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Phase diagram of N = 2 superconformal field theories and bifurcation sets in catastrophe theory
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989
PHASE DIAGRAM OF N= 2 SUPERCONFORMAL FIELD THEORIES A N D B I F U R C A T I O N S E T S I N C A T A S T R O P H E T H E O R Y ~r Kei ITO
International Centrefor TheoreticalPhysics,Strada Costiera11, Miramare,1-34100Trieste,Italy Received 11 August 1989
Phase diagrams of N= 2 superconformal field theories are mapped out. It is shown that they coincide with bifurcation sets in catastrophe theory. The results are applied to the determination of renorrnalization group flows triggered by a combination of two or more relevant operators.
Recently it has become increasingly clear that there are deep connections between N = 2 superconformal field theories and the geometry (topology) of the singularity of the corresponding algebraic varieties, through N = 2 super L a n d a u - G i n z b u r g ( L G ) field theories [ 1-8 ]. In a series o f previous papers [ 5-7 ], we have investigated the relationship between tachyons in N = 2 superconformal models and topology changing of the defining polynomials of the corresponding Calabi-Yau manifolds [ 5 ], and the changing of one conformal field theory to another triggered by a relevant operator (tachyon fields) [6 ]. We also found a relation between a set o f physical quantities associated to a conformal model and a set ot topological quantities associated to the singularity of the corresponding algebraic variety [ 7 ]. In this paper, we would like to map out the phase diagram o f N = 2 superconformal field theories. The infrared fixed point of the renormalization group ( R G ) flow of the N = 2 super LG field theories are identified with N = 2 superconformal field theories in refs. [3,4,8,9]. For example the fixed point of the LG field theory with the action [3,4,8 ]
S= f d2zd4Ofbq~+ (g f dZzd20(gn+' +c.c.)
(1)
is the N = 2 discrete series of A, type [ 10,11 ]. Then how does the infrared fixed point change if interaction terms ~, ~2, ..., 0 n - ~ are added to the L G action? To each interaction term, there is a coupling constant g~, g2, ..., g , - J . Therefore, the infrared fixed point changes depending on the set of coupling constants g~, g2, ..., g , - 1. We add lower power polynomial interaction terms to the LG potential:
W=
dZzd:O
~n+ + ~--1
""+
=~d2zd2Og(H-~(~n+l-. l-°ln-l~)n-l..~-..-~-202-~o[,O) n-1 oti=gi/g(i=l,2
.... , n - 1 ) , g ¢ 0 .
(2)
Then, in the ( n - 1 ) dimensional coupling constant space, (a~, a2, ..., a,_~ ), we can draw a phase diagram. Let us first consider the simplest case, i.e. n = 3. ¢r The main results of this work were presented at the Workshop on Superstrings (ICTP Trieste, July 1989). 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
125
Volume 231, number 1,2
PHYSICS LETTERSB
S~_ f d2z d400~..b ( f dZzd20g(14 3
2
2 November 1989
)
If oq =or2=0, then the infrared fixed point describes the A3 conformal model [ 3,4,8 ]. On the other hand, if or1 # 0 and or2 # 0, but oq and or2 satisfy the relation (½c~l)2+ (~0t2)3=0,
(4)
then the fixed point describes the A2 model. We will show why this is the case. In this case, 0 = 0 is not a critical point of W. The critical point of W is ~ = a, where a satisfies 2a3=oq,
- 3a2=ot2.
(5)
We shift the LG field 0 by a constant superfield a which does not depend on the two dimensional coordinates z, q~' = 0 - a .
(6)
The kinetic term of the LG action is not changed under this constant shift of the field, up to a constant, f d2zd400'~'
= ; dZz d40 q~q~+constant.
(7)
The potential term is g( ~g~4+ ½a2q)2 ..~ al ~) =g(ag~, 3 + 10, 4+ ] a ' ) .
(8)
~ga 4 is a constant superfield and the !,~, 4v- 4 term is irrelevant which means that the LG action with this term has the same infrared stable fixed point as that without this term. Therefore, the infrared fixed point of the LG action describes the A2 theory. Ifcq ~ 0, a2 ¢ 0 and a l and otz do not satisfy the above condition, then it describes the A1 theory which is a trivial conformal field theory. We can map the phase diagram in the coupling constant space, ( a l , 0~2), which is depicted in fig. 1: OLI = O~2 --~-0 ,
a l ~0, a 2 # O ,
A3,
(½0L1)2-b(10~2)3=0,
otherwise,
A2,
(9)
A~.
It is remarkable that this diagram coincides with the bifurcation sets in catastrophe theory [ 12,13 ]. Now we proceed to the next example, n = 4:
S= f d2z d40 (a~+ ( f d2z d2O g( ½Os + ~ct3Os + ½0~z(b2+ Otl(b) +c.c. ) .
(10)
If oq = o~2= or3 = 0, then it describes the Aa theory [ 3,4,8 ]. On the other hand, if the oti satisfy cq=-3a
4, 0~2=8a 3, 0~3_--6a 2
(11)
for some a # 0, the LG potential is
W=g(afg'4+½fb '5 - 34a5~/,
~b'=q~-a.
(12)
Again 30 ~ ' 5 is irrelevant and the theory corresponds to the A 3 model. Now if
oq=ca z, ot2=2a3-2ac,
ots=-3aZ+c,
(13)
then
W=g[(a2+~c)~'3+adp'4+½0'5+(laS+~ca3)] , O'=q~-a, 126
(14)
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989 O( I
A2
! &
;
3
~A3 Ccq2+(~213>O'
A/
...... /--
Fig. 1. Phase diagram of N= 2 super LG field theory eq. (3) in the coupling constant space (a~, a2). It coincides with the bifurcation sets in catastrophe theory. (For the bifurcation sets in catastrophe theory, see fig. 5.4 in ref. 112] or fig. 25 in ref, [ 13], and references therein. ) A2, A3 stand for minimal N=2 superconformal field theories corresponding to these Lie algebras.
I~A~
Fig. 2. Phase diagram of N=2 super LG field theory eq. (10) in the coupling constant space (a~, a2, a3). It coincides with the bifurcation sets in catastrophe theory, the so-called swallow tail. (For this bifurcation set in catastrophe theory, see fig. 7.4 in ref. [ 12 ] or fig. 26 in ref. [ 13 ], and references therein. )
then it describes the A2 model. If cq = a 4, o~2 = 0, 0/3 = - 2a z for some a # 0, then there exist two critical points o f W, which are ¢ = a and 0 = - a . F o r the critical point 0 = a , we shift the field as 0' = ¢ - a so the L G potential in the shifted field becomes
W=g(4a2¢'3Wa¢ '4+ 301,s + ~a8 5~j.
(15)
F o r the critical p o i n t O= - a , ¢}, ~ 0 . . . ~ a ,
4 2 ¢ , 3-a0'4+ W=g(ga
1 ,' 30 5 - ~ a S ) .
(16)
Therefore, the conformal field theory is A2 + A2. N o w we m a p out the phase diagram in the three d i m e n s i o n a l coupling constant space (cq, a2, a 3 ) . T h e phase d i a g r a m is depicted in fig. 2. It is r e m a r k a b l e that it coincides with a bifurcation set in catastrophe theory [ 12,13 ]. Also, we can d e t e r m i n e the phase diagram starting from Ak+l, Dk/2+2, E6, E7 or E 8. One o f the interesting applications o f the phase d i a g r a m s o f N = 2 superconformal field theories is the changing o f one conformal m o d e l to a n o t h e r triggered by a c o m b i n a t i o n o f two or more relevant operators. In one o f the previous papers [6 ], we d e r i v e d the changing o f one conformal model to another ( r e n o r m a l i z a t i o n group flow) triggered by one relevant operator. N o w we can extend this analysis to a more general case, that is, a conformal m o d e l is p e r t u r b e d by a c o m b i n a t i o n o f two or m o r e relevant operators with adjusted relative coupling constants. F o r example, i f we perturb the A3 conformal field theory by one relevant operator, the only renormalization group flow is A3~Al.
(17)
However, if we perturb the A3 theory by a c o m b i n a t i o n o f two relevant operators with adjusted coupling constant ratio
127
Volume 23 1, number 1,2
( &l!l)2+
PHYSICS LETTERS B
2 November 1989
( fa2)3=0,
(18)
the renormalization group flow is A3-+A2.
(19)
Also there is a flow A,+A, by another coupling constant ratio. More generally, in the case of one relevant operator, there are flows Ak+,~Ak_,rAk-2,...,A2,Al,
(20)
but there is no flow to the conformal model Ak which corresponds to the “adjacent” singularity. On the contrary, if the Ak+I theory is perturbed by a combination of two or more relevant operators, there always exists a coupling constant ratio which triggers the flow to the “adjacent” conformal field theory Ak. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References [l] D. Gepner, Nucl. Phys. B 296 (1988) 757; Phys. Lett. B 199 (1987) 380.
[ 2 ] D. Kastor, E. Martinet and S. Shenker, preprint EFI 88-3 1. [ 31 E.J. Martinet, preprint EFI 88-76. [4] B. Greene, C. Vafa and N.P. Warner, preprint HUTP-88/A047; C. Vafa and N.P. Warner, preprint HUTP-88/A037. [ 51K. Ito, Phys. Lett. B 226 (1989) 264. [ 61 K. Ito, Renormalization group flows in N=2 superconformal models, ICTP preprint IC/89/52 ( 1989). [7] K. Ito, Phys. Lett. B 229 (1989) 379. [8] P.S. Howe and P.C. West, Phys. Lett. B 223 (1989) 377. [9] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730; Sov. J.Nucl. Phys. 44 (1986) 529; A.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 [FS19] (1987) 687; Y. Kitazawaet al., preprint UT-522 (1987). [ lo] Z. Qiu, Modular invariant partition functions for N= 2 superconformal field theories, preprint IASSNS-HEP-87/26 ( 1987). [ 1 l] A. Kato, Mod. Phys. Lett. A 2 (1987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) 1. [ 121 R. Gilmore, Catastrophe for scientists and engineers (Wiley, New York, 198 1). [ 131 V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differential maps, Vols. I, II (Birkhiiuser, Basel).
128
PHYSICS LETTERS B
2 November 1989
PHASE DIAGRAM OF N= 2 SUPERCONFORMAL FIELD THEORIES A N D B I F U R C A T I O N S E T S I N C A T A S T R O P H E T H E O R Y ~r Kei ITO
International Centrefor TheoreticalPhysics,Strada Costiera11, Miramare,1-34100Trieste,Italy Received 11 August 1989
Phase diagrams of N= 2 superconformal field theories are mapped out. It is shown that they coincide with bifurcation sets in catastrophe theory. The results are applied to the determination of renorrnalization group flows triggered by a combination of two or more relevant operators.
Recently it has become increasingly clear that there are deep connections between N = 2 superconformal field theories and the geometry (topology) of the singularity of the corresponding algebraic varieties, through N = 2 super L a n d a u - G i n z b u r g ( L G ) field theories [ 1-8 ]. In a series o f previous papers [ 5-7 ], we have investigated the relationship between tachyons in N = 2 superconformal models and topology changing of the defining polynomials of the corresponding Calabi-Yau manifolds [ 5 ], and the changing of one conformal field theory to another triggered by a relevant operator (tachyon fields) [6 ]. We also found a relation between a set o f physical quantities associated to a conformal model and a set ot topological quantities associated to the singularity of the corresponding algebraic variety [ 7 ]. In this paper, we would like to map out the phase diagram o f N = 2 superconformal field theories. The infrared fixed point of the renormalization group ( R G ) flow of the N = 2 super LG field theories are identified with N = 2 superconformal field theories in refs. [3,4,8,9]. For example the fixed point of the LG field theory with the action [3,4,8 ]
S= f d2zd4Ofbq~+ (g f dZzd20(gn+' +c.c.)
(1)
is the N = 2 discrete series of A, type [ 10,11 ]. Then how does the infrared fixed point change if interaction terms ~, ~2, ..., 0 n - ~ are added to the L G action? To each interaction term, there is a coupling constant g~, g2, ..., g , - J . Therefore, the infrared fixed point changes depending on the set of coupling constants g~, g2, ..., g , - 1. We add lower power polynomial interaction terms to the LG potential:
W=
dZzd:O
~n+ + ~--1
""+
=~d2zd2Og(H-~(~n+l-. l-°ln-l~)n-l..~-..-~-202-~o[,O) n-1 oti=gi/g(i=l,2
.... , n - 1 ) , g ¢ 0 .
(2)
Then, in the ( n - 1 ) dimensional coupling constant space, (a~, a2, ..., a,_~ ), we can draw a phase diagram. Let us first consider the simplest case, i.e. n = 3. ¢r The main results of this work were presented at the Workshop on Superstrings (ICTP Trieste, July 1989). 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
125
Volume 231, number 1,2
PHYSICS LETTERSB
S~_ f d2z d400~..b ( f dZzd20g(14 3
2
2 November 1989
)
If oq =or2=0, then the infrared fixed point describes the A3 conformal model [ 3,4,8 ]. On the other hand, if or1 # 0 and or2 # 0, but oq and or2 satisfy the relation (½c~l)2+ (~0t2)3=0,
(4)
then the fixed point describes the A2 model. We will show why this is the case. In this case, 0 = 0 is not a critical point of W. The critical point of W is ~ = a, where a satisfies 2a3=oq,
- 3a2=ot2.
(5)
We shift the LG field 0 by a constant superfield a which does not depend on the two dimensional coordinates z, q~' = 0 - a .
(6)
The kinetic term of the LG action is not changed under this constant shift of the field, up to a constant, f d2zd400'~'
= ; dZz d40 q~q~+constant.
(7)
The potential term is g( ~g~4+ ½a2q)2 ..~ al ~) =g(ag~, 3 + 10, 4+ ] a ' ) .
(8)
~ga 4 is a constant superfield and the !,~, 4v- 4 term is irrelevant which means that the LG action with this term has the same infrared stable fixed point as that without this term. Therefore, the infrared fixed point of the LG action describes the A2 theory. Ifcq ~ 0, a2 ¢ 0 and a l and otz do not satisfy the above condition, then it describes the A1 theory which is a trivial conformal field theory. We can map the phase diagram in the coupling constant space, ( a l , 0~2), which is depicted in fig. 1: OLI = O~2 --~-0 ,
a l ~0, a 2 # O ,
A3,
(½0L1)2-b(10~2)3=0,
otherwise,
A2,
(9)
A~.
It is remarkable that this diagram coincides with the bifurcation sets in catastrophe theory [ 12,13 ]. Now we proceed to the next example, n = 4:
S= f d2z d40 (a~+ ( f d2z d2O g( ½Os + ~ct3Os + ½0~z(b2+ Otl(b) +c.c. ) .
(10)
If oq = o~2= or3 = 0, then it describes the Aa theory [ 3,4,8 ]. On the other hand, if the oti satisfy cq=-3a
4, 0~2=8a 3, 0~3_--6a 2
(11)
for some a # 0, the LG potential is
W=g(afg'4+½fb '5 - 34a5~/,
~b'=q~-a.
(12)
Again 30 ~ ' 5 is irrelevant and the theory corresponds to the A 3 model. Now if
oq=ca z, ot2=2a3-2ac,
ots=-3aZ+c,
(13)
then
W=g[(a2+~c)~'3+adp'4+½0'5+(laS+~ca3)] , O'=q~-a, 126
(14)
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989 O( I
A2
! &
;
3
~A3 Ccq2+(~213>O'
A/
...... /--
Fig. 1. Phase diagram of N= 2 super LG field theory eq. (3) in the coupling constant space (a~, a2). It coincides with the bifurcation sets in catastrophe theory. (For the bifurcation sets in catastrophe theory, see fig. 5.4 in ref. 112] or fig. 25 in ref, [ 13], and references therein. ) A2, A3 stand for minimal N=2 superconformal field theories corresponding to these Lie algebras.
I~A~
Fig. 2. Phase diagram of N=2 super LG field theory eq. (10) in the coupling constant space (a~, a2, a3). It coincides with the bifurcation sets in catastrophe theory, the so-called swallow tail. (For this bifurcation set in catastrophe theory, see fig. 7.4 in ref. [ 12 ] or fig. 26 in ref. [ 13 ], and references therein. )
then it describes the A2 model. If cq = a 4, o~2 = 0, 0/3 = - 2a z for some a # 0, then there exist two critical points o f W, which are ¢ = a and 0 = - a . F o r the critical point 0 = a , we shift the field as 0' = ¢ - a so the L G potential in the shifted field becomes
W=g(4a2¢'3Wa¢ '4+ 301,s + ~a8 5~j.
(15)
F o r the critical p o i n t O= - a , ¢}, ~ 0 . . . ~ a ,
4 2 ¢ , 3-a0'4+ W=g(ga
1 ,' 30 5 - ~ a S ) .
(16)
Therefore, the conformal field theory is A2 + A2. N o w we m a p out the phase diagram in the three d i m e n s i o n a l coupling constant space (cq, a2, a 3 ) . T h e phase d i a g r a m is depicted in fig. 2. It is r e m a r k a b l e that it coincides with a bifurcation set in catastrophe theory [ 12,13 ]. Also, we can d e t e r m i n e the phase diagram starting from Ak+l, Dk/2+2, E6, E7 or E 8. One o f the interesting applications o f the phase d i a g r a m s o f N = 2 superconformal field theories is the changing o f one conformal m o d e l to a n o t h e r triggered by a c o m b i n a t i o n o f two or more relevant operators. In one o f the previous papers [6 ], we d e r i v e d the changing o f one conformal model to another ( r e n o r m a l i z a t i o n group flow) triggered by one relevant operator. N o w we can extend this analysis to a more general case, that is, a conformal m o d e l is p e r t u r b e d by a c o m b i n a t i o n o f two or m o r e relevant operators with adjusted relative coupling constants. F o r example, i f we perturb the A3 conformal field theory by one relevant operator, the only renormalization group flow is A3~Al.
(17)
However, if we perturb the A3 theory by a c o m b i n a t i o n o f two relevant operators with adjusted coupling constant ratio
127
Volume 23 1, number 1,2
( &l!l)2+
PHYSICS LETTERS B
2 November 1989
( fa2)3=0,
(18)
the renormalization group flow is A3-+A2.
(19)
Also there is a flow A,+A, by another coupling constant ratio. More generally, in the case of one relevant operator, there are flows Ak+,~Ak_,rAk-2,...,A2,Al,
(20)
but there is no flow to the conformal model Ak which corresponds to the “adjacent” singularity. On the contrary, if the Ak+I theory is perturbed by a combination of two or more relevant operators, there always exists a coupling constant ratio which triggers the flow to the “adjacent” conformal field theory Ak. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References [l] D. Gepner, Nucl. Phys. B 296 (1988) 757; Phys. Lett. B 199 (1987) 380.
[ 2 ] D. Kastor, E. Martinet and S. Shenker, preprint EFI 88-3 1. [ 31 E.J. Martinet, preprint EFI 88-76. [4] B. Greene, C. Vafa and N.P. Warner, preprint HUTP-88/A047; C. Vafa and N.P. Warner, preprint HUTP-88/A037. [ 51K. Ito, Phys. Lett. B 226 (1989) 264. [ 61 K. Ito, Renormalization group flows in N=2 superconformal models, ICTP preprint IC/89/52 ( 1989). [7] K. Ito, Phys. Lett. B 229 (1989) 379. [8] P.S. Howe and P.C. West, Phys. Lett. B 223 (1989) 377. [9] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730; Sov. J.Nucl. Phys. 44 (1986) 529; A.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 [FS19] (1987) 687; Y. Kitazawaet al., preprint UT-522 (1987). [ lo] Z. Qiu, Modular invariant partition functions for N= 2 superconformal field theories, preprint IASSNS-HEP-87/26 ( 1987). [ 1 l] A. Kato, Mod. Phys. Lett. A 2 (1987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) 1. [ 121 R. Gilmore, Catastrophe for scientists and engineers (Wiley, New York, 198 1). [ 131 V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differential maps, Vols. I, II (Birkhiiuser, Basel).
128