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. 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)

(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 !

(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

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 ) • Relates strong and weak coupling in string theory: S - duality

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". same structure (conveniently generalized) is also The first example of this type of symmetry in field found. This target-space modular lnvariance strongly theory was discovered by Cardy [ 2 ] who showed that constrains the form of the low-energy effective action Freitag, 15. Märzthe 13 phase structure of the abelian Higgs model on the as a function of the compactlfication moduli [ 5 ]. It lattice exhibits such a type of invariance under rever- can also gwe interesting information about the pos- sion of couplings and shift of the 0-parameter. These sible form of non-perturbative stnng corrections (like transformations generate an infinite discrete group e.g. supersymmetry breaking [6,7 ] ) if duality were SL(2, Z). In the context of string theory such a sort an exact symmetry of string theory (possibly broken of symmetry seems also ublquous. The one loop par- spontaneously but not explicitly) [ 6 ]. tltion functions in terms of the world sheet modular In the present letter we conjecture the existence of parameter z must be exphcltly modular invanant. a further modular invarlance symmetry m string the- More recently [ 3 ] It has been realized that the spec- ory. This includes a duahty invariance under which trum and interactions of compactified strings are in- the string dilaton (whose VEV yields the coupling variant under the inversion of the compactlfiCatlon constant in heterotic strings) gets inverted. In four length R-~ oe'/R (c~' denotes the string tension) ffone dimensions the dllaton comes along [ 8 ] with a pseu- simultaneously replaces quantlzed momenta by doscalar ("axlon") field 0. Both degrees of freedom winding modes (target space duality). This symme- form a complex scalar try survives order by order m perturbation theory [ 4 ]. Therefore all mequlvalent compactificatIons are un- 1 s= --; +10 (1) ambiguously characterized by radii R larger (smaller) g~ than the mlmmal (maximal) length ~, the scale set by the extension of the string itself. In the case of which is the lowest component of a chiral superfield a two-dimensional compactlfication the duality sym- in supersymmetrlc 4D strings. Thus we conjecture an metry is extended to the full modular invarlance due mvariance under the modular symmetry

Freitag, 15. März 13 In this paper we conjectured strong-weak coupling S-duality in the compactiﬁed heterotic string. • Olive & Montonen electric - magnetic duality.

Freitag, 15. März 13 In this paper we conjectured strong-weak coupling S-duality in the compactified heterotic string. • Olive & Montonen electric - magnetic duality. • Elementary string - wrapped NS 5-brane duality. • Geometric duality from compactified 11- dimensional supergravity - we speculated that the heterotic string can obtained from a 11- dimensional membrane, compactified to 10 dimensions.

➠ Formulation of S-duality invariant effective action.

Freitag, 15. März 13 Freitag, 15. März 13 N UCLEAR Nuclear Physics B 382 (1992) 305—361 P H Y S I CS B North-Holland ______

Duality-anomaly cancellation, minimal string unification and the effective low-energy lagrangian of 4D strings

Luis E. Ibáflez and Dieter Lust * CERN, Genera, Switzerland

Received 19 February 1992 (Revised 22 May 1992) Accepted for publication 1 June 1992

We present a systematic study of the constraints coming from target-space duality and the associated duality-anomaly cancellations on orbifold-like 4D strings. A prominent role is played by the modular weights of the massless fields. We present a general classification of all possible modular weights of massless fields in abelian orbifolds. We show that the cancellation of modular anomalies strongly constrains the massless fermion content of the theory, in close analogy with the standard ABJ anomalies. We emphasize the validity of this approach not only for (2, 2) orbifolds but for large classes of (0, 2) models with and without Wilson lines. As an application one can show that one cannot build a or ~ orbifold whose massless charged sector with respect to the (level one) gauge group SU(3)XSU(2)XU(1) is that of the minimal Freitag, 15. März 13 supersymmetric standard model, since any such model would necessarily have duality anomalies. A general study of those constraints for abelian orbifolds is presented. Duality anomalies are also related to the computation of string threshold corrections to gauge coupling constants. We present an analysis of the possible relevance of those threshold Corrections to the computation of 2O~and a sin 3 for all abelian orbifolds. Some particular minimal scenarios, namely those based on all ~N orbifolds except Z6 and ~, are ruled Out on the basis of these constraints. Finally we discuss the explicit dependence of the SUSY-breaking soft terms on the modular weights of the physical fields. We find that those terms arc in general not universal. In some Cases specific relationships for gaugino and scalar masses are found.

1. Introduction

Four-dimensional superstrings [1] constitute at the moment the best candidates for unification of all interactions. In trying to use these theories to describe the observed physics, it is of fundamental importance to obtain the effective low-energy field theory of each given 4D string. Much progress along these lines has been achieved in the last few years and there is at present a good knowledge of the form of the effective lagrangian for the case of orbifold-like 4D strings.

* Heisenberg Fellow.

0550-32t3/92/$05.00 © 1992 — Elsevier Science Publishers B.V. All rights reserved • In this paper we obtained stringy constraints on the massless particle spectrum in compactiﬁed string from the requirement of absence of sigma- model, duality anomalies.

• Model independent discussion of the soft parameters in the 4D effective heterotic string action after spontaneous supersymmetry breaking.

Freitag, 15. März 13 Further outline: II) Non-geometric ﬂux compactiﬁcations III) Non-commutative/non-associative geometries from non-geometric string backgrounds

D. Lüst, JEHP 1012 (2011) 063, arXiv:1010.1361, R. Blumenhagen, E. Plauschinn, arXiv:1010.1263, R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arXiv:1106.0316, C. Condeescu, I. Florakis, D. Lüst, arXiv:1202.6366, D. Andriot, M. Larfors, D. Lüst, P. Patalong, arXiv:1204.1979. IV) (Intersecting) Q- and R-branes

(F. Haßler, D.Lüst, arXiv:1303.1413) V) Outlook & open problems

Freitag, 15. März 13 II) Non-geometric ﬂux compactiﬁcations

12 Freitag, 15. März 13 II) Non-geometric ﬂux compactiﬁcations

Recall standard Riemannian geometry:

12 Freitag, 15. März 13 II) Non-geometric ﬂux compactiﬁcations

Recall standard Riemannian geometry: - Flat space: Triangle: + ⇥ + ⇤ = ⌅ - Curved space: Triangle: + ⇥ + ⇤ > ⌅(< ⌅)

12 Freitag, 15. März 13 II) Non-geometric ﬂux compactiﬁcations

Recall standard Riemannian geometry: - Flat space: Triangle: + ⇥ + ⇤ = ⌅ - Curved space: Triangle: + ⇥ + ⇤ > ⌅(< ⌅)

Manifold: need different coordinate charts, which are patched together by coordinates transformations, i.e. group of diffeomorphisms: Di(M): f : U U 12 Freitag, 15. März 13 Properties of Riemannian manifolds:

● distances between two points can be arbitrarily short.

● coordinates commute with each other:

[Xi,Xj]=0

This is the situation, if one is using point particles to probe distance and the geometry of space.

Now we want to understand, how extended closed strings may possibly see the (non)-geometry of space.

Freitag, 15. März 13 (Hellerman, McGreevy, Williams (2002); Non-geometric string backgrounds: Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005, Aldazabal, Camara, Font, Ibanez (2006))

14 Freitag, 15. März 13 (Hellerman, McGreevy, Williams (2002); Non-geometric string backgrounds: Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005, Aldazabal, Camara, Font, Ibanez (2006)) - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. Transition functions between two coordinate patches are given in terms of O(D,D) T-duality transformations: Di↵(M ) O(D, D) D ! Q-space will become non-commutative: [Xi,Xj] =0 - Non-geometric R-fluxes: spaces that are even 6 locally not anymore manifolds. R-space will become non-associative: [Xi,Xj,Xk]:=[[Xi,Xj],Xk]+cycl. perm. = =(Xi Xj) Xk Xi (Xj Xk)+ =0 · · · · ···6 14 Freitag, 15. März 13 Example: Three-dimensional flux backgrounds: Fibrations: 2-dim. torus that varies over a circle:

2 3 1 T 1 2 , M , S 3 X ,X ! ! X The ﬁbration is speciﬁed by its monodromy properties.

2 3 3 3 T : ij(X )=Gij(X )+Bij(X ) E S1 O(2,2) monodromy: (X3 +2⇡)=g (X3) E O(2,2)E

Freitag, 15. März 13 Example: Three-dimensional ﬂux backgrounds: Fibrations: 2-dim. torus that varies over a circle:

2 3 1 T 1 2 , M , S 3 X ,X ! ! X The ﬁbration is speciﬁed by its monodromy properties.

2 3 3 3 T : ij(X )=Gij(X )+Bij(X ) E S1 O(2,2) monodromy: (X3 +2⇡)=g (X3) E O(2,2)E a⌧(X3)+b 2 ⌧(X3 +2⇡)= Complex structure ⌧ of T : c⌧(X3)+d a ⇢(X3)+b 2 ⇢(X3 +2⇡)= 0 0 Kähler parameter ⇢ of : 3 T c0⇢(X )+d0 15

Freitag, 15. März 13 (Shelton, Taylor, Wecht, 2005; Chain of four T-dual examples: Dabholkar, Hull, 2005) (i) Geometric space: 3-dimensional torus with H - ﬂux

2 R1 00 3 2 , B12 = HXH ,H123 = @3B12 = H Gij = 0 R2 0 0 21 00R3 ⇢(X3 )=iRR HX3 @ A H 1 2 H X3 X3 +2⇡R g : ⇢(X3 +2⇡R )=⇢(X3 )+2⇡HR H ! H 3 ) O(2,2) H 3 H 3

16 Freitag, 15. März 13 (Shelton, Taylor, Wecht, 2005; Chain of four T-dual examples: Dabholkar, Hull, 2005) (i) Geometric space: 3-dimensional torus with H - ﬂux

2 R1 00 3 2 , B12 = HXH ,H123 = @3B12 = H Gij = 0 R2 0 0 21 00R3 ⇢(X3 )=iRR HX3 @ A H 1 2 H X3 X3 +2⇡R g : ⇢(X3 +2⇡R )=⇢(X3 )+2⇡HR H ! H 3 ) O(2,2) H 3 H 3 T-duality in X1 : (ii) Geometric spaces: twisted 3-torus with f - ﬂux (f H)

3 ⌘ 1 fXf R2 R2 0 1 1 , Bij =0 3 3 2 Gij = 0 fXf 2 fXf 1 R2 R2 + R 0 3 3 1 1 ⌧(X )=iRR fX B 00R2C f 1 2 f B ⇣ ⌘ 3C 3 @ 3 A 3 3 Xf Xf +2⇡R3 gO(2,2) : ⌧(Xf +2⇡R3)=⌧(Xf )+2⇡fR3 ! ) 16 Freitag, 15. März 13 T-duality in X2 : (iii) Non-geometric space: T-fold with Q-ﬂux (Q f H) ⌘ ⌘ QX3 F Q 2 1 R2 00 0 R2R2 0 3 1 1 2 QX F 3 Q QXQ Gij = 0 R2 0 ,Bij = F 0 1 ,F= 1+ 0 2 1 R2R2 00 2 1 2 0 R1R2 ! 1 00R3 B 000C B C B C @ A @ A @ A 1 ⇢(X3 ) ⇢(X3 )= g : ⇢(X3 +2⇡R )= Q Q QX3 iR R ) O(2,2) Q 3 1+2⇡R Q ⇢(X3 ) Q 1 2 3 Q This does not correspond to a standard diffeomorphism but to a T-duality transformation.

Freitag, 15. März 13 T-duality in X2 : (iii) Non-geometric space: T-fold with Q-flux (Q f H) ⌘ ⌘ QX3 F Q 2 1 R2 00 0 R2R2 0 3 1 1 2 QX F 3 Q QXQ Gij = 0 R2 0 ,Bij = F 0 1 ,F= 1+ 0 2 1 R2R2 00 2 1 2 0 R1R2 ! 1 00R3 B 000C B C B C @ A @ A @ A 1 ⇢(X3 ) ⇢(X3 )= g : ⇢(X3 +2⇡R )= Q Q QX3 iR R ) O(2,2) Q 3 1+2⇡R Q ⇢(X3 ) Q 1 2 3 Q This does not correspond to a standard diffeomorphism but to a T-duality transformation. T-duality in X3 : (iv) Non-geometric space with R-flux Now the Buscher rules for T-duality cannot be applied. There exist no locally defined metric and B-field. 17

Freitag, 15. März 13 III) World sheet non-commutativity/non-associativity

18 Freitag, 15. März 13 III) World sheet non-commutativity/non-associativity Open strings with F-ﬂux:

18 Freitag, 15. März 13 III) World sheet non-commutativity/non-associativity Open strings with F-ﬂux: Coordinates of open string end-points are non-commutative: 2-dimensional D-branes with 2-form F-ﬂux ⇒ 2⇡i↵ F [Xi(⌧),Xj(⌧)] = ✏ij⇥ , ⇥ = 0 1+F 2 (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); J. Fröhlich, K. Gawedzki (1993); F. Lizzi, ER. Szabo (1997); A.Connes, M. Douglas, A. Schwarz (1997), V. Schomerus (1999); .... )

18 Freitag, 15. März 13 III) World sheet non-commutativity/non-associativity Open strings with F-ﬂux: Coordinates of open string end-points are non-commutative: 2-dimensional D-branes with 2-form F-ﬂux ⇒ 2⇡i↵ F [Xi(⌧),Xj(⌧)] = ✏ij⇥ , ⇥ = 0 1+F 2 (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); constant J. Fröhlich, K. Gawedzki (1993); F. Lizzi, ER. Szabo (1997); A.Connes, M. Douglas, A. Schwarz (1997), V. Schomerus (1999); .... )

➢ Non-commutative gauge theories.

(N. Seiberg, E. Witten (1999); J. Madore, S. Schraml, P. Schupp, J. Wess (2000); .... )

18 Freitag, 15. März 13 III) World sheet non-commutativity/non-associativity Open strings with F-ﬂux: Coordinates of open string end-points are non-commutative: 2-dimensional D-branes with 2-form F-ﬂux ⇒ 2⇡i↵ F [Xi(⌧),Xj(⌧)] = ✏ij⇥ , ⇥ = 0 1+F 2 (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); constant J. Fröhlich, K. Gawedzki (1993); F. Lizzi, ER. Szabo (1997); A.Connes, M. Douglas, A. Schwarz (1997), V. Schomerus (1999); .... )

➢ Non-commutative gauge theories.

(N. Seiberg, E. Witten (1999); J. Madore, S. Schraml, P. Schupp, J. Wess (2000); .... )

Remark: In the T-dual picture (D1-brane at angle) the coordinates are commutative!

18 Freitag, 15. März 13 We considered two different types of models:

Freitag, 15. März 13 We considered two different types of models: (i) (Non-)geometric backgrounds with elliptic monodromy and non-constant ﬂuxes. D. L., JHEP 1012 (2011) 063, arXiv:1010.1361, They can be described in terms of an (a)symmetric freely acting Z 4 - orbifold. C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arXiv:1202.6366

Twisted Non- Non- Flat torus Tx1 Tx2 Tx torus with geometric 3 geometric with H-flux f-flux space with space with Q-flux R-flux

i j [XH,f ,XH,f ]=0

Twisted Non- Non- Flat torus Tx1 Tx2 Tx torus with geometric 3 geometric with H-flux f-flux space with space with Q-flux R-flux

i j [X1 ,X2 ] Q p˜3 [XH,f ,XH,f ]=0 R R '

Twisted Non- Non- Flat torus Tx1 Tx2 Tx torus with geometric 3 geometric with H-flux f-flux space with space with Q-flux R-flux

i j [X1 ,X2 ] Q p˜3 [XH,f ,XH,f ]=0 R R ' 1 2 3 [[XR,XR],XR] R ' 17 Freitag, 15. März 13 - model of geometric H- or f-ﬂux background: 1 S = d2 G (X) ⌘↵ + B (X) "↵ @ Xi@ Xj 4⇡↵ ij ij ↵ 0 Z⌃

20 Freitag, 15. März 13 - model of geometric H- or f-ﬂux background: 1 S = d2 G (X) ⌘↵ + B (X) "↵ @ Xi@ Xj 4⇡↵ ij ij ↵ 0 Z⌃ Quantize at linear order in the ﬂux H or f ,

20 Freitag, 15. März 13 - model of geometric H- or f-ﬂux background: 1 S = d2 G (X) ⌘↵ + B (X) "↵ @ Xi@ Xj 4⇡↵ ij ij ↵ 0 Z⌃ Quantize at linear order in the ﬂux H or f , i i i XH (⌧, )=X(H0)(⌧, )+HX(H1)(⌧, ) obeying the closed string boundary (monodromy) conditions: X3 (⌧, +2⇡)=X3 (⌧, )+2⇡ p˜3 = H H ) 1 1 XH (⌧, +2⇡)=XH (⌧, ) , X2 (⌧, +2⇡)=X2 (⌧, ) , O(2, 2) H H X˜ (⌧, +2⇡)=X˜ (⌧, ) 2⇡ p˜3 HX2 (⌧, ) , H1 H1 H ˜ ˜ 3 1 { XH2(⌧, +2⇡)=XH2(⌧, )+2⇡ p˜ HXH (⌧, ) .

20 Freitag, 15. März 13 - model of geometric H- or f-ﬂux background: 1 S = d2 G (X) ⌘↵ + B (X) "↵ @ Xi@ Xj 4⇡↵ ij ij ↵ 0 Z⌃ Quantize at linear order in the ﬂux H or f , i i i XH (⌧, )=X(H0)(⌧, )+HX(H1)(⌧, ) obeying the closed string boundary (monodromy) conditions: X3 (⌧, +2⇡)=X3 (⌧, )+2⇡ p˜3 = H H ) 1 1 XH (⌧, +2⇡)=XH (⌧, ) , winding X2 (⌧, +2⇡)=X2 (⌧, ) , O(2, 2) H H number X˜ (⌧, +2⇡)=X˜ (⌧, ) 2⇡ p˜3 HX2 (⌧, ) , H1 H1 H ˜ ˜ 3 1 { XH2(⌧, +2⇡)=XH2(⌧, )+2⇡ p˜ HXH (⌧, ) .

20 Freitag, 15. März 13 - model of geometric H- or f-ﬂux background: 1 S = d2 G (X) ⌘↵ + B (X) "↵ @ Xi@ Xj 4⇡↵ ij ij ↵ 0 Z⌃ Quantize at linear order in the ﬂux H or f , i i i Xf (⌧, )=X(f0)(⌧, )+fX(f1)(⌧, ) obeying the closed string boundary (monodromy) conditions:

X3(⌧, +2⇡)=X3(⌧, )+2⇡ p˜3 = f f ) X1(⌧, +2⇡)=X1(⌧, ) 2⇡ p˜3 fX2(⌧, ) , f f f winding 2 2 Xf (⌧, +2⇡)=Xf (⌧, ) , number O(2, 2) X˜f1(⌧, +2⇡)=X˜f1(⌧, ) , ˜ ˜ 3 ˜ {Xf2(⌧, +2⇡)=Xf2(⌧, )+2⇡ p˜ f Xf1(⌧, ) .

X3(⌧, +2⇡)=X3(⌧, )+2⇡ p˜3 = f f ) X1(⌧, +2⇡)=X1(⌧, ) 2⇡ p˜3 fX2(⌧, ) , f f f winding 2 2 Xf (⌧, +2⇡)=Xf (⌧, ) , number O(2, 2) X˜f1(⌧, +2⇡)=X˜f1(⌧, ) , 3 X˜f2(⌧, +2⇡)=X˜f2(⌧, )+2⇡ p˜ f X˜f1(⌧, ) .

[Xi (⌧, ),Xj (⌧, )] = 0 Result:{ H,f (H,f) 0 (P = L ) i i @⌧ X [Pi(⌧, ),Pj(⌧, 0)] = 0 i i [X (⌧, ),Pj(⌧, 0)] = i ( 0) H,f j 20 Freitag, 15. März 13 Quantization of non-geometric Q-ﬂux background: i i i XQ(⌧, )=XQ0(⌧, )+QXQ1(⌧, )

Freitag, 15. März 13 Quantization of non-geometric Q-ﬂux background: i i i XQ(⌧, )=XQ0(⌧, )+QXQ1(⌧, ) Two consistency requirements: (i) Canonical T-duality: (E. Alvarez, L. Alvarez-Gaume, Y. Lozano, 1994) T d. along i =2: @ X2 = @ X2 QX3@ X1 ⌧ Q f f f @ X2 = @ X2 QX3@ X1 Q ⌧ f f ⌧ f

X3 (⌧, +2⇡)=X3 (⌧, )+2⇡ p˜3 = (Q f H) Q Q ) ⌘ ⌘ X1 (⌧, +2⇡)=X1 (⌧, ) 2⇡ p˜3 Q X˜ (⌧, ) , Q Q Q2 2 2 3 ˜ X (⌧, +2⇡)=X (⌧, )+2⇡ p˜ Q XQ1(⌧, ) , (Similar to DN O(2, 2) Q Q X˜ (⌧, +2⇡)=X˜ (⌧, ) , boundary conditions Q1 Q1 of open string.) X˜Q2(⌧, +2⇡)=X˜Q2(⌧, ) . Mix{ coordinates with Non-geometric background. dual coordinates. () 21 Freitag, 15. März 13 Quantization of non-geometric Q-ﬂux background: i i i XQ(⌧, )=XQ0(⌧, )+QXQ1(⌧, ) Two consistency requirements: (i) Canonical T-duality: (E. Alvarez, L. Alvarez-Gaume, Y. Lozano, 1994) T d. along i =2: @ X2 = @ X2 QX3@ X1 ⌧ Q f f f 2 2 3 1 @XQ = @⌧ Xf QXf @⌧ Xf (ii) Closed string boundary conditions:

X3 (⌧, +2⇡)=X3 (⌧, )+2⇡ p˜3 = (Q f H) Q Q ) ⌘ ⌘ X1 (⌧, +2⇡)=X1 (⌧, ) 2⇡ p˜3 Q X˜ (⌧, winding) , Q Q Q2 2 2 3 ˜ number X (⌧, +2⇡)=X (⌧, )+2⇡ p˜ Q XQ1(⌧, ) , (Similar to DN O(2, 2) Q Q X˜ (⌧, +2⇡)=X˜ (⌧, ) , boundary conditions Q1 Q1 of open string.) X˜Q2(⌧, +2⇡)=X˜Q2(⌧, ) . Mix{ coordinates with Non-geometric background. dual coordinates. () 21 Freitag, 15. März 13 Then we derive the following result for the commutator of the coordinates: D. Andriot, M. Larfors, D. L., P. Patalong, arXiv:1211.6437 1 2 [XQ(⌧, ),XQ(⌧, 0)] =

i 3 1 in(0 ) 1 in(0 ) i 2 Q p˜ e (0 ) e + (0 ) 2 n2 n 2 ✓n=0 n=0 ◆ X6 X6

Freitag, 15. März 13 Then we derive the following result for the commutator of the coordinates: D. Andriot, M. Larfors, D. L., P. Patalong, arXiv:1211.6437 1 2 [XQ(⌧, ),XQ(⌧, 0)] =

i 3 1 in(0 ) 1 in(0 ) i 2 Q p˜ e (0 ) e + (0 ) 2 n2 n 2 ✓n=0 n=0 ◆ X6 winding X6 number

Freitag, 15. März 13 Then we derive the following result for the commutator of the coordinates: D. Andriot, M. Larfors, D. L., P. Patalong, arXiv:1211.6437 1 2 [XQ(⌧, ),XQ(⌧, 0)] =

i 3 1 in(0 ) 1 in(0 ) i 2 Q p˜ e (0 ) e + (0 ) 2 n2 n 2 ✓n=0 n=0 ◆ X6 X6

0 : ! ⇡2 [X1 (⌧, ),X2 (⌧, )] = i Q p˜3 Q Q 6

The non-commutativity of the torus (ﬁbre) coordinates is determined by the winding in the circle (base) direction. 22

Freitag, 15. März 13 Corresponding uncertainty relation:

(X1 )2(X2 )2 L6 Q2 p˜3 2 Q Q s h i

The spatial uncertainty in the X 1 ,X 2 - directions grows with the dual momentum in the third direction: non-local strings with winding in third direction.

23 Freitag, 15. März 13 R-flux background: T-duality in x 3 -direction ⇒ R-flux p˜3 p , X˜ X3 ! 3 Q,3 ⌘ R ⇒ For the case of non-geometric R-fluxes one gets: ⇡2 [X1 ,X2 ]= i Rp R Q R R 6 3 ⌘

24 Freitag, 15. März 13 R-flux background: T-duality in x 3 -direction ⇒ R-flux p˜3 p , X˜ X3 ! 3 Q,3 ⌘ R ⇒ For the case of non-geometric R-fluxes one gets: ⇡2 [X1 ,X2 ]= i Rp R Q R R 6 3 ⌘ Use [X3 ,p ]=i = R 3 ) ⇡2 [[X1 (⌧, ),X2 (⌧, )],X3 (⌧, )] + perm. = R R R R 6 Non-associative algebra! (It is called twisted Poisson structure, and these commutation relations also appear for a point particle in the field of a magnetic monopole.) (R. Jackiw, 1985; C. Klimcik, T. Strobl, (2002); A. Alekseev, T. Strobl, (2005); C. Saemann, R. Szabo, arXiv:1106.1890) 24 Freitag, 15. März 13 V) (Intersecting) Q- and R-branes (F. Haßler, D.L., arXiv:1303.1413)

Freitag, 15. März 13 V) (Intersecting) Q- and R-branes (F. Haßler, D.L., arXiv:1303.1413) Standard NS action:

10 2 2 1 ijk S = d xe G + 4(@) H H | | R 12 ijk Z ✓ ◆ (i) NS 5-brane solutionp - source of H-ﬂux: x0 x1 x2 x3 y1 y2 y3 y4 y5 y6 NS5

2N N N i 2 kN2 N N dsNS5 = (dx ) + h(r) (dx ) k ? i X Xk Hmnp = ✏mnpq@qh(r)

Freitag, 15. März 13 V) (Intersecting) Q- and R-branes (F. Haßler, D.L., arXiv:1303.1413) Standard NS action:

10 2 2 1 ijk S = d xe G + 4(@) H H | | R 12 ijk Z ✓ ◆ (i) NS 5-brane solutionp - source of H-ﬂux: x0 x1 x2 x3 y1 y2 y3 y4 y5 y6 NS5

2N N N i 2 kN2 N N dsNS5 = (dx ) + h(r) (dx ) k ? i X Xk Hmnp = ✏mnpq@qh(r) (ii) T-duality in y 1-direction KK-monopole - source of f-ﬂux:

x0 x1 x2 x3 y y2 y3 y4 y5 y6 KK • N N N N 2 N N 1 ds2 = (dxµ)2 + (dyi)2 + dy + A dyi + h(r) (dx3)2 + (dyi)2 KK h(r) i µ=0,1,2 i=4,5,6 i=2,3 i=2,3 X X ✓ X ◆ ✓ X ◆ 25

Freitag, 15. März 13 (iii) T-duality in y -direction2 Q-brane - source of Q-ﬂux:

26 Freitag, 15. März 13 (iii) T-duality in y -direction2 Q-brane - source of Q-flux: 0 1 2 3 3 4 5 6 x x x x y y0 y y y y Q • • 2 µ 2 i 2 h(r) 2 2 3 2 3 2 ds = (dx ) +N N(dy N) + (dy + dy0 )+N h(rN) (dxN) +(dy ) Q h(r)2 + A2 µ=0,1,2 i=4,5,6 2 ✓ ◆ X X See also: Bergshoeff, Ortin, Riccioni (2011); A2 de Boer, Shigemori (2010,2012) By,y0 = 2 2 h(r) + A2 This is a non-geometric configuration and hence it is globally ill defined with respect to the standard NS action.

10 2˜ 2 1 2 S˜ = d x G˜ e ˜ + 4(@˜) Q | | R 4 Z q ✓ (Andriot, Hohm,Larfors, D.L.◆ Patalong (2011/12))

2 µ 2 1 2 2 3 2 3 2 i 2 ds˜ = (dx ) + dy + dy0 + h(r) (dx ) +(dy ) + (dy ) , Q h(r) µ=0,1,2 i=4,5,6 X ✓ ◆ ✓ ◆ X y,y0 = A y,y0 y,y0 Q 2 Q-ﬂux: Q = @y3 = Q 3 Q 26 Freitag, 15. März 13 (iv) T-duality in y 3-direction R-brane - source of R-ﬂux:

0 1 2 3 4 5 6 x x x x y y0 y00 y y y R • • • HoweverN thereN doesN not exist a local metricN inN termsN of the original coordinates. One has to use a dual coordinate:

2 µ 2 1 2 2 2 3 2 i 2 ds˜ = (dx ) + dy + dy0 + dy˜00 + h(r)(dx ) + (dy ) , R h(r) µ=0,1,2 i=4,5,6 X ✓ ◆ X y,y0 = Ry˜00 , R R-ﬂux: Ry,y0,y00 = @ y,y0 = R y˜00 R

Freitag, 15. März 13 The intersection of NS 5-branes, KK-monopoles, Q- branes and R-branes leads to supersymmetric type IIA/B backgrounds of the form: H,f,Q,R M10 = AdS4 M6 ⇥See also: Kounnas, D.L., Petropoulos, Tsimpis (2007) H,f,Q,R M6 are compact, 6-dimensional flux backgrounds with SU(3) x SU(3) group structure. 4 2 The spaces M 6 are torus fibrations of T over T . Corresponding closed string boundary conditions: Y m(⌧, +2⇡)=Y m(⌧, )+2⇡ p˜m i i i m j ij m ˜ Y (⌧, +2⇡)=Y (⌧, )+fjm p˜ Y (⌧, )+Qm p˜ Yj(⌧, ) , Y˜ (⌧, +2⇡)=Y˜ (⌧, ) f j p˜m Y˜ (⌧, )+Hijm p˜m Y j(⌧, ) . i i im j This leads to the following closed string commutators: i j ij m [Y (⌧, ),Y (⌧, )] Qm p˜ ' 28 Freitag, 15. März 13 (i) IIA with four H-fluxes: x0 x1 x2 x3 y1 y2 y3 y4 y5 y6 NS5 NS50 N N N N N N NS500 N N N N N N NS5000 N N N N N N NEffectiveN N 4D flux superpotential:N N N IIA WH = H2,4,6S + H2,3,5U1 + H1,4,5U2 + H1,3,6U3

29 Freitag, 15. März 13 (i) IIA with four H-fluxes: x0 x1 x2 x3 y1 y2 y3 y4 y5 y6 NS5 NS50 N N N N N N NS500 N N N N N N NS5000 N N N N N N NEffectiveN N 4D flux superpotential:N N N IIA WH = H2,4,6S + H2,3,5U1 + H1,4,5U2 + H1,3,6U3 (ii) IIA with four f-fluxes (Iwasawa manifold):

y1 y2 y3 y4 y5 y6 KK IIA 2 2 • Wf = f4,6ST1 + f3,5T1U1 KK0 1 1 N • N N + f4,5T1U2 + f3,6T1U3 KK00 N• N N KK000 • N N N N N N 29 Freitag, 15. März 13 (iii) IIA with four Q-ﬂuxes: y1 y2 y3 y4 y5 y6

Q IIA 2,4 2,3 • • WQ = Q6 ST1T2 + Q5 T1T2U1 Q0 1,4 1,3 N • N• N + Q5 T1T2U2 + Q6 T1T2U3 Q00 N• • N N Q000 • N N• N N N N

30 Freitag, 15. März 13 (iii) IIA with four Q-ﬂuxes: y1 y2 y3 y4 y5 y6

Q IIA 2,4 2,3 • • WQ = Q6 ST1T2 + Q5 T1T2U1 Q0 1,4 1,3 N • N• N + Q5 T1T2U2 + Q6 T1T2U3 Q00 N• • N N Q000 • N N• N N N N (iv) IIBA with four R-ﬂuxes: y1 y2 y3 y4 y5 y6 IIA 2,4,6 2,3,5 R WR = R ST1T2T3 + R T1T2T3U1 • • • + R1,4,5T T T U + R1,3,6T T T U R0 1 2 3 2 1 2 3 3 N • N• N• R00 N• • N N• R000 • N N• N• N N N

30 Freitag, 15. März 13 (v) IIB with two H-fluxes and two f-fluxes: y1 y2 y3 y4 y5 y6 NS5 NS50 N N N KK00 N• N N KK000 • N N N (vi) IIB with two f-fluxesN Nand two Q-fluxes:N y1 y2 y3 y4 y5 y6 KK • Q0 N • N• N Q00 N• • N N KK000 • N N N (vii) IIB with two Q-fluxesN N and two f-fluxes:N y1 y2 y3 y4 y5 y6 Q • • R0 N • N• N• Q00 N• • N N R000 • N N• N• 31 Freitag, 15. März 13 N N N V) Outlook & open questions

32 Freitag, 15. März 13 V) Outlook & open questions ● Mixed closed string bound conditions (in analogy to mixed D-N boundary conditions for D-branes) lead to closed string non-commutativity.

32 Freitag, 15. März 13 V) Outlook & open questions ● Mixed closed string bound conditions (in analogy to mixed D-N boundary conditions for D-branes) lead to closed string non-commutativity. ● Investigation of the phase space of doubled geometry (I.Bakas, D. Lüst, work in progress) Uniﬁed description of H,f,Q,R-ﬂuxes in double geometry needs a 4D-dimensional phase space. ● Relation to non-associative generalized diffeomorphisms (O. Hohm, B. Zwiebach (2012); in double geometry O. Hohm, D.L., B. Zwiebach, work in progress) ● Is there are non-commutative (non-associative) (Non-commutative geometry & gravity: P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess (2005), theory of gravity? L. Alvarez-Gaume, F. Meyer, M. Vazquez-Mozo (2006))

How to represent this algebra? 32 Freitag, 15. März 13 Happy Birthday and all the best wishes to you and your family, Luis!

Freitag, 15. März 13