plane. If we set x2 +ix3 =reiθ, then the most general possible rotation-invariant ansatz is
( )
( )
r dr r f d r a A=θ
+ ,φ
( )
c( )
r dθ
r dr r b − = . (2.3)Setting f
( )
r =0 by a gauge transformation and introducing a new variable s=−lnr , we can write the supersymmetry equations (2.2) in the form of Nahm’s equations:
[ ]
b c ds da , = ,[ ]
c a ds db , = ,[ ]
a b ds dc , = . (2.4)The most general conformally invariant solution is obtained by setting a ,,b c to constant elements
γ
β
α
, , of the Lie algebra g of G . The equations imply thatα
,β
, andγ
must commute, so we can conjugate them to the Lie algebra t of a maximal torus Τof G . The resulting singular solutionof Hitchin’s equations then takes the simple form
A=αdθ ,
φ
β
γ
dθ
r dr −= . (2.5)
The definition of the surface operator is that A and
φ
have singularities proportional toα
,β
,γ
modulo terms that are less singular than 1/r. Generically, forα
,β
,γ
→0, we conclude that A andφ
are less singular than 1/r. In fact, Hitchin’s equations do have a rotationally symmetric solution that is singular at r=0 but less singular than 1/r. The Nahm equations (2.4) are solved withf s t a / 1 1 + − = , f s t b / 1 2 + − = , f s t c / 1 3 + − = (2.6)
where t1,t2, and t are elements of the Lie algebra g , which satisfy the usual 3 su
( )
2 commutationrelations,
[ ]
t1,t2 =t3, etc. Moreover, f is an arbitrary non-negative constant. Since we are taking( )
2 SUG= , the matrices t , if nonzero, correspond to the two-dimensional representation of i
( )
2SU . So the surface operator that we get from the ansatz (2.6), with f allowed to fluctuate, is
actually conformally invariant.
A convenient way to describe this surface operator is to say that the fields behave near r =0 as
... ln 1 + = r d t A
θ
, ... ln ln 3 2 − + = r d t r r dr tθ
φ
, (2.7)where the ellipses refer to terms that are less singular (at most of order 1/rln2r) at r =0. The complex-valued flat connection Α= A+i
φ
is invariant under part of the supersymmetry preserved by the surface operator. Hence the conjugacy class of the monodromy=
( )
−∫
Αl
exp P
U (2.8)
is a supersymmetric observable. Here l is a contour surrounding the singularity. Hitchin’s equations imply that the curvature of Α, namely F =dΑ+Α∧Α, is equal to zero. So if Hitchin’s equations are obeyed, then the conjugacy class of U is invariant under deformations of l . Of
course, U is an element of G , the complexification of G . For a generic surface operator with C parameters
α
,β
,γ
,we setξ
=α
−iγ
. Then Α=ξ
dθ
, and the monodromy is henceU =exp
(
−2πξ
)
. (2.9) Thence, from the eq. (2.8), we can write also:=
( )
−∫
Αl
exp P
