Adelic conformal field theory

Volume 2 IS, number

2

PHYSICS

ADELIC CONFORMAL

FIELD THEORY

LETTERS

B

15 December

1988

*

Bernard GROSSMAN Rockefeller University, New York, NY 10021, USA Received

16 November

I987

We stress the use of modular forms in obtaining adelic formulations of field theoretical problems. Supersymmetry then appears in the real section with the p-adic parts as arithmetic completions. We first show how the Casimir effect is naturally interpreted adelically and the coefficient arises from dimensional analysis. We then suggest looking at the zero slope limit of adelic string amplitudes. Finally, we interpret the rationality of the critical exponents for conformal field theories in terms ofp-adic analyticity of correlation functions.

1. Introduction Most of the results of string theory can be interpreted mathematically in terms of the properties of two-dimensional Riemann surfaces over the complex numbers. The Riemann surface is interpreted as the world-sheet swept out by a string moving through spacetime. However, insofar as there exist Riemann surfaces over the p-adic numbers (the closure of the rational numbers with respect to p-adic norms) there is no reason why the world-sheet, which is not physically observable, cannot be p-adic instead of complex. A p-adic interpolation of string theory based upon the p-adic interpolation of the Veneziano amplitude and its number-theoretical significance was first proposed independently by Volovich [ 1 ] and by the author [ 21. However, this p-adic formulation was problematical insofar as the amplitudes themselves were p-adic numbers. (This was intimately connected with their number-theoretical significance and relation to the Weil conjectures.) A beautiful formulation of the p-adic string was proposed by Freund and Olson [ 3 1, where the world-sheet is p-adic, but the amplitudes are complex-valued. This formulation is intimately connected with modular representations over the p-adic numbers [ 41. There exists an adelic formulation that can be seen to be related to

an idea of Manin [ 5 1, i.e. that in addition to a superspace extension of the ordinary real space, there should be an arithmetic extension. The gamma function will be seen to have a built-in supersymmetry, while the zeta function is the arithmetic extension. However, the most beautiful application of these ideas is to the Casimir effect. We see that the adelic way of looking at things leads to an immediate understanding of the arithmetic form of the answer. In this paper, we shall show that the Veneziano amplitude has not only a p-adic formulation, but also an adelic formulation, i.e. a natural formulation taking into account all the primes including the co-prime which is related to the absolute value norm, a point stressed first by Freund and Witten [ 6 1. The adelic formulation of the Veneziano amplitudes is important insofar as the crossing symmetric amplitude [ 6,7] #’ is shown to be naturally related to the functional relation for the Riemann zeta function. However, there is a problem with defining this adelic product for general values of the momentum [ 6-8 1. We suggest using the zero slope limit which may be related to an adelic form of gauge theories. In addition there are further implications for more general conformal field theories [ 91. Because the critical exponents of the unitary conformal theories in the discrete series are rational, one can use p-adic analysis

* Work supported in part under the Department of Energy Contract Grant Number DE-AC02-87ER-40325. B.

” For the adelic point of view for dynamical 181.

260

systems,

0370-2693lggf$O3.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

see ref.

B.V.

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PHYSICS LETTERSB

to relate the critical exponents to the radius ofp-adic convergence of the series expansions of physical parameters [ I 0 ]. The adelic formulation of the conformal field theory then indicates how to obtain the convergence properties with respect to the absolute value norm. In this paper we shall first review the p-adic Veneziano amplitude in section 2. In section 3 we shall describe the functional relation of the Riemann zeta function and its relation to the Casimir effect and to the adelic Veneziano amplitude. Finally in section 4 we shall discuss the p-adic convergence properties of physical parameters in conformal field theories as inspired by the work of Pearson. For excellent textbooks on p-adic numbers see in addition to ref. [ 4 ] ref. [11].

2. p-adic amplitude for the open bosonic string The 4-point function A T of the open bosonic string has the form [ 12 ]

AT(s, t, u)

15 December1988

P(x)=2F(x) cos(½nx)

(5)

with F(x) the usual gamma function.

Similarly, the p-adic Veneziano amplitude has the integral representation [ 3,4 ]

A~(s, t, u) =Kpz J d2p 12l~-"~s)-I I 1 - 2 l y =")-~ , or, d2p = ( 1 - p - ' )

12p Ip d*2p,

(6a) (6b)

such that the natural invariant measure d*2p satisfies f

d*2p=l

(6c)

12lp=l

=K~ I-[ 1-P-"~x)-I .... t,, 1_p,¢X)

(6d)

This amplitude has one pole per channel, one channel at a time, at m o2 = - 8 , the familiar tachyon and its repetitions m]=-8+--

16nin lnp '

n~7.

I

= K ~ j dt Itl~ ~ " ) - I I I - t l ~ =~')-1 ,

(1)

0

where c~(s)=l+½s, s + t + u = - 8 , o~(s)+o~(t)+ o ¢ ( u ) = - 1. The relation we are interested in involves the crossing symmetric Veneziano amplitude

A~(s, t, u)=AT(s, t, u)+AT(t, u, s)

F(~l ) = f ~(X)~l (X)Ixl- 1/2 dx

+AT(u,s,t) .

A~(s,t,u)

be the Mellin transform of ~ (x) which transforms like

T.(g)(b(x)

=0 (ax÷ fl~ ~(~x+6)I~x+61-1/2 \~,x+~/

= K ~ i dt I t l - ~ c ~ ) - ~ l l - t l ~ ° ) - ~

(3)

Moreover, the crossing symmetric amplitude can be represented as a product formula analogous to the Veneziano amplitude: 1

=-zK~

(7)

(2)

The crossing symmetric amplitude also has an integral representation [ 13 ]:

A~(s,t,u)

Because of the periodicity, with period 16ni/ln p, of the amplitude [ 3 ]. We mention that the Veneziano amplitude has an interpretation as the integral kernel of a representation of SL2 in the space of MeUin transforms of a Hilbert space of modular functions. If we let

2

l~(-°~(s))P(-°~(t)) /~(-o~(s)-o~(t))

(8)

where g = (~ g), and n is a multiplicative character, then

T,(g)F(nl)=fK(glz~lrqrc2)F(n2) d7~2,

(9)

where '

(4)

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Volume 215, number 2

K(g[ nl nl n2) o~x+ fl = nn2 laxWflll/2nSl

PHYSICS LETTERSB

15 December 1988

d*2 = 1-I d*2p,

x n~ (x) Ixl - ~/2 d x .

(13c)

P

yx+J l y x + J I t/2 (10)

2s/21"(S)= i

IxlS-I exp(--½x2) d x .

(13d)

--oo

If we let

If we define

g=(1n n 2-(1x1)) =' l x l - ' ~ ( ~ , - ' / 2 0 nl (x) = Ix[-.
q~(0o, s ) = ~

3. Adelic amplitudes and functional relations

The form obtained by Freund and Olson for the padic Veneziano amplitude is clearly reminiscent of the Euler factors that appear in the familiar form of the Riemann zeta function,

n=l

n"

= I-I ( 1 - p - s )

(12a) •

(lZb)

P

Moreover, the Riemann zeta function can be placed together with the gamma function to satisfy a functional relation

2"/2F(½s)~(s)=2'-S/2F(½(1-s))~(1-s).

(12c)

This can be proved using an adelic representation of ~(s) as a moment of an adelic measure I

, 1 [2plpd*2p= l - p - S '

(13a)

12/, It~ ~< I

~ ( s ) = J I,tl s d*2 with 262

(13b)

e x p ( - x 2) Ixl s-t dx

--oo

(1 1)

then K gives the Veneziano amplitude. Moreover, since absolute values factorize, the kernel K gives a measure on U p SL2(Qp)/SLz(Q). Perhaps there is a relation with the Tamagawa measure. We note that if we define the symplectic matrix J = (_o o~), then gJ=Jg, where g = (~ l ), which gives the same crossing symmetric amplitude by changing x to - x . J is precisely the transformation which yields the functional relation for the zeta function as we shall see below.

~(s)=

1 i

x I-[ P

J"

12pl~d2p,

(14)

121pp ~< 1

one can obtain the functional relation for the Riemann zeta function as a consequence of the Poisson summation formula a~Q

q~(2a) =

1

1-~

o~QE ~ ( ' ~ - t a )

,

(15)

where q~ is the Fourier transform of a function ¢ on the group of adeles Q and 2 is a rational number. We can think of the gamma function as having a built-in supersymmetry. Consider, the two-dimensional gamma function

i i dzdglzln-'exp(-½lz'2)'

(16)

--c~ --co

Iz 12= zg, is the K~ihler potential for the harmonic oscillator with creation and annihilation operators at=z, a=g. Then a t and a define a derivations since [a t, a t ] = 0 [a, a ] = 0 . Moreover, H=ata+½= ½{a t,a} just as in supersymmetry [14]. The functional relation for the zeta function follows from the arithmetic completion necessary to obtain a modular form ~. The symplectic transformation J = (_o o') then is the dual relation switching a and a t. This suggests choosing the Euler factors as the Mellin transforms of p-adic harmonic oscillators on which algebraic extensions of SL2 (Qp) act [ 15 ]. The adelic form of the zeta function, i.e. 2 ,/2 ×F(½s)~(s), has an immediate application to the Casimir effect, a result which should have wide applicationssuch as to the cosmological constant. If we consider the free energy for a single bosonic state for a string theory in four dimensions compactified on a circle, which is equivalent to the energy of the vac-

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15 December 1988

uum of two condenser plates separated by a distance

a=fl/2n,

\ 1

M~O

f

Therefore, up to a finite renormalization of coupling constants (17a)

- 2fl1 3 ~ f dt t_4/2_ l e x p ( - n n Z t )

(lYb)

=4n2 B4 1 4! f13.

(17c)

However, since F(fl) can be written as the moment o f a theta function, it also has an adelic expression:

plY__2K-~zp= 1 .

~

1

q)(4) .

(18)

The Casimir effect is obtained by purely dimensional analysis! By the functional relation, q)(4) is related to • ( - 3 ), the dimension necessary so that the moments scale like fl-3. We note that our calculation has implicitly used zeta function regularization. One can similarly take I-I~=2A~ to obtain an adelic Veneziano amplitude including the usual Veneziano amplitude at p---or, which depends upon gamma functions. Moreover, the functional relation for the Riemann zeta function is clearly related to the adelic amplitudes in the following way:

AP(s,t, w)-_K,,2.,-=,.,,,I] 1-p~C~)-~ l_p,(~) , A~

~(-a(x) )

HK-~= [[ ¢(l+a(x))"

A~-~oKp(~)

n

t,u

/~(I[ 1 "~-OL(X) ] '

=

U

This may be related to p-adic gluon scattering. We note that l

l-I Ag (( - 1 ) - '~ ~ o K ~ - ~(2) - ~(2)

1 -

(23c)

27~ 2"

We suggest that even though the higher N-point amplitudes do not seem to factorize, perhaps the zero slope limit does. Moreover, this product formula can be easily related to the earlier product formula for the norms, i.e. ~ l x l p = 1. The Green's function for conformal field theory G~(z, w) = l o g ] z - wl. We then have the Ward identity

for rational conformal field theory, i.e., a conformal field theory over rational numbers where Gp (z, w) = log Iz - w lp and z, w are rational numbers [ 16 ]. One then obtains a p-adic version of the Veneziano amplitude. In addition, by having the Green's function p-adically, one obtains the p-adic conformal algebra since the Green's function defines a quadratic form. Moreover, the central extension splits as in ref. [ 17 ] because of the Ward identity ~ Gp= O.

(21a,

F(-~(x)) X = S , I , lt

(23b)

(20)

P ....

1

~5 ( P + 1 ) .

~, Gp(z, w)=0

Therefore, we obtain from the functional relation for the Riemann zeta function that

0

-Kp2 -

(23a)

(19)

x=s,l,u

p#~

(22,

It is interesting to consider the zero slope limit of the amplitude as a regularization because the product is ill defined [ 6-8 ]. We obtain

_

F(fl) =

(21d)

K~ u)j

r(~il-~(x)l)r(~[l+~(x,]) (21b) (2n)-3/2cos[½na(x)]r ( - a ( x ) )

(24)

4. Critical e x p o n e n t s a n d a d e l i c field t h e o r y [10]

x=s.&zt

(21c)

In two-dimensional conformal field theory, there exists a discrete series whose unitary representations 263

Volume 215, number 2

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15 December 1988

have been conjectured to be d e t e r m i n e d by a set of central terms c = 1 - 6 / m ( m + 1 ) for positive integral m a n d critical exponents d e t e r m i n e d by rational m-dependent n u m b e r hp.q. The critical c o m p o n e n t s are all rational so we claim that the radius of convergence for the series near the critical point can be determined. The generic singular behavior near a critical point is

before the W i t t e n - F r e u n d paper was written. I would also like to t h a n k the Aspen Institute for Physics where part of this work was carried out a n d H. G a r l a n d for discussions on adeles. I especially t h a n k Mike M c G u i g a n for suggesting the more elegant way of calculating the Casimir effect. I also t h a n k V. Rittenberg for encouragement and discussions.

f ( x ) ~ (1 - x ) P g ( x )

References

.

(25)

If the radius of convergence o f g ( x ) is at least that of ( 1 - x ) p we can determine the radius of convergence of f(x) = ~f,x

(26)

~.

Suppose that asymptotically (27)

I f ~ l p ~ p ''~,' .

T h e n since (28)

[n!l~,~p -'/p-'

we have that in order for I(1-x)alP
,

(29a)

E IxlT, lfllg [n!lp

y~ Ixlj!fl__l~ < 1 p-n/p-I '

(29b)

1 Ifll,,= (P .... , / p _ , ) _ , .

(29C)

Since we know fl, we can determine the p-adic radius of convergence for any p-adic conformal field theory in the discrete series.

Acknowledgement The author's work grew out of a E. Witten on functional relations tion for an elliptic curve. I would for generously suggesting to look

264

conversation with for the zeta funclike to t h a n k h i m at F r e u n d ' s work

[ 1] I.V. Volovich,Class. Quantum Grav. 4 ( 1987 ) L83. [2] B. Grossman, Phys. Lett. B 197 (1987) 101. [3] P.G.O. Freund and M. Olson, University of Chicago preprints (1987). [4]I.M. Gel'fan& M.I. Graev and I.I. Piatetskii-Shapiro, Representation theory and automorphic functions (Saunders, Philadelphia, 1969). [ 5 ] Y.I. Manin, Bonn Arbeitstag Conference (Bonn, 1984). [6 ] P.G.O. Freund and E. Witten, Phys. Len. B 199 ( 1987) 191. [ 7 ] L. Brekke, P.G.O. Freund and E. Witten, Chicago preprint; E. Marinari and G. Parisi, Rome preprint; Z. Hlousek and D. Spector, Cornell preprint; I.V. Volovich, Steklov preprint. [8 ] Y. Meurice, Fermilab preprint. [9] A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Nucl. Phys. B 241 (1980) 333; D. Friedan, S. Shenker and Z. Qiu, in: Vertex operators in mathematics and physics, eds. J. Lepowsky et al., MSRI publication #3 ( 1985). [ 10] R. Pearson, Phys. Rev. B 22 (1980) 3465. [11] N. Koblitz, p-adic numbers, p-adic analysis and zeta functions (Springer, Berlin, 1977); J.W.A. Cassels, Local fields (Cambridge U.P., Cambridge, 1986). [ 12] G. Veneziano,Nuovo Cimento 57A (1986) 190. [ 13 ] M. Ademollo,A. d'Adda, R. d'Auriau, E. Napolitano, P. di Vecchia, F. Gliozzi and S. Sciuto, Nucl. Phys. B 77 (1974) 189. [ 14] M. De Crombruggheand V. Rittenberg, Ann. Phys. (NY) 151 (1983) 99. [ 15 ] B. Grossman, in preparation. [ 16] K. Ueno, Kyoto preprint; J.K. Smit, University of Utrecht preprint. [17] E. Witten, Quantum field theory, grassmannians and algebraic curves, Princeton preprint ( 1987).