Narrow Results

We show a physical realization of the Langlands duality in correlation functions of WZNW model. We derive a dual version of the Stoyanovky–Riabult–Teschner (SRT) formula that relates the correlation function of the WZNW and the dual Liouville theory to investigate the level duality k - 2 → (k - 2)-1 in the WZNW correlation functions. Then, we show that such a dual version of the -Liouville relation can be interpreted as a particular case of a biparametric family of nonrational conformal field theories (CFT's) based on the Liouville correlation functions, which was recently proposed by Ribault. We study symmetries of these new nonrational CFT's and compute correlation functions explicitly by using the free field realization to see how a generalized Langlands duality manifests itself in this framework. Finally, we suggest an interpretation of the SRT formula as realizing the Drinfeld–Sokolov Hamiltonian reduction. Again, the Hamiltonian reduction reveals the Langlands duality in the WZNW model. Our new identity for the correlation functions of WZNW model may yield a first step to understand quantum geometric Langlands correspondence yet to be formulated mathematically.

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra $E_{11}$ and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of $E_{11}$ we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the $E_{11}$ conjecture given many years ago.

We argue that the exceptional field theory is a truncation of the nonlinear realisation of the semi-direct product of $E_{11}$ and its first fundamental as proposed in 2003. Evaluating the simple equations of the $E_{11}$ approach, and using the commutators of the $E_{11}$ algebra, we find the local variations of the fields of exceptional field theory after making a radical truncation. This procedure does not respect any of the higher level $E_{11}$ symmetries and so these are lost. We suggest that the need for the section condition in the exceptional field theory could be a consequence of the truncation.

From the underlying nonlinear realisation, we compute the complete E${}_{11}$ invariant equations of motion in eleven dimensions, at the linearised level, up to and including level four in the fields. Thus, we include the metric, the three and six forms, the dual graviton and three fields at level four. The fields are linked by a set of duality equations, which are first-order in derivatives and transform into each other under the E${}_{11}$ symmetries. From these duality relations, we deduce second-order equations of motion, including those for the usual supergravity fields. As a result the on-shell degrees of freedom are those of the eleven-dimensional supergravity. We also show that the level four fields provide an eleven-dimensional origin of Romans theory and lead to a novel duality relation.

We consider the nonlinear realisation of the semi-direct product of $E_{11}$ and its vector representation which leads to a space-time with tangent group that is the Cartan involution invariant subalgebra of $E_{11}$. We give an alternative derivation of the invariant tangent space metric that this space–time possesses and compute this metric at low levels in eleven, five and four dimensions. We show that one can gauge fix the nonlinear realisation in an $E_{11}$ invariant manner.

Following arXiv:hep-th/0412336 we use the nonlinear realisation of the semi-direct product of $E_{11}$ and its vector representation to construct brane dynamics. The brane moves through a space-time which arises in the nonlinear realisation from the vector representation and it contains the usual embedding coordinates as well as the worldvolume fields. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. Each brane carries the full $E_{11}$ symmetry and so the Cremmer–Julia duality symmetries. We apply this theory to find the dynamics of the IIA and IIB strings, the M2 and M5 branes, the IIB D3 brane as well as the one and two branes in seven dimensions.

Using the general properties that have emerged from $E$ theory we sketch the generic features of the dynamics of branes in seven and eight dimensions. The dynamical equations are a set of duality equations involving the coordinates of the vector representation of $E_{11}$.

We construct the $E$ theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of $E_{11}$ with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in $E$ theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of $E_{11}$. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

We construct the nonlinear realisation of the semi-direct product of $E_{11}$ and its vector representation in its decomposition into the subalgebra $GL(7)\otimes SL(5)$ to find a seven-dimensional theory. The resulting equations of motion essentially follow from the Dynkin diagram of $E_{11}$ and if one restricts them to contain only the usual fields of supergravity and the derivatives with respect to the usual coordinates of space–time then these are the equations of motion of seven-dimensional supergravity.

We construct the nonlinear realisation of the semidirect product of the very extended algebra $A_1^{+++}$ and its vector representation. This theory has an infinite number of fields that depend on a space–time with an infinite number of coordinates. Discarding all except the lowest level field and coordinates the dynamics is just Einstein’s equation for the graviton field. We show that the gravity field is related to the dual graviton field by a duality relation and we also derive the equation of motion for the dual gravity field.

We study in detail the irreducible representation of $E$ theory that corresponds to massless particles. This has little algebra $I_c(E_9)$ and contains 128 physical states that belong to the spinor representation of $\mathrm{\text{SO}}(16)$. These are the degrees of freedom of maximal supergravity in eleven dimensions. This smaller number of the degrees of freedom, compared to what might be expected, is due to an infinite number of duality relations which in turn can be traced to the existence of a subaglebra of $I_c(E_9)$ which forms an ideal and annihilates the representation. We explain how these features are inherited into the covariant theory. We also comment on the remarkable similarity between how the bosons and fermions arise in $E$ theory.

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra E₁₁ and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of E₁₁ we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E₁₁ conjecture given many years ago.