Duality and Strings Dieter Lüst, LMU and MPI München Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Duality of 4 - dimensional string constructions: • Covariant lattices ⇔ (a)symmetric orbifolds (1986/87: W. Lerche, D.L., A. Schellekens ⇔ L. Ibanez, H.P. Nilles, F. Quevedo) • Intersecting D-brane models ☞ SM (?) (2000/01: R. Blumenhagen, B. Körs, L. Görlich, D.L., T. Ott ⇔ G. Aldazabal, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, A. Uranga) Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Duality of 4 - dimensional string constructions: • Covariant lattices ⇔ (a)symmetric orbifolds (1986/87: W. Lerche, D.L., A. Schellekens ⇔ L. Ibanez, H.P. Nilles, F. Quevedo) • Intersecting D-brane models ☞ SM (?) (2000/01: R. Blumenhagen, B. Körs, L. Görlich, D.L., T. Ott ⇔ G. Aldazabal, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, A. Uranga) ➢ Madrid (Spanish) Quiver ! Freitag, 15. März 13 Luis made several very profound and important contributions to theoretical physics ! Often we were working on related subjects and I enjoyed various very nice collaborations and friendship with Luis. Duality of 4 - dimensional string constructions: • Covariant lattices ⇔ (a)symmetric orbifolds (1986/87: W. Lerche, D.L., A. Schellekens ⇔ L. Ibanez, H.P. Nilles, F. Quevedo) • Intersecting D-brane models ☞ SM (?) (2000/01: R. Blumenhagen, B. Körs, L. Görlich, D.L., T. Ott ⇔ G. Aldazabal, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, A. Uranga) Duality of burning, common interests! Freitag, 15. März 13 Duality has always played an important role in our common work. Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality • It is a stringy symmetry: needs momentum M i and winding (dual momentum) modes: p =(˜p ,pi) Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality • It is a stringy symmetry: needs momentum M i and winding (dual momentum) modes: p =(˜p ,pi) winding momentum Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality • It is a stringy symmetry: needs momentum M i and winding (dual momentum) modes: p =(˜p ,pi) • Suggests a doubling of space coordinates: M i Doubled geometry: X =(X˜i,X ) Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality • It is a stringy symmetry: needs momentum M i and winding (dual momentum) modes: p =(˜p ,pi) • Suggests a doubling of space coordinates: M i Doubled geometry: X =(X˜i,X ) • Relates strong and weak coupling in string theory: S - duality Freitag, 15. März 13 Duality has always played an important role in our common work. • Relates different string geometries: large and small backgrounds: T - duality • It is a stringy symmetry: needs momentum M i and winding (dual momentum) modes: p =(˜p ,pi) • Suggests a doubling of space coordinates: M i Doubled geometry: X =(X˜i,X ) • Relates strong and weak coupling in string theory: S - duality • Relates conventional (Riemannian) geometries to new stringy geometries (non-commutative & non-associative) Freitag, 15. März 13 Volume 245, number 3, 4 PHYSICS LETTERS B 16 August 1990 Supersymmetry breaking from duality invariant gaugino condensation A. Font a, L.E. Ib~ifiez b, D. Liist b and F. Quevedo c a Departamento de Fisica, Universidad Central de Venezuela, Aptdo. 20513, Caracas 1020-A, Venezuela b CERN, CH-1211 Geneva 23, Switzerland c Theoretical Division LANL, Los Alamos, NM 87545, USA Received 3 May 1990 It is known that the formation of gaugino condensates can be a source of supersymmetry breaking in string theory. We study the constraints imposed by target space modular invariance on the formation of such condensates. We find that the dependence of the vacuum energy on the moduli of the internal variety is such that the theory is forced to be compactified. The radius of compactification is of the order of the string scale and in the process target space duality is spontaneously broken. One of the major open problems in four- crete shifts of the axionic background B, and the T dimensional string theories is the breaking of space- moduli space has to be restricted to the fundamental time (N = 1) supersymmetry. Because ofphenomeno- region SU(1, 1)/[U(1) x PSL(2, Z)]. logical reasons it is desirable that space-time super- Although duality respectively target space modular symmetry is broken spontaneously below the Planck invariance is only shown [12] to be an unbroken scale, and the most promising scenario realizing this symmetry at any order of string perturbation theory, requirement is the mechanism of gaugino condensa- one also expects that non-perturbative string effects tion in the so-called hidden sector [1-7]. Gaugino respect these discrete symmetries. Adopting this point condensation cannot, up to now, be directly analyzed of view, the effective action describing the spon- in string theory; however, there are strong arguments taneous supersymmetry breaking via non-perturba- that it actually occurs at the level of the low-energy tive effects, like gaugino condensates must be also Freitag, 15. Märzeffective 13 field theory. More recently it was shown invariant under the modular transformation on T. In [8-10] that the effective supergravity action following fact it was shown in ref. [8] that the non-perturbative, from string compactification on orbifolds or even purely T-dependent effective superpotential must be Calabi-Yau manifolds is severely constrained by an a modular form like underlying string symmetry, the so-called target space W(T) ~ ~9 ( T)-6, (2) modular invariance. The target space modular group PSL(2, Z) acts on the complex scalar T as where ~(T)=ql/24I]n(1-q" ) is the well-known Dedekind function, q~ exp(-2~rT), and the result- aT-ib ing gravitino mass is of the form m~/2- T-~-- a,b,c,d~Z, ad-bc=l, (1) icT+d' 1/(T+T*)3]~(T)] '2. We will show that gaugino condensation provides a natural dynamical reali- where (T) is the background modulus associated to zation for exactly this kind of effective actions. the overall scale of the internal six-dimensional space Using the relation W ~ (AA) ~ exp[(3/2bo)f( T)], this on which the string is compactified. Specifically, T = could be seen as if the gauge kinetic function of the R2+iB with R being the "radius" of the internal N = 1 supergravity action is of the form f(T)= space and B an internal axion. The target space -bolog[B(T)4]. Interestingly enough, this kind of modular transformations contain the well-known expression for the gauge kinetic function was recently duality transformation R~ 1/R [11] as well as dis- derived from a direct one-loop string calculation [ 13 ]. 0370-2693/90/$ 03.50 O 1990 - Elsevier Science Publishers B.V. (North-Holland) 401 • In this paper we considered T-duality covariant non-perturbative superpotentials from gaugino condensation in the 4D effective string action: 1 6 W (T ) n.p. ' ⌘(T ) ✓ ◆ Freitag, 15. März 13 • In this paper we considered T-duality covariant non-perturbative superpotentials from gaugino condensation in the 4D effective string action: 1 6 W (T ) n.p. ' ⌘(T ) ✓ ◆ ➠ Spontaneous supersymmetry breaking and stabilization of T-modulus. Freitag, 15. März 13 • In this paper we considered T-duality covariant non-perturbative superpotentials from gaugino condensation in the 4D effective string action: 1 6 W (T ) n.p. ' ⌘(T ) ✓ ◆ ➠ Spontaneous supersymmetry breaking and stabilization of T-modulus. (Nilles, Olechowski; Ferrara, Magnoli, Taylor, Veneziano (1990)) Freitag, 15. März 13 Volume 249, number 1 PHYSICS LETTERS B 11 October 1990 Strong-weak coupling duality and non-perturbative effects in string theory A. Font a L.E. Ib~fiez b, D. [,fist b and F. Quevedo c a Departamento de Ftstca, Umverstta Central de Venezuela, Aptdo 20513, Caracas I020-A, Venezuela b CERN, CH-1211 Geneva 23, Swttzerland c TheoretwalDtvtston LANL, LosAlamos, NM87545, USA Received 13 July 1990 We conjecture the existence of a new discrete symmetry of the modular type relating weak and strong coupling in string theory. The existence of thxs symmetry would strongly constrain the non-perturbat~ve behawour m string partlt~on functmns and intro- duces the notion of a maximal (minimal) couphng constant. An effective lagrangmn analysxs suggests that the ddaton vacuum expectatmn value is dynamically fixed to be of order one In supersymmetnc heteroUc strings, supersymmetry (as well as thxs modular symmetry itself ) is generically spontaneously broken Modular lnvariance appears in a variety of physi- to the existence of the Bran antisymmetric tensor cal problems [ 1 ]. These symmetries involve an in- which acts as a 0-parameter. In more reahstlc six-di- varmnce under the inversion of coupling constants mensional compactifications (like e.g. orblfolds) the along with the discrete translations of a "theta term".