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Milnor monodromy of singularities and chiral primary fields of N=2 superconformal field theories
Volume 229. number 4
PHYSICS LETTERS B
19 October 1989
MILNOR MONODROMY OF SINGULARITIES AND C H I R A L P R I M A R Y F I E L D S O F N = 2 S U P E R C O N F O R M A L F I E L D T H E O R I E S Kei ITO
Internattonal Centrefor TheorettcalPhysws.Strada Co.sttera I I. Mirarnare. 1.34100 Trteste. haly Received 5 June 1989
The relation between chiral primary fields of a N= 2 supcrconformal model and the Milnor monodromy of the corresponding target space singularity is investigated. It is shown that the conformal weightsof the chiral primary fields are identical to one half of the difference between spectral numbers and the smallest spectral number.
Recently it has become increasingly clear that there arc deep connections between N = 2 superconformal field theories and geometry (topology) of the singularity of the corresponding algebraic varieties [ 1-8 ]. In a series of prcvious paper [ 5,6 ], we have investigated the relationship between tachyons in N = 2 superconformal models and topology changing of the defining polynomials o f the corresponding CalabiYau manifolds [ 51, and the changing of one conformal field theory to another triggered by relevant operators (tachyon fields) [ 6 ]. In this paper we would like to geometrize chiral primary, fields of N = 2 superconformai field theories. In other words, we would like to relate a sct of physical quantities associated to a conformai model to a set o f topological quantities associated to the singularity of corresponding algebraic variety. We will find in this paper that conformal weights h, of the chiral primary fields are exactly the same as one half of the difference between the spectral numbers a, and the smallest spectral number a,,,,,: h, --- ~ ( a , - a
..... ) .
quoting an example in Arnold, Gusein-Zade and Varchenko's book [ 9 ]. The function f ( x ) = x ~ has a degenerate critical point o f A2-type at the origin which can be morsified to)r(x) by adding a perturbation term 32x ().> 0), T(x) = x ~ + 3;.x.
(I)
The spcctral numbcrs [9] are topological invariant characteristic numbers associated to a singularity and are defined by logarithm of eigenvalues A, of the Milnor m o n o d r o m y divided by 2hi: o4 = ( I / 2 n i ) In A,.
(2)
The Milnor m o n o d r o m y is a geometrical operation associated to a singularity which we will explain by 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishcrs B.V. ( North-Holland Physics Publishing Division )
(3)
The critical points x, (x2) and the critical values z, (z2) are given by x, = iv/).,
x2=-iv/2~,
z~ =)r(x, ) = -
2).,,//2 i.
z2=f(x~)=2).~'2i.
(4)
The set o f points which satisfy f ( x ) = z is called the lcvel manifold of level z, and in the present case it consists o f three points, xT, x [ and x~'. The homology bases of the level manifold arc called vanishing cycles; they are
~J, ={x~}-{xr},
,~={x;}-{x~}.
(5)
The m o n o d r o m y operator fl, (H2) is defined by the transformation of the vanishing cycles if the level z goes around the critical values z, (z2), starting from the non-critical value zc~and ending at the same point. In the present case, II, and Hz act on A, and ,62 as follows:
HILt I = - L l , , H2,6, =Lll + 4 2 , The
product
HtA2=A, +A2. !12A2=--LI~. of all the
monodromy
(6) operators
H.=II,H~ is called the Milnor monodromy, and 379
Volume 229. number 4
PHYSICS LET rERS B
H.A, =d2 , H . A 2 = - A I -A2.
(7)
19 October 1989
l 2(k+2)
Ah+l:
The eigenvalues of the Milnor monodromy At and A~ are DJ,÷2 ~2:
A, = e x p ( ~ z i ) ,
A2=exp(-~ni).
(8)
The spectral numbcrs at, otz ofthe singularity arc defined by o~,= ( I / 2 n i ) In A,.
(9)
The Riemann sheet is taken in such a way that y ,~,= ( ~ n - l)/~,
(10)
t
n being the number of variables and/x being the multiplicity (n = I. p = 2 in this case). Therefore a,=-~,
c~2=-] .
(11)
This calculation can be extended to a general simple singularity classified according to simply-laced Lie algebra A - D - E [9 ]. The spectral numbers of other types of singularities can be calculated in the same way: and for the n = 3 case. they are listed in ref. [ 9 ]. In the n variable case they are o'/
4.~,"
~n-~
D,/2,2:
k+2
( r e = l , 2 ..... k ) ,
~n- 1 , ~n- I +
2rn-~k k+----~
(O<~m<~~k)
2(k+2)
( l = 0 , 1 , 2 ..... k ) . (l= ~k, 0, 2,4 ..... k ) ,
Eo: {41 (l=0, 3, 4, 6, 7, 10) , E,"
~l
(/=0,4,6,8,10.12,16),
Es: {0l ( l = 0 , 6 , 10, 12, 16, 18, 22, 28) .
(13)
Comparing thcse conformal weights h, of N = 2 superconformal models with spectral numbers a, ofthc corresponding singularities, we find that the following relation holds for all thc cases:
h,=~(ot,-ot .... ) ,
(14)
where a,,,, is lhc smallest spcctral number. Now we prove that the relation we have found holds for tcnsor product models [ I ] made out of N= 2 discrcte series. The LG potential of the tensor product theory made out of two N = 2 superconformal models which arc the infrared fixed points of LG potentials Wand W', respectively, is It,'+ W'. Let the spectral numbers ofthe critical point of W ( W' ) be a, (6',). Then the spectral numbers of the critical points of IV+ 14" are a, +/3,+ 1 [9 ]. One halfofthe difference betwccn the spectral numbers and the smallest spectral number is
~ l(o~, + / s , + 1 ) - ( a .... +/;,,,,n + 1 ) 1 156: ~ n - l + ~ m ET:
~n-i+'m
Es:
~n-l+~m
(m=-5,-2,-1.1,2,5),
= ~ (or, -- Otto,n) + ~ (fl, --,Bin,,, ) .
(m=-8.-4,-2,0,2,4,8),
{rn=-14,-8,-4.-2,2,4,8,14).
(12)
There is a set of N = 2 supcrconformal field theories corresponding to A - D - E type simple singularities [ 10,11 ]. It is the set of A - D - E classified N = 2 superconformal discrete series. These models are conjectured to be describcd by infrared fixed points of Landau-Ginzburg ( L G ) field theories with LG potcntial which arc identical to thc "'defining polynomial" of A - D - E type simple singularities [ 2-4,12 ]. The primary fields of the N = 2 discrete series arc labelled by 3 integers (/, q, .s). The conformal weights of the chiral primary fields ( q = / ) in the NS sectors ( s = 0 ) of these models are [ 1,4 ] 380
(15)
On the other hand, let the conformal weight of the ith chiral primary field of the conformal model corresponding to 14"be h, and the jth chiral p r i m a l field of the conformal model corrcsponding to 14" be h~, then the conlormal weight of the product ofthe chiral primary ficlds in the tensor product theory, is h,+h~. Therefore, if the identity we have just found holds for the sub theories W and W', then it also holds for the tensor product model corresponding to 14'+ W'. We would like to rcmind the reader that Milnor monodromy ofthc singularity is a concept associated to the target space and not to be confused with thc worldsheet monodromy which appears in the amplitudes calculations in conformal field theories. Nevertheless it is interesting to note the formal similarity of the identity which we have found (which relates
Volume 229, number 4
PHYSICS LETTERS B
19 October 1989
the Milnor monodromy of the target space with primary. fields) and the well-known identity which relates conformal weights to worldsheet monodromy. Now a question arises: why are these two sets of quantities of completely diffcrent nature the same? Under natural assumptions this can be verified. The conformal weights h of the chiral fields can be read off from the asymptotic behavior of the LG path integral under the rescaling of the 2-dimensional mettic [ 4 ] g,/~ by g,,a-o2 Zg,,/t,
f I]1 ~XtP(XI) e -s f ]7, fSXi e- s"
.f H~ qX~ P,(xl) e -s
-2h, =- (a,-am,.) ,
f lq~ c / X l e - S
, .~
, 3.- 2~,
l-b (2 j/2 dx/) P,(xl) exp[i2W(xl) l f 1Ii (X j/2 dXl) exp[i2W(xl) ]
t ~ - t~, -
1 + n/2
,~ - t l t m m -
I +n/2
,~--
In the limit ). -,oo, the action is dominated by the superpotential H, which scales likc [4] j14'--,J.f W. The path integral measure scales as [4 ] ).~/: for each chiral superfields Xt. Therefore the asymptotic behaviour of thc LG path integral is the same as that of the following oscillatory integral: (17)
Following ref. [ 9 ], ifwe take sufficiently general and linearly independent holomorphic differential nforms o~, ( i = I, 2 ..... /1) and take the contour [F] in such a way that the power of the asymptotic behavtour of the integral is in its maximal value (which will be denoted by [ F.,a~ ] ), then ',
(20)
or, equivalently,
S=~d2zd4OK(Xi,)(~)+(~d2zd2OW(Xi)+c.c.).
e'~'~'oJ,~,;.
(19)
Hence,
h,= ~ ( ~ , - a m , . ) .
f Iq/(2 ~/2 dx/) P,(xr) expIi2W(x/) 1 .f l-It (2 i/2 dx/) exp[ i)tW(xl) ]
(~a - ¢ . r m m )
(16)
where
f
J
(18)
[/'ma~l
where o6 are the spectral numbers of the critical point. Therefore, if we assume that the set of n-forms dx, ^ ... ^ dx,, P,(x~) corresponding to chiral primary fields arc sufficiently general and linearly indcpendcnt and also assume that [Fm,,~ ] is the correct choicc of the contour to identify with the LG path integral. then
(21)
The spectral numbers of unimordal and bimordal singularities have already been calculated by mathematicians and are listcd in ref. [9]. Combining these numbers and the identity we have found, we can gucss the conformal weights of the chiral primary fields of yet unknown or unidentified N = 2 supcrconformal field theories which correspond to these kinds of singularities. 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 [I ]1). Gcpner, Nucl. Phys B 296 (19881 757; Phys. Left. B 199 (1987) 380. [2] D. Kastor, E, Martinec and S. Shenker, Nucl. Phys. B 316 (1989) 590. [3] E.J. Martinec, Phys. Lett. B 217 (1989) 431. [4] B. Greene, C. Vafa and N.P. Warner. preprint HUTP-88/ A047; C. Vafa and N. Warner, Phys. Lctt. B 218 ( 1989 ) 51. [5] K. ho, Phys. Lett. B 226 (1989) 264. [6] K. ho, "Renormallzation Group Flows m N = 2 Superconformal Models" ICTP preprint IC/89/52 ( 1989 ). [7] D. Gepner, preprint PUPT-I 115 (1988). [8] W. Lerche, C. Vafa and N.P. Warner, preprmt HI)'TP-88/ A065 (CALT-68-1540 ). [9]V.I. Arnold. S.M. Gusetn-Zade and A N . Varchenko, Singularities of differential maps. Vols. 1, II (Birkhhuser, Basel ).
381
Volume 229, number 4
PHYSICS LETTERS B
[10] Z. Qiu, Phys. Lett. B 198 (1987) 497. [I 1 ] A. Kato, Mod. Phys. Lett. A 2 (1987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) I.
382
19 October 1989
[ 12] ~.B. Zamolodchikov, J ETP Left. 43 (1986) 730; Sov. J. Nucl. Phys. 44 11986) 529; A.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285[FSI9] (1987) 687: Y. Kitazawa et al., prcprint 15"1--522(1987).
PHYSICS LETTERS B
19 October 1989
MILNOR MONODROMY OF SINGULARITIES AND C H I R A L P R I M A R Y F I E L D S O F N = 2 S U P E R C O N F O R M A L F I E L D T H E O R I E S Kei ITO
Internattonal Centrefor TheorettcalPhysws.Strada Co.sttera I I. Mirarnare. 1.34100 Trteste. haly Received 5 June 1989
The relation between chiral primary fields of a N= 2 supcrconformal model and the Milnor monodromy of the corresponding target space singularity is investigated. It is shown that the conformal weightsof the chiral primary fields are identical to one half of the difference between spectral numbers and the smallest spectral number.
Recently it has become increasingly clear that there arc deep connections between N = 2 superconformal field theories and geometry (topology) of the singularity of the corresponding algebraic varieties [ 1-8 ]. In a series of prcvious paper [ 5,6 ], we have investigated the relationship between tachyons in N = 2 superconformal models and topology changing of the defining polynomials o f the corresponding CalabiYau manifolds [ 51, and the changing of one conformal field theory to another triggered by relevant operators (tachyon fields) [ 6 ]. In this paper we would like to geometrize chiral primary, fields of N = 2 superconformai field theories. In other words, we would like to relate a sct of physical quantities associated to a conformai model to a set o f topological quantities associated to the singularity of corresponding algebraic variety. We will find in this paper that conformal weights h, of the chiral primary fields are exactly the same as one half of the difference between the spectral numbers a, and the smallest spectral number a,,,,,: h, --- ~ ( a , - a
..... ) .
quoting an example in Arnold, Gusein-Zade and Varchenko's book [ 9 ]. The function f ( x ) = x ~ has a degenerate critical point o f A2-type at the origin which can be morsified to)r(x) by adding a perturbation term 32x ().> 0), T(x) = x ~ + 3;.x.
(I)
The spcctral numbcrs [9] are topological invariant characteristic numbers associated to a singularity and are defined by logarithm of eigenvalues A, of the Milnor m o n o d r o m y divided by 2hi: o4 = ( I / 2 n i ) In A,.
(2)
The Milnor m o n o d r o m y is a geometrical operation associated to a singularity which we will explain by 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishcrs B.V. ( North-Holland Physics Publishing Division )
(3)
The critical points x, (x2) and the critical values z, (z2) are given by x, = iv/).,
x2=-iv/2~,
z~ =)r(x, ) = -
2).,,//2 i.
z2=f(x~)=2).~'2i.
(4)
The set o f points which satisfy f ( x ) = z is called the lcvel manifold of level z, and in the present case it consists o f three points, xT, x [ and x~'. The homology bases of the level manifold arc called vanishing cycles; they are
~J, ={x~}-{xr},
,~={x;}-{x~}.
(5)
The m o n o d r o m y operator fl, (H2) is defined by the transformation of the vanishing cycles if the level z goes around the critical values z, (z2), starting from the non-critical value zc~and ending at the same point. In the present case, II, and Hz act on A, and ,62 as follows:
HILt I = - L l , , H2,6, =Lll + 4 2 , The
product
HtA2=A, +A2. !12A2=--LI~. of all the
monodromy
(6) operators
H.=II,H~ is called the Milnor monodromy, and 379
Volume 229. number 4
PHYSICS LET rERS B
H.A, =d2 , H . A 2 = - A I -A2.
(7)
19 October 1989
l 2(k+2)
Ah+l:
The eigenvalues of the Milnor monodromy At and A~ are DJ,÷2 ~2:
A, = e x p ( ~ z i ) ,
A2=exp(-~ni).
(8)
The spectral numbcrs at, otz ofthe singularity arc defined by o~,= ( I / 2 n i ) In A,.
(9)
The Riemann sheet is taken in such a way that y ,~,= ( ~ n - l)/~,
(10)
t
n being the number of variables and/x being the multiplicity (n = I. p = 2 in this case). Therefore a,=-~,
c~2=-] .
(11)
This calculation can be extended to a general simple singularity classified according to simply-laced Lie algebra A - D - E [9 ]. The spectral numbers of other types of singularities can be calculated in the same way: and for the n = 3 case. they are listed in ref. [ 9 ]. In the n variable case they are o'/
4.~,"
~n-~
D,/2,2:
k+2
( r e = l , 2 ..... k ) ,
~n- 1 , ~n- I +
2rn-~k k+----~
(O<~m<~~k)
2(k+2)
( l = 0 , 1 , 2 ..... k ) . (l= ~k, 0, 2,4 ..... k ) ,
Eo: {41 (l=0, 3, 4, 6, 7, 10) , E,"
~l
(/=0,4,6,8,10.12,16),
Es: {0l ( l = 0 , 6 , 10, 12, 16, 18, 22, 28) .
(13)
Comparing thcse conformal weights h, of N = 2 superconformal models with spectral numbers a, ofthc corresponding singularities, we find that the following relation holds for all thc cases:
h,=~(ot,-ot .... ) ,
(14)
where a,,,, is lhc smallest spcctral number. Now we prove that the relation we have found holds for tcnsor product models [ I ] made out of N= 2 discrcte series. The LG potential of the tensor product theory made out of two N = 2 superconformal models which arc the infrared fixed points of LG potentials Wand W', respectively, is It,'+ W'. Let the spectral numbers ofthe critical point of W ( W' ) be a, (6',). Then the spectral numbers of the critical points of IV+ 14" are a, +/3,+ 1 [9 ]. One halfofthe difference betwccn the spectral numbers and the smallest spectral number is
~ l(o~, + / s , + 1 ) - ( a .... +/;,,,,n + 1 ) 1 156: ~ n - l + ~ m ET:
~n-i+'m
Es:
~n-l+~m
(m=-5,-2,-1.1,2,5),
= ~ (or, -- Otto,n) + ~ (fl, --,Bin,,, ) .
(m=-8.-4,-2,0,2,4,8),
{rn=-14,-8,-4.-2,2,4,8,14).
(12)
There is a set of N = 2 supcrconformal field theories corresponding to A - D - E type simple singularities [ 10,11 ]. It is the set of A - D - E classified N = 2 superconformal discrete series. These models are conjectured to be describcd by infrared fixed points of Landau-Ginzburg ( L G ) field theories with LG potcntial which arc identical to thc "'defining polynomial" of A - D - E type simple singularities [ 2-4,12 ]. The primary fields of the N = 2 discrete series arc labelled by 3 integers (/, q, .s). The conformal weights of the chiral primary fields ( q = / ) in the NS sectors ( s = 0 ) of these models are [ 1,4 ] 380
(15)
On the other hand, let the conformal weight of the ith chiral primary field of the conformal model corresponding to 14"be h, and the jth chiral p r i m a l field of the conformal model corrcsponding to 14" be h~, then the conlormal weight of the product ofthe chiral primary ficlds in the tensor product theory, is h,+h~. Therefore, if the identity we have just found holds for the sub theories W and W', then it also holds for the tensor product model corresponding to 14'+ W'. We would like to rcmind the reader that Milnor monodromy ofthc singularity is a concept associated to the target space and not to be confused with thc worldsheet monodromy which appears in the amplitudes calculations in conformal field theories. Nevertheless it is interesting to note the formal similarity of the identity which we have found (which relates
Volume 229, number 4
PHYSICS LETTERS B
19 October 1989
the Milnor monodromy of the target space with primary. fields) and the well-known identity which relates conformal weights to worldsheet monodromy. Now a question arises: why are these two sets of quantities of completely diffcrent nature the same? Under natural assumptions this can be verified. The conformal weights h of the chiral fields can be read off from the asymptotic behavior of the LG path integral under the rescaling of the 2-dimensional mettic [ 4 ] g,/~ by g,,a-o2 Zg,,/t,
f I]1 ~XtP(XI) e -s f ]7, fSXi e- s"
.f H~ qX~ P,(xl) e -s
-2h, =- (a,-am,.) ,
f lq~ c / X l e - S
, .~
, 3.- 2~,
l-b (2 j/2 dx/) P,(xl) exp[i2W(xl) l f 1Ii (X j/2 dXl) exp[i2W(xl) ]
t ~ - t~, -
1 + n/2
,~ - t l t m m -
I +n/2
,~--
In the limit ). -,oo, the action is dominated by the superpotential H, which scales likc [4] j14'--,J.f W. The path integral measure scales as [4 ] ).~/: for each chiral superfields Xt. Therefore the asymptotic behaviour of thc LG path integral is the same as that of the following oscillatory integral: (17)
Following ref. [ 9 ], ifwe take sufficiently general and linearly independent holomorphic differential nforms o~, ( i = I, 2 ..... /1) and take the contour [F] in such a way that the power of the asymptotic behavtour of the integral is in its maximal value (which will be denoted by [ F.,a~ ] ), then ',
(20)
or, equivalently,
S=~d2zd4OK(Xi,)(~)+(~d2zd2OW(Xi)+c.c.).
e'~'~'oJ,~,;.
(19)
Hence,
h,= ~ ( ~ , - a m , . ) .
f Iq/(2 ~/2 dx/) P,(xr) expIi2W(x/) 1 .f l-It (2 i/2 dx/) exp[ i)tW(xl) ]
(~a - ¢ . r m m )
(16)
where
f
J
(18)
[/'ma~l
where o6 are the spectral numbers of the critical point. Therefore, if we assume that the set of n-forms dx, ^ ... ^ dx,, P,(x~) corresponding to chiral primary fields arc sufficiently general and linearly indcpendcnt and also assume that [Fm,,~ ] is the correct choicc of the contour to identify with the LG path integral. then
(21)
The spectral numbers of unimordal and bimordal singularities have already been calculated by mathematicians and are listcd in ref. [9]. Combining these numbers and the identity we have found, we can gucss the conformal weights of the chiral primary fields of yet unknown or unidentified N = 2 supcrconformal field theories which correspond to these kinds of singularities. 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 [I ]1). Gcpner, Nucl. Phys B 296 (19881 757; Phys. Left. B 199 (1987) 380. [2] D. Kastor, E, Martinec and S. Shenker, Nucl. Phys. B 316 (1989) 590. [3] E.J. Martinec, Phys. Lett. B 217 (1989) 431. [4] B. Greene, C. Vafa and N.P. Warner. preprint HUTP-88/ A047; C. Vafa and N. Warner, Phys. Lctt. B 218 ( 1989 ) 51. [5] K. ho, Phys. Lett. B 226 (1989) 264. [6] K. ho, "Renormallzation Group Flows m N = 2 Superconformal Models" ICTP preprint IC/89/52 ( 1989 ). [7] D. Gepner, preprint PUPT-I 115 (1988). [8] W. Lerche, C. Vafa and N.P. Warner, preprmt HI)'TP-88/ A065 (CALT-68-1540 ). [9]V.I. Arnold. S.M. Gusetn-Zade and A N . Varchenko, Singularities of differential maps. Vols. 1, II (Birkhhuser, Basel ).
381
Volume 229, number 4
PHYSICS LETTERS B
[10] Z. Qiu, Phys. Lett. B 198 (1987) 497. [I 1 ] A. Kato, Mod. Phys. Lett. A 2 (1987) 585; A. Capelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) I.
382
19 October 1989
[ 12] ~.B. Zamolodchikov, J ETP Left. 43 (1986) 730; Sov. J. Nucl. Phys. 44 11986) 529; A.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285[FSI9] (1987) 687: Y. Kitazawa et al., prcprint 15"1--522(1987).