On the Lebesgue integral and the Lebesgue measure: mathematical applications in some sectors of Chern-Simons theory and Yang-Mills gauge theory and mathematical connections with some sectors of String Theory and Number Theory

On the Lebesgue integral and the Lebesgue measure: mathematical applications in some sectors of Chern-Simons theory and Yang-Mills gauge theory and mathematical connections with some sectors of String Theory and Number Theory

Michele Nardelli1,2

1_{Dipartimento di Scienze della Terra}

Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy

2_{Dipartimento 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

In this paper, in the Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In the Section 2, we have described the possible mathematical applications, of Lebesgue integration, in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory, precisely the two dimensional quantum Yang-Mills theory. In conclusion, in the Section 3, we have described also the possible mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings.

1. On the Lebesgue integral and the Lebesgue measure [1] [2] [3] [4]

In this paper we use Lebesgue measure to define the of functions

, where is the Euclidean space.

If is an unsigned simple function, the integral is defined

, (1)

thus will take values in .

Let be natural numbers, , and let

be Lebesgue measurable sets such that the identity

(2)

holds identically on . Then one has

. (3)

A complex-valued simple function is said to be absolutely integrable of

. If is absolutely integrable, the integral is defined for real

signed by the formula

(4)

where and (we note that these are

unsigned simple functions that are pointwise dominated by and thus have finite integral), and

for complex-valued by the formula

. (5)

Let be an unsigned function (not necessarily measurable). We define the lower

unsigned Lebesgue integral

(6)

where ranges over all unsigned simple functions that are pointwise bounded

by . One can also define the upper unsigned Lebesgue integral

(7)

but we will use this integral much more rarely. Note that both integrals take values in , and

Let be measurable. Then for any , one has

, (7b)

that is the Markov’s inequality.

An almost everywhere defined measurable function is said be absolutely integrable if

the unsigned integral

(8)

is finite. We refer to this quantity as the norm of , and use or

to denote the space of absolutely integrable functions. If is real-valued and

absolutely integrable, we define the Lebesgue integral by the formula

(9)

where and are the positive and negative components of

. If is complex-valued and absolutely integrable, we define the Lebesgue integral by the formula

(10)

where the two integrals on the right are interpreted as real-valued absolutely integrable Lebesgue integrals.

Let . Then

. (11)

This is the Triangle inequality.

If is real-valued, then . When is complex-value one cannot argue quite so

simply; a naive mimicking of the real-valued argument would lose a factor of 2, giving the inferior bound

To do better, we exploit the phase rotation invariance properties of the absolute value operation and

of the integral, as follows. Note that for any complex number , one can write as for

some real . In particular, we have

, (13)

for some real . Taking real parts of both sides, we obtain

. (14)

Since

,

we obtain the eq. (11)

Let be a measure space, and let be measurable. Then

. (15)

It suffices to establish the sub-additivity property

. (15b)

We establish this in stages. We first deal with the case when is a finite measure (which means

that ) and are bounded. Pick an and be rounded down to the nearest

integer multiple of , and be rounded up to nearest integer multiple. Clearly, we have the pointwise bound

(16)

and

. (17)

Since is bounded, and are simple. Similarly define . We then have the pointwise

bound

, (18)

hence, from the properties of the simple integral,

. (19)

From the following equation

, (19b)

we conclude that

. (20)

Letting and using the assumption that is finite, we obtain the claim. Now we continue

to assume that is a finite measure, but now do not assume that are bounded. Then for any

natural number (also the primes) , we can use the previous case to deduce that

. (21)

Since , we conclude that

. (22)

Taking limits as using horizontal truncation, we obtain the claim.

Finally, we no longer assume that is of finite measure, and also do not require to be bounded. By Markov’s inequality, we see that for each natural number (also the primes) , the set

,

has finite measure. These sets are increasing in , and are supported on , and

so by vertical truncation

. (23)

From the previous case, we have

. (24)

Let be a measure space, and let be a monotone non-decreasing

sequence of unsigned measurable functions on . Then we have

Write , then is measurable. Since the are

non decreasing to , we see from monotonicity that are non decreasing and bounded

above by , which gives the bound

. (26)

It remains to establish the reverse inequality

. (27)

By definition, it suffices to show that

, (28)

whenever is a simple function that is bounded pointwise by . By horizontal truncation we may assume without loss of generality that also is finite everywhere, then we can write

(29)

for some and some disjoint -measurable sets , thus

. (30)

Let be arbitrary (also ). Then we have

(31)

for all . Thus, if we define the sets

(32)

then the increase to and are measurable. By upwards monotonicity of measure, we conclude that

. (33)

On the other hand, observe the pointwise bound

(34)

. (35)

Taking limits as , we obtain

; (36)

sending we then obtain the claim.

Let be a measure space, and let be a sequence of measurable

functions that converge pointwise -almost everywhere to a measurable limit . Suppose

that there is an unsigned absolutely integrable function such that are

pointwise -almost everywhere bounded by for each . Then we have

. (37)

By modifying on a null set, we may assume without loss of generality that the converge to

pointwise everywhere rather than -almost everywhere, and similarly we can assume that are bounded by pointwise everywhere rather than -almost everywhere. By taking real and

imaginary parts we may assume without loss of generality that are real, thus

pointwise. Of course, this implies that pointwise also. If we apply Fatou’s lemma to

the unsigned functions , we see that

, (38)

which on subtracting the finite quantity gives

. (39)

Similarly, if we apply that lemma to the unsigned functions , we obtain

; (40)

negating this inequality and then cancelling again we conclude that

. (41)

A probability space is a measure space

(

Ω,F,P

)

of total measure1: P

( )

Ω = 1. The measure P is known as a probability measure. If is a (possibility infinite) nonempty set with the discrete -algebra , and if

( )

pω _ω∈Ω are a collection of real numbers in

[ ]

0 with ,1

∑

ω∈Ω pω = 1, then the probability measure P defined by P :=

∑

_ω∈Ω pωδω , or in other words

( )

∑

∈ = E p E P ω ω : _{, (42)}

is indeed a probability measure, and

(

Ω ,2Ω,P

)

is a probability space. The function ω  pω is known as the (discrete) probability distribution of the state variable ω . Similarly, if Ω is a Lebesgue measurable subset of _{R of positive (and possibly infinite) measure, and} d _{f :}Ω →

[

_0,+ ∞

]

is a Lebesgue measurable function on Ω (where of course we restrict the Lebesgue measure space

on _{R to} d _{Ω in the usual fashion) with}

∫

( )

Ω f x dx= 1, then

(

[ ]

R P

)

d _,

, ↓_Ω

Ω L is a probability space,

where P:= mf is the measure

P

( )

E :=

∫

_Ω1E

( ) ( )

x f x dx=

∫

_E f

( )

x dx. (43)

The function f is known as the (continuous) probability density of the state variable ω . Theorem 1

(Connes’ Trace Theorem) Let M be a compact n -dimensional manifold, ξ a complex vector bundle on M , and P a pseudo-differential operator of order n− acting on sections of ξ . Then the corresponding operator P in _{H = L}2

(

_M,ξ

)

_{belongs to L}1,∞

( )

_{H and one has:}

( )

n P

Tr_ω = 1Res

( )

P (44) for any ω .

Here Res is the restriction of the Adler-Manin-Wodzicki residue to pseudo-differential operators of order n− . Let ξ be the exterior bundle on a (closed) compact Riemannian manifold M , vol the 1-density of M , f ∈ C∞

( )

M , M the operator given by f acting by multiplication on smooth f

sections of ξ , ∆ the Hodge Laplacian on smooth sections of ξ , and P= M_f

(

1+ ∆

)

−n/2, which is a pseudo-differential operator of order n− . Using Theorem 1, we have that

( )

(

)

_{( )}

∫

( ) ( )

      + Γ = = − ∆ n _M n f f f x vol x n T M Tr M 1 2 2 1 2 1_π φω ω , f ∈ C∞

( )

M (45) where we set _∆ =

(

+ ∆

)

− /2∈ 1,∞ 1 : n L

T . This has become the standard way to identify φω with the

Corollary 1

Let M be a n-dimensional (closed) compact Riemannian manifold with Hodge Laplacian ∆ . Set

(

)

(

L

( )

M

)

T n/2 1, 2 1 : − ∞ ∆ = + ∆ ∈ L . Then φω

( )

Mf := Trω

(

MfT∆

)

= c

∫

_M f

( ) ( )

x vol x , ∀f ∈ L∞

( )

M (46)

where c> 0 is a constant independent of ω ∈ DL2.

Theorem 2

Let M,∆,T∆ be as in Corollary 1. Then, M T M T

(

L

( )

M

)

s f s T f s 2 1 2 / 2 / _{∈ L} = _{∆ ∆}

∆ for all s> 1 if and

only if f _∈ L1

( )

M _{. Moreover, setting}

( )

          = + ∆ k T f f _kTr M M 1 1 1 : ξ ψ ξ (47) for any ξ ∈ BL,

( )

= _ + _ =

∫

( ) ( )

∆ ∞ → f T M k f Tr M c f x vol x k M _k1 1 1 lim : ξ ψ , _∀f _∈ L1

( )

M ₍₄₈₎

for a constant c> 0 independent of ξ ∈ BL.

Thus ψ ξ , as the residue of the zeta function

(

/2 f s/2

)

s _{M T}

T

Tr _{∆ ∆} at s= 1, is the value of the Lebesgue integral of the integrable function f on M . This is the most general form of the identification

between the Lebesgue integral and an algebraic expression involving M , the compact operator f

(

₂

)

1

1+ ∆ − and a trace.

Theorem 3

Let ₀< _G

( )

_D ∈ _L1,∞ _{and ξ ∈ BL∩ DL. Then}

_{( )}

( )

_{( )}

(

( )

)

( )

( )

    = =

∫

+ x d x f k D G T Tr T F k f f 1 1 1 : ξ µ φLξ Lξ , ∀f ∈ L20

(

F,µ1,∞

)

. (49) Moreover, if _→∞ −

∫

( )

₊ −

( )

F k k k h x d ₁ 1 x 1

lim µ exists for all h∈ L∞

(

F,µ1,∞

)

, then

( )

T Tr

(

T G

( )

D

)

_{k F} f

( )

x d

( )

x k k f f = = _→∞

∫

₁₊1 1 lim : µ φ_{ω ω ,} ∀f ∈ L20

(

F,µ1,_∞

)

(50)

and all ω ∈ DL2.

We note that it is possible to identify φω with the Lebesgue integral.

Now we consider an arbitrary manifold X with a fixed continuous non-negative finite Borel measure m . The construction of the integral models of representations of the current groups _{G is} X

based on the existence, in the space D

( )

X of Schwartz distributions on X , of a certain measure L

which is an infinite-dimensional analogue of the Lebesgue measure. Furthermore, we have that ξ runs over the points of the cone

( )

      _{= > < ∞ ∈} =

∑

∞

∑

= + k k k k k k x X x r r r X l k 0, , 1 1 _{ξ δ} ,

on which the infinite-dimensional Lebesgue measure L is concentrated. With each finite partition of X into measurable sets,



n k k X X 1 : = = α , m

( )

Xk = λk, k = 1,...,n,

we associate the cone F_α = R₊n of piecewise constant positive functions of the form

( )

∑

( )

= = n k k k x f x f 1 χ , f_k > 0,

where χk is the characteristic function of X , and we denote by k Φ α =

( )

R+n ' the dual cone in the space distributions. We define a measure L on α Φ α by

(

)

∏

_{( )}

= − = n k k k k n dx x x x d k 1 1 1,..., _{γ λ} λ α L , where λk = m

( )

Xk . (51)

Let D₊

( )

X ⊂ D

( )

X be the set (cone) of non-negative Schwartz distributions on X , and let

( )

X D

( )

X

l₊1 ⊂ _{+ be the subset (cone) of discrete finite (non-negative) measures on X , that is,}

( )

      ∈ ∞ < > = =

∑

∞

∑

= + 1 1 _{0, ,} k k k k k x k r r x X r X l ξ δ _k .

There is a natural projection D+

( )

X → Φ α . Theorem-definition

There is a σ -finite (infinite) measure L on the cone D₊

( )

X that is finite on compact sets, concentrated on the cone l1

( )

X

+ , and such that for every partition α of the space X its projection

on the subspace Φ α has the form (51). This measure is uniquely determined by its Laplace

( )

∫

_{( )}

∑

( )



( )

=

(

−

∫

( ) ( )

)

     − ≡ + X X l k k kf x d f x dm x r f

F 1 exp L ξ exp log , (52)

where f is an arbitrary non-negative measurable function on

(

X ,m

)

which satisfies

( ) ( )

∫

< ∞

Xlog f x dm x .

Elements of l1

( )

X

+ will be briefly denoted by ξ =

{

rk,xk

}

k∞=1, or even just ξ =

{

r ,k xk

}

(sequences that

differ only by the order of elements are regarded as identical).

Let us apply the properties of the measure L to computing the integral

∫

_{( )}

∏

(

)

( )

+     = ∞ = X l k k k x d r I 1 1 , ξ ϕ L , (53) where ϕ

( )

r,x is a function on R∗ × X

+ satisfying the conditions

ϕ

( )

0,x ≡ 1 and

∫ ∫

∞

(

( )

− −

)

−

( )

< ∞ X r _{r drdm x} e x r 0 1 , ϕ . (54) Theorem

The following equality holds:

∫

_{( )}

∏

(

)

( )

+     ∞ = X l k k k x d r 1 1 , ξ ϕ L ₌

∫ ∫

∞

(

( )

₋ −

)

−

( )

X r _{r drdm x} e x r 0 1 ₎ , (exp ϕ . (55)

Proof. Under the projection D+

( )

X → Φ α (recall that Φ α is the finite-dimensional space associated with a partition α :X =

_

n_k=1X_k) the left-hand side of (53) takes the form

∏

= = n k k I I 1 α α ,

( )

∫

( )

∞ − Γ = 0 , 1 1 k k k k k k _{r r dr} I λk α α _λ ϕ , (56) where λk = m

( )

Xk and

( )

∫

( ) ( )

− = k X k k k k r r ,x dm x 1 , λ ϕ

ϕ_{α . Thence the eq. (56) can be rewritten also as}

follows

∏

= = n k I 1 α

( )

∫

∫

( ) ( )

∞ ₋ − Γ 0 1 , 1 1 k k X k k k dr r x dm x r k k λ ϕ λ λ . (56b)

The original integral I is the inductive limit of the integrals I over the set of partitions αα . Since

( )

∫

∞ − − = Γ 0 1 ₁ 1 k k t k dr r e λk

λ , the integral Iαk can be written in the form

= + _Γ

_{( )}

∫

₀∞

(

,

( )

− −

)

−1 1 1 r _{k k} k k k k _{r e r dr} I k λk α α _λ ϕ . (57)

It follows that = +

∫

∞

(

( )

− −

)

− +

( )

0 2 1 , 1 r _{k k k} k k k k _{r e r dr O} I λ ϕ k λ α α , (58) whence exp ₀

(

,

( )

)

k1 k

( )

2k r k k k k _{r e r dr O} I λ ϕ k λ α α = 

∫

−  + ∞ _{− −} .

The eq. (58) can be rewritten also as follows

= +

∫

∞

(

( )

− −

)

− +

( )

0 2 1 , 1 r _{k k k} k k k k _{r e r dr O} I λ ϕ k λ α α

(

( )

)

( )

2 0 1 , exp r _{k k k} k k k ϕ r e k r dr Oλ λ _{α } +   ₋ =

∫

∞ − − . (58b)

Thus, up to terms of order greater than 1 with respect to λk,

(

( )

)

      ₋ ≅

∑ ∫

= ∞ _{− −} n k r k k r e r dr I 1 0 1 , exp _α α λ ϕ . Since

∑

_kn₌

(

(

( )

− −

)

)

=

∫

(

( )

− −

)

( )

X r r k k r e r x e dm x 1 λ ϕα, ϕ , ,

the expression obtained can be written in the following form

I ≅ 

∫ ∫

_X ∞

(

( )

r x − e−r

)

r− drdm

( )

x  0 1 , exp ϕ α . (59)

The proof is completed by taking the inductive limit over the set of partitions α .

Corollary

If ϕ

( )

r,x =

∑

_in₌₁ciϕi

( )

r,x , where ci > 0,

∑

ci = 1, and the functions ϕi

( )

r,x satisfy (54), then

∫

_{( )}

∏

(

)

( )

∏

∫ ∫

(

( )

)

( )

+ ₌ ∞ _{− −} ∞ =     ₋ =     X l n i X r i i k k k x d c r x e r drdm x r 1 1 0 1 1 , exp , ξ ϕ ϕ L . (60)

Let _ϕ

( )

_{r,x = e}−rσa( )x _{, where σ ≥ 1 and Rea}

( )

_{x > 0. In this case we obtain}

∫

_{( )}

∏

( )

( )

∫ ∫

(

( )

)

( )

+     ₋ =     ∞ ∞ _{− − −} = − X l X r x a r k x a r _{d e e r drdm x} e k k 1 ₀ 1 1 exp σ σ ξ L . (61)

Let us integrate with respect to r . We have

∫

∞

(

( )

)

∫

(

( )

)

− −

( ) ( )

→ ∞ _{− − −} → − − −     _{− Γ}       Γ =     ₋ = − 0 / 1 0 0 1 0 1 _{lim lim λ} σ λ σ λ σ λ λ λ σ σ x a dr r e e dr r e e r a x r r a x r . (62)

Since Γ

( )

λ ≈ λ−1+ γ _{as λ → 0, where} γ _{is the Euler constant, it follows that}

∫

∞

(

(

−

( )

)

−

( )

−

)

− = − −

( )

+

(

− −

)

0 1 1 1 _{log 1} exp exp rσa x r r dr σ a x σ γ . Hence,

∫

_{( )}

∏

(

( )

)

( )

(

(

)

)

(

∫

( ) ( )

)

+ − − ∞ = − − =     ₋ X l X k k k a x d a x dm x r

1 exp exp 1 exp log

1 1 1 σ γ σ ξ σ _L . (63)

In particular, for σ = 1 we recover the original formula for the Laplace transform of the measure L

:

∫ ∏

_{( )}

(

∑

( )

)

( )

(

∫

( ) ( )

)

+ ∞ = − = − X l k X k ka x d a x dm x r 1 1 log exp exp L ξ . (63b)

2. Mathematical applications in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory

2.1 Chern-Simons theory [5]

The typical functional integral arising in quantum field theory has the form Z−1

∫

_Α f

( )

AeβS( )ADA (64)

where S is an action functional,

( )

⋅ β a physical constant (real or complex), f is some function of the field A of interest, DA signifies “Lebesgue integration” on an infinite-dimensional space Α of field configurations, and Z a “normalizing constant”. Now we shall describe the Chern-Simons theory over _{R , with gauge group a compact matrix group} 3 _{G whose Lie algebra is denoted L}

( )

_{G .} The formal Chern-Simons functional integral has the form

∫

( )

( ) Α f Ae DA Z A iCS 1 (65)

where f is a function of interest on the linear space Α of all L

( )

G -valued 1-forms A on _{R , and} 3

( )

⋅

CS is the Chern-Simons action given by

( )

∫

      _{∧ + ∧ ∧} = 3 3 2 4 R Tr A dA A A A A CS π κ , (66)

involving a parameter κ . We choose a gauge in which one component of A= a0dx1+ a1dx1+ a2dx2 vanishes, say a₂ = 0. This makes the triple wedge term A∧ A∧ A disappear, and we end up with a quadratic expression

( )

=

∫

3

(

∧

)

4 R Tr A dA A CS π κ for A= a0dx0+ a1dx1. (67)

Then the functional integral has the form

∫

( )

( ) Α ∧ ∫ 0 3 4 1 DA e A Z RTr A dA i π κ φ (68)

where Α0 consists of all A for which a2 = 0. As in the two dimensional case, the integration element remains DA after gauge fixing. The map

( )

Ae ( )DA Z R Tr A dA i CS

∫

_Α ∧ ∫ = 0 3 4 1 κ_π φ φ φ  , (69)

whatever rigorously, would be a linear functional on a space of functions φ over Α0. Now for 0 1 1 0 0 + ∈ Α = a dx adx

A , decaying fast enough at infinity, we have, on integrating by parts,

(

0 0+ 1 1

)

= −

∫

3

(

0 1

)

0 1 2 2 R Tr a f dx dxdx dx a dx a CS π κ ₍₇₀₎ where f1= ∂2a1. (71)

So now the original functional integral is reformulated as an integral of the form

= 1

∫

_e4 0, 1

(

_a0,f1

)

_Da0Df1 Z f a i CS φ φ π κ (72) where a,f = −

∫

_R3Tr

( )

af dvol, (73)

and Z always denotes the relevant formal normalizing constant. Taking φ to be of the special form

(

)

0( )0 1( )1 1 0, j if j ia j a f e + = φ (74)

where j and 0 j are, say, rapidly decreasing 1 L

( )

G -valued smooth functions on R , we find, from a 3 formal calculation i2 (j0,j1) CS j e κ π φ = − . (75)

In the paper “Non-Abelian localization for Chern-Simons theory” of Beasley and Witten (2005), the Chern-Simons partition function is

( )

( )

∫



∫

     _{∧ + ∧ ∧}       = ∆ XTr A dA A A A k i A k Vol k Z 3 2 4 exp 4 1 2 _π π D G G . (76)

We note that, in this equation, AD signifies “Lebesgue integration” on an infinite-dimensional space of field configurations. If X is assumed to carry the additional geometric structure of a Seifert manifold, then the partition function of eq. (76) does admit a more conventional interpretation in terms of the cohomology of some classical moduli space of connections. Using the additional Seifert structure on X , decouple one of the components of a gauge field A , and introduce a new partition function

( )

_∫

_∫

_∫

(

)

_∫

( )

          _{− ∧ Φ + ∧ Φ}       _{∧ + ∧ ∧} Φ ⋅ = XTr A dA A A A X Tr FA X d Tr k i A K k Z ₂ 2 3 2 4 exp κ κ κ π D D ,

(77), then give a heuristic argument showing that the partition function computed using the alternative description of eq. (77) should be the same as the Chern-Simons partition function of eq. (76). In essence, it is possible to show that

Z

( )

k = Z

( )

k , (78)

by gauge fixing Φ = 0 using the shift symmetry. The Φ dependence in the integral can be eliminated by simply performing the Gaussian integral over Φ in eq. (77) directly. We obtain the alternative formulation

( )

( )

∫

∫

∫

[

(

)

]

_          _∧ ∧ −       _{∧ + ∧ ∧} ⋅ = = X A XTr A dA A A A d Tr F k i A K k Z k Z 1 2 3 2 4 exp ' κ κ κ π D , (79) where

( ) ( )

2 / 2 4 1 1 :' G S G ∆       − = π ik Vol Vol

K . thence, we can rewrite the eq. (79) also as follows:

( )

( )

( ) ( )

 ⋅     − = = ∆ /2 2 4 1 1 G S G π ik Vol Vol k Z k Z

(

)

[

]

∫

∫

∫

          _∧ ∧ −       _{∧ + ∧ ∧} ⋅ X A XTr A dA A A A _d Tr F k i A 1 2 3 2 4 exp κ κ κ π D . (79b)

We restrict to the gauge group U

( )

1 so that the action is quadratic and hence the stationary phase approximation is exact. A salient point is that the group U

( )

1 is not simple, and therefore may have non-trivial principal bundles associated with it. This makes the U

( )

1 -theory very different from the

( )

2

SU -theory in that one must now incorporate a sum over bundle classes in a definition of the

( )

1

U -partition function. As an analogue of eq. (76), our basic definition of the partition function for

( )

1

U -Chern-Simons theory is now

( )

(

)

( )

(

)

( )

∑

∈ = Z X TorsH p U U X k Z X p k Z ; 1 1 2 , , , ₍₈₀₎

where ( )

(

)

=

_{( )}

∫

_Α ( ) P P X A ikS P U X p k _Vol Ae Z _{, ,} 1 , 1 π D G . (81)

Thence, the eq. (80) can be rewritten also as follows

( )

(

)

_{( )}

( ) ( )

∑

_∫

∈ Α = Z X TorsH p A ikS P U P X Ae Vol k X Z ; 1 2 , 1 , D π G . (80b)

The main result is the following: Proposition 1

Let

(

X,φ,ξ ,κ ,g

)

be a closed, quasi-regular K -contact three manifold. If,

_{( )}

(

)

( ) ( )

∫

( )

Μ       _{−∗ + ∫ ∧} = P X P X X d C d R D i A ikS n U X p k k e e T Z 512 1/2 1 4 1 2 0 , , , η κ κ π π ₍₈₂₎

where R∈ C∞

( )

X = the Tanaka-Webster scalar curvature of X , and ( )

(

)

( ) ( ) ( )

( )

∫

Μ     + ∗ − = P g P X X d RS A CS d i A ikS m U X p k k e e T Z 12 2 1/2 1 4 1 , , , 0 π η π π ₍₈₃₎ then Z_{U( )1}

(

X,k

)

= Z_{U( )1}

(

X,k

)

as topological invariants.

Now our starting point is the analogue of eq. (79) for the U

( )

1 -Chern-Simons partition function: _{( )}

(

)

( )

( ) ( )

∫

Α

∫

∫

(

)

_           ∧ ∧ − ∧ = P P X X X P A ikS U d dA dA A ik DA Vol Vol e k p X Z κ κ κ π π 2 1 4 exp , , 0 , G S (84)

where SX,P

( )

A0 is the Chern-Simons invariant associated to P for A a flat connection on P . Also 0 here, DA signifies “Lebesgue integration” on an infinite-dimensional space Α Pof field

configurations. The eq. (84) is obtained by expanding the U

( )

1 analogue of eq. (79) around a critical point A of the action. Note that the critical points of this action, up to the action of the shift 0 symmetry, are precisely the flat connections. In our notation, A∈ TA0Α P. Let us define the notation

( )

∫

∫

(

)

∧ ∧ − ∧ = X X d dA dA A A S κ κ κ 2 : (85)

( )

∫

(

)

∧ ∧ = X d dA A S κ κ κ 2 : (86)

so that we may write

S

( )

A = CS(A)− S

( )

A . (87) Thence, we can rewrite the eq. (87) also as follows:

∫

∫

(

)

∫

 −      _{∧ + ∧ ∧} = ∧ ∧ − ∧ X X X d Tr A dA A A A dA dA A 3 2 2 κ κ κ

(

)

∫

X ∧_∧ d dA κ κ κ 2 . (87b)

The primary virtue of eq. (84) above is that it is exactly equal to the original Chern-Simons partition function of eq. (81) and yet it is expressed in such a way that the action S

( )

A is invariant under the shift symmetry. This means that S

(

A+ σ κ

) ( )

= S A for all tangent vectors A T_A

( )

_P 1

( )

X

0 Α ≅ Ω ∈ and

( )

X 0 Ω ∈

σ . We may naturally view A_{∈ Ω}1

( )

H _{, the sub-bundle of Ω} 1

( )

_{X restricted to the contact} distribution H ⊂ TX . Equivalently, if ξ denotes the Reeb vector field of κ , then

( )

{

1

( )

0

}

1 _{= ∈ Ω =}

Ω H ω X ιξω . The remaining contributions to the partition function come from the

orbits of S in Α P, which turn out to give a contributing factor of Vol

( )

S . We thus reduce our

integral to an integral over Α_P := Α _P/S and obtain:

( )

(

)

( )

( )

(

)

_ =          ∧ ∧ − ∧ =

∫

∫

∫

ΑP P X X X P A ikS U d dA dA A ik A D Vol e k p X Z κ κ κ π π 2 1 4 exp , , 0 , G ( )

( )

∫

Α

( )

    = P P X A S ik A D Vol e P A ikS π π 4 exp 0 , G , (88)

where AD denotes an appropriate quotient measure on Α P, i.e. the “Lebesgue integration” on an

infinite-dimensional space Α _Pof field configurations. We now have _{( )}

(

)

( )

( )

( )

( )

( )

( )

( )

( )

(

)

[

]

∫

Α

∫

Α ∗         = = = P P P P X P X H H A S ik P A ikS P A S ik P A ikS U e d d Vol e H Vol Vol Ae D Vol e k p X Z G G G G / 2 / 1 4 4 1 , , det' 0 , 0 , µ π π π π ( )

( )

( )

(

)

[

]

∫

Α ∗     = P P P X H H A S ik A ikS d d e H Vol e G / 2 / 1 4 _det' 0 , µ π π (89)

where µ is the induced measure on the quotient space Α _P/G_P and det' denotes a regularized

determinant. Since

( )

_, 1 1 κ A D A A

S = − ∗_H is quadratic in A , we may apply the method of stationary

( )

(

)

( )

( )

∫

Μ ( )

[

[

₍

(

)

]

₎

]

∗ ∗ − ∗ − = P H P X D k d d e H Vol e k p X Z H H H D i A ikS U ν π π 2 / 1 1 2 / 1 sgn 4 1 det' det' , , 1 0 , (90)

We will use the following to define the regularized determinant of _{k D}1

H ∗ − :

(

)

( )

[

(

)

]

( )

[

]

1/2 2 / 1 2 1 det' det' , : det' ∗ ∗ + ⋅ = ∗ − TT TT S J k C D k _H (90b) where 2 ₊ ∗ ₌ 2

(

( )

1 _∗ 1₊

(

∗

)

2

)

H Hd d D D k TT S , ∗ ₌ 2

(

∗

)

2 H Hd d k TT and

_{( )}

     _{− ∫ ∧} = XR d k J k C κ κ 2 1024 1 : , (90c)

is a function of R∈ C∞

( )

X , the Tanaka-Webster scalar curvature of X , which in turn depends only on a choice of a compatible complex structure J End∈

( )

H . The operator

_{∆ :=}

( )

1 ∗ 1₊

(

∗

)

2 H Hd d D D (91)

is equal to the middle degree Laplacian and is maximally hypoelliptic and invertible in the Heisenberg symbolic calculus. We define the regularized determinant of ∆ via its zeta function

( )( )

( )

∑

∆ ∈ − ∗ = ∆ spec s s λ λ ζ : ₍₉₂₎

Also, ζ

( )( )

∆ s admits a meromorphic extension to C that is regular at s= 0. Thus, we define the regularized determinant of ∆ as

_det'

( )

_{∆ := e}−ζ'( )( )∆ 0 _{. (93)} Let _∆0:₌

(

d_H∗ d_H

)

2 _{on Ω} 0

( )

_{X , ∆ := ∆}

1 on Ω1

( )

H and define ζ i

( )

s := ζ

( )( )

∆i s . We claim the

following Proposition 2

For any real number 0< c∈ R,

det'

( )

c∆i := cζi( )0 det'

( )

∆i (94)

for i= 0,1. Proposition 3

For _∆0:₌

(

d_H∗ d_H

)

2 _{on Ω} 0

( )

_{X , ∆ := ∆}

1 on Ω 1

( )

H defined as above and ζ i

( )

s := ζ

( )( )

∆i s , we have

0

( )

1

( )

3 2 dim 1 dim 0 8 1 0 0  + ∆ − ∆     _{− ∧} = −

∫

R d Ker Ker X κ κ ζ ζ , (95)

( )

( )

_XR d dimH

(

X,dH

)

dimH

(

X,dH

)

512 1 0 0 2 1 0 1 0  + −     _{− ∧} = − ζ

_∫

κ κ ζ , (96)

where R∈ C∞

( )

X is the Tanaka-Webster scalar curvature of X and _{κ ∈ Ω} 1

( )

X _{is our chosen}

contact form as usual. Let

ζˆ₀

( )

s := dimKer∆₀ + ζ ₀

( )

s , ζˆ₁

( )

s := dimKer∆₁+ ζ ₁

( )

s (97)

denote the zeta functions. We have that ζˆ₁

( )

0 = 2ζˆ₀

( )

0 for all 3-dimensional contact manifolds. We know that on CR-Seifert manifolds that

( )

=

( )( )

∆ =

( )

∆

( )

=

∫

∧ XR κ dκ ζ ζ ζ 2 2 0 0 0 512 1 0 ˆ 0 ˆ 0 ˆ _{. (98)} Thus,

( )

=

∫

∧ XR κ dκ ζ 2 1 256 1 0 ˆ _{. (99)}

By our definition of the zeta functions, we therefore have

( )

=

∫

∧ − ∆ XR d Ker 0 2 0 ₅₁₂ dim 1 0 κ κ ζ ,

( )

=

∫

∧ − ∆ XR d Ker 1 2 1 ₂₅₆ dim 1 0 κ κ ζ . (100) Hence,

( )

−

( )

= _

∫

∧ − ∆ _ − _

∫

∧ − ∆ _ = X XR d Ker R d Ker 1 2 0 2 1 0 dim 256 1 dim 512 1 0 0 ζ κ κ κ κ ζ  + ∆ − ∆ =     _{− ∧} =

∫

2 dim 1 dim 0 512 1 Ker Ker d R X κ κ _{512 X}R d dimH

(

X,dH

)

dimH

(

X,dH

)

1 2 _{ +} 1 ₋ 0     _{− ∧} =

_∫

κ κ . (101)

2.2Yang-Mills Gauge Theory [6]

Let Σ be an oriented closed Riemann surface of genus g . Let E be an H bundle over Σ . The adjoint vector bundle associated with E will be called ad

( )

E . Let Α be the space of connections on E , and G be the group of gauge transformations on E . The Lie algebra G of G is the space of ad

( )

E -valued two-forms. G acts symplectically on Α , with a moment map given by the map

( )

₂

4π

from the connection A to its ad

( )

E -valued curvature two-form F = dA+ A∧ A. _µ −1

( )

0 _therefore consists of flat connections, and _µ −1

( )

0 /G _{is the moduli space M of flat connections on E up to}

gauge transformation. M is a component of the moduli space of homomorphisms ρ :π1

( )

Σ → H ,

up to conjugation. The partition function of two dimensional quantum Yang-Mills theory on the surface Σ is formally given by the Feynman path integral

( )

_{( )}

∫

(

)

Α       − = DA F F G vol Z , 2 1 exp 1 ε ε , (103)

where ε is a real constant, DA is the symplectic measure on the infinite dimensional function space Α , i.e. the “Lebesgue integration”, and vol

( )

G is the volume of G .

For any BRST invariant operator Ο (we want remember that the BRST (i.e. the

Becchi-Rouet-Stora-Tyutin) invariance is a nilpotent symmetry of Faddeev-Popov gauge-fixed theories, which

encodes the information contained in the original gauge symmetry) , let Ο be the expectation

value of Ο computed with the following equation concerning the cohomological theory

{

}

∫

(

)

[ ] [

]

Σ       _{− − − ∗ + + + +} = − = i i i i i i i i D D iD D i f f H Tr d h V Q i L µ χ ψ η ψ λ φ χ χ ,φ ψ ,λ ψ 2 2 1 2 1 1 , 2 2 2 , (104)

and let Ο ' be the corresponding expectation value concerning the following equation

( )

∫

(

[

]

)

Σ − + = d Tr DifDi if i i iDl l ijDi j u h i u L'' ₂ µ φ ψ ,ψ ψ ε ψ _{. (105)}

We will describe a class of Ο ’s such that the higher critical points do not contribute, and hence '

Ο =

Ο . Two particular BRST invariant operators will play an important role. The first, related

to the symplectic structure of M, is

∫

Σ       _{+ ∧} = φ ψ ψ π ω 2 1 4 1 2 Tr i F . (106) The second is

∫

Σ = Θ 2 2 8 1 φ µ π d Tr . (107) We wish to compute exp

(

ω + εΘ

)

⋅β ' (108)

with ε a positive real number, and β an arbitrary observable with at most a polynomial dependence on φ . This is

(

)

_{( )}

∫

_

∫

  +       ⋅ ⋅ = ⋅ Θ + Σ f D d Q u h D DAD G vol k k µ ψ β φ ψ β ε ω ' 1 exp 1 , exp ₂   +       _{+ ∧} +

_∫

_∫

Σ Σ 2 2 2 _{2 8} 1 4 1 φ µ π ε ψ ψ φ π Tr i F d Tr . (109)

Thus, we can simply set u= ∞ in eq. (109), discarding the terms of order 1/u, and reducing to

(

)

_{( )}

_∫

_∫

_∫

⋅     +       _{+ ∧} = ⋅ Θ + Σ Σ β φ µ π ε ψ ψ φ π φ ψ β ε ω 2 2 2 _{2 8} 1 4 1 exp 1 ' exp DAD D Tr i F d Tr G vol . (110)

Thence, we have passed from “cohomological” to “physical” Yang-Mills theory. Also here DA is the “Lebesgue integration”. With regard the eq. (110), if assuming that β = 1, in this case, the only

ψ _{dependent factors are in}

    ∧

∫

Σ ψ ψ π ψ Tr DAD ₂ 4 1 exp . (111)

Let us generalize to ε ≠ 0, but for simplicity β = 1. In this case, by integrating out φ , we get

(

)

_{( )}

∫

∫

    = Θ + Σ 2 2 2 exp 1 ' exp DA d Trf G vol ε µ π ε ω . (112)

This is the path integral of conventional two dimensional Yang-Mills theory. Now, at ε ≠ 0, we cannot claim that the and ' operations coincide, since the higher critical components M α contribute. However, their contributions are exponentially small, involving the relevant values of

∫

Σ − = _{d Trf}2 I µ . So we get

(

)

_{( )}

∫

∫

+

(

(

−

)

)

    = Θ + Σ ε π µ ε π ε ω 1 exp 2 exp 2 / exp _DA 2 _{d Trf}2 _O 2_c G vol , (113)

where c is the smallest value of the Yang-Mills action I on one of the higher critical points. We consider the topological field theory with Lagrangian

∫

Σ − = i Tr F L φ π 2 4 , (114)

which is related to Reidemeister-Ray-Singer torsion. The partition function is defined formally by

( )

Σ =

_{( )}

∫

DAD

( )

− L G

Vol

Z 1 φ exp . (115)

Here if E is trivial, G is the group of maps of Σ to H ; in general G is the group of gauge transformations. Now we want to calculate the 'H partition function

( )

Σ =

_{( )}

∫

DAD

( )

− L G Vol u Z ' exp ' 1 ; ~ _φ . (116)

Thence, with the eq. (114) we can rewrite the eq. (116) also as follows

( )

_{( )}

∫

∫

    = Σ Σ F Tr i D DA G Vol u Z φ π φ ₂ 4 exp ' ' 1 ; ~ . (116b)

First we calculate the corresponding H partition function for connections on Σ − P with monodromy u around P . This is

( )

Σ =

_{( )}

∫

DAD

( )

− L G

Vol u

Z ; 1 φ exp . (117)

Also here, we can rewrite the above equation also as follows

( )

_{( )}

∫

∫

    = Σ Σ F Tr i DAD G Vol u Z φ π φ ₂ 4 exp ' 1 ; . (117b)

A is the lift of 'A . From what we have just said, this is given by the same formula as the following

( )

( )

( )

∑

(

)

− − ⋅     = Σ α α π 2 2 2 2 dim dim 1 2 g g H H Vol Z , (118)

but weighting each representation by an extra factor of

( )

_u−1 α λ . So

( )

( )

( )

∑

(

( )

)

− − − ⋅     = Σ α α α λ π 2 2 1 2 2 dim dim 2 ; _g g H u H Vol u Z . (119)

We now use the following equation

_Vol

( )

_{G =}#_Γ1−2g_Vol

( )

_G' _{, (120)} to relate Z ;

( )

Σ u to Z ;~

( )

Σ u , and also

Vol

( )

H =#Γ ⋅Vol

( )

H' ; #Z

( )

H =#Γ⋅#Z

( )

H' ; #π₁

( )

H' =#Γ ; (121) to express the result directly in terms of properties of 'H . Using also (121), we get

( )

( )

( )

( )

∑

(

( )

)

− − − ⋅     = Σ α α α λ π π 2 2 1 2 2 ' dim 1 2 dim ' ' # 1 ; ~ g g H u H Vol H u Z . (122)

Note that in this formula, the sums runs over all isomorphism classes of irreducible representations of the universal cover H of 'H . We can immediately write down the partition function, with gauge

₍

₎

( )

( )

( )

( )

( )

(

)

∑

− − −           ₊ − ⋅ ⋅     = Σ α α α α ε λ π π ε 2 2 2 1 2 2 ' dim 1 dim 2 ' exp 2 ' ' # 1 ; , ~ g g H t C u H Vol H u Z . (122a)

Furthermore, with the (119) and (122) we can rewrite the eqs. (116b) and (117b) also as follows:

( )

_{( )}

=     = Σ

∫

∫

Σ F Tr i D DA G Vol u Z φ π φ ₂ 4 exp ' ' 1 ; ~

( )

( )

( )

∑

(

( )

)

− − − ⋅     α α α λ π π 2 2 1 2 2 ' dim 1 2 dim ' ' # 1 g g H u H Vol H ; (122b)

( )

_{( )}

∫

∫

    = Σ Σ F Tr i DAD G Vol u Z φ π φ ₂ 4 exp ' 1 ;

( )

( )

∑

(

( )

)

− − − ⋅     = α α α λ π 2 2 1 2 2 dim dim 2 g g H u H Vol . (122c)

We consider the case of H = SU

( )

2 . Then _Vol

(

_SU

( )

₂

)

_{= 2}5/2_π 2 _{with our conventions, and so}

( )

_{( )}

∑

(

)

∞ = − − − = Σ 1 2 2 2 1 2 4 / ' exp 2 1 , n g g _n n Z ε π ε . (123)

On the other hand, for a non-trivial SO

( )

3 bundle with u= −1, we have

( )

−1 ₌

( )

_{− 1}n+1 n u λ ,

( )

' 2 #π₁ H = and _Vol

(

_SO

( )

₃

)

_{= 2}3/2_π 2_{, so}

(

)

( )

∑

( )

(

)

∞ = − + − − − ⋅ = − Σ 1 2 2 2 1 1 2 4 / ' exp 1 8 2 1 1 ; , ~ n g n g _n n Z ε π ε . (124)

We will now show how (124) and (123) can be written as a sum over critical points. First we consider the case of a non-trivial SO

( )

3 _{bundle. It is convenient to look at not Z~ but}

( )

(

)

∑

( )

(

)

∞ = − − − − − ⋅ − = ∂ ∂ 1 2 1 2 1 1 4 / ' exp 1 32 2 1 ' ~ n n g g g g n Z ε π ε . (125) We write

∑

( )

(

)

∑

( )

(

)

∞ = ∈ − − + − = − − 1 2 2 _{1 exp ' /4} 2 1 2 1 4 / ' exp 1 n n Z n n n n ε ε . (126)

The sum on the right hand side of (126) is a theta function, and in the standard way we can use the Poisson summation formula to derive the Jacobi inversion formula:

( )

(

)

(

)

(

(

)

)

∑

∑ ∫

∑

∈ ∈ ∞ ∞ − ∈    ₋ + = − + = − − Z n m Z m Z n _{n dn inm i n n} m ' 2 / 1 2 exp ' 4 4 / ' 2 exp 4 / ' exp 1 2 2 2 ε π ε π ε π π ε (127)

Putting the pieces together,

( )

(

)

(

(

)

)

_         ₋ + + − ⋅ ⋅ − = ∂ ∂

_∑

∈ − − − Z m g g g g _{Z m} ' 2 / 1 2 exp ' 4 1 32 4 1 ' ~ 2 1 2 1 1 ε π ε π π ε . (128)

The eq. (128) shows that _∂ g−1_Z~_/∂ε'g−1 _{is a constant up to exponentially small terms, and hence}

( )

ε

Z~ is a polynomial of degree g− 1 up to exponentially small terms. The terms of order _ε k_,

2

−

≤ g

k that have been annihilated by differentiating g− 1 times with respect to ε' are most easily computed by expanding (124) in powers of ε :

( )

( )

∑

(

) (

)

(

)

( )

− = − + − − − − − + − = 2 0 1 2 2 3 2 1 2 _! 1 2 2 2 2 8 2 1 ~ g k g k g k g _k g k O Z π ε ζ ε π ε . (129)

Using Euler’s formula expressing ζ

( )

2n for positive integral n in terms of the Bernoulli number

n B₂ ,

( ) ( ) ( )

_{( )}

! 2 2 1 2 2 2 1 2 n B n n n n ₋ + = π ζ , (130) eq. (129) implies

(

) ( )

(

₍

)

₎

( )

∫

∑

− − = − − − − − + − − − − = Θ + 1 ' 1 0 3 1 2 2 2 2 2 2 1 ! 2 2 2 2 2 2 ! 1 exp M g k g k g k g k g k g B k ε ε ω . (131)

Thence, we obtain the following relationship:

( )

( )

(

−

) (

−

)

(

− −

)

+

( )

⇒ =

∑

− = − + − − 2 0 1 2 2 3 2 1 2 _! 1 2 2 2 2 8 2 1 ~ g k g k g k g _k g k O Z π ε ζ ε π ε

(

) ( )

(

₍

)

₎

( )

∫

∑

− − = − − − − − + − − − − = Θ + ⇒ 1 ' 1 0 3 1 2 2 2 2 2 2 1 ! 2 2 2 2 2 2 ! 1 exp M g k g k g k g k g k g B k ε ε ω . (132)

With regard the link between the Bernoulli number and the Riemann’s zeta function, we remember that

( )

(

)

( )

1 1 1 ₁ 1 + − + = + + k B x B x S k k k .

As B'_m

( )

x = mB_m−1

( )

x for all m , we see that

∫

(

)

∫

(

)

( ) ( )

+ − = + − + = − + + + 1 0 1 0 1 1 1 1 1 1 1 1 1 k B k B x B dx x S k k k k k .

Thence, we have that:

( )

∫

(

)

( )

+ − = − = − 1 + 0 1 1 1 1 k B dx x S k k k k ζ .

The cohomology of the smooth SO

( )

3 moduli space M'

( )

−1 is known to be generated by the classes ω and Θ , whose intersection pairings have been determined in equation (131) above, along

with certain non-algebraic cycles, which we will now incorporate. The basic formula that we will use is equation (110):

(

)

_{( )}

_∫

_∫

_∫

⋅     +       _{+ ∧} = ⋅ Θ + Σ Σ β φ µ π ε ψ ψ φ π φ ψ β ε ω 2 2 2 _{2 8} 1 4 1 exp 1 ' exp DAD D Tr i F d Tr G vol . (133)

We recall that ' coincides with integration over moduli space, up to terms that vanish exponentially for ε → 0. Note that ψ is a free field, with a Gaussian measure, and the “trivial” propagator

( ) ( )

x y ab

(

x y

)

ij b j a i = − − 2 2 4π ε δ δ ψ ψ . (134)

For every circle C⊂ Σ there is a quantum field operator

=

∫

C C Tr V φ ψ π 2 4 1 . (135)

It represents a three dimensional class on moduli space; this class depends only on the homology class of C. As the algebraic cycles are even dimensional, non-zero intersection pairings are possible only with an even number of the V ’s. The first case is C exp

(

ω + εΘ

)

⋅VC1VC2 ', with two

oriented circles C1,C2 that we can suppose to intersect transversely in finitely many points. So we consider

(

)

_{( )}

∫

∫

∫

⋅     +       _{+ ∧} = ⋅ Θ + Σ Σ 2 2 2 _{2 8} 1 4 1 exp 1 ' exp 2 1 π µ φ ε ψ ψ φ π φ ψ ε ω DAD D Tr i F d Tr G vol V V_{C C} ⋅

∫

⋅

∫

1 2 2 2 ₄ 1 4 1 C C Tr Tr φ ψ π φ ψ π . (136)

Upon performing the ψ integral, using (134), we see that this is equivalent to

( )

∫

∫

Σ

∫

Σ ∈

∑

∩

( )

( )

− ⋅     +       _{+ ∧} 2 1 2 2 2 2 2 ₄ 1 8 2 1 4 1 exp 1 C C P P Tr P Tr d F i Tr D DAD G vol π µ φ π σ φ ε ψ ψ φ π φ ψ (137)

Here P runs over all intersection points of C and 1 C , and 2 σ

( )

P = ±1 is the oriented intersection number of C and 1 C at P . Since the cohomology class of 2 Trφ2

( )

P is independent of P , and equal to that of

∫

_ΣdµTrφ2, eq. (137) implies

(

)

_{( )}

∫

∫

∫

⋅     +       _{+ ∧} = ⋅ Θ + Σ Σ 2 2 2 _{2 8} 1 4 1 exp 1 ' exp _{1 2} φ π ε ψ ψ φ π φ ψ ε ω DAD D Tr i F Tr G vol V V_{C C}

(

)

   ₋ ∩ ⋅

_∫

Σ 2 2 2 1 4 # φ π Tr C C , (138)