Susanne Reffert CERN Based on Work with with D. Orlando and S

BPS States in the Duality Web of the Omega deformation

Susanne Reffert CERN

based on work with with D. Orlando and S. Hellerman (arXiv:1106.2097, 1108.0644, 1111.4811, 1204.4192, 1210.7805) Motivation The Omega background was introduced by Nekrasov as a way of regularizing the 4d instanton partition function and reproducing the results of Seiberg and Witten. Many applications: In limit , partition fn is the same as for topological string (for , refined top. string) In the Nekrasov Shatashvili limit , related to quantum integrable models with Compactification of 6D (2,0) theory in Omega BG leads to AGT All of the above can be understood via string theory by placing branes into a geometrical deformation of the bulk corresponding to the Omega BG ! Overview Gauge/Bethe correspondence relates susy gauge theories in 2d/4d to (quantum) integrable systems. Find/study string theory realization! fluxbrane BG T duality fluxtrap BG + D2, NS5 + D4, NS5 2d N=2 gauge theory 4d N=2 gauge theory w. twisted masses in Omega BG (2d Gauge/Bethe corr.) (4d Gauge/Bethe corr.) M Theory lift The same string theory background can give rise to different deformations (twisted masses/Omega deformation) depending on how we place branes in it! Overview Start from M theory lift of background. Study different limits of (2,0) theory in 6d. Why (2,0) theory? (2,0) in 6d on compactify on compactify on

4d SYM in Omega BG Liouville field theory (2d) AGT Here: (2,0) in 6d on in fluxtrap compactify on compactify on angular directions of 4d N=4 SYM in Omega BG reciprocal field theory (4d) Theories are realized via a chain of dualities. Study BPS states in this duality web. Overview

Bulk Probe Gauge Theory six-dimensional gauge theory with Wilson type iib in Melvin space D5 line boundary conditions in two directions

T–duality inm u˜1 and u˜2

Ω–deformed = 4 SYM type iib in complex ﬂuxtrap D3 N

T–duality inm x˜6 and lift

M–theory ﬂuxtrap M5 (2, 0) six-dimensional theory

reduction inm σ1 and σ2

type iib in deformed D5/NS5 D3 Reciprocal gauge theory (reciprocal background)

Table 1: The chain of dualities among the different string frames and the corresponding effective gauge theories.

Adding the two spectator directions x6 and x7, we have the following variables:

ρ1, θ1, ρ2, θ2, ρ3, θ3, x6, x7, u1, u2.(2.2)

In the notation of [10] we want to set up a ﬂuxbrane with two independent deformation parameters, one of which being purely real, the other being purely imaginary. Shifts are induced in the θ1, θ2–directions, which for supersymmetry preservation need to be 1 compensated by a shift in the θ3–directions. We impose the monodromies

u u + 2π , u u + 2π , 1 1 2 2 θ1 θ1 + 2π1R1 , θ2 θ2 + 2π2R2 , (2.3)   θ3 θ3 2π1R1 , θ3 θ3 2π2R2 , − −   and we change to new angular coordinates φi which are 2π periodic:

θ1 = φ1 + R11u1 ,(2.4)

θ2 = φ2 + R22u2 ,(2.5) θ3 = φ3 R11u1 R22u2 .(2.6) − − 1 The two parameters 1 and 2 are real. In the habitual conventions for the Ω–deformation they correspond to a real and a purely imaginary ε. See [10, 11] for comparison.

3 Outline

* Introduction * A Chain of Dualities * Bulk * Branes * BPS States in the Duality Web * Summary A Chain of Dualities A Chain of Dualities

Bulk Probe Gauge Theory six-dimensional gauge theory with Wilson IIB FB type iib in Melvin space D5 line boundary conditions in two directions

T–duality inm u˜1 and u˜2

IIB FT Ω–deformed = 4 SYM type iib in complex ﬂuxtrap D3 N T–duality inm x˜ and lift IIA FT 6

(2, 0) six-dimensional theory MFT M–theory ﬂuxtrap M5

reduction inm σ1 and σ2

type iib in deformed D5/NS5 IIB rec D3 Reciprocal gauge theory (reciprocal background)

Table 1: The chain of dualities among the different string frames and the corresponding effective gauge theories.

Adding the two spectator directions x6 and x7, we have the following variables:

ρ1, θ1, ρ2, θ2, ρ3, θ3, x6, x7, u1, u2.(2.2)

θ1 = φ1 + R11u1 ,(2.4)

3 A Chain of Dualities: Bulk Fluxbrane background with two independent deformation IIB FB parameters, one purely real, the other purely imaginary

ρ1, θ1, ρ2, θ2, ρ3, θ3, x6,x7, u1, u2

IIB FT x8 =R1 u1 , x9 = R2 u2 θ1 θ1 +2πk2 ,θ2 θ2 +2πk3 IIA FT impose identifications fluxbrane parameter MFT u u +2π , u u +2π , 1 1 2 2 θ θ +2π R , θ θ +2π R ,  1 1 1 1  2 2 2 2 θ3 θ3 2π1R1 , θ3 θ3 2π2R2 , − − IIB rec   This corresponds to the well known Melvin or fluxbrane background. Introduce new angular variables θ1 = φ1 + R11u1 ,

θ2 = φ2 + R22u2 , θ3 = φ3 R11u1 R22u2 . − − A Chain of Dualities: Bulk T dualize u 1 and u 2 into u 1 and u 2 Fluxtrap background

IIB FB Coordinate change: φ1 + φ2 + φ3 = ψ

6 7 α α ρ1, φ1, ρ2, φ2, ρ3, ψ, x ,x ,x8 = u1,x9 = u2 R1 R2

IIB FT ρ3 << ρ1, ρ2 Study the limit

IIA FT Before T duality, locally, the metric was still flat, but some of the rotation symmetries were broken globally.

MFT not anymore flat Bulk fields 2 2 2 2 2 2 2 2 ρ1 dφ1 +dx8 2 ρ2 dφ2 +dx9 2 2 2 2 2 ds =dρ1 + 2 2 +dρ2 + 2 2 +dρ3 + ρ3 dψ +dx6 +dx7 , 1+1ρ1 1+2ρ2 IIB rec ρ2 ρ2 B field has appeared 1 2 B = 1 2 2 dφ1 dx8 + 2 2 2 dφ2 dx9 , 1+1ρ1 ∧ 1+2ρ2 creates∧ a potential that localizes Φ 2 2 2 2 e− = (1 + 1ρ1)(1+2ρ2) instantons A quarter of the original supersymmetries are preserved. 1 φ1 R2 1 r R S1 ×

Figure 1: Cartoon of the geometry of theA base Chain of the manifold ofM 3(Dualities:1): a cigar with Bulk asymptotic radius 1/1.

6 7 α α The space splits into a product ρ1, φ1, ρ2, φ2, ρ3, ψ, x ,x ,x8 = u1,x9 = u2 IIB FB R1 R2 M = M ( ) M ( ) R3 S1 ,(2.14) 10 3 1 × 3 2 × × (ρ3, ψ,x7) 3 Space splits1 into where R is generated by (ρ3, ψ, x7), the S is generated by (x6, x7), and M3 is a three- dimensional manifold which is obtained as a R foliation (generated by x8 or x9) over x6 ( ) ( ) the cigar withIIB asymptotic FT radius 1/i described by ρ1, φ1 or ρ2, φ2 (see the cartoon in Figure 1): R x8 M3(1) IIA FT

MFT cigar ρ , φ 1 1 (2.15) This shows that the effect of the Ω–deformation is to regularize the rotations generated by ∂φ1 and ∂φ2 in the sense that the operators become bounded: IIB rec 1 2 2 φ1 2 ρ1 1 2 2 ρ2 1 1 ∂φ1 = 2 2 < 2 , R ∂φ2 = 2 2 < 2 .(2.16) 1 + 1ρ1 1 1 + 2ρ2 2 r In a different frame this will translate into a bound on the asymptotic coupling of the R S1 effective gauge theory for the motion of a D–brane. × As a ﬁnal remark we observe that even though the background in Equation (2.13) where the contributions of the two i are decoupled was obtained as a limit, it is by itself a solution of the ten-dimensionalFigure 1: Cartoon supergravity of the equations geometry of of motion the base for any of value the manifold M3(1): a cigar with of ρi. asymptotic radius 1/1.

What we have obtained is the starting point of the chain of dualities leading eventually to the reciprocalThe space background, splits into as detailed a product in Table 1.

M = M ( ) M ( ) R3 S1 ,(2.14) M–theory. As a ﬁrst step we dualize in x6 to type10 iia and3 then1 × lift to3 M–theory.2 × × A remarkable feature of the M–theory background is the fact that it is symmetric under the exchange whereρ , φ , xR, 3 is generatedρ , φ , x , by .( Thisρ , ψ is, x the), origin the S of1 is the generated S–duality by (x , x ), and M is a three- { 1 1 8 1} ↔ { 2 2 9 2} 3 7 6 7 3 covariance of the ﬁnaldimensional type iib background. manifold This which has to is be obtained contrasted as with a R thefoliation fact (generated by x8 or x9) over that the directions x6 and the M–circle x10 appear in a non-symmetric fashion. This is the cigar with asymptotic radius 1/i described by (ρ1, φ1) or (ρ2, φ2) (see the cartoon in Figure 1): 5 R x M ( ) 8 3 1

cigar ρ , φ 1 1 (2.15) This shows that the effect of the Ω–deformation is to regularize the rotations

generated by ∂φ1 and ∂φ2 in the sense that the operators become bounded:

2 2 2 ρ1 1 2 ρ2 1 ∂φ1 = 2 2 < 2 , ∂φ2 = 2 2 < 2 .(2.16) 1 + 1ρ1 1 1 + 2ρ2 2

In a different frame this will translate into a bound on the asymptotic coupling of the effective gauge theory for the motion of a D–brane. As a ﬁnal remark we observe that even though the background in Equation (2.13)

where the contributions of the two i are decoupled was obtained as a limit, it is by itself a solution of the ten-dimensional supergravity equations of motion for any value

of ρi.

What we have obtained is the starting point of the chain of dualities leading eventually to the reciprocal background, as detailed in Table 1.

M–theory. As a ﬁrst step we dualize in x6 to type iia and then lift to M–theory. A remarkable feature of the M–theory background is the fact that it is symmetric under the exchange ρ , φ , x , ρ , φ , x , . This is the origin of the S–duality { 1 1 8 1} ↔ { 2 2 9 2} covariance of the ﬁnal type iib background. This has to be contrasted with the fact that the directions x6 and the M–circle x10 appear in a non-symmetric fashion. This is

5 A Chain of Dualities: Bulk

Now we want to lift to M theory. First T dualize in x 6 to IIA, IIB FB then lift: ρ3 << ρ1, ρ2 2 2 2 2 2 2 2 2/3 2 1ρ1 2 dx8 2 2ρ2 2 dx9 ds =(∆1∆2) dρ1 + 2 2 dσ1 + 2 2 +dρ2 + 2 2 dσ2 + 2 2 1+1ρ1 1+1ρ1 1+2ρ2 1+2ρ2 IIB FT 2 2 2 2 2 4/3 2 +dρ3 + ρ3 dψ +dx6 +dx7 +(∆1∆2)− dx10 , 2 2 2 2 1ρ1 2ρ2 IIA FT A3 = 2 2 dσ1 dx8 dx10 + 2 2 dσ2 dx9 dx10 1+1ρ1 ∧ ∧ 1+2ρ2 ∧ ∧

MFT φi 2 2 2 σi = ∆i =1+i ρi i Symmetric under exchange IIB rec Origin of S duality covariance in final BG.

Return to IIA by reducing on σ 1 : backreaction of the near horizon limit of a D6 brane in the fluxtrap A Chain of Dualities: Bulk

Last step: T duality in σ2 d˜σ2 IIB FB ds2 = ρ 1+2ρ2 dρ2 +dρ2 + 2 +dρ2 + ρ2 dψ2 +dx2 +dx2+ 1 1 2 2 1 2 2ρ22ρ2 3 3 6 7 1 1 2 2 dx2 dx2 dx2 + 8 + 9 + 10 , 1+2ρ2 1+2ρ2 (1 + 2ρ2)(1+2ρ2) 1 1 2 2 1 1 2 2 IIB FT 2ρ2 B = 1 1 dx dx , 2 2 8 10 from NS5 1+1ρ1 ∧ IIA FT 2 2 Φ 2ρ2 1+1ρ1 e− = 2 2 , 1ρ1 1+2ρ2 from D5 MFT 2 2 2ρ2 C2 = 2 2 dx9 dx10 1+2ρ2 ∧ The bulk is the backreaction of the near horizon limit of an NS5 IIB rec and a D5 brane. String coupling constant: Under S duality, is exchanged with , which amounts to swapping the D5 with the NS5. A Chain of Dualities: Branes

Bulk Probe Gauge Theory IIB FB six-dimensional gauge theory with Wilson type iib in Melvin space D5 line boundary conditions in two directions

T–duality inm u˜1 and u˜2 IIB FT

Ω–deformed = 4 SYM type iib in complex ﬂuxtrap D3 N IIA FT T–duality inm x˜6 and lift

MFT M–theory ﬂuxtrap M5 (2, 0) six-dimensional theory

reduction inm σ1 and σ2

IIB rec type iib in deformed D5/NS5 D3 Reciprocal gauge theory (reciprocal background)

Table 1: The chain of dualities among the different string frames and the corresponding effective gauge theories.

Adding the two spectator directions x6 and x7, we have the following variables:

ρ1, θ1, ρ2, θ2, ρ3, θ3, x6, x7, u1, u2.(2.2)

θ1 = φ1 + R11u1 ,(2.4)

3 A Chain of Dualities: Branes

IIB FB Start from M5 brane extended in (ρ1, ρ2, σ1, σ2,x6,x10) 2 2 2 R T T × ×

IIB FT Ω =4 deformed N SYM: Reciprocal gauge theory: IIA FT D3 branes extended in D3 branes in reciprocal BG ( ρ 1 , φ 1 , ρ 2 , φ 2 ) in fluxtrap extended in (ρ1, ρ2,x6,x10) MFT Effective 4D gauge theories on D3 branes in different duality

IIB rec frames/bulk backgrounds. A Chain of Dualities: Branes

Ω =4 (ρ1, φ1, ρ2, φ2) deformed N SYM: D3 branes extended in in IIB FB fluxtrap bg 3 cplx scalars describing motion of D3: i ψ ρ3 e x6 +ix7 x8 +ix9 IIB FT w(ξ)= ,z(ξ)= , ϕ(ξ)= πα πα πα

IIA FT DBI action for one D3 brane:

1 ij 1 i k i ¯ k 1 ¯ i i k ¯ l 2 LΩ = 2 FijF + ∂ ϕ + V Fk ∂iϕ¯ + V Fki (V ∂iϕ V ∂iϕ¯ + V V Fkl) MFT 4gym 2 −8 − 1 1 + δij + V iV¯ j (∂ z ∂ z¯ +c.c.)+ δij + V iV¯ j (∂ w ∂ w¯ +c.c.) 4 i j 4 i j IIB rec new 1 i i 1 2 + 3V¯ +¯3V (¯w ∂iw c.c.) + 3 ww¯ kinetic term 2i − 2 | | V = ξ0 ∂ ξ1 ∂ +i ξ2 ∂ ξ3 ∂ g2 =2πgΩ mass term 1 1− 0 2 3− 2 ym iib one derivative term breaks Lorentz invariance A Chain of Dualities: Branes Reciprocal gauge theory: D3 branes in reciprocal BG extended in (ρ1, ρ2,x6,x10) IIB FB Supersymmetric non Lorentz invariant gauge theory

Embedding ρ1 = y1 , ρ2 = y2 ,x6 = y3 ,x10 = y4

IIB FT Dynamics described by i ψ ρ3 e x7 σ˜2 x8 x9 U1 +iU2 = ,U3 = ,U4 = ,U5 = ,U6 = IIA FT 2πα 2πα 2πα 2πα 2πα Effective action for D3 brane: from B field from C field MFT 2 3 3 2 y2 1y1y2 2 1 2 y2 Lrec = FklFkl+ (Fk4 ∂kU5) + 2 i ( F )k4 ∂kU6 8πy1 4π − ∆2 1 y1 ∗ − k=1 k=1 IIB rec kl ij 2 2 2 2 2 2 2 2 +τ (ξ)h (ξ) ∂kUi ∂lUj +∆2(∂4U5) +∆1(∂4U6) + y1− + y2− U1 + U2 1 1 1 1 τ kl(ξ)= ,hij(ξ)=  1   1  2 2 2 2  ∆ ∆   ( y )− ( y )−   1 2  1 1 2 2      A Chain of Dualities: Branes

Define gauge kinetic tensor: IIB FB ijkl Lg = M FijFkl Define scalar gauge coupling: IIB FT 1 2 2 = M = M ijij M klkl g2 3|| ||K 3 ijkl ijkl IIA FT eff Gauge coupling in reciprocal gauge theory: MFT 1 1 y 1+2y2 = 2 1 1 g2 2π 2 2 rec y11+2y2 IIB rec Far away from the branes (large y limit), this reduces to 2 2 grec 2π y ,y −−1 − −2→∞ − −→ 1 A Chain of Dualities: Branes The S dual of the Lagrangian is defined as follows:

1 1 ijkl IIB FB L = (M − ) ( F ) ( F ) dual 16π2 ∗ ij ∗ kl

y1 ( F ) ( F ) L ( , )= ∗ 4k ∗ k4 + F F 1+2y2 dual 1 2 4πy 1+2y2 k4 k4 2 2 2 1 1 IIB FT By comparison we find IIA FT Ldual(1, 2)=Lg(2, 1)

2 MFT Define Liouville parameter as b2 = 1 The reciprocal gauge theory displays several features of Liouville

IIB rec field theory: Asymptotic coupling constant is proportional to b2

S duality exchanges b 1 /b , like Liouville duality which ↔ exchanges perturbative and instanton spectrum BPS States in the Duality Web BPS States in the Duality Web In Omega deformed SYM: 2 sets of BPS objects IIB FB BPS instanton configurations perturbative modes of the in IIB and particles in IIA, fundamental string stretching localized at the origin between two D4 branes carrying IIB FT oscilloids angular momentum along the U(1) U(1) IIA FT × rotational isometries Φ 1 2 2 2 2 L = µ e− = (1 + ρ )(1+ ρ ) 0 Ω 1 1 2 2 localized at infinity − −gIIAlΩ MFT DOZZoids nD0 2 2 2 2 ED0 = Ω (1 + 1ρ1)(1+2ρ2) 2 gIIAlΩ 2 L2 J2 E = 2J2 + 2 − 2πα ρ2 IIB rec Give rise to the modular form defining the holomorphic Reproduce the holomorphic factor of the Liouville DOZZ factors of the Liouville partition function partition function M theory BPS States in the Duality Web Follow BPS states along chain of dualities to reciprocal frame.

IIB FB oscilloids DOZZoids D0 branes in fluxtrap type IIA fundamental string at origin extended in x 9 with IIB FT momentum in φ2

x10 IIA FT momentum mode in M theory M2 brane with momentum in φ2 MFT reciprocal D3 brane in (x6,x9,x10, σ˜2) momentum mode in x10 frame with el. flux in σ˜2

IIB rec Now only one kind of BPS state carrying both types of charge and living at a finite radius. BPS States in the Duality Web Follow the BPS states along the chain of dualities: IIB FB angular momentum extended in direction

frame object ρ1 φ1 ρ2 φ2 ρ3 ψ x6 x7 x8 x9 x10 D4 IIB FT × × × × × × iia ﬂuxtrap F1 × × D0 IIA FT × M5 × × × × × × dynamical M–theory M2 × × × branes MFT momentum D3 × × × × D3 × × × × reciprocal frame momentum IIB rec D5 × × × × × × NS5 × × × × × × Table 2: Extended objects in the various different frames. The objects are extended in momentum BPS the direction of the crosses ( ). Angular momentumnongeometrical in a direction direction is marked as and × excitationsmomentum as . The direction marked with a square () in type ii is not geometrical. The white background is for the dynamical branes described by the gauge theories; the light greynon () dynamical for the branes that correspond to the bps excitations and the dark grey background () for non-dynamical objects in the bulk.

particles in ﬁve dimensions, that are static in the (Euclidean) timelike x6 direction. We need go into this aspect no further; this description has been discussed in detail in [1, 5]. We note simply that the localization of these objects to the origin by the Ω–deformation is easy to understand at the ﬁeld-theoretic level: the scalar effective gauge coupling that controls the mass of a small instanton has a global maximum at the origin, and so the instanton’s action is globally minimized there. Further discussion of this effect can be found in [9].

Field-theoretic description of the BPS DOZZoids. These are linearized eigen- modes of the 4D massive vector multiplet in the Ω–deformed gauge theory, with or without angular momenta J in the U(1) U(1) angular directions. The eigenvalue of 1,2 × the Laplace operator on these modes χ obeys a bps condition

L 2 2χ = + J + J χ ,(4.1) −∇ 2πα 1 1 2 2 where L is the distance between the branes in string frame, and so L is the vev of 2πα the scalar in the vector multiplet. The angular momenta J1,2 include both orbital contributions depending on the profile of the mode and intrinsic contributions depending on the representation of the field. Treated as particles in the five-dimensional gauge theory on a circle, these are bps particle excitations of the vector multiplet, with a mass determined by the free-field equation of motion:

L m = + J + J .(4.2) BPS 2πα 1 1 2 2

13 M theory BPS States in the Duality Web Extended objects in the reciprocal frame: IIB FB

NS5 (bulk) IIB FT D3 (dynamical)

IIA FT

D3 (bps) D3 (dynamical) MFT

IIB rec x9

x10

ρ2 M theory BPS States in the Duality Web

Introduce a D3 brane extended in ( x 6 ,x 9 ,x 10 , σ˜ 2 ) with an IIB FB electric field in σ˜ 2 , velocity v in x 10 and another component of

the electric field in x 10 , which is required by the coupling of v in the DBI action. IIB FT L2 ζ2 x6 = iζ0 ,x9 = ζ1 , σ˜2 =2πα2 ,x10 = ζ3 + vζ0 ,x8 =0, πRWS κ IIA FT ρ1 = const. , ρ2 = const. , ρ3 =0,x7 =0

1 1 MFT F = f ,F= f 02 κ 02 03 κ 03

4 Φ S = µ d ζe− det(g + B +2πα F )+µ exp(B +2παF ) C − 3 − 3 ∧ q IIB rec q ρ2 (f + vf ) (1 + 2ρ2)(1+ρ2f 2 (1 + 2ρ2)) ρ2 (f + vf )2 1L2R10 2 2 02 03 − 2 2 2 03 1 1 − 2 02 03 = 2 2 2πα 1+2ρ2 δS δS δS J2 = ,D3 = ,P= δf02 δf03 δv M theory BPS States in the Duality Web From the Hamiltonian, we find R 2 J 2 P 2 IIB FB E2 = 1 10 L J + 2 1+ρ2 1+2ρ2 . 2πα 2 − 2 2 ρ 2 1 1 J 2 2 This is minimized for J 2 ρ2 = 2 IIB FT 2 1R10 P L2 2J2 2πα − IIA FT Energy of the BPS states: R E = 1 10 L J + P MFT BPS 2πα 2 − 2 2 L2 = J2 =0 P =0 IIB rec oscilloids DOZZoids

Eosc = P R E = 1 10 L J dozz 2πα 2 − 2 2 EBPS = Eosc + Edozz BPS States in the Duality Web States are only marginally stable with respect to their decay

IIB FB into separate oscilloids and DOZZoids. Any decay would have to tunnel over a barrier, since a state that is broken apart locally would have an energy that is

IIB FT strictly larger than the one of the bound state. Any such decay process would have to be nonperturbatively suppressed. IIA FT

MFT

IIB rec Summary Summary

We described the Omega BG and its various limits via a fluxtrap BG in string theory. Allows us to study different gauge theories of interest via string theoretic methods. Omega deformation and twisted mass deformation have same origin. Gives a geometrical interpretation for the Omega BG and its properties, such as localization etc. Understanding of relation between deformation parameters and quantization of spectral curve. Summary

Progress towards understanding the (2,0) theory in 6D in Omega BG. Studied two limits of (2,0) theory linked by a chain of dualities.

4 Compactify on torus: Omega deformed N=4 SYM on R

2 2 R T Compactify on angular directions: Reciprocal gauge theory on + × Reciprocal gauge theory displays properties of Liouville field theory: Asymptotic coupling constant is proportional to b2 = 2 1 S duality exchanges b 1 /b , like Liouville duality which ↔ exchanges perturbative and instanton spectrum Important step towards realization of Liouville theory from string theory. Summary

BPS states appearing on the SYM side of the AGT correspondence: fundamental strings and D0 branes in type IIA fluxtrap Reciprocal frame: D3 branes carrying both electric fields and momentum in the 10 direction, localized at a finite value of the radial direction Two distinct types of states localized in different positions (the origin and infinity) in one duality frame. States carrying both charges localized in one place in the other duality frame. Presence of new phenomena which are only accessible via a microscopic description. Outlook The reciprocal theory is intrinsically fourdimensional. Aim: construct Liouville field theory as compactification of an M5 brane in M theory fluxtrap background.

S4 Σ Start from a geometry of the type Ω × to be able to reduce on the 4d part and realize a 2d theory on the Riemann surface. Thank you for your attention!