Proceedings A Kaluza–Klein Approach to Double and Exceptional Field Theory LMS/EPSRC Durham Symposium on Higher Structures in M-Theory a, David S. Berman ∗ We examine the challenge of viewing all the fields in su- duced on a d-dimensional torus was shown to exhibit an pergravity as arising from a Kaluza–Klein like dimensional Ed exceptional group of global symmetries. Later with the advent of M-theory [12, 13] this E -symmetry was reduction of some higher-dimensional theory. This gives d extended to a duality of string theories known as U- rise to what is known as exceptional field theory or dou- duality that combined T-duality with S-duality. Various ble field theory. A particular emphasis is placed on fol- works then attempted to reformulate supergravity the- lowing the Kaluza–Klein intuition leading to the identifica- ories such that exceptional symmetry would become a tion of charged states and a reinterpretation of the cen- manifest symmetry of the theory, notably in the numer- tral charges. We further give a description of the novel ex- ous works of West [14], Nicolai [15] and others. In [16], by tended geometry as a generalised phase space and the extending the number of dimensions, the group E7 was relationship to string and M-theory theory and the notion made into a manifest symmetry and in [17] the group SL(5) was similarly made manifest again by extending of quantization. the number of dimensions. There have been other nu- merous developments, perhaps most notably the role of generalised geometry in this scheme has been developed 1 Introduction in [18,19].The mainthrust ofthis paper is to focusonthe extra coordinates and so this will not be our approach. 1.1 Some small history Crucially, as we can see from the above narrative, the duality symmetries in string and M-theory were at the This paper as a whole follows a very non-historically ac- heart of the developments for these theories. We wish curate approach to double and exceptional field theory to emphasize that in the approach described here this based on a Kaluza–Klein approach to supergravity. Be- is only a by-product of these theories and one should fore beginning this chain of logic let us first briefly de- not see DFT or exceptional field theory (EFT) as a the- scribe some of the history of the subject. Almost 30 years ory to make duality symmetries manifest. To do so would ago Michael Duff [1] developed a string world-sheet the- make these theories redundant for backgrounds without ory where T-duality appeared as a manifest symmetry. isometries (T-duality in its usual form requires isome- arXiv:1903.02860v1 [hep-th] 7 Mar 2019 Subsequently Arkady Tseytlin [2,3] described different as- tries of the background) and this is not true. pects of a string in a doubled space-time with manifest T- duality before Warren Siegel [4] described a sophisticated world-sheet theory with manifest O(d,d) symmetry. Af- 1.2 Kaluza–Klein theory ter a break of some years, in 2009, Hull and Zwiebach [5, 6] and then with Hohm [7, 8] examined a truncation Let us begin with a review of traditional Kaluza–Klein the- of closed string field theory keeping only the momentum ory. This is so we can give a prescription for a series of and winding modes of the string field and produced a steps that we will emulate later for all the bosonic fields string background with twice the number of coordinates. This theory is known as double field theory (DFT). Some- what in parallel, the Korean group [9,10] developed what is now called the semi-covariant formalism. ∗ Corresponding author e-mail: [email protected] In parallel developments, in the work of Cremmer, Ju- a School of Physics and Astronomy, Queen Mary University of lia and Scherk [11], eleven-dimensional supergravity re- London, 327 Mile End Road, London E1 4NS, United Kingdom 1 D. S. Berman: A Kaluza–Klein Approach to Double and Exceptional Field Theory in supergravity and so produce double or exceptional reproduce this we need the constraint (6) to apply to the field theory. The starting point is Einstein–Maxwell the- parameters generating the local transformations. ory in four dimensions coupled to a scalar field. The field Thus, the five-dimensional diffeomorphisms restric- content is thus the metric gµν, the one-form vector po- ted to four dimensions by (6) are equivalent to four- Proceedings tential Aµ and the scalar field φ. The action for these dimensional diffeomorphisms and one-form gauge trans- fields is given by formations. Strictly speaking we have only seen this in- finitesimal transformations and have not examined so S d4xp geφ R(g) 1 F F µν 1 ∂ φ∂µφ , (1) called ‘large’ transformations i.e. diffeomorphisms or = − − 4 µν − 2 µ Z gauge transformations that are finite and not connected ¡ ¢ where F ∂ A ∂ A is the field strength for A . The to the identity. µν = µ ν − ν µ µ local symmetries of the theory are given by diffeomor- In some sense we were lucky in that the combina- phisms, as described by the Lie derivative, tion of diffeomorphisms and one-form transformations nicely combine into five-dimensional diffeomorphisms L U µ V ρ∂ U µ ∂ V µU ρ , (2) (it is hard to imaginean a prioriargument that this had to V = ρ + ρ be the case without already knowing about Kaluza–Klein and the gauge transformations of Aµ, theory). In the sections that follow we will have to be a lit- tle more creative since to combine diffeomorphism with δA ∂ χ. (3) µ = µ p-form gauge transformations will not produce usual diffeomorphisms in a higher-dimensional theory. Before Thus there are five parameters for the local symmetries: doing this, let us examine the Kaluza–Klein approach the four vector that generates the diffeomorphisms, V µ, some more as this will provide a guiding hand later. and the single scalar χ that generates the gauge transfor- Now that we have the higher-dimensional theory (5), mations of A . µ the metric ansatz (4) and the constraint for the reduc- The Kaluza–Klein idea may then be expressed as fol- tion (6) the next step is to look for the origin of the elec- lows: given there are five parameters for the local sym- trically charged states in the theory. So far the action metries, can one combine them to form a five vector (1) describes Einstein–Maxwell theory with no currents. Vˆ µˆ such that diffeomorphisms generated by this five How can we add electric sources from the new higher- vector acting on some five-dimensional metric, gˆ will µˆνˆ dimensional perspective? reproduce the four-dimensional local transformations The answer is given quickly by calculating the geodesic described above. (We take hatted objects to be five- equations for a probe particle in the five-dimensional dimensional and µˆ (µ,5).) = theory with the ansatz (4). Doing this, one discovers that The challenge then is to find the five-dimensional the Lorentz force law for electric charges is recovered pro- metric that meets this criteria, and the answer can be vided one identifies the four-dimensional electric charge found to be: with the derivative in the fifth dimension. One then uses g φ2 A A φ2 A the usual relationship between the probe particle wave gˆ µν µ ν ν . (4) µˆνˆ + 2 2 function and the momentum operator as a derivative to = " φ Aµ φ # write: For the five-dimensional theory to have five-dimensional i∂5 P5 Qe . (7) diffeomorphism invariance implies that it should be de- = = scribed by the five-dimensional Einstein–Hilbert action, Thus objects with momentum in the fifth direction will appear as electric charges from the four-dimensional per- S d5x gRˆ (gˆ). (5) spective. A light-like object in five dimensions will obey: 5 = − µˆ 5 µ Z q P Pµˆ 0 P P5 P Pµ 0. (8) Inserting the ansatz (4) into this five-dimensional ac- = + = tion reproduces the four-dimensional action (1) pro- Then, since the four-dimensional on-shell relation is P µP M 2, this implies that P M and thus the BPS vided that the fields are independent of the new fifth di- µ =− 5 = mension i.e. for all fields and gauged transformations: condition: M Q . (9) = e ∂x5 0. (6) = At this point one may ask about electric charge quanti- Another way to think of this is that the original local trans- sation. If one requires Qe to be quantised then this im- formations only depended on four dimensions and so to plies that the extra Kaluza–Klein direction is compact, 2 Proceedings typically one takes this direction to be a circle of ra- and Gross and Perry [22]: dius R. Then the momentum operator P5 will have a dis- 2 2 1 2 i 2 2 ds dt H − (dz Ai dy ) Hdy . (11) crete spectrum with P n where n is an integer. This =− + + + 5 = ħ R is the usual way Kaluza–Klein quantises electric charge, The field Ai which controls the twist is related to the har- which differs slightly from the usual Dirac quantisation monic function as follows: in that there is no mentioning of magnetic charges. The 1 k g ∂[i A j ] 2 ǫij ∂k H , H 1 . (12) alert reader may feel slightly dissatisfied with the fact = = + r that although the electromagnetic field is geometric the From the perspective of four dimensions and the gauge charges are not.