- Email: [email protected]
Non-perturbative two-dimensional quantum gravity, again
Nuclear Physics B (l'r~e. Suppl.) 25? (1992) 87-91 No rth- I loll,rod
PROCEEDINGS SUPPLEMENTS
NON-PERTU RBATIVE TWO-DIMENSIONAl, QUANTUM GRAVITY, AGAIN Simon Dalley, Clifford J o h n s o n a n d T i m Morris
Departrael,t o] Physics, Universit~ o] Southampton, Southampton S09 5NIl, U.K. This is an updated review of recent work done with C.Jolmsonand T.Mor¢is concerning a proposal for nonperlurbat.ively stable 2D tn!anturq gravity coupled to c < I mailer, hamed on the flows of the (Eeneralis~t) [~dV hkrarch; [1,2].
S i , c e tile discovery of the c o n t i u u u m limit of 21) q u a n t u m gravity coupled to minimal m a t ~or [3.4] to all orders in the genus expansion, the question of a n o n - p e r t n r b a t i v e definitio, of these theories h a s remained. In refs. [1.2] we have presented a 'physical' n o n - p e r t u r b a t i v e definition by elevating the f u n d a m e n t a l (generalised) K d V tk,w s y m m e t r y , known to exi:t perturbatively, to a n o n - p e r t u r b a t i v e principle. C o n c e n t r a t i n g on KdV for sm~plieity, these flows are [,5]:
0_.~ = I~,
/)It
[,,] = ,~ ~, .,'.
(1)
~oa
~tt
I
(4)
k=a
+ II2)l~n~
- ~= ~ = o
('z)
Using (1) and the recurrence re!ation [6];
k
1 ,,
Ilere tile string suseeplibi['_ty u = r " , prhne is d/dz where : = p/u, while p a n d v are the ren~rmalised 'cosntologica[ con~lant' and 1IN. T h e Imramett'rs I k / v couple to the m a t r i x - m o d e l inl,'grated local o p e r a t o r s O~: on the surfaces a n d R ~ are the Celfand-Dikii differentials [6]. T h e fl~ws I~tt.rmir~e correlators of" the lo~a! '~peratots: Oz.z
J
L , Ck+ I/~) t ~ , . + : : ~ + u = 0
where ~(k
Tile striug e q u a t i o n (2) is the usual cond;tton imposed by tile h e r m i t i a n m a t r i x m o d e l ( H M M ) [3], leading to non-perturb~±ive instability in particular for pure gravity anti p r o b a b l y all u n i t a r y c < 1 m a t t e r . Let us relax this condition a n d ask f~)r the most gene:a] string e q u a t i o n c o m p a t ible with the flows (1). To derive it note t h a t diuteusiouafly if In] = 1 ~hen [z] = - 1 / 2 a n d [/~:] = - ( k + 1/2) ~ Ibllows from (1). T h e n u as a ftmclrion ~i" its dimensionful a r g u m e n t s niust have tile scaling s y m m e t r y #t ;
< 0 , , . . . 0 ~ . > = a t e , . . • Ot~. u
(3)
I
f .
this becomes: ~Tz_l , , - n ~ ' - l u ' l ~ = 0
(6)
Multiplyiug by ,~. and i n t e g r a t i n g gives the strir.g equatios [1]: u~ ~ -
~'R.'R." -I-~ ( ' g ' ) '
=0
(7)
#1 We a.~sume iultially that no new dimenslonful paxametel arises ~t the Iloll-peftllrbatJve level•
0920-5c,37102/$05 0~) (~)1.°,92- El~evier grien~' P.hli*hera R.V, All rigl~ts r~,rved•
S. DM/ey eg a/. / Non-Perturbative 2D Quantum Gra¢it¥, Again
88
The constant of integration (which is an order v ~ term) has been set to z~ro by hand L+ order that the d M M equation ~ = 0 satisfies (7) asymptotically. This is necessary to reproduce the perturbative (genus) expansion at z --* +co. Note that the asymptotic expansion of the solution to (7) is uniquely determined once we have fixed the spherical approximation. As an illustrative ease we can consider pure gravity t m = t,~6rn2. In the spherical approximation, where we can ncglect derivatives, (7) becomes: u ( . ~ - z) ~- = 0
. = - ~ c .
(9)
= ; z ~
-co
It is highly likely t h a t all other non-pc, turbative solutions to (7) reproducing the standard genus expansion at + c o are complex or have poles on the real axis and are thus physically unacceptable[7]. The z ~ 4-0o behaviour is now enough to fix completely the boundary conditions for (7) [2] and the (unique) numerical solution we found for pure gravity g~ shown in fignre 1. On the question of uniqueness we note also t h a t at t,n = t m / ~ l , (7) is reducible to Painleve II in hy the transformation u = 2X 2 q- z. With our physical boundary conditions it has already been #2 The same asymptotic behaviour has also recently been found in the unitary matrix model with 'external fields' [9]
_J: Fig..I.
(8)
As z ~ + c o we take u = v ~ and the corrections will y'.'cld s*~altd~d genus expanc,ion, by construction. If we require n to have a real asymptotic expansion at z ~ - c o then we must take the root u = 0. The susceptibility then beg;us at the 'torus' level and in fact ha~" non-vanishing contributioas only ~'very 2rid ge,+us. More generally it is every mth genus #2 [1]:
Co=1/4
,10 +
shown analytically t h a t there is a unique, real, pole-free solution [10]. Thus, starting from the KdV flows and requiring physicaUy acceptable behaviour in choosing boundary conditions, we are led to the solution for pure gravity shown in figure 1. Considering u as the potential for a +:~miltoi~ian operator H, whose eigenvalues govern the non-perturbative positions of charges in a Dyson gas, we see t h a t the spectrum is continuous and bounded below. In fact our analysis is equivalent to taking the contiuuum limit of an appropriate (critical) Dyson gas on the positive real line, as we show in a moment. ~Ve can compare our stabi:isa:ion with results provided by the SUSY D = I / stochastic quap:isation of the IIMM investigated in refs. [SJ. The non-perturbative difference between the two at positive z is obvious from the manifostly different behaviour of the solutions a t z ~ - c o . For the model investigated in [8], sin~e n is real and cx ~ at z ~ - c o [11,12], the susceptihility cannot satisfy (7) and so violates the KdV flows non-perturbatively. Also there still appear to be eon-perturbativc ambiguities in the approach of [8] in its most general form, while there are none for our approach. O u r anaiysis so far has dealt directly with the
S. DMley et al. / Non-Perturbatlve 2D Quantum Gravity, Aga/n
(al
•
(b)
Fig. 2. continuum limit and it is natural to ask whether there is an appropriate formulation in terms of a Dyson gas. The universality class of the ~ritical b,~haviour in such a gas can be conveniently characterised by the structure of the eigenvalue density in the scaled neighbourhood of the end of an arc of eigenvalues, in the spherical approximation[13]. Figure 2(a) shows this region as appropriate for pure gravity (m = 2) at z > 0. Eigenva!nes are concentrated on the cut (wavy line) ended by a squa: e root branch point, and formally extending the expression for the density off this cut, its first integral has the interpretation of effective potential for one eigenvalue, Ven [12]. For the ruth critical point there are m - 1 extra zeros of the density in the scaled neighbourhood of the branch point, shown as dots in the figure. The behaviour as a function of z is easily calculated by using the W K B approximation for the spect-um of the co-ordinate operator H , with the appropriate "potential' u from (8). This also facilitates a comparison of the extremal values of Vefr with the leading exponential corrections to the asymptotic solutions to (7) [2]. As z ~ 0 the branch point and the e x t r a zero in figure 2(a) collide and at first it is difficult to see how a real, sensible answer is maintained for z < 0. A solution is shown in figure 2(b) whereby the density now diverges, rather than vanishes, like a square root at the branch point, thus generating an extra zero. This scenario" is thoroughly
80
natural if there is a 'wall' in the problem, which tile branch point hits at z = 0. In this ease the two zeros depart into the complex plane perpendicular to the real axis as z becomes more negative. The structure shown in figure 2(a) to the left of this wall is then in the sense of analytic coutinuation (V~tt -- oo strictly). Simply speaking tile wall will prevent eigenvalues leaking out of the arc and this is in fact the origin of nonperturbative stability in our formulation, as we nOW show.
We can derive the full continuum limit h la Dougb.~ [4] which also indicates the inclusion of the generalised KdV hierarchy. Insertions of the elgenvalue operators - A , - d / d A have, for the ruth critical point of the one-matrix model, the continuum limits: Q = -II ~d z-u; 2m
P =
-
d=d/dz
(10)
1
Z aid/ i=0
(11)
The eigenvalue space of the HMM is IF'~ and the canonical momentum operator is P (translaticu~s) The c.c.r. [P,Q] = I determines the ol : i < 2m - 1 e-iquely (~2m-! is an undeternfined non-universal constant), the order d o of this relation being "/~' = 0. Our stabilisation scheme is equivalent f ~ the restriction to 1~.+ i.e. the imposition of a 'wall' in the scaling region. To sitow this we note that the canonical momentum on ]i~.+ ts /3, the generator of scale transformations, satisfying the new e.c.r. [ / 3 Q ] = Q. If Q is conjugat~ to macroscopic loop length then t h e ~ local .,tale transformations are related to physical scale transformations of the string/onedimensional universe. We can find/3 by introducing the fractional-power w~udo-differential operator and its differemial part: Q , , + t p . = d.~,~+t + ( m +
l / 2 ) { u , d ~m-l}
+...{tt.:+:[ul,a -~} + . . .
~12)
S. Dailey el at. / Non-Perlurhative 2D Quantum Gravity, Again
90
[Q~.,+vz, Q] = p~+~
(13)
We expect /5 = z.~i=oX"~+la'd~, but Q,~+ll2 d o ~ not quite do the job. To obtain a term d 2 on the r.h.s, of the c.c.r, we must use most generally: /5 ~" O'2m'l'll~'l" r~,-+l/2 -- _z 2d
(14)
formal Virasoro constraints [16]. O u r differentiated string equation (7) follows from the constraint L 0 r = 0 ( w h e r e u = - 2 d 2 1 o g r and z ~ z + to) on using the KdV flow (I). This is not suprising in t h a t it is precisely the expression of scale invariance. The higher Virasoro constraints:
giving the (differential of the) string equation; Lnr=0
s
[/5, Q] = a~m+lR'm+l + ~
+ d ~ = d2 - - u . ( 1 3 )
This is (6) at the ruth critical voint, mouulo a2,n+1 which may be absorbed into the string coupling. By explicitly taking the continuum limit of the Dyson gas on R+, realized for example by the complex matrix model [I], one finds that the constant arising from an integration of (15) is zero as in (7) Moreover the physical boundary conditions we chose earlier are precisely the ones appropriate to this gas. In the spherical approxiw.ation u is the scaling part of the end of the eigenvalue density. For z > 0 the 'wall' has no affect on the W K B expansion, in particular the ieading term is ~ as in the IIMM. At z < 0 the end of the density remains fixed at the wall ~ u = 0 in the spherical approximation. It is natural to suppose t h a t these considerations have analogues for the generalised KdV hierarchy, related to the full set of (p, q) minimal models [4,14]. One would generalise to: q-2
Q=d,+~u,~'
P=~/'
(16)
i=0
/5 = Ql++pIq _ z d q
(.17)
Some details of this will be explored in [15]. Let us finally consider the Dyson-Sehwinger equations #3 of the one-matrix models, at least that part of them which mv.y be represented as #3 Some of what follows was elaboratec" wh'le these notes were being prepared.
: n> 1
(18)
where L. =
k+
t k a~-T~--"
k=0 l .n
02
+~ ~ a~k-~l,.-k
(10)
folios by applying the recursion operator which generates higher symmetries of the KdV hierarchy [17] # 4 Tile IlMM string equation derives from L _ t r = 0 and one might ask what happens to this constraint in our formulation. Since we are working on ~,+ there is no translation invariance and tile L - t constraint is naively absent. More precisely the relevant DS equation picks up a bmntdary term from the 'wall': ar L_ ~r = O-~
(~0)
where # is tire scaled position of the 'wall', which up 1o no~ we have taken as the origin. In fact there is no reason why we should choose # = 0 aud ntore correctly we might consider it as an ext r a (non-I)erturbative) parameter u = u(z, tk, tr). 'File correct canonical momentum gets shifted now t o / 3 = /5 + # P and thus
[Po,Q+ ~] =Q+~
(~1)
implying #4 II.Kawal irdormed me that the DS constraints {Lar --- 0 : n _> O} should in turn imply that ¢ is a r-ftmction of the KdV hierarchy subject to L0¢ = 0 (see iris contribution to these proceedings).
S. DMley et al. /Non-Pert.rbative 2D Quantum Gravity, Again 1 ..,. ,~
1 - ,t~' - ~
i
~ + ~
,
= 0
(2~)
which m a y now be i n t e g r a t e d to give the s t r i n g equation. B u t a , which h a s the s a m e dimen[ion as n, now contributes to the scaling equation (4) a t e r m o'8u/i)a. C o m p a r i n g with (22) we identify 0u = n' 0~,
--
(23)
which is equivale,3, to (20). T h e o t h e r Virasoro constraints also pick up b o u n d a r y t e r m s a t general a L . ," - a "+~ ~
(24)
as follows s i m p l y from v a r y i n g the b o u n d a r y of the eigenvalue integration as a ~
a + e a "+1.
T h e p a r a m e t e r a can be set to zero by an analytic redefinition of t h e tt t o g e t h e r with u a ~
n. In this ~ n s e it is r e d u n d a n t a n d in
fact if we a s s u m e , following the I t M M , t h a t the non-perturbative
loop e x p e c t a t i o n is q~(l) : <
e - H ! > , a corresponds to a b o u n d a r y cosmological c o n s t a n t [18]. T h e loop wavefunction in the scaling limit is
@(I) oc
p(tb)e-~¢ d¢
(25)
< x [/~(~', - t t ) [ x > d.r
(26)
where p(¢,) 0¢
5
Hence for a > 0 t h e wavefunction is g u a r a n t e e d to have sensible exponentially decreasing large l behaviour.
91
Eeferenees [I] S.Dal|ey, C.John~on and T.Morrls, Sou~.hampton Univ. preprint SIIEP 90/91-1fi. [2] S.Dalley, C.Johs~on and T.Morris, Southampton Univ. preprint S|IEP 90191-28. [3] D.J.Gross atnq A.A.MiKdal , t~hys, t~v. I~-Vtt.tt4 (1990) 127; M.R.Douglas and S.H.Shenker, Nucl. Phys. B33S (1990) 135; E.Br~in trod V.A.Kazakov, Phys. Left. B236 (1990} 144. [4] ~.LR.Doug|a~, Phys. Left. B238 (1990) 176. [5] I ) . J . ( ; r ~ and A.A.Migdal, Nucl. Phys. B340 (19~{:) ~33: T.Banks, N.S~iberg, M.[l.Doug|as and S.II.Shenker, Phys. Lett. 11t238 (1990) 279. [6] I.R.Gel/and and l,.A.Dikii, Russlam Math. Surveys 30 (1975) 77. [7] F David, Mod. Phys. L~tt. AS (1990) 1019. iF| E.Marinaxi and G.Parisi, Phys. Lett. B240 (1990) 375. ~.Air,i)jern, J.Grer~ite emd S.Vatsted, Phys. Lctt. ][1249 (1¢./9~J) 411; E.Me,rinari and G.Patisi, Phys. I,etL 11247 (I'.YJO) 537; M.Karliner anti A.A.Migdal, Mtxl. Phys. l~ett. A5 (!990) 2565; J.Ambj~srn and J.Grepnsite, Phys. left. B254 ( t ~ ) ] ) e~.
[9] I).J.f;ross and M.J.Newman, Princeton preprint PUPT-1257. [IO] S.P.ll~,lings a:~d J.B.Ma~le~)d, Arch. Rat. Mech. and Anal. 13 11980) 31. ill| J.Ambj~rn, C.Johnson aswl T.Morris, preprlnt SIIEP 90/91-29 / NBI-|IE-91-27. [12] F.l)avid. In Proceedings of workshop op [{and~m Surfaces, Quantum Gravity and Strings C a r g o , May 28-Jime I 1990. [13] S.Dalley, P|lys. Let(. 11253 (1991) 292; P.G;rL,paig at,,| J.Zi¢lq-J~Lstin, Phys. [,eft. 11255 (19f)l) 189; II.NeuberKer, Nuc|. Phys. B352 (lr.:VJI) 6~9; F.l)avid, Nud. Ph:~s. B348 11991) 507. [I.I] P. di Francesco and D.Kut~m)v, Nud. Phys. B342 { t~N)) ~ 9 .
Acknowledgements S D would like to t h a n k the o r g a n | s e t s for the o p p o r t u n i t y to present this work and workshop p a r t i c i p a n t s for their interest. Financial s u p p o r t f r o m the S.E.R.C. is acknowledged by S D and CJ.
[1.5] C.Join~scm, T.Morrls and B.Spence, in preparation. [16] M.Fukuma, I|.Kaw~i and R.Nakayama, Int. J. Mod. Phys. AG (19ql) 1385; R.Dijkgra,M', E.Verlinde and I|.Verli!!de, Nucl. Phys. B34g (1991) 435. [17] P.J.Olver, Applieatio,t* o] Lie Group~ ~,, Di.Drre~t. ttal Eqealion$, Sl~;nger-Verlag, 1986. [18] E.,M~rllnec, G.Moore a.~l N.Selherg, preprint HU14-91 ] YCTP-PIO~91 / EFI-91-14.
PROCEEDINGS SUPPLEMENTS
NON-PERTU RBATIVE TWO-DIMENSIONAl, QUANTUM GRAVITY, AGAIN Simon Dalley, Clifford J o h n s o n a n d T i m Morris
Departrael,t o] Physics, Universit~ o] Southampton, Southampton S09 5NIl, U.K. This is an updated review of recent work done with C.Jolmsonand T.Mor¢is concerning a proposal for nonperlurbat.ively stable 2D tn!anturq gravity coupled to c < I mailer, hamed on the flows of the (Eeneralis~t) [~dV hkrarch; [1,2].
S i , c e tile discovery of the c o n t i u u u m limit of 21) q u a n t u m gravity coupled to minimal m a t ~or [3.4] to all orders in the genus expansion, the question of a n o n - p e r t n r b a t i v e definitio, of these theories h a s remained. In refs. [1.2] we have presented a 'physical' n o n - p e r t u r b a t i v e definition by elevating the f u n d a m e n t a l (generalised) K d V tk,w s y m m e t r y , known to exi:t perturbatively, to a n o n - p e r t u r b a t i v e principle. C o n c e n t r a t i n g on KdV for sm~plieity, these flows are [,5]:
0_.~ = I~,
/)It
[,,] = ,~ ~, .,'.
(1)
~oa
~tt
I
(4)
k=a
+ II2)l~n~
- ~= ~ = o
('z)
Using (1) and the recurrence re!ation [6];
k
1 ,,
Ilere tile string suseeplibi['_ty u = r " , prhne is d/dz where : = p/u, while p a n d v are the ren~rmalised 'cosntologica[ con~lant' and 1IN. T h e Imramett'rs I k / v couple to the m a t r i x - m o d e l inl,'grated local o p e r a t o r s O~: on the surfaces a n d R ~ are the Celfand-Dikii differentials [6]. T h e fl~ws I~tt.rmir~e correlators of" the lo~a! '~peratots: Oz.z
J
L , Ck+ I/~) t ~ , . + : : ~ + u = 0
where ~(k
Tile striug e q u a t i o n (2) is the usual cond;tton imposed by tile h e r m i t i a n m a t r i x m o d e l ( H M M ) [3], leading to non-perturb~±ive instability in particular for pure gravity anti p r o b a b l y all u n i t a r y c < 1 m a t t e r . Let us relax this condition a n d ask f~)r the most gene:a] string e q u a t i o n c o m p a t ible with the flows (1). To derive it note t h a t diuteusiouafly if In] = 1 ~hen [z] = - 1 / 2 a n d [/~:] = - ( k + 1/2) ~ Ibllows from (1). T h e n u as a ftmclrion ~i" its dimensionful a r g u m e n t s niust have tile scaling s y m m e t r y #t ;
< 0 , , . . . 0 ~ . > = a t e , . . • Ot~. u
(3)
I
f .
this becomes: ~Tz_l , , - n ~ ' - l u ' l ~ = 0
(6)
Multiplyiug by ,~. and i n t e g r a t i n g gives the strir.g equatios [1]: u~ ~ -
~'R.'R." -I-~ ( ' g ' ) '
=0
(7)
#1 We a.~sume iultially that no new dimenslonful paxametel arises ~t the Iloll-peftllrbatJve level•
0920-5c,37102/$05 0~) (~)1.°,92- El~evier grien~' P.hli*hera R.V, All rigl~ts r~,rved•
S. DM/ey eg a/. / Non-Perturbative 2D Quantum Gra¢it¥, Again
88
The constant of integration (which is an order v ~ term) has been set to z~ro by hand L+ order that the d M M equation ~ = 0 satisfies (7) asymptotically. This is necessary to reproduce the perturbative (genus) expansion at z --* +co. Note that the asymptotic expansion of the solution to (7) is uniquely determined once we have fixed the spherical approximation. As an illustrative ease we can consider pure gravity t m = t,~6rn2. In the spherical approximation, where we can ncglect derivatives, (7) becomes: u ( . ~ - z) ~- = 0
. = - ~ c .
(9)
= ; z ~
-co
It is highly likely t h a t all other non-pc, turbative solutions to (7) reproducing the standard genus expansion at + c o are complex or have poles on the real axis and are thus physically unacceptable[7]. The z ~ 4-0o behaviour is now enough to fix completely the boundary conditions for (7) [2] and the (unique) numerical solution we found for pure gravity g~ shown in fignre 1. On the question of uniqueness we note also t h a t at t,n = t m / ~ l , (7) is reducible to Painleve II in hy the transformation u = 2X 2 q- z. With our physical boundary conditions it has already been #2 The same asymptotic behaviour has also recently been found in the unitary matrix model with 'external fields' [9]
_J: Fig..I.
(8)
As z ~ + c o we take u = v ~ and the corrections will y'.'cld s*~altd~d genus expanc,ion, by construction. If we require n to have a real asymptotic expansion at z ~ - c o then we must take the root u = 0. The susceptibility then beg;us at the 'torus' level and in fact ha~" non-vanishing contributioas only ~'very 2rid ge,+us. More generally it is every mth genus #2 [1]:
Co=1/4
,10 +
shown analytically t h a t there is a unique, real, pole-free solution [10]. Thus, starting from the KdV flows and requiring physicaUy acceptable behaviour in choosing boundary conditions, we are led to the solution for pure gravity shown in figure 1. Considering u as the potential for a +:~miltoi~ian operator H, whose eigenvalues govern the non-perturbative positions of charges in a Dyson gas, we see t h a t the spectrum is continuous and bounded below. In fact our analysis is equivalent to taking the contiuuum limit of an appropriate (critical) Dyson gas on the positive real line, as we show in a moment. ~Ve can compare our stabi:isa:ion with results provided by the SUSY D = I / stochastic quap:isation of the IIMM investigated in refs. [SJ. The non-perturbative difference between the two at positive z is obvious from the manifostly different behaviour of the solutions a t z ~ - c o . For the model investigated in [8], sin~e n is real and cx ~ at z ~ - c o [11,12], the susceptihility cannot satisfy (7) and so violates the KdV flows non-perturbatively. Also there still appear to be eon-perturbativc ambiguities in the approach of [8] in its most general form, while there are none for our approach. O u r anaiysis so far has dealt directly with the
S. DMley et al. / Non-Perturbatlve 2D Quantum Gravity, Aga/n
(al
•
(b)
Fig. 2. continuum limit and it is natural to ask whether there is an appropriate formulation in terms of a Dyson gas. The universality class of the ~ritical b,~haviour in such a gas can be conveniently characterised by the structure of the eigenvalue density in the scaled neighbourhood of the end of an arc of eigenvalues, in the spherical approximation[13]. Figure 2(a) shows this region as appropriate for pure gravity (m = 2) at z > 0. Eigenva!nes are concentrated on the cut (wavy line) ended by a squa: e root branch point, and formally extending the expression for the density off this cut, its first integral has the interpretation of effective potential for one eigenvalue, Ven [12]. For the ruth critical point there are m - 1 extra zeros of the density in the scaled neighbourhood of the branch point, shown as dots in the figure. The behaviour as a function of z is easily calculated by using the W K B approximation for the spect-um of the co-ordinate operator H , with the appropriate "potential' u from (8). This also facilitates a comparison of the extremal values of Vefr with the leading exponential corrections to the asymptotic solutions to (7) [2]. As z ~ 0 the branch point and the e x t r a zero in figure 2(a) collide and at first it is difficult to see how a real, sensible answer is maintained for z < 0. A solution is shown in figure 2(b) whereby the density now diverges, rather than vanishes, like a square root at the branch point, thus generating an extra zero. This scenario" is thoroughly
80
natural if there is a 'wall' in the problem, which tile branch point hits at z = 0. In this ease the two zeros depart into the complex plane perpendicular to the real axis as z becomes more negative. The structure shown in figure 2(a) to the left of this wall is then in the sense of analytic coutinuation (V~tt -- oo strictly). Simply speaking tile wall will prevent eigenvalues leaking out of the arc and this is in fact the origin of nonperturbative stability in our formulation, as we nOW show.
We can derive the full continuum limit h la Dougb.~ [4] which also indicates the inclusion of the generalised KdV hierarchy. Insertions of the elgenvalue operators - A , - d / d A have, for the ruth critical point of the one-matrix model, the continuum limits: Q = -II ~d z-u; 2m
P =
-
d=d/dz
(10)
1
Z aid/ i=0
(11)
The eigenvalue space of the HMM is IF'~ and the canonical momentum operator is P (translaticu~s) The c.c.r. [P,Q] = I determines the ol : i < 2m - 1 e-iquely (~2m-! is an undeternfined non-universal constant), the order d o of this relation being "/~' = 0. Our stabilisation scheme is equivalent f ~ the restriction to 1~.+ i.e. the imposition of a 'wall' in the scaling region. To sitow this we note that the canonical momentum on ]i~.+ ts /3, the generator of scale transformations, satisfying the new e.c.r. [ / 3 Q ] = Q. If Q is conjugat~ to macroscopic loop length then t h e ~ local .,tale transformations are related to physical scale transformations of the string/onedimensional universe. We can find/3 by introducing the fractional-power w~udo-differential operator and its differemial part: Q , , + t p . = d.~,~+t + ( m +
l / 2 ) { u , d ~m-l}
+...{tt.:+:[ul,a -~} + . . .
~12)
S. Dailey el at. / Non-Perlurhative 2D Quantum Gravity, Again
90
[Q~.,+vz, Q] = p~+~
(13)
We expect /5 = z.~i=oX"~+la'd~, but Q,~+ll2 d o ~ not quite do the job. To obtain a term d 2 on the r.h.s, of the c.c.r, we must use most generally: /5 ~" O'2m'l'll~'l" r~,-+l/2 -- _z 2d
(14)
formal Virasoro constraints [16]. O u r differentiated string equation (7) follows from the constraint L 0 r = 0 ( w h e r e u = - 2 d 2 1 o g r and z ~ z + to) on using the KdV flow (I). This is not suprising in t h a t it is precisely the expression of scale invariance. The higher Virasoro constraints:
giving the (differential of the) string equation; Lnr=0
s
[/5, Q] = a~m+lR'm+l + ~
+ d ~ = d2 - - u . ( 1 3 )
This is (6) at the ruth critical voint, mouulo a2,n+1 which may be absorbed into the string coupling. By explicitly taking the continuum limit of the Dyson gas on R+, realized for example by the complex matrix model [I], one finds that the constant arising from an integration of (15) is zero as in (7) Moreover the physical boundary conditions we chose earlier are precisely the ones appropriate to this gas. In the spherical approxiw.ation u is the scaling part of the end of the eigenvalue density. For z > 0 the 'wall' has no affect on the W K B expansion, in particular the ieading term is ~ as in the IIMM. At z < 0 the end of the density remains fixed at the wall ~ u = 0 in the spherical approximation. It is natural to suppose t h a t these considerations have analogues for the generalised KdV hierarchy, related to the full set of (p, q) minimal models [4,14]. One would generalise to: q-2
Q=d,+~u,~'
P=~/'
(16)
i=0
/5 = Ql++pIq _ z d q
(.17)
Some details of this will be explored in [15]. Let us finally consider the Dyson-Sehwinger equations #3 of the one-matrix models, at least that part of them which mv.y be represented as #3 Some of what follows was elaboratec" wh'le these notes were being prepared.
: n> 1
(18)
where L. =
k+
t k a~-T~--"
k=0 l .n
02
+~ ~ a~k-~l,.-k
(10)
folios by applying the recursion operator which generates higher symmetries of the KdV hierarchy [17] # 4 Tile IlMM string equation derives from L _ t r = 0 and one might ask what happens to this constraint in our formulation. Since we are working on ~,+ there is no translation invariance and tile L - t constraint is naively absent. More precisely the relevant DS equation picks up a bmntdary term from the 'wall': ar L_ ~r = O-~
(~0)
where # is tire scaled position of the 'wall', which up 1o no~ we have taken as the origin. In fact there is no reason why we should choose # = 0 aud ntore correctly we might consider it as an ext r a (non-I)erturbative) parameter u = u(z, tk, tr). 'File correct canonical momentum gets shifted now t o / 3 = /5 + # P and thus
[Po,Q+ ~] =Q+~
(~1)
implying #4 II.Kawal irdormed me that the DS constraints {Lar --- 0 : n _> O} should in turn imply that ¢ is a r-ftmction of the KdV hierarchy subject to L0¢ = 0 (see iris contribution to these proceedings).
S. DMley et al. /Non-Pert.rbative 2D Quantum Gravity, Again 1 ..,. ,~
1 - ,t~' - ~
i
~ + ~
,
= 0
(2~)
which m a y now be i n t e g r a t e d to give the s t r i n g equation. B u t a , which h a s the s a m e dimen[ion as n, now contributes to the scaling equation (4) a t e r m o'8u/i)a. C o m p a r i n g with (22) we identify 0u = n' 0~,
--
(23)
which is equivale,3, to (20). T h e o t h e r Virasoro constraints also pick up b o u n d a r y t e r m s a t general a L . ," - a "+~ ~
(24)
as follows s i m p l y from v a r y i n g the b o u n d a r y of the eigenvalue integration as a ~
a + e a "+1.
T h e p a r a m e t e r a can be set to zero by an analytic redefinition of t h e tt t o g e t h e r with u a ~
n. In this ~ n s e it is r e d u n d a n t a n d in
fact if we a s s u m e , following the I t M M , t h a t the non-perturbative
loop e x p e c t a t i o n is q~(l) : <
e - H ! > , a corresponds to a b o u n d a r y cosmological c o n s t a n t [18]. T h e loop wavefunction in the scaling limit is
@(I) oc
p(tb)e-~¢ d¢
(25)
< x [/~(~', - t t ) [ x > d.r
(26)
where p(¢,) 0¢
5
Hence for a > 0 t h e wavefunction is g u a r a n t e e d to have sensible exponentially decreasing large l behaviour.
91
Eeferenees [I] S.Dal|ey, C.John~on and T.Morrls, Sou~.hampton Univ. preprint SIIEP 90/91-1fi. [2] S.Dalley, C.Johs~on and T.Morris, Southampton Univ. preprint S|IEP 90191-28. [3] D.J.Gross atnq A.A.MiKdal , t~hys, t~v. I~-Vtt.tt4 (1990) 127; M.R.Douglas and S.H.Shenker, Nucl. Phys. B33S (1990) 135; E.Br~in trod V.A.Kazakov, Phys. Left. B236 (1990} 144. [4] ~.LR.Doug|a~, Phys. Left. B238 (1990) 176. [5] I ) . J . ( ; r ~ and A.A.Migdal, Nucl. Phys. B340 (19~{:) ~33: T.Banks, N.S~iberg, M.[l.Doug|as and S.II.Shenker, Phys. Lett. 11t238 (1990) 279. [6] I.R.Gel/and and l,.A.Dikii, Russlam Math. Surveys 30 (1975) 77. [7] F David, Mod. Phys. L~tt. AS (1990) 1019. iF| E.Marinaxi and G.Parisi, Phys. Lett. B240 (1990) 375. ~.Air,i)jern, J.Grer~ite emd S.Vatsted, Phys. Lctt. ][1249 (1¢./9~J) 411; E.Me,rinari and G.Patisi, Phys. I,etL 11247 (I'.YJO) 537; M.Karliner anti A.A.Migdal, Mtxl. Phys. l~ett. A5 (!990) 2565; J.Ambj~srn and J.Grepnsite, Phys. left. B254 ( t ~ ) ] ) e~.
[9] I).J.f;ross and M.J.Newman, Princeton preprint PUPT-1257. [IO] S.P.ll~,lings a:~d J.B.Ma~le~)d, Arch. Rat. Mech. and Anal. 13 11980) 31. ill| J.Ambj~rn, C.Johnson aswl T.Morris, preprlnt SIIEP 90/91-29 / NBI-|IE-91-27. [12] F.l)avid. In Proceedings of workshop op [{and~m Surfaces, Quantum Gravity and Strings C a r g o , May 28-Jime I 1990. [13] S.Dalley, P|lys. Let(. 11253 (1991) 292; P.G;rL,paig at,,| J.Zi¢lq-J~Lstin, Phys. [,eft. 11255 (19f)l) 189; II.NeuberKer, Nuc|. Phys. B352 (lr.:VJI) 6~9; F.l)avid, Nud. Ph:~s. B348 11991) 507. [I.I] P. di Francesco and D.Kut~m)v, Nud. Phys. B342 { t~N)) ~ 9 .
Acknowledgements S D would like to t h a n k the o r g a n | s e t s for the o p p o r t u n i t y to present this work and workshop p a r t i c i p a n t s for their interest. Financial s u p p o r t f r o m the S.E.R.C. is acknowledged by S D and CJ.
[1.5] C.Join~scm, T.Morrls and B.Spence, in preparation. [16] M.Fukuma, I|.Kaw~i and R.Nakayama, Int. J. Mod. Phys. AG (19ql) 1385; R.Dijkgra,M', E.Verlinde and I|.Verli!!de, Nucl. Phys. B34g (1991) 435. [17] P.J.Olver, Applieatio,t* o] Lie Group~ ~,, Di.Drre~t. ttal Eqealion$, Sl~;nger-Verlag, 1986. [18] E.,M~rllnec, G.Moore a.~l N.Selherg, preprint HU14-91 ] YCTP-PIO~91 / EFI-91-14.