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Trajectories of renormalization group flows on the phase diagram of N = 2 superconformal field theories
Volume 237, number 3,4
PHYSICS LETTERS B
22 March 1990
TRAJECTORIES OF RENORMALIZATION GROUP FLOWS ON THE PHASE DIAGRAM OF N=2 SUPERCONFORMAL FIELD THEORIES Kei ITO
International Centrefor TheoreticalPhysics, Strada Costiera 11, Miramare, 1-34100Trieste, Italy Received 13 December 1989
Trajectories of renormalization group flows of N= 2 super Landau-Ginzburg field theories are mapped out on the phase diagram of N= 2 superconformal field theories (SCFTs). It is shown that the coupling constants flowon one and the same bifurcation set due to the non-renormalization theorem and the scaling properties of the bifurcation sets.
Relations between N = 2 superconformal field theories (SCFTs) and catastrophe (singularity) theory through two-dimensional L a n d a u - G i n z b u r g ( L G ) field theories have attracted much interest recently [ l-11 ]. In one [ 9 ] o f the prcvious papers [ 6-10 ], wc mapped out the phase diagram o f N = 2 SCFTs and showed that they coincide with the bifurcation sets in catastrophe theory. In this paper, we would like to map out the trajectories of the renormalization group ( R G ) flow o f coupling constants o f twodimensional N - - 2 super LG field theories on the phase diagram [ 9 ] of N = 2 SCFTs. We find that the coupling constant ratios flow on the bifurcation set corresponding to one and the same N = 2 SCFT due to the non-renormalization theorem o f two-dimensional N = 2 super LG field theories and the scaling properties o f the bifurcation sets. First we prove this fact. Although the proof is given below only for An-type LG field theories, the fixed point of which describes [ 1-4] An-type N = 2 SCFTs [12,14] the extension to other types of theories is straightforward. The LG potential of A~-type LG field theories is given by [ 1-4 ]
W ( O ; a ~ , a 2 , ..., a , _ ~)
=g( a '1- ' ~'0+"' -+n-1 '+'''+°t'0)~
where the a,=g,/g are coupling constant ratios, O is the LG field and g, ( i = l, 2 ..... n - l ) (g) are coupling constants corresponding to the ith ( (n + l )th ) order polynomial interaction term. The renormalization constants for wave functions Z and coupling constants Zg(Zg, ) are defined by [ I 1 ] ~=Z-1/2¢o ,
g=,u-IZ~lgo,
with Address after 1 January 1990: Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA.
gi=P
- i Z - g, i gi,o,
(2)
with q)o (gi,o, go) being the bare wave function (coupling constants) and (9(gi, g) being the renormalized ones. The renormalization point is denoted by/t. The non-renormalization theorem implies that (accordingtoref.[lt])
ZgZ (n+ l Zg, Zi/2=l
f d20 d2z( W + W) ,
(1)
= 1, ( i = 1 , 2 .... , n - l ) ,
(3)
which shows that coupling constant renormalizations are merely reflections of the wave function renormalization [ l,l 1 ]. The definitions of the renormalization constants together with the non-renormalization theorem lead to the following renormalization o f coupling constant ratios: 397
Volume 237, number 3,4
PHYSICS LETTERS B
Z- I zi/2 O~i~ Zg--W gi a,,o = Z (n+ t)/2 0li, 0 = z-tn+
(4)
t-i)/20li,O ,
where ot~,o=gi.o/go are bare coupling constant ratios. On the other hand, the bifurcation sets of the deformation of the function W is defined [ 13,15 ] by the set of constants ( a l , or2, ..., otn_t) for which W has a degenerate critical point at 0 = a of Ak+t-type which satisfies
°21
22 March 1990
Instead of considering the coupling constant flow in the original LG field 0, we consider it in the shifted LG field 0'. There is an infrared (IR) fixed point of the RG flow and near the fixed point the coupling constant of 0' 3 (i.e. ( g a ) ) becomes independent of /~. Transformed back to the coupling constants in the original LG field 0, this implies that gll3g 2/3 is independent of #. From the definitions of the renormalization constants and the non-renormalization theorem, it turns out that Z t/2 scales as #~/3, and the coupling constants scale as gl ~]./--2/3,
g~flt/3,
g2 ~ # - - W 3
0
~-~-
or oq ~ # -t,
=0
( i = 2 , 3 ..... k ) ,
2<~k<~n-1.
(5)
~=a
It is clear from the above definition that the bifurcation sets have a scaling invariance [ 13,15 ] that if {o~} ( / = 1..... n - 1 ) is on a bifurcation set, {;t"+ ~-%} is on the bifurcation set of the same criticality (accordingto refs. [ 13,151 ). Comparing the non-renormalization theorem of the coupling constant ratios and the scaling properties of the bifurcation sets, we find that if bare coupling constants are on a bifurcation set, then the renormalized coupling constants are on the bifurcation set of the same criticality which implies that the coupling constants flow on one and the same bifurcation set. Now we draw the trajectories of RG flows on the phase diagram of N = 2 SCFTs. First we draw the trajectories in the simplest case, n = 3. The LG potential is given by
W=g( 1 0 4 + 12Ot202 +Otl 0 ) •
OL2~U -2/3 .
(11)
Now, the changing of the effectivc coupling constant ratios aTfr(t), under the change of two-dimensional momentum scale T:p--*e-tp can be derived by the standard RG argument [ 16 ] a~ff(t)=e' aCf(O) ,
(12)
a ~ff( t ) = e 2t/ 3 a~ff ( O ) .
It is clear that if the o~fr(0) ( i = 1,2) satisfy the condition (7) so do a7 fr (t). The trajectories of the effective coupling constant flows are depicted in fig. 1. Next, we consider the n = 4 case. The LG potential W in this case is W=g(½Os+ ~0t303 • . + ~l a 2 0 2 +ott0) •
0(2
4'
(6)
If a t ~:0, a 2 ¢ 0 , but a t and a2 satisfy (,a,)~+
~ 3 (~ot2) =0,
(7)
then the critical point of Wis at 0 = a where a satisfies 2a3=at,
-3a2=ot2 .
(8)
So, we expand W around the critical point 0 = a by shifting the LG field by a constant superfield a, 0' = 0 - a -
(9)
,,/___:i_ ', l _ i
•
~2..
In terms of the shifted LG field 0 ' , W is expressed as W = g ( a O ,3+ ~0,4+ ¼a4) .
398
(10)
Fig. 1. Trajectories of the RG flows in the n = 3 case.
(13)
Volume 237, number 3,4
PHYSICS LETTERS B
22 March 1990
If the a, satisfy
at=-3a
4,
0~2---8a 3,
ot3=-6a 2
(14)
for some a S 0, then W h a s a critical point of A3-type. A similar argument shows that Z ~/2 scales a s # ~/4 near the IR-fixed point and
a ~ ~ l t -~,
a z ~ l t -3/4,
a3,,-/t -~/2
(15)
which implies that effective coupling constant ratios scale as
a~ff( t ) = c ' a~rr( O ) , a~fr(t) = e 3//4 a~rr (0) ,
Az+ Az~
ot~fr(t) = e t/2 a~ff(0) ,
~z
(16)
under the change o f m o m e n t u m scale T : p ~ e - ' p . On the other hand, if the a~ satisfy oti = ca 2, or2 = 2a 3 _ 2ac and a 3 = - 3 a 2 + c for some a, c, the LG potential W in terms o f the shifted field ¢' is
W = g [ (aZ+ ~c)~'3-baO'4-t-t(b'5 + (]as.-I-~ca3) ] . (17) It has a critical point o f Az-type at O=a. If oh = a 4, a 2 = 0 , a 3 = - 2 a z for some a ¢ 0 , then the LG potential Whas two critical points o f Az-type: one at O = a and the other at ~ = - a . In both cases, Z ~/2 scales as/~/3 near the IR-fixed point and
a~N/1-4/3,
a 2 ~ g -~,
a 3 ~ / z -2/3.
(18)
The effective coupling constant ratios scale as
a]rf ( t) = e 4t/3 ot~fr( O ) , a~ff(t) = e ' a[ff(0) , a~ff(t) = e 2'/3 a]ff(0) .
(19)
The trajectories of the R G flow on the phase diagram o f N = 2 SCFTs are depicted in fig. 2. In this paper we have drawn the trajectories o f the RG flows for N = 2 super LG field theories on the phase diagram [ 9 ] o f N = 2 SCFTs and have shown that it flows in the bifurcation sets in catastrophe theory. By this observation, the fact that the R G flow itself does not change one conformal model to another, as was noted in ref. [2], became quite manifest. Of course, the addition of polynomial interaction terms corresponding to the relevant operators, changes one conformal model to another, which was conjectured in refs. [ 1,2], and how it does change in
Fig. 2. Trajectories oftlae RG flows in the n=4 case. Flows from {a~,fr(0)} to {aq/fr(t)} are shown by arrows. general, in the A - D - E series by the addition o f a single relevant operator was demonstrated in ref. [7 ] and for two or more relevant operators in ref. [ 9 ]. The fact that the coupling constants flow on the bifurcation set corresponding to one and the same N = 2 SCFT, implies that, in order to describe phase transitions o f N--- 2 SCFTs, one-parameter families o f LG potentials which are not contained in the same bifurcation set must be considered. Actually, we found that the fiber bundle called the Milnor fibration of the deformation of the singularities [ 15 ] is an appropriate mathematical framework in describing phase transitions of N - - 2 SCFTs. We will describe this in one of the forthcoming papers [ 17]. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and U N E S C O for hospitality at the International Centre for Theoretical Physics, Trieste.
References [ 1 ] D. Kastor, E. Martinec and S. Shenker, Nucl. Phys. B 316
(1989) 590. 399
Volume 237, number 3,4
PHYSICS LETTERS B
[ 2 ] C. Vafa and N. Warner, Phys. Left. B 2 t 8 (1989) 51. [3] Be Green, C. Vafa and N.P. Warner, NucL Phys. B 324 (1989) 371. [4] E.J. Martinec, Phys. Lctt. B 217 (1989) 43I. [5] D. Gepner, Nucl. Phys. B 322 (1989) 65. [61K~ lto, Phys. Lett. B 226 (1989) 264. [7] K. Ito, ICTP preprint IC/89/52 ( 1989), [8] K. Ito, Phys. Lett. B 229 (1989) 379. [9] K. Ito, Phys. Left. B 231 ( 1989( 125. [ 10 ] K. Ito, Topology of singularities and phase diagram of N= 2 superconformal field theories, talk presented at Workshop on Superstrings (ICTP, Trieste, July 1989 ). [ 11 ] P.S. Howe and P.C. West, Phys. Lett. B 223 (1989) 377.
400
22 March 1990
[ 12 ] Z. Qiu, Phys. Lett. B 198 ( 1989 ) 493. [ 13 ] R. Gilmore, Catastrophe for scientists and engineers (Wiley, New York, 1981 ). [ 14] A. Kato, Mod. Phys. Lett. A 2 (t987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) 1. [!5]V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differential maps, Vols. I, II (Birkh~iuser, Basel). [16] D.J. Gross, in: Les Houchcs Session 28 (1975) (NorthHolland, Amsterdam, 1976). [ 17] K. lto, in preparation.
PHYSICS LETTERS B
22 March 1990
TRAJECTORIES OF RENORMALIZATION GROUP FLOWS ON THE PHASE DIAGRAM OF N=2 SUPERCONFORMAL FIELD THEORIES Kei ITO
International Centrefor TheoreticalPhysics, Strada Costiera 11, Miramare, 1-34100Trieste, Italy Received 13 December 1989
Trajectories of renormalization group flows of N= 2 super Landau-Ginzburg field theories are mapped out on the phase diagram of N= 2 superconformal field theories (SCFTs). It is shown that the coupling constants flowon one and the same bifurcation set due to the non-renormalization theorem and the scaling properties of the bifurcation sets.
Relations between N = 2 superconformal field theories (SCFTs) and catastrophe (singularity) theory through two-dimensional L a n d a u - G i n z b u r g ( L G ) field theories have attracted much interest recently [ l-11 ]. In one [ 9 ] o f the prcvious papers [ 6-10 ], wc mapped out the phase diagram o f N = 2 SCFTs and showed that they coincide with the bifurcation sets in catastrophe theory. In this paper, we would like to map out the trajectories of the renormalization group ( R G ) flow o f coupling constants o f twodimensional N - - 2 super LG field theories on the phase diagram [ 9 ] of N = 2 SCFTs. We find that the coupling constant ratios flow on the bifurcation set corresponding to one and the same N = 2 SCFT due to the non-renormalization theorem o f two-dimensional N = 2 super LG field theories and the scaling properties o f the bifurcation sets. First we prove this fact. Although the proof is given below only for An-type LG field theories, the fixed point of which describes [ 1-4] An-type N = 2 SCFTs [12,14] the extension to other types of theories is straightforward. The LG potential of A~-type LG field theories is given by [ 1-4 ]
W ( O ; a ~ , a 2 , ..., a , _ ~)
=g( a '1- ' ~'0+"' -+n-1 '+'''+°t'0)~
where the a,=g,/g are coupling constant ratios, O is the LG field and g, ( i = l, 2 ..... n - l ) (g) are coupling constants corresponding to the ith ( (n + l )th ) order polynomial interaction term. The renormalization constants for wave functions Z and coupling constants Zg(Zg, ) are defined by [ I 1 ] ~=Z-1/2¢o ,
g=,u-IZ~lgo,
with Address after 1 January 1990: Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA.
gi=P
- i Z - g, i gi,o,
(2)
with q)o (gi,o, go) being the bare wave function (coupling constants) and (9(gi, g) being the renormalized ones. The renormalization point is denoted by/t. The non-renormalization theorem implies that (accordingtoref.[lt])
ZgZ (n+ l Zg, Zi/2=l
f d20 d2z( W + W) ,
(1)
= 1, ( i = 1 , 2 .... , n - l ) ,
(3)
which shows that coupling constant renormalizations are merely reflections of the wave function renormalization [ l,l 1 ]. The definitions of the renormalization constants together with the non-renormalization theorem lead to the following renormalization o f coupling constant ratios: 397
Volume 237, number 3,4
PHYSICS LETTERS B
Z- I zi/2 O~i~ Zg--W gi a,,o = Z (n+ t)/2 0li, 0 = z-tn+
(4)
t-i)/20li,O ,
where ot~,o=gi.o/go are bare coupling constant ratios. On the other hand, the bifurcation sets of the deformation of the function W is defined [ 13,15 ] by the set of constants ( a l , or2, ..., otn_t) for which W has a degenerate critical point at 0 = a of Ak+t-type which satisfies
°21
22 March 1990
Instead of considering the coupling constant flow in the original LG field 0, we consider it in the shifted LG field 0'. There is an infrared (IR) fixed point of the RG flow and near the fixed point the coupling constant of 0' 3 (i.e. ( g a ) ) becomes independent of /~. Transformed back to the coupling constants in the original LG field 0, this implies that gll3g 2/3 is independent of #. From the definitions of the renormalization constants and the non-renormalization theorem, it turns out that Z t/2 scales as #~/3, and the coupling constants scale as gl ~]./--2/3,
g~flt/3,
g2 ~ # - - W 3
0
~-~-
or oq ~ # -t,
=0
( i = 2 , 3 ..... k ) ,
2<~k<~n-1.
(5)
~=a
It is clear from the above definition that the bifurcation sets have a scaling invariance [ 13,15 ] that if {o~} ( / = 1..... n - 1 ) is on a bifurcation set, {;t"+ ~-%} is on the bifurcation set of the same criticality (accordingto refs. [ 13,151 ). Comparing the non-renormalization theorem of the coupling constant ratios and the scaling properties of the bifurcation sets, we find that if bare coupling constants are on a bifurcation set, then the renormalized coupling constants are on the bifurcation set of the same criticality which implies that the coupling constants flow on one and the same bifurcation set. Now we draw the trajectories of RG flows on the phase diagram of N = 2 SCFTs. First we draw the trajectories in the simplest case, n = 3. The LG potential is given by
W=g( 1 0 4 + 12Ot202 +Otl 0 ) •
OL2~U -2/3 .
(11)
Now, the changing of the effectivc coupling constant ratios aTfr(t), under the change of two-dimensional momentum scale T:p--*e-tp can be derived by the standard RG argument [ 16 ] a~ff(t)=e' aCf(O) ,
(12)
a ~ff( t ) = e 2t/ 3 a~ff ( O ) .
It is clear that if the o~fr(0) ( i = 1,2) satisfy the condition (7) so do a7 fr (t). The trajectories of the effective coupling constant flows are depicted in fig. 1. Next, we consider the n = 4 case. The LG potential W in this case is W=g(½Os+ ~0t303 • . + ~l a 2 0 2 +ott0) •
0(2
4'
(6)
If a t ~:0, a 2 ¢ 0 , but a t and a2 satisfy (,a,)~+
~ 3 (~ot2) =0,
(7)
then the critical point of Wis at 0 = a where a satisfies 2a3=at,
-3a2=ot2 .
(8)
So, we expand W around the critical point 0 = a by shifting the LG field by a constant superfield a, 0' = 0 - a -
(9)
,,/___:i_ ', l _ i
•
~2..
In terms of the shifted LG field 0 ' , W is expressed as W = g ( a O ,3+ ~0,4+ ¼a4) .
398
(10)
Fig. 1. Trajectories of the RG flows in the n = 3 case.
(13)
Volume 237, number 3,4
PHYSICS LETTERS B
22 March 1990
If the a, satisfy
at=-3a
4,
0~2---8a 3,
ot3=-6a 2
(14)
for some a S 0, then W h a s a critical point of A3-type. A similar argument shows that Z ~/2 scales a s # ~/4 near the IR-fixed point and
a ~ ~ l t -~,
a z ~ l t -3/4,
a3,,-/t -~/2
(15)
which implies that effective coupling constant ratios scale as
a~ff( t ) = c ' a~rr( O ) , a~fr(t) = e 3//4 a~rr (0) ,
Az+ Az~
ot~fr(t) = e t/2 a~ff(0) ,
~z
(16)
under the change o f m o m e n t u m scale T : p ~ e - ' p . On the other hand, if the a~ satisfy oti = ca 2, or2 = 2a 3 _ 2ac and a 3 = - 3 a 2 + c for some a, c, the LG potential W in terms o f the shifted field ¢' is
W = g [ (aZ+ ~c)~'3-baO'4-t-t(b'5 + (]as.-I-~ca3) ] . (17) It has a critical point o f Az-type at O=a. If oh = a 4, a 2 = 0 , a 3 = - 2 a z for some a ¢ 0 , then the LG potential Whas two critical points o f Az-type: one at O = a and the other at ~ = - a . In both cases, Z ~/2 scales as/~/3 near the IR-fixed point and
a~N/1-4/3,
a 2 ~ g -~,
a 3 ~ / z -2/3.
(18)
The effective coupling constant ratios scale as
a]rf ( t) = e 4t/3 ot~fr( O ) , a~ff(t) = e ' a[ff(0) , a~ff(t) = e 2'/3 a]ff(0) .
(19)
The trajectories of the R G flow on the phase diagram o f N = 2 SCFTs are depicted in fig. 2. In this paper we have drawn the trajectories o f the RG flows for N = 2 super LG field theories on the phase diagram [ 9 ] o f N = 2 SCFTs and have shown that it flows in the bifurcation sets in catastrophe theory. By this observation, the fact that the R G flow itself does not change one conformal model to another, as was noted in ref. [2], became quite manifest. Of course, the addition of polynomial interaction terms corresponding to the relevant operators, changes one conformal model to another, which was conjectured in refs. [ 1,2], and how it does change in
Fig. 2. Trajectories oftlae RG flows in the n=4 case. Flows from {a~,fr(0)} to {aq/fr(t)} are shown by arrows. general, in the A - D - E series by the addition o f a single relevant operator was demonstrated in ref. [7 ] and for two or more relevant operators in ref. [ 9 ]. The fact that the coupling constants flow on the bifurcation set corresponding to one and the same N = 2 SCFT, implies that, in order to describe phase transitions o f N--- 2 SCFTs, one-parameter families o f LG potentials which are not contained in the same bifurcation set must be considered. Actually, we found that the fiber bundle called the Milnor fibration of the deformation of the singularities [ 15 ] is an appropriate mathematical framework in describing phase transitions of N - - 2 SCFTs. We will describe this in one of the forthcoming papers [ 17]. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and U N E S C O for hospitality at the International Centre for Theoretical Physics, Trieste.
References [ 1 ] D. Kastor, E. Martinec and S. Shenker, Nucl. Phys. B 316
(1989) 590. 399
Volume 237, number 3,4
PHYSICS LETTERS B
[ 2 ] C. Vafa and N. Warner, Phys. Left. B 2 t 8 (1989) 51. [3] Be Green, C. Vafa and N.P. Warner, NucL Phys. B 324 (1989) 371. [4] E.J. Martinec, Phys. Lctt. B 217 (1989) 43I. [5] D. Gepner, Nucl. Phys. B 322 (1989) 65. [61K~ lto, Phys. Lett. B 226 (1989) 264. [7] K. Ito, ICTP preprint IC/89/52 ( 1989), [8] K. Ito, Phys. Lett. B 229 (1989) 379. [9] K. Ito, Phys. Left. B 231 ( 1989( 125. [ 10 ] K. Ito, Topology of singularities and phase diagram of N= 2 superconformal field theories, talk presented at Workshop on Superstrings (ICTP, Trieste, July 1989 ). [ 11 ] P.S. Howe and P.C. West, Phys. Lett. B 223 (1989) 377.
400
22 March 1990
[ 12 ] Z. Qiu, Phys. Lett. B 198 ( 1989 ) 493. [ 13 ] R. Gilmore, Catastrophe for scientists and engineers (Wiley, New York, 1981 ). [ 14] A. Kato, Mod. Phys. Lett. A 2 (t987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) 1. [!5]V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differential maps, Vols. I, II (Birkh~iuser, Basel). [16] D.J. Gross, in: Les Houchcs Session 28 (1975) (NorthHolland, Amsterdam, 1976). [ 17] K. lto, in preparation.