Duality-symmetric linearized gravity in PST approach Alexei J. Nurmagambetov A.I. Akhiezer Institute for Theoretical Physics NSC KIPT Kharkov, Ukraine Points to get over: Why do we need to dualize gravity? What is the hidden symmetry of gravitational theory? What is the algebraic structure of A(n)+++, what are levels and low levels fields? Why do we need the duality-symmetric (=democratic) formulation? How to realize the dynamics of low levels and what is the role of locality? What is next?

Why do we need to dualize gravity? Long ago it was realized that D=4 gravity reduced to lower (D=3, D=2) dimensions possesses unexpected symmetries (Ehlers SL(2,R) and Matzner-Misner SL(2,R) symmetries). Their interplay leads to an infinite-dimensional group (Geroch group) which acts on the solutions to the Einstein equation in the background with two commuting Killing vectors. The structure of the Geroch group was established by Julia, who showed that the infinitesimal form of the Geroch group corresponds to the affine Kac-Moody algebra SL(2,R)+. The way of establishing the Ehlers group includes the dualization of the Kaluza-Klein vector to a scalar field upon the reduction from D=4 to D=3. This scalar (axion) together with the other scalar (dilaton) forms SL(2,R)/SO(2) coset space. The Matzner-Misner SL(2,R) arises upon the reduction from D=4 to D=2 and represents the global symmetry group of the internal two-torus (moduli space group). This situation with dualizing fields was quit similar to that of realizing “hidden symmetries” in dimensionally reduced supergravities. Reducing SUGRAs to D=4 and D=3 one gets just finite-dimensional hidden symmetry groups. Further reduction to D=2 results in an infinite-dimensional analog of the Geroch “hidden symmetry” group. The next important point was realized by Nicolai who demonstrated arising the SL(2,R)++ hyperbolic Kac-Moody algebra upon the reduction of D=4 SUGRA to D=1.

Why do we need to dualize gravity? The structure of hidden symmetries in reduced maximal SUGRAs comes from the U duality consideration (Morrison, TASI04) as follows: Consider both type IIA/IIB theories in space-times. In type IIA picture we have the SL(n,R) to O(n,n) enhancement due to T-duality, and the SL(n,R) to SL(n+1,R) enlargement through the M-theory interpretation (S-duality). Together, these symmetries generate the larger U-duality group. A convenient way to realize its structure is to consider Dynkin diagrams. The diagram of SL(n,R) corresponds to A(n-1) diagram with n-1 nodes The enlargement to O(n,n) has D(n) Dynkin diagram And the enlargement to SL(n+1,R) is A(n) (with n nodes) An entanglement of two diagrams is realized in E(n+1) (with n+1 nodes)

Why do we need to dualize gravity? The interpretation of E(n+1) for n<2 is subtle, rather it is a unifying notation for global symmetry groups of the moduli spaces in the reduced theories. The moduli space of reduced M-theory is A conjecture by West: E(11) is a hidden symmetry of M-theory, i.e. hidden symmetries of the reduced supergravity are already present in the unreduced theory. Clearly, the symmetry groups get extended from finite-dimensional groups and algebras to infinite-dimensional Kac-Moody-type algebras. Hence M-theory constructed in such a way contains infinitely many fields. Some of them may be auxiliary fields, so a problem is to relate the fields corresponding to the generators of the Kac-Moody-type algebra to the perturbative string spectrum.

Why do we need to dualize gravity? E(11) is a one of the so called very-extended algebras. E(9) affine Kac-Moody algebra is the extended (with one additional node in the Dynkin diagram) classical E(8), E(10) hyperbolic Kac-Moody-type algebra is the over-extended (with two additional nodes in the Dynkin diagram) E(8), and E(11) is as in the figure The deletion of the node 11 results in the A(10)~SL(11) algebra related to the gravity sector of D=11 SUGRA (the gravity line). The simple roots of E(11) are those of A(10) and with x to be the orthogonal to the A(10) roots, and is one of the fundamental weights of A(10). Any root of E(11) can be written as and is in the weight space of A(10). The integer is called the level, and defines the number of times the simple root appears in the root decomposition. Put it differently, we are considering the adjoint reps. of E(11) in terms of the reps. of A(10).

Why do we need to dualize gravity? We will use the following basic facts to define the reps. of E(11) at first three levels: Then Having in mind the above we get the following generators at levels 0, 1, 2 and 3 which correspond to the fields of graviton, 3 rd rank tensor field of D=11 SUGRA, its 6 th rank dual counterpart, and the dual to graviton field, as well as which does not occur in E(11).

Why do we need the duality-symmetric formulation? To realize M-theory within the E(11) conjecture one has to take into account that The standard fields of D=11 SUGRA (graviton, 3 rd rank tensor field) have to be included together with their dual fields. Recall, there is not a formulation of D=11 SUGRA solely in terms of a 6 th rank (dual) tensor field. It is only possible to construct the duality-symmetric formulation of D=11 SUGRA with 3 rd and 6 th rank tensor fields dual to each other (Bandos, Berkovits, Sorokin 1998). The duality-symmetric formulation is realized either in non-covariant (Hamiltonian) or covariant ways. Advantages of the latter are more controlled and flexible modifications of duality relations between fields in presence of additional sources (branes, orbifold planes, Domain Walls etc.). There is a simple argument why the fully-fledged theory of the graviton dual field should be duality-symmetric. As for bosonic fields the dual to graviton field has to be described by a second order in space-time derivatives equation of motion. Its structure in a curved background is where … corresponds to possible self-interactions and interactions with the background gravity field. The box is the D’Alembertian, and the latter is constructed out of space-time derivatives and of the background metric. The dynamics of the dual field in the full non-linear self-consistent theory, which takes into account the backreaction of the graviton dynamics, will contain the D’Alembertian with the dynamical (i.e. non-background) metric. Hence, we have to deal with the duality-symmetric theory which manages the dynamics both the graviton and its dual partner.

What is the hidden symmetry of gravity? On the background of the previously discussed SL(2,R)+ algebra of the Geroch group in the reduction of D=4 gravity to D=2, and its subsequent extention to SL(2,R)++ upon the reduction to D=1, one may expect the very-extended Kac-Moody-type algebra SL(2,R)+++ in the end. Following the West’s proposal, this algebra should be the hidden symmetry algebra of D=4 gravity. For D-dimensional gravity this algebra becomes SL(D-2,R)+++. Recall, that SL(2,R)+++ ~ A(1)+++, then in arbitrary dimensions (D>3) the hidden symmetry algebra is A(D-3)+++ The relevance of the algebra to gravitational theory was demonstrated by West within the slightly modified Ogievetsky-Borisov approach to gravity as a non-linearly realized theory. As for E(11) one may figure our the generators of A(D-3)+++ classifying them w.r.t. reps. of the “gravity line” which corresponds in the case to A(D-1) ~ SL(D). Clearly, we have A(D-1) generators at level 0, and It is easy to recognize the dual to graviton field generator, while the last generator do not enter the algebra.

How to realize the dynamics of the low levels fields? There are many ways to describe the dynamics of the low levels fields. For instance, one may follow the non-linear realization of the appropriate Kac-Moody algebra in a manner proposed by West. We will not follow this way. Another way is to try to construct a sigma-model-like action in D=1 which would be invariant under the appropriate very-extended hidden symmetry algebra (see details in the de Buyl’s PhD hep-th/06…). We will not follow this way too. The next way is the Hamiltonian approach to the duality-symmetric theories. We also refuse this way on the background mentioned in the above. The way we will follow is the covariant formulation of the duality-symmetric theories. To construct such a duality-symmetric gravity we have to introduce new auxiliary fields. It has to be done to avoid the Marcus-Schwarz no-go theorem. We would not like to spoil the content of the original theory (# of d.o.f.), therefore we should find new symmetries to eliminate auxiliary fields on-shell, and to reduce the # of d.o.f. extended by dual fields by use of duality relations.

Duality-symmetric formulations: auxiliary fields One may formulate the duality-symmetric theory by use of either infinitely many or the single auxiliary fields. For example in D=3 we (the Hodge) duality between Maxwell field and a scalar. The duality-symmetric theory is described by eqs.o.m. and the duality relation, which reduces the # of d.o.f. to the initial value Naively one may wonder to get this set from the action like this Indeed, we get eqs.o.m. and the duality relation However the Largange multiplier becomes propagating Introducing new Lagrange multiplier to set the first multiplier to zero does not improve the situation since the new Lagrange multiplier becomes propagating, and so on. Then we end up with the following action (McClain, Yu, Wu, Berkovits, Henneaux,…)

Duality-symmetric formulations: auxiliary fields One may cut the infinite tail of Lagrange multipliers. At an N+1th step the Nth Lagrange multipliers has to be replaced with Taking for instance N=1 the action becomes Solving for the Lagrange multipliers algebraic eqs.o.m. we arrive at This is the duality-symmetric action in Pasti-Sorokin-Tonin (PST) approach. Eqs.o.m. from the PST action If we reproduce the duality relation and the standard eqs.o.m. In fact, it is enough to set since another form of the above eqs.o.m. is Then these equations hold the eq.o.m. of the PST scalar a(x) is automatically satisfied. It means that the PST scalar is non-dynamical, and we have an additional symmetry which gauges this field away. There is also another symmetry which reduce eqs.o.m. to

Duality-symmetric formulation of gravity We proposed the following action for D-dim. gravity within the PST approach In the first order formalism encode the duality relations between vielbein and the Hodge dual field The duality relations generate as the Einstein equation as well as a second order equation of the dual field To reduce the dual field eq.o.m. coming from the PST action to we use the following symmetry of the PST action

Duality-symmetric formulation of linearized gravity There are numerous indications that the linearized dual gravity admits the local formulation. What does it mean from the point of view of the PST-like approach? When we expand over the flat space we have In the linearized limit the generalized “field strengths” become hence the PST-like action is also local But what about the PST symmetries? When we require locality of the PST symmetries of the linearized action we find that the dual field has to be constrained to this end, Surprisingly, this constraint is the same as it is required to cut the infinite tail of A(D-3)+++ generators to separate the subalgebra corresponding to the graviton and to its dual partner fields generators!

Resume’ To summarize we point out that 1. Duality-symmetric approach to gravity and supergravity is natural in searching for the dynamical realization of M-theory 2. Constructing the PST-like action for the lower levels generators of A(D-3)+++, we deal with non-local action and duality relations, but with local equations of motion of fields. 3. When we require locality of the linearized duality-symmetric action of pure gravity and its special (PST) symmetries, we restrict the index structure of the dual to graviton field. This constraint is also responsible for eliminating the antisymmetric part of the vielbein, thus leading to the Fierz-Pauli description of spin-2 duality- symmetric theory. 4. On the side of non-linear realization of gravity in terms of the corresponding Kac- Moody algebra the restriction we observed separates the subalgebra of the low- level generators (corresponding to graviton and its dual field) from the infinite- dimensional Kac-Moody algebra. 5. However, constructing the linearized duality-symmetric gravity with a cosmological constant, or with matter fields still requires non-locality of the action, as well as non-locality of the PST symmetries. We have mentioned that some of the infinitely many fields of the non-linearly realizing gravity could be unphysical. It is an interesting problem to construct the duality- symmetric gravity action within the formulation with infinitely many auxiliary fields since we may expect that a part of the generators of the gravity Kac-Moody algebra would be in a correspondence to the auxiliary fields.