# nLab S-duality

duality

## In QFT and String theory

The term S-duality can mean two different things:

# Contents

## Idea

In the original and restricted sense, S-duality refers to the conjectured Montonen-Olive duality auto-equivalence of (super) Yang-Mills theory in 4 dimensions under which the coupling constant is inverted, and more generally under which the combined coupling constant and theta angle tranform under an action of the modular group. At least for super Yang-Mills theory this conjecture can be argued for in detail.

There is also a duality in string theory called S-duality. Specifically in type IIB superstring theory/F-theory this is given by an action of the modular group on the axio-dilaton, hence is, via the proportionality of the dilatorn to the string coupling constant, again a weak-strong coupling duality.

Indeed, at least for super Yang-Mills theory Montonen-Olive S-duality may be understood as a special case of the string duality (Witten 95a, Witten 95b): one may understand N=2 D=4 super Yang-Mills theory as the KK-compactification of the M5-brane 6d (2,0)-superconformal QFT on the F-theory torus (Johnson 97) to get the D3-brane worldvolume theory, and the remnant modular group action on the compactified torus is supposed to be the 4d Montonen-Olive S-duality (Witten 07).

### In (super) Yang-Mills theory

#### General idea

In its original form, S-duality refers to Montonen-Olive duality , which is about the following phenomenon:

The Lagrangian of Yang-Mills theory has two summands,

$S_{YM} : \nabla \mapsto \int_X \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle + \int_{X} i \theta \langle F_\nabla \wedge F_\nabla \rangle \,,$

each pairing the curvature 2-form with itself in an invariant polynomial, but the first involving the Hodge star operator dual, and the second not. One can combine the coefficients $\frac{1}{e^2}$ and $i \theta$ into a single complex number

$\tau = \frac{\theta}{2 \pi} + \frac{4 \pi i}{e^2} \,.$

Montonen-Olive duality asserts that the quantum field theories induced from one such parameter value and another one obtained from it by an action of $SL(2,\mathbb{Z})$ on the upper half plane are equivalent.

This is actually not quite true for ordinary Yang-Mills theory, but seems to be true for N=2 D=4 super Yang-Mills theory.

#### From compactification of the 6d (2,0)-SCFT and AGT correspondence

In (Witten 95a, Witten 95b, Witten 07) it was suggested that the above S-duality of N=2 D=4 super Yang-Mills theory may be understood geometrically by regarding the super Yang-Mills theory as the Kaluza-Klein compactification of the 6d (2,0)-superconformal QFT – that instead of a gauge field given by a principal bundle with connection involves a principal 2-bundle with 2-connection – on a complex torus. The $SL(2,\mathbb{Z})$-invariance of the resulting 4-dimensional theory is then the modular group remnant of the conformal invariance of the 6-dimensional theory under conformal transformations of that torus.

Moreover, Witten has suggested that this S-duality secretly drives a host of other subtle phenomena, notably that the geometric Langlands duality (see there for more) is just an aspect of a special case of this.

The AGT correspondence refines this further and regards the 6d (2,0)-superconformal QFT as something like a “2d SCFT with values in 4d super-Yang-Mills theories”. This way the whole mapping class group of general 2d Riemann surfaces acts as a generalized S-duality on 4d super-Yang-Mills theory

### In string theory

In string theory, S-duality is supposed to apply to whole string theories and make type II string theory be S-dual to itself and make heterotic string theory be S-dual to type I string theory.

#### Type IIB S-duality

##### General idea

Type IIB string theory is obtained by KK-compactification of M-theory on a torus bundle followed by T-dualizing one of the torus cycles. This perspective – referred to as F-theory – exhibits the axio-dilaton of type IIB string theory as the fiber of an elliptic fibration (essentially the torus bundle that M-theory was compactified on (Johnson 97)).

The modular group acts on this elliptic fibration, and this is S-duality for type IIB-strings. In particular the transformation $\tau \mapsto - \frac{1}{\tau}$ inverts the type II coupling constant. See at F-theory for more.

The type IIB F1-string and the D1-brane appear this way by double dimensional reduction from the M2-brane wrapping (either) one of the two cycles of the compactifying torus. S-duality mixes these strings by the evident modular group action on the $(p,q)\in \mathbb{Z}^2$ labels of the (p,q)-strings. Here at least part of the S-duality action on $(p,q)$-strings may be seen as a system of autoequivalences of the super L-infinity algebras which defines the extended super spacetime constituted by the type II superstring (Bandos 00, FSS 13, section 4.3).

Similarly the D5-brane and the NS5-brane are the double dimensional reduction of the M5-brane wrapping one of the two cycles of the compactifying torus, and hence the S-duality modular group also acts on $(p,q)$-5-branes, exchanging them.

Finally, the D3-brane is instead the double dimensional reduction of the M5-brane, wrapping both compactifying dimensions. Accordingly the worldvolume theory of the D3, which is super Yang-Mills theory in $d = 4$ has an S-self-duality. That is supposed to be the Montonen-Olive duality discussed above, which is thereby unified with type IIB S-duality.

##### Cohomological nature of type II fields under S-duality

While F-theory does capture much of this non-perturbative S-duality, there currently remains a puzzle as to the correct differential cohomology nature of all the fields under S-duality: by the above S-duality mixes the Kalb-Ramond field $\hat B_{NS}$ with the degree-3 component $\hat B_{RR}$ of the RR-field. But the best available description of the fine-structure of these fields is (see also at orientifold) that $\hat B_{NS}$ is a cocycle in (twisted) ordinary differential cohomology while $\hat B_{RR}$ is (only) one component of a cocycle in (twisted) KU (or really: KR-theory).

This issue was first highlighted in (DMW 00, section 11). In (DFM 03, section 9) it was observed that taking into account the cubical structure in M-theory on the 11-dimensional Chern-Simons term of the supergravity C-field the conceptual mismatch is alleviated, but not quite resolved. See also (BEJVS 05)

On the other hand, as discussed at cubical structure in M-theory, this structure plausibly relates to a generalized cohomology theory beyond ordinary cohomology and beyond K-theory, namely to elliptic cohomology/tmf. Hints like this led in (KrizSati 05) to the conjecture that the right cohomology theory to capture the S-duality of type IIB/F-theory is modular equivariant elliptic cohomology.

#### Heterotic/type I duality

Something substantial should go here, for the moment the following is copied from a discussion forum comment by some Olof here:

For the Het/I relation, the first observation is that the massless spectra of the two models agree. Moreover, if we make the identification

$G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad \Phi^I = - \Phi^h , \qquad \tilde{F}^I_3 = \tilde{H}^h_3 , \qquad A^I_1 = A^h_1$

the low energy effective supergravity actions of the two models match. Since the string coupling constants $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling:

$g^I_s = \frac{1}{g^h_s} .$

From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by

$l^I_s = l^h_s \sqrt{g^h_s}.$

As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula

$T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2}$

where I’ve used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string

$T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}.$

This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.

#### For type IIA

A priori type IIA superstring theory does not have S-duality, but by compactifying M-theory on a torus one can sort of read off what the non-perturbative additions to type IIA should be that make it have S-duality after all, see

• Gottfried Curio, Boris Kors, Dieter Lüst, Fluxes and Branes in Type II Vacua and M-theory Geometry with G(2) and Spin(7) Holonomy, Nucl.Phys.B636:197-224,2002 (arXiv:hep-th/0111165)

## Overview

S-duality in string theory

reduction from 11delectric σ-modelweak/strong coupling dualitymagnetic σ-model
M2-brane in 11d sugra EFT$\leftarrow$electric-magnetic duality$\rightarrow$M5-brane in 11d sugra EFT
HW reduction
$\downarrow$ on orientifold K3$\times S^1//\mathbb{Z}_2$$\downarrow$ on orientifold K3$\times S^1//\mathbb{Z}_2$
F1-brane in heterotic supergravity$\leftarrow$S-duality$\rightarrow$black string in heterotic sugra
HW reduction
$\downarrow$ on orientifold T4$\times S^1//\mathbb{Z}_2$$\downarrow$ on orientifold T4$\times S^1//\mathbb{Z}_2$
F1-brane in heterotic supergravity$\leftarrow$S-duality$\rightarrow$black string in type IIA sugra
KK reduction
$\downarrow$ on K3$\times S^1$$\downarrow$ on K3$\times S^1$
F1-brane in IIA sugra$\leftarrow$S-duality$\rightarrow$black string in heterotic sugra
KK reduction
$\downarrow$ on $T^4\times S^1$$\downarrow$ on $T^4 \times S^1$
F1-brane in IIA sugra$\leftarrow$S-duality$\rightarrow$black string in type IIA sugra
F-reduction$\updownarrow$ T-duality on $S^1$
F1-brane in IIB sugra$\leftarrow$S-duality$\rightarrow$D1-brane in 10d IIB sugra
U-duality$\updownarrow$ T-duality on $T^2$
D3-brane in IIB sugra$\leftarrow$S-duality$\rightarrow$D3-brane in IIB sugra

gauge theory induced via AdS-CFT correspondence

11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

## References

### In (super-)Yang-Mills theory

It was originally noticed in

• P. Goddard, J. Nuyts, and David Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B125 (1977) 1-28.

that where electric charge in Yang-Mills theory takes values in the weight lattice of the gauge group, then magnetic charge takes values in the lattice of what is now called the Langlands dual group.

This led to the electric/magnetic duality conjecture formulation in

According to (Kapustin-Witten 06, pages 3-4) the observaton that the Montonen-Olive dual charge group coincides with the Langlands dual group is due to

The insight that the Montonen-Olive duality works more naturally in super Yang-Mills theory is due to

and that it works particularly for N=4 D=4 super Yang-Mills theory is due to

• H. Osborn, Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1, Phys. Lett. B83 (1979) 321-326.

The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

• John Cardy, E. Rabinovici, Phase Structure Of Zp Models In The Presence Of A Theta Parameter, Nucl. Phys. B205 (1982) 1-16;

• John Cardy, Duality And The Theta Parameter In Abelian Lattice Models, Nucl. Phys. B205 (1982) 17-26.

• A. Shapere and Frank Wilczek, Selfdual Models With Theta Terms, Nucl. Phys. B320 (1989) 669-695.

The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional principal 2-bundle-theory – the 6d (2,0)-superconformal QFT – compactified on a torus is described in

### In type II superstring theory

The suggestion of an $SL(2,\mathbb{Z})$-duality action in type II superstring theory goes back to

• John Schwarz, Ashoke Sen, Duality Symmetries Of $4D$ Heterotic Strings, Phys. Lett. 312B (1993) 105-114,

Duality Symmetric Actions, Nucl. Phys. B411 (1994) 35-63 (arXiv:hep-th/9304154)

The geometric understanding of S-duality in type II superstring theory via M-theory/F-theory goes maybe back to

A textbook account is in

A 2-loop test is in

S-duality acting on the worldsheet theory if (p,q)-strings is discussed for instance in

• Igor Bandos, Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane, Nucl.Phys.B599:197-227,2001 (arXiv:hep-th/0008249)

Closely related to this, S-duality in type II string theory as an operation on the extended super spacetime super L-infinity algebra is

The cohomological problem of the type II S-duality action on the 3-form flux was originally highlighted in

The conjecture that with combined targetspace/worldsheet modular transformations the type IIB S-duality is reflected in modular equivariant elliptic cohomology is due to