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Timoshkin A. V. The asymptotic conformai invariance in Chern-Sirnons theory; with matter in..
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Timoshkin A. V.
THE ASY!V‘ FOriC CONFORIVIAL INVARIANCE IN CHERN-SIMONS THEORY WITH I./-.; *IN
CURVED SPACE-TIME
Tomsk Pedagogical University, 834041 Tomsk,Russia
1. The study of renormalizable field theories in curved space [1,2,3,4] proved the existance of the phenomenon of asymptotic conformal invariance. In this short review we discuss the asymptotic behaviour of 3d CS theories in curved space. Let us consider renormalizable abelian CS theory with scalar and spinor in three dimensions [5]. The Lagrangian looks like:
L = —epv"AA 9 Ax+1 D O f +
2 (1)
+i'¥DW + af ¥#*# - A(®M>)3.
Here Dil=dtl-ieAli, <&,¥ -complex scalar and dirac spinor consequently, coupling constants e,h, a are dimensiionless. The theory with Lagrangian (I) is multiplicatively renormalizable.
The two-loop RG equation for coupling constants has the form [6,7]:
dt
- 0
(8ji)2 = 34a(r)e4(t) - 24/(t) (2)
dt 3
(8tc)2 = 168ft2 (0 - mh(t)cr(t) + 36e\t) +
dt
+8a(t)e6 (t) + 4a2 (t)e6 (?) - 4a4 (r).
It has been shown in paper [6] that for the theory with Lagrangian (1) exists finite four cases in which the theory is finite at two - loop level:
TSFU Vestnik. 2004. Выпуск 7 (44). Volume: NA TURAL AND EXA CT SCIENCES
1.a = 3e2, A = -—e4
4
2. a = 3e2, h = e4;
3. a ~ -2.23e2, h ~ -0.59e4;
4. a = -2.23e\ A = 0.62/.
(3)
In regime of finiteness the effective coupling constants are:
e{t) ~ e, a(f) - a, h(t) - h,
(4)
hit)
ait) ■
- 3e2.
(5)
a : a(0) = 3e~
Then
19 ,
a (i) = 3e2
and
(6)
where -gamma-function for mass of the scalar
field in two - loop approximation. Then with supposition (6), two-loop RG equation for £;(/) has the
form:
(7)
where is given in paper [7]. Then the solution of equation (7) in regimes of finiteness (3) has the form:
where values a, h give one from four variants (3). Let us consider the limit t —> °° (infrared) for solutions of equations (2). in this limit independently of initial
values
(8)
Therefore the theory becomes finite and supersymmetric in the asymptotic [8]. Consequently supersymmetry is infrared stable as far as in four dimensions [8].Let us study now the theory in the limit I +OQ _ jn this case we fix the initial value for
at
t —» +cc, k(i) —> —~ e“ independent of initial value 4
h{0) itly in ultraviolet as
fixes.? ' -= 3e~ the theory etre<
finite (asymptotic fimtness [811 in i value hit) is arbitrary. Fix now a(0)-at f —> oo , h(i) — > 0,62 ’ ' " seridentl)
/*(()>, The theory agat s to be asymptotic finite.
In ultraviolet limit tw. is,.,,ir; of asymptotic finitness exist.
2. Let us consider the behavior of effective charge £(/) in abelian CS theory with matter in curved space -time, the beta-funetion on two-loop level has the form [10].
where y^2,1 = 0 (for N = 2 supersymmetric theory) or
f’ «—^-11.23-6.77.
24k
Therefore at a~ -2.23e2 the theory appears to be asymptotic conformally invariant in d- 3 at / —>«>,
|(0—independent of initial value. At i-»-8
\m-+ ». For N = 2 there are 18 regimes of finiteness for f2,’ can be positive, negative or zero, Thee at t —> » the following situations are possible:
J. £,(f) —> - (asymptotic supersymmetry);
8
2, J;(f ) = Ç (the asymptotic supersymmetry);
S. | Cjt) |.-> °o .
for r -4 « the behavior of £,(/) in I and 3 is exchanged. Thus, it is shown that in d3 gauge theories also can be asymptotic conformai invariante of exponential type, like in [9]. This short review is mainly based on ref. [10].
Acknowledgements.
A.V.T. thanks S.D. Odintsov for help at preparation this article.
References
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Noguchi A„ Sugamoto A. Dynamical origin of duality between gauge theory and gravity
9. Buchbinder I.L., Odintsov S.D., Lichtzier I.L. //Theor. Math. Phys. 1989. V. 79. P. 314; Class. Grav. 1989. V. 8. P. 605; Zaripov R.Sh., Odintsov S.D. // Theor. Math. Phys. 1990. V. 83. P. 399; Mod. Phys. Lett. 1989. V. A4. P, 1955.
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Noguchi A,, Sugamoto A.
DYNAMICAL ORIGIN OF DUALITY BETWEEN GAUGE THEORY AND GRAVITY
Department of Physics and Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610,
Japan
1 Gravity as collective excitaion of dual gauge fields
We are very happy to submit this paper to the special issue of Gravitation and Cosmology on the occasion of 70 years anniversary of Faculty of Physics and Mathematics of Tomsk State Pedagogical University.
It is popular now that by the AdS/CFT correspondence developed by I. Maldacena [1], gauge theory and gravity become dual with each other. In other words, the strong coupling regime of the former theory corresponds to the weak coupling regime of the latter, and vice versa. In this corresponcence D-branes and extra dimension play essential roles. This AdS/CFT correspondece may be considered within the context of the ‘t Hooft-Mandelstam duality [2].
In the late 70's and the early 80’s, similar correspondence was known in which the closed string theory (with the Kalb-Ramond field as a. gauge field of string) is dual to the gauge theory (being massive with or without Higgs field), and the extension of membranes is also considered [3,4,5,6].
Malcacena’s AdS/CFT correspondence is, of cource, the more sophisticated, and provides an extremely powerful tool in studying the realistic hadron physics in QCD. Examples of such study can be found in a review article [7] and in a recent study of pentaquark baryons using the AdS/CFT correspondence [8].
In this paper, we study an origin of the dualiy between gauge theory and gravity (AdS/CFT) within the local field theory using the old-fashioned duality. The idea which our study is based on is as follows. Starting from a gauge theory and applying a dual transformation to it, we obtain a dual theory. (We may start from a manifestly self-dual theory also.) If the gauge coupling of the original theory is e, then the gauge coupling of the dual theory is g -2%/e . When e is small, g is large, so that bound states (or collective excitations) are formed in the dual theory by the exchange of strongly coupled dual gauge bosons. Among them we have a graviton as the collective excitaion of dual gauge bosons. Then, the graivity theory becomes dual to the original gauge theory, since
the former is the low energy manifestation of the dual gauge theory in the strong coupling regime.
This idea is cpite consistent with the fact that in
string theories, a gauge boson is represented by an open string's mode, while a graviton is by a closed
string's mode, so that a closed string can be considered as a bound state of two open strings.
We examine this idea in the usual {/(1) gauge theory, and also in its manifestly self-dual formulation by Zwanziger [9]. More detailed description of this paper can be found in the master tfaeis written by one of the authors (A.N.) f 10],
2 Duality between Ufl) gauge theory and gravity
We start from the 1/(1) gauge theory with a coupling <?. The parition function of this theory reads,
— f f*iv +-F.vypv !>.(
2 2 ' I;'
Z\J)« JlMj, expj i j'(tx
where F = d A, ~dvA^ is the field strength of the gauge field \ , and Jil is an external source.
Using an auxiliary field W/(1V, the partition function becomes [3,4],
Z(J]oc jlIMj)PW/lv expji |ii4X
X
where
1
ip (a = _,p0123=])
Path integration over Д, leads (1) to
(2)
(3)
expj/ji/4*
_1
4g
2 GMVCr
+І-С /•" +-У 2 2
(4)
where Gm, =d Bv -dvBjL is the field strenght of the dual gauge field Bnv, and eg = 2% holds.
