Kac-Moody Symmetries in Reductions to Two Dimensions String and - PowerPoint PPT Presentation

Kac-Moody Symmetries in Reductions to Two Dimensions String and M-Theory Approaches to Particle Physics and Cosmology Galileo Institute, Firenze, 20’th June 2007

Plan • Toroidal Reduction of Eleven-Dimensional Supergravity to 3 ≤ D ≤ 10 • Borel-Gauge Description of the Scalar Coset Manifolds • Reduction of Pure D = 4 Gravity to D = 2 • Reduction of Eleven-Dimensional Supergravity to D = 2 • Construction of Infinity of Conserved Currents The description of the reductions to 3 ≤ D ≤ 10 and the Borel gauge coset construction summarises results in hep-th/9710119, Cremmer, Julia, L¨ u and Pope. The approach described here to understanding the Kac-Moody symmetries of the reductions to D = 2 is work in progress by L¨ u, Perry, Pope and Stelle– “Demystification of Kac-Moody Symmetries in D = 2”

Kaluza-Klein Reduction on S 1 Reduction on T n can be broken up into a step-by-step reduction on a sequence of circles. Consider the reduction of gravity and a p -form potential, l − 1 L = ˆ ˆ ∗ ˆ F ( p +1) ∧ ˆ R ˆ ∗ 1 2 ˆ F ( p +1) in the step from D + 1 to D dimensions: e 2 αφ ds 2 + e 2 βφ ( dz + A (1) ) 2 s 2 d ˆ = ˆ A ( p ) = A ( p ) + A ( p − 1) ∧ dz where all quantities on the RHS are independent of the circle coordinate z . The constants α and β are chosen such that 1 α 2 = β = − ( D − 2) α 2( D − 1)( D − 2) , with the latter ensuring the lower-dimensional metric is in the Einstein frame, and the former fixing a canonical normalisation for the kinetic term of the KK scalar φ : 2 e − 2( D − 1) αφ ∗F (2) ∧ F (2) l − 1 2 ∗ dφ ∧ dφ − − 1 = R ∗ 1 L 2 e − 2 pαφ ∗ F ( p +1) ∧ F ( p +1) − 1 2 e 2( D − p − 1) αφ ∗ F ( p ) ∧ F ( p ) − 1 Here F (2) = d A (1) , F ( p +1) = dA ( p ) − dA ( p − 1) ∧A (1) and F ( p ) = dA ( p − 1) .

Kaluza-Klein Reduction of Pure Gravity on T n At each successive step of reduction on S 1 , the metric gives rise to a metric and a new KK scalar (dilaton) and KK vector. Each existing p -form potential gives rise to a p -form and a ( p − 1)- form potential. Note that a 1-form potential (such as an already existing KK vector) gives a 1-form and a 0-form potential, and that the latter is an axionic scalar. Pure gravity reduced on T n will therefore have a set of n 1-forms (1) ; a set of 1 A i 2 n ( n − 1) axionic scalars A i (0) j (with j > i ) from the reductions of the 1-forms at subsequent steps; and a set of n dilatonic scalars � φ = ( φ 1 , φ 2 , · · · , φ n ). The kinetic term for each form field will have an exponential prefactor of the form e � b · � φ , where the constant “dilaton vector” � b characterises the coupling of the dilatonic scalars to that partic- ular form field: � � e � b i · � e � b ij · � φ ∗F i φ ∗F i l − 1 2 ∗ d� φ ∧ d� φ − 1 (2) ∧ F i (2) − 1 (1) j ∧ F i L grav = R ∗ 1 (1) j 2 2 i i<j

Reduction of D = 11 Supergravity on T n to D = 11 − n l − 1 F (4) + 1 D = 11 supergravity ˆ L = ˆ ∗ ˆ F (4) ∧ ˆ 6 ˆ F 4 ∧ ˆ F (4) ∧ ˆ R ˆ ∗ 1 2 ˆ A (3) reduced on T n then gives � � e � b i · � e � b ij · � φ ∗F i φ ∗F i l − 1 (2) ∧ F i (2) − 1 (1) j ∧ F i 2 ∗ d� φ ∧ d� L = R ∗ 1 φ − (1) j 2 i i<j � a · � a i · � φ ∗ F (4) ∧ F (4) − 1 φ ∗ F (3) i ∧ F (3) i − 1 2 e � e � 2 i � � a ijk · � a ij · � φ ∗ F (1) ijk ∧ F (1) ijk + L FFA φ ∗ F (2) ij ∧ F (2) ij − 1 − 1 e � e � 2 2 i<j i<j<k where the dilaton vectors are given by ˆ F (4) Metric 4 − form : � a a − � 3 − forms : � a i = � b i a − � b i − � � 2 − forms : � a ij = � b j b i a − � b i − � b j − � � b ij = � b i − � 1 − forms : a ijk = � � b k b j � 2 a = 1 � b i · � � b j = 2 δ ij + D − 2 , � b i 3 i

Global Symmetry of Toroidally-Reduced Theory Any theory including gravity, reduced on T n , will have at least an SL ( n, R ) global symmetry that acts “internally” (i.e. it leaves the lower-dimensional Einstein-frame metric invariant). It corresponds to the subgroup of general coordinate transforma- tions of the original theory comprising rigid SL ( n, R ) transforma- tions in the torus T n : δx µ = 0 , δy i = Λ ij y j If the original theory has an overall scaling symmetry (“trombone symmetry”), such as pure gravity or D = 11 supergravity: → λ 9 ˆ → λ 2 ˆ → λ 3 A MNP , ˆ ˆ g MN − ˆ g MN , A MNP − ⇒ L − L , then volume-changing transformations are included too and this global internal symmetry becomes GL ( n, R ). If there are other form fields in the higher-dimensional theory, the global symmetry may be enhanced further. The global symmetry G is non-linearly realised on the scalar fields (dilatons plus axions) in the reduced theory. These scalars lie in a coset space K = G / H . The group G acts linearly on the other form fields.

Global Symmetry of Toroidally-Reduced Pure Gravity The global symmetry can therefore conveniently be studied by first focusing on the scalar sector. Consider first the reduction of pure gravity from D + n to D ≥ 4. The scalar sector comprises n dilatons � φ and 1 (0) j with dilaton vectors � 2 n ( n − 1) axions A i b ij = � b i − � b j , where � b i · � b j = 2 δ ij + 2 / ( D − 2). These dilaton vectors are in one-to-one correspondence with the positive roots of the A n − 1 = SL ( n, R ) algebra. The simple roots are � b i,i +1 , for 1 ≤ i ≤ n − 1: � � � � b 12 b 23 b n − 2 ,n − 1 b n − 1 ,n — — · · · · · · — — � � � � If reduced to D = 3, the KK 1-forms A i (1) , (with dilaton vectors � b i ), can be dualised to give an additional n axions, with dilaton vectors − � b i . The symmetry enhances to A n = SL ( n + 1 , R ), with { � b ij , − � b i } as positive roots, and − � b 1 the extra simple root: − � � � � � b 1 b 12 b 23 b n − 2 ,n − 1 b n − 1 ,n — — — · · · · · · — — � � � � �

Global Symmetry of T n -Reduced D = 11 Supergravity In a reduction on T n to D = 11 − n , we have n dilatons � φ , 1 2 n ( n − 1) (0) j from the metric and 1 axions A i 6 n ( n − 1)( n − 2) axions A (0) ijk A (3) . These have dilatons vectors � b ij = � b i − � from the 3-form ˆ b j � a = 1 a − � b i − � b j − � ℓ � and � a ijk = � b k respectively ( � b ℓ ). 3 In 3 ≤ D ≤ 5 we obtain further axions by dualising form fields: D = 5 : ∗ A (3) Dilaton vector − � a 1 D = 4 : Dilaton vectors 8 ∗ A (2) i − � a i ( ∗A i ( − � D = 3 : (1) , ∗ A (1) ij ) Dilaton vectors b i , − � a ij ) 8 + 28 In all dimensions 3 ≤ D ≤ 10, the full set of axion dilaton vectors (including those coming from dualisation when 3 ≤ D ≤ 5) are in one-to-one correspondence with the postive roots of E n , where, for n ≤ 5 we have = R , E 2 = GL (2 , R ) , E 3 = SL (3 , R ) × SL (2 , R ) E 1 (1) SL (5 , R ) , E 4 = E 5 = O (5 , 5) a 123 and � The simple roots are � b i,i +1 for 1 ≤ i ≤ n − 1.

The E n Symmetry of D = 11 Supergravity on T n � � � � � � � b 12 b 23 b 34 b 45 b 56 b 67 b 78 o — o — o — o — o — o — o | o � a 123 � b i,i +1 with i ≤ 7 and � a 123 generate the E 8 Dynkin diagram Vertices with indices exceeding n are to be deleted for n < 8. We have exhibited the root structure of the dilaton vectors char- acterising the couplings of the dilatons � φ in the exponential pref- actors of the axionic kinetic terms. We still need to show exactly why this implies that the scalars are described by the coset man- ifold E n /K ( E n ), where K ( E n ) is the maximal compact subgroup of E n . The construction is extremely simple, by virtue of the fact that the step-by-step reduction scheme naturally leads to a parame- terisation of the coset representative in the Borel gauge.

SL (2 , R ) /O (2) Scalar Coset in Borel Gauge First consider a toy model, namely an SL (2 , R ) /O (2) scalar coset model: 2 e 2 φ ∗ dχ ∧ dχ L = − 1 2 ∗ dφ ∧ dφ − 1 Defining � � � � � � 1 0 0 1 0 0 H = , E + = , E − = 0 − 1 0 0 1 0 the coset K = G / H with G = SL (2 , R ) and H = O (2) has gener- ators as follows: K : H and ( E + + E − ) (Non-Compact) H : ( E + − E − ) (Compact) It is convenient to use the Borel gauge for writing the coset representative:   1 1 2 φ 2 φ χ 1  e e 2 φH e χE + =   V = e  e − 1 2 φ 0

in terms of which we find d VV − 1 1 2 Hdφ + E + e φ dχ = 1 2 Hdφ + 1 2 ( E + + E − ) e φ dχ + 1 2 ( E + − E − ) e φ dχ = Since d VV − 1 = P + Q , where P is the projection into the Lie alge- bra of the coset K and Q is the projection into the denominator algebra H , we have P χ = e φ dχ Q → e φ dχ P φ = dφ , The Cartan-Maurer equation d ( d VV − 1 ) − ( d VV − 1 ) ∧ ( d VV − 1 ) = 0 implies dQ − Q ∧ Q − P ∧ P = 0 , DP ≡ dP − Q ∧ P − P ∧ Q = 0 2 ( P φ ) 2 − 1 The Lagrangian can be written as L = − 1 2 ( P χ ) 2 , and the equations of motion are D ∗ P = 0 The (right-acting) SL (2 , R ) global symmetry is V − → OV Λ, where O is a local O (2) compensator that restores V to Borel gauge.