- Email: [email protected]
Ten dimensional origin of string dualities
UCLEAR PHYSICS
F:I,SEVIER
Nuclear Physics B (Proc. Suppl.) 52A (1997) 301-304
PROCEEDINGS SUPPLEMENTS
TEN DIMENSIONAL ORIGIN OF STRING DUALITIES Ashoke Sen a • t ~Mehta Research Institute of Mathematics and Mathematical Physics 10 K a s t u r b a Gandhi Marg, Allahabad 211002, INDIA We show how the various known dualities in string theory can all be understood in terms of dualities involving ten dimensional string theories.
During the last two years we have discovered m a n y examples of string dualities. We have also learned that m a n y of these dualities can, in turn, be 'derived' from other conjectured dualities in string theory. A notable example is that of Sduality of heterotic string theory compactified on 7"6; it can be shown to be a consequence[I,2] of the T-duality of type II string theory compactifled on T 2, and the conjectured equivalence between the heterotic string theory on T 4 and type IIA string theory on K313]. In view of the interrelation between the various dualities, one could pose the question: ' W h a t is the minimal set of duality conjectures from which all other string dualities can be derived'? This question does not seem to have a unique answer; just as string dualities have taught us that no single theory is more fundamental than the other - different string theories being different formulations of the same underlying theory - we learn that no particular (set of) dulity(ies) is more fundamental than any other set. Nevertheless it might be useful to find out at least one simple minimal set of dualities from which every other duality can be derived. In this talk I shall argue that such a minimal set is provided by T-duallty symmetries, and the set of conjectured dualities involving ten dimensional string theories (e.g. self-duality of type IIB string theory[3] and the duality between the type I and the SO(32) heterotic string theory[2]). If we add M-theory to the list of known consistent theories, then we also need to add a duality conjecture involving M-theory to this list. *On leave of absence from Tata Institute of Fundamental Research, Homi Bhabha Road, B o m b a y 400005, INDIA E-mail: [email protected], [email protected]
0920-5632(97)/$17.00 © 1997 Elsevier Science B.~/~ All rights reserved. Plh S0920-5632(96)00581-6
The question now is: how can we derive all other existing duality conjectures from this minimal set of dualities. We shall illustrate this by example. In particular, we shall derive the conjectured duality between type IIA theory on K 3 and heterotic string theory on T 4, - also known as the string-string duality conjecture - from the set of ten dimensional duality conjectures. The reason that we have chosen this particular exampie is that m a n y other duality conjectures can be derived from this string-string duality conjecture; e.g. the S-duality of heterotic string theory on T 6, or dualities[4] involving type II theories compactified on a Calabi-Yau manifold and heterotic string theory compactified on K 3 × T215,6]. Before going into the details, let us outline the general procedure. Suppose a duality conjecture relates theory A compactified on a manifold KA to theory B compactified on a manifold B. In that case if M is another smooth manifold, then we can conclude that the theory A compactified on KA × M should be dual to the theory B compactified on KB x M. Now let us suppose t h a t the first theory has a discrete global s y m m e t r y group GA; then by duality this should m a p to a discrete global s y m m e t r y group GB in the second theory isomorphic to GA. Now let us construct a new pair of theories by modding out the first theory by the group GA and the second theory by the group GB. Then naively one would expect that this new pair of theories are also dual to each other. Unfortunately this naive expectation does not always give right answer. So we first need to study under what condition we expect the resulting pair of theories to be dual to each other, and then construct new dual pair of theories by applying this procedure.
A. Sen~Nuclear Physics13(Proc. Suppl.) 52A (1997) 301 304
302
From now on we shall focus on cases where GA
(GB) is a Z2 group generated by gA (gB). We shall also decompose the action of ga (gB) into a component s that acts geometrically on the manifold M that is common to both the theories and a component hA (hB) which describes geometric action on the manifold KA (KB) as well as some possible internal symmetry transformation. We shall distinguish three possible cases: 1. s acts freely on M. 2. s does not act freely on M but has nontrivial action on M. 3. s is the identity transformation, i.e. leaves the whole of M fixed.
it
We shall see that in the first two cases we expect the resulting pair of coset theories, obtained by modding out the first theory by GA and the second theory by GB, to be dual to each other, whereas in the last case we do not expect the resulting pair of coset theories to be dual. s a c t s f r e e l y o n M: This case was analyzed by Vafa and Witten[6]. In this case M / s has no fixed points and hence it is a smooth manifold 3. As a result, the manifold KA × M/hA • s has a fibered structure with base M / s and fiber KA, with hA denoting the twist on the fiber along the closed cycle in M / s connecting a point P to s(P) in M. KB × M / h B . s has a similar structure. While comparing theory A on the first manifold with the theory B on the second manifold, we can simply replace the theory A on KA at a given point P on M / s by theory B on KB by the original duality between theory A on KA and theory B on KB. This way we can establish the duality between the two coset theories. This argument is valid when the fiber KA (KB) varies slowly over the base, which in turn can be achieved by going to the region of the moduli space where M / s has large volume. Once duality between two theories is established in one region of the moduli space, it must be valid at all points in the moduli space since we can go from one point in the moduli space to another by switching on appropriate background fields. 3By an abuse of notation we shall denote a Z~ group by its generator.
s d o e s n o t a c t f r e e l y o n M: In this case one can apply the same argument; however now M has fixed points under s and hence M / s is an orbifold. As a result KA x M/hA • s still has a fibered structure with base M / s and fiber KA, but at a fixed point Po on M / s , the fiber degenerates to KA/hA. Similar structure holds for KB x M/hB • s. Since a priori there is no reason why A on KA/hA should be dual to B on KB/hB, we see that the 'proof' of duality breaks down at the fixed point Po. However, since fixed points constitute a set of 'measure zero', it is reasonable to expect that they will not spoil the duality between the two coset theories. This has in fact been verified in many different examples; and we shall proceed with the assumption that this is indeed the case. s leaves t h e w h o l e o f M fixed: In this case s represents the identity transformation. Thus K A x M/hA • s now represents the product space (KA ~hA) x M; similarly KB x M / h B . s represents the product space (KB/hB) X M. Since there is no a priori reason why A on KA/hA should be dual to B on KB/hB, there is no reason for A on ( g A / h a ) × M to be dual to B on (KB/hB) × M. We shall now show how the string-string duality conjecture in six dimensions can be 'derived' from ten dimensional dualities by using these resuits[7]. We start with type IIB theory in ten dimensions. This has two perturbatively realized Zz symmetries, ( - 1 ) F~ which changes the sign of all the Ramond sector states on the left, and Q, which corresponds to the world-sheet parity transformation. It also has a conjectured SL(2,Z) S-duality symmetry, whose non-trivial Z2 element S exchanges the anti-symmetric tensor field arising in the Neveu-Schwarz NeveuSchwarz (NS) sector with the one arising in the Ramond-Ramond (RR) sector. By studying the action of various transformations on the massless fields in the theory, one can verify that S conjugates ( - 1 ) F~ to ~. We now construct a new dual pair of theories using the method of quotienting discussed before by choosing KA and KB to be points, hA and hB to be ( - 1 ) FL and i'~ respectively, M to be a four dimensional torus T 4 labelled by y 6 , . . . y g , and s to be the transformation ym __~ _ym for 6 < m < 9. We shall denote this transformation
A. Sen~Nuclear Physics B (Proc. Suppl.) 52A (1997) 301 304 by 2-4. This gives the equivalence between type IIB on T4/(--1)FL.74 and type IIB on T4/f~.Z4. If we now perform an R ---* 1 / R T-duality transformation on the y6 coordinate in the first theory, it converts the type IIB theory into a type IIA theory. Also by studying the action of various synlmetry transformations on the massless fields in the theory one can verify that it conjugates the transformation ( - 1 ) F~ .2"4 to 2-4. Thus the first theory is equivalent by T-duality transformation to type IIA on T4/Z4, which is a particular point in the moduli space of type IIA theory on K3. On the other hand, by making an R ~ 1 / R duality transformation on all the four coordinates of T 4 in the second theory we can conjugate the transformation f~ • 2-4 to just fL Thus the second theory is equivalent via T-duality transformation to type IIB theory on T 4 modded out by ft, which is just the type I theory on T 4. Using the conjectured equivalence between the type I theory and the SO(32) heterotic string theory in ten dimensions[2], we arrive at the equivalence between type IIA theory on K3 and heterotic string theory on T 4, which is precisely the string-string duality conjecture. The same procedure can be used to derive m a n y other duality conjectures[7,8]. Among these are dualities between 1. type IIA on TS/2-s and type I on T 8, 2. heterotic string theory on TS/2-s,24 and type l i b on K 3 x K3, where 2-s,24 denotes the change of sign of all coordinates of the (8, 24) signature Narain lattice, 3. M - t h e o r y on K3 and type I on T 3, 4. M-theory on T s / Z 5 and type IIB on K3, 5. M-theory on T 8 / Z s and type I on T 7, 6. M - t h e o r y on T9/2-9 and type IIB on TS/2-s, 7. F - t h e o r y on K 3 and heterotic on T 2, etc. Each of these conjectures can be tested by comparing the spectrum of massless states in the two resulting theories and in each case they agree. I shall conclude the talk by discussing how the duality between F-theory[9] on K3 and heterotic string theory on T 2 can be derived using this procedure[10]. We begin with a brief introduction
303
to F-theory. Type IIB theory in ten dimensions has two scalar fields, - the dilaton • and the RR scalar a sometimes known as the axiom The complex field
=_ a + ie -~/2 ,
(1)
transforms under the SL(2,Z) S-duality of the type IIB theory as a modular paramter. Conventional conformal field theory (CFT) compactification of type IIB theory sets I to be a constant on the internal manifold. F-theory is a novel way of compactifying type IIB theory with non-trivial background ;~ field. The starting point for such compactifications is an elliptically fibered manifold M - a manifold which has the structure of a two dimensional torus T 2 fibered over a base manifold B. The complex structure modulus r of the torus is a function of the coordinate y of the base. By definition F-theory on M is type IIB theory compactified on B with ~(y) = ~(y).
(2)
Vafa conjectured that F-theory on an ellptically fibered K3 (which has the structure of T 2 fibered over the base C P 1) is equivalent to beterotic string theory on T219]. We shall now derive this by going to the T4/Z4 orbifold limit of K3. (As argued before, once the duality is established at one point in the moduli space, it must hold at all points in the moduli space). T 4 / I 4 can be rewritten as (T2) ' x T2/Z~ • 2-2. By definition, F-theory on (T2) I x T 2 is type IIB theory on T 2, with ), being set equal to the modular parameter of (T2) '. The transformation I~ corresponds to the transformation ( - 1
- 1 ) of the
SL(2,Z) S-duality group. By studying the action of this transformation on the massless fields of the theory one can show that this is just the global s y m m e t r y transformation ( - 1 ) EL • f~ of the type IIB theory. From this we conclude that F - t h e o r y on K3 at this special point in the moduli space is equivalent to type IIB on T 2 / ( - 1 ) EL • f~ • Z2. By making R ~ 1 / R T-duality transformation on both the coordinates of T 2 one can convert ( - l ) EL • ~ • :Z-2 to f2. This shows that F - t h e o r y on K3 is equivalent to type IIB on T 2 / f L This in turn is just type I on T 2 which, by the ten dimensional duality conjecture, is equivalent to
304
A. Sen~Nuclear Physics B (Proc. Suppl.) 52A (1997)301-304
heterotic string theory on T 2. This extablishes the duality between F-theory on K3 and heterotic string theory on T 2. This way all conjectured dualities in string theory can be traced to a few duality conjectures involving ten / eleven dimensional theories. Understanding the origin of these dualities still remains a mystery; however reducing the number of independent duality conjectures might be a step towards a deeper understanding of all duality symmetries involving string theory. REFERENCES
1. M. Duff, hep-th/9501030. 2. E. Witten, Nucl. Phys. B443 (1995) 85 [hepth/9503124]. 3. C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109 [hep-th/9410167]. 4. S. Kachru and C. Vafa, Nucl. Phys. B450 (1995) 69 [hep-th/9605105]. 5. A. Klemm, W. Lerche and P. Mayr, Phys. Lett. B357 (1995) 313 [hep-th/9506112]. 6. C. Vafa and E. Witten, hep-th/9507050. 7. A. Sen, hep-th/9604070. 8. A. Sen, hep-th/9603113. 9. C. Vafa, hep-th/9602022. 10. A. Sen, hep-th/9605150.
F:I,SEVIER
Nuclear Physics B (Proc. Suppl.) 52A (1997) 301-304
PROCEEDINGS SUPPLEMENTS
TEN DIMENSIONAL ORIGIN OF STRING DUALITIES Ashoke Sen a • t ~Mehta Research Institute of Mathematics and Mathematical Physics 10 K a s t u r b a Gandhi Marg, Allahabad 211002, INDIA We show how the various known dualities in string theory can all be understood in terms of dualities involving ten dimensional string theories.
During the last two years we have discovered m a n y examples of string dualities. We have also learned that m a n y of these dualities can, in turn, be 'derived' from other conjectured dualities in string theory. A notable example is that of Sduality of heterotic string theory compactified on 7"6; it can be shown to be a consequence[I,2] of the T-duality of type II string theory compactifled on T 2, and the conjectured equivalence between the heterotic string theory on T 4 and type IIA string theory on K313]. In view of the interrelation between the various dualities, one could pose the question: ' W h a t is the minimal set of duality conjectures from which all other string dualities can be derived'? This question does not seem to have a unique answer; just as string dualities have taught us that no single theory is more fundamental than the other - different string theories being different formulations of the same underlying theory - we learn that no particular (set of) dulity(ies) is more fundamental than any other set. Nevertheless it might be useful to find out at least one simple minimal set of dualities from which every other duality can be derived. In this talk I shall argue that such a minimal set is provided by T-duallty symmetries, and the set of conjectured dualities involving ten dimensional string theories (e.g. self-duality of type IIB string theory[3] and the duality between the type I and the SO(32) heterotic string theory[2]). If we add M-theory to the list of known consistent theories, then we also need to add a duality conjecture involving M-theory to this list. *On leave of absence from Tata Institute of Fundamental Research, Homi Bhabha Road, B o m b a y 400005, INDIA E-mail: [email protected], [email protected]
0920-5632(97)/$17.00 © 1997 Elsevier Science B.~/~ All rights reserved. Plh S0920-5632(96)00581-6
The question now is: how can we derive all other existing duality conjectures from this minimal set of dualities. We shall illustrate this by example. In particular, we shall derive the conjectured duality between type IIA theory on K 3 and heterotic string theory on T 4, - also known as the string-string duality conjecture - from the set of ten dimensional duality conjectures. The reason that we have chosen this particular exampie is that m a n y other duality conjectures can be derived from this string-string duality conjecture; e.g. the S-duality of heterotic string theory on T 6, or dualities[4] involving type II theories compactified on a Calabi-Yau manifold and heterotic string theory compactified on K 3 × T215,6]. Before going into the details, let us outline the general procedure. Suppose a duality conjecture relates theory A compactified on a manifold KA to theory B compactified on a manifold B. In that case if M is another smooth manifold, then we can conclude that the theory A compactified on KA × M should be dual to the theory B compactified on KB x M. Now let us suppose t h a t the first theory has a discrete global s y m m e t r y group GA; then by duality this should m a p to a discrete global s y m m e t r y group GB in the second theory isomorphic to GA. Now let us construct a new pair of theories by modding out the first theory by the group GA and the second theory by the group GB. Then naively one would expect that this new pair of theories are also dual to each other. Unfortunately this naive expectation does not always give right answer. So we first need to study under what condition we expect the resulting pair of theories to be dual to each other, and then construct new dual pair of theories by applying this procedure.
A. Sen~Nuclear Physics13(Proc. Suppl.) 52A (1997) 301 304
302
From now on we shall focus on cases where GA
(GB) is a Z2 group generated by gA (gB). We shall also decompose the action of ga (gB) into a component s that acts geometrically on the manifold M that is common to both the theories and a component hA (hB) which describes geometric action on the manifold KA (KB) as well as some possible internal symmetry transformation. We shall distinguish three possible cases: 1. s acts freely on M. 2. s does not act freely on M but has nontrivial action on M. 3. s is the identity transformation, i.e. leaves the whole of M fixed.
it
We shall see that in the first two cases we expect the resulting pair of coset theories, obtained by modding out the first theory by GA and the second theory by GB, to be dual to each other, whereas in the last case we do not expect the resulting pair of coset theories to be dual. s a c t s f r e e l y o n M: This case was analyzed by Vafa and Witten[6]. In this case M / s has no fixed points and hence it is a smooth manifold 3. As a result, the manifold KA × M/hA • s has a fibered structure with base M / s and fiber KA, with hA denoting the twist on the fiber along the closed cycle in M / s connecting a point P to s(P) in M. KB × M / h B . s has a similar structure. While comparing theory A on the first manifold with the theory B on the second manifold, we can simply replace the theory A on KA at a given point P on M / s by theory B on KB by the original duality between theory A on KA and theory B on KB. This way we can establish the duality between the two coset theories. This argument is valid when the fiber KA (KB) varies slowly over the base, which in turn can be achieved by going to the region of the moduli space where M / s has large volume. Once duality between two theories is established in one region of the moduli space, it must be valid at all points in the moduli space since we can go from one point in the moduli space to another by switching on appropriate background fields. 3By an abuse of notation we shall denote a Z~ group by its generator.
s d o e s n o t a c t f r e e l y o n M: In this case one can apply the same argument; however now M has fixed points under s and hence M / s is an orbifold. As a result KA x M/hA • s still has a fibered structure with base M / s and fiber KA, but at a fixed point Po on M / s , the fiber degenerates to KA/hA. Similar structure holds for KB x M/hB • s. Since a priori there is no reason why A on KA/hA should be dual to B on KB/hB, we see that the 'proof' of duality breaks down at the fixed point Po. However, since fixed points constitute a set of 'measure zero', it is reasonable to expect that they will not spoil the duality between the two coset theories. This has in fact been verified in many different examples; and we shall proceed with the assumption that this is indeed the case. s leaves t h e w h o l e o f M fixed: In this case s represents the identity transformation. Thus K A x M/hA • s now represents the product space (KA ~hA) x M; similarly KB x M / h B . s represents the product space (KB/hB) X M. Since there is no a priori reason why A on KA/hA should be dual to B on KB/hB, there is no reason for A on ( g A / h a ) × M to be dual to B on (KB/hB) × M. We shall now show how the string-string duality conjecture in six dimensions can be 'derived' from ten dimensional dualities by using these resuits[7]. We start with type IIB theory in ten dimensions. This has two perturbatively realized Zz symmetries, ( - 1 ) F~ which changes the sign of all the Ramond sector states on the left, and Q, which corresponds to the world-sheet parity transformation. It also has a conjectured SL(2,Z) S-duality symmetry, whose non-trivial Z2 element S exchanges the anti-symmetric tensor field arising in the Neveu-Schwarz NeveuSchwarz (NS) sector with the one arising in the Ramond-Ramond (RR) sector. By studying the action of various transformations on the massless fields in the theory, one can verify that S conjugates ( - 1 ) F~ to ~. We now construct a new dual pair of theories using the method of quotienting discussed before by choosing KA and KB to be points, hA and hB to be ( - 1 ) FL and i'~ respectively, M to be a four dimensional torus T 4 labelled by y 6 , . . . y g , and s to be the transformation ym __~ _ym for 6 < m < 9. We shall denote this transformation
A. Sen~Nuclear Physics B (Proc. Suppl.) 52A (1997) 301 304 by 2-4. This gives the equivalence between type IIB on T4/(--1)FL.74 and type IIB on T4/f~.Z4. If we now perform an R ---* 1 / R T-duality transformation on the y6 coordinate in the first theory, it converts the type IIB theory into a type IIA theory. Also by studying the action of various synlmetry transformations on the massless fields in the theory one can verify that it conjugates the transformation ( - 1 ) F~ .2"4 to 2-4. Thus the first theory is equivalent by T-duality transformation to type IIA on T4/Z4, which is a particular point in the moduli space of type IIA theory on K3. On the other hand, by making an R ~ 1 / R duality transformation on all the four coordinates of T 4 in the second theory we can conjugate the transformation f~ • 2-4 to just fL Thus the second theory is equivalent via T-duality transformation to type IIB theory on T 4 modded out by ft, which is just the type I theory on T 4. Using the conjectured equivalence between the type I theory and the SO(32) heterotic string theory in ten dimensions[2], we arrive at the equivalence between type IIA theory on K3 and heterotic string theory on T 4, which is precisely the string-string duality conjecture. The same procedure can be used to derive m a n y other duality conjectures[7,8]. Among these are dualities between 1. type IIA on TS/2-s and type I on T 8, 2. heterotic string theory on TS/2-s,24 and type l i b on K 3 x K3, where 2-s,24 denotes the change of sign of all coordinates of the (8, 24) signature Narain lattice, 3. M - t h e o r y on K3 and type I on T 3, 4. M-theory on T s / Z 5 and type IIB on K3, 5. M-theory on T 8 / Z s and type I on T 7, 6. M - t h e o r y on T9/2-9 and type IIB on TS/2-s, 7. F - t h e o r y on K 3 and heterotic on T 2, etc. Each of these conjectures can be tested by comparing the spectrum of massless states in the two resulting theories and in each case they agree. I shall conclude the talk by discussing how the duality between F-theory[9] on K3 and heterotic string theory on T 2 can be derived using this procedure[10]. We begin with a brief introduction
303
to F-theory. Type IIB theory in ten dimensions has two scalar fields, - the dilaton • and the RR scalar a sometimes known as the axiom The complex field
=_ a + ie -~/2 ,
(1)
transforms under the SL(2,Z) S-duality of the type IIB theory as a modular paramter. Conventional conformal field theory (CFT) compactification of type IIB theory sets I to be a constant on the internal manifold. F-theory is a novel way of compactifying type IIB theory with non-trivial background ;~ field. The starting point for such compactifications is an elliptically fibered manifold M - a manifold which has the structure of a two dimensional torus T 2 fibered over a base manifold B. The complex structure modulus r of the torus is a function of the coordinate y of the base. By definition F-theory on M is type IIB theory compactified on B with ~(y) = ~(y).
(2)
Vafa conjectured that F-theory on an ellptically fibered K3 (which has the structure of T 2 fibered over the base C P 1) is equivalent to beterotic string theory on T219]. We shall now derive this by going to the T4/Z4 orbifold limit of K3. (As argued before, once the duality is established at one point in the moduli space, it must hold at all points in the moduli space). T 4 / I 4 can be rewritten as (T2) ' x T2/Z~ • 2-2. By definition, F-theory on (T2) I x T 2 is type IIB theory on T 2, with ), being set equal to the modular parameter of (T2) '. The transformation I~ corresponds to the transformation ( - 1
- 1 ) of the
SL(2,Z) S-duality group. By studying the action of this transformation on the massless fields of the theory one can show that this is just the global s y m m e t r y transformation ( - 1 ) EL • f~ of the type IIB theory. From this we conclude that F - t h e o r y on K3 at this special point in the moduli space is equivalent to type IIB on T 2 / ( - 1 ) EL • f~ • Z2. By making R ~ 1 / R T-duality transformation on both the coordinates of T 2 one can convert ( - l ) EL • ~ • :Z-2 to f2. This shows that F - t h e o r y on K3 is equivalent to type IIB on T 2 / f L This in turn is just type I on T 2 which, by the ten dimensional duality conjecture, is equivalent to
304
A. Sen~Nuclear Physics B (Proc. Suppl.) 52A (1997)301-304
heterotic string theory on T 2. This extablishes the duality between F-theory on K3 and heterotic string theory on T 2. This way all conjectured dualities in string theory can be traced to a few duality conjectures involving ten / eleven dimensional theories. Understanding the origin of these dualities still remains a mystery; however reducing the number of independent duality conjectures might be a step towards a deeper understanding of all duality symmetries involving string theory. REFERENCES
1. M. Duff, hep-th/9501030. 2. E. Witten, Nucl. Phys. B443 (1995) 85 [hepth/9503124]. 3. C. Hull and P. Townsend, Nucl. Phys. B438 (1995) 109 [hep-th/9410167]. 4. S. Kachru and C. Vafa, Nucl. Phys. B450 (1995) 69 [hep-th/9605105]. 5. A. Klemm, W. Lerche and P. Mayr, Phys. Lett. B357 (1995) 313 [hep-th/9506112]. 6. C. Vafa and E. Witten, hep-th/9507050. 7. A. Sen, hep-th/9604070. 8. A. Sen, hep-th/9603113. 9. C. Vafa, hep-th/9602022. 10. A. Sen, hep-th/9605150.