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N = 8 Supergravity, and beyond

Or: what symmetry can teach us about quantum gravity

and the unification of physics.

IISER Pune, 26 February 2011

Hermann Nicolai

MPI fur Gravitationsphysik, Potsdam

(Albert Einstein Institut)

A projection of the 240 roots of E8 (an eight-dimensional‘diamond’) onto a two-dimensional plane.

Main theme: Symmetry

... arguably the most successful principle of physics!

• Space-time symmetries

– Rotations and translations in Newtonian physics

– Special Relativity: unifying space and time [Einstein (1905)]

– General Relativity and general covariance [Einstein (1915)]

• Internal symmetries

– Isospin symmetry: mneutron = 1.00135mproton [Heisenberg]

– Flavor symmetry, strong interactions and the quark model

– Standard Model and (non-abelian) Gauge Theories

• The two fundamental theories of modern physics,General Relativity and the Standard Model of Par-ticle Physics, are based on and largely determinedby symmetry principles!

Where we stand

Known laws of physics successfully describe observedphenomena over a huge range of distances from 10−18mall the way to the visible horizon of our universe!

• General Relativity : gravity from space-time curva-ture (general covariance and equivalence principle)

• Standard Model of Particle Physics: combines quan-tum mechanics and special relativity to describe

Matter = 3 generations of 16 spin-12 fermions

Forces = electromagnetic, weak and strong

via G = SU (3)c × SU (2)w × U (1)Y gauge symmetry.

Where we stand

Known laws of physics successfully describe observedphenomena over a huge range of distances from 10−18 cmall the way to the visible horizon of our universe!

• General Relativity : gravity from space-time curva-ture (general covariance and equivalence principle)

• Standard Model of Particle Physics: combines quan-tum mechanics and special relativity to describe

Matter = 3 generations of 16 spin-12 fermions

Forces = electromagnetic, weak and strong

via G = SU (3)c × SU (2)w × U (1)Y gauge symmetry.

... but both theories are incomplete and possibly eveninconsistent: singularities and infinities!

Symmetry and Unification

Like a ferromagnet: symmetry is broken more andmore with decreasing temperature as universe expands.

But where do we go from here?

Idea: symmetry enhancement as a guiding principle!

• Grand Unification: look for a ‘simple’ group

SU(3)c × SU(2)w × U(1)Y ⊂ SU(5) ⊂ SO(10) ⊂ E6 ⊂ . . .?

→ quark lepton unification, proton decay, ...

• ‘Fusion’ of space-time and internal symmetries?

E.g. from pure gravity in higher dimensions?[Kaluza,Klein (1921)]

• (Electromagnetic) Dualities:

E + iB→ eiα(E + iB) , q + ig → eiα(q + ig)

• Quantum symmetry and quantum space-time?

But where do we go from here?

Idea: symmetry enhancement as a guiding principle!

• Grand Unification: look for a ‘simple’ group

SU(3)c × SU(2)w × U(1)Y ⊂ SU(5) ⊂ SO(10) ⊂ E6 ⊂ . . .?

→ quark lepton unification, proton decay, ...

• ‘Fusion’ of space-time and internal symmetries?

E.g. from pure gravity in higher dimensions?[Kaluza, Klein (1921)]

• (Electromagnetic) Dualities:

E + iB→ eiα(E + iB) , q + ig → eiα(q + ig)

• Quantum symmetry and quantum space-time?

QUESTION: is it possible to pin down the ‘right’ the-ory simply by imposing a symmetry principle?

Supersymmetry

• A new kind of symmetry relating Bosons↔ Fermions.

Or: Forces (vector bosons) ↔ Matter (quarks & leptons)?

• Probably one of the most important developmentsin mathematical physics over the last 40 years.

• Supersymmetry is generally believed to be essentialfor constructing a perturbatively consistent (finite)theory of quantum gravity.

• Experimental signatures of (low energy) supersym-metry might show up at LHC accelerator at CERN.

Supersymmetry in a Nutshell

Simple example: supersymmetric quantum mechanicswith supercharge Q,Q† and Hamiltonian H:

H =1

2{Q , Q†} ⇒ [Q,H ] = [Q†, H ] = 0

Hence for any eigenstate |E〉 (with E > 0) we have

H|E〉 = E|E〉 ⇒ H(Q|E〉) = E(Q|E〉)

⇒ degeneracy of bosonic/fermionic energy levels ⇒

• SUSY particles come in pairs of opposite statistics,

e.g. Photon γ (s = 1) ←→ Photino γ (s = 12)

• Equal masses for bosons and fermions in supersym-metric models of elementary particles (unless su-persymmetry is broken).

Supersymmetric (Quantum) Field Theory

For (semi-)realistic field theories need to ‘marry’ su-persymmetry with other symmetries of relativistic quan-tum field theory (Poincare and internal symmetries).

Supercharges Qiα and Qαj are now space-time spinors.

The key relation of the relativistic superalgebra is

{Qiα , Qβj} = 2δijσ

µ

αβPµ

N-extended supersymmetry (for i, j = 1, . . . , N) mergesspacetime and internal symmetries when N ≥ 2.

In QFT, supersymmetry entails (partial) cancellationsof UV infinities in Feynman diagrams →

Supersymmetry softens the infinities of QFT!

But can it remove them altogether?

Supermultiplets

... a general analysis of what types of particles (ofdifferent spin s) can be put together in ‘superfamilies’.

There are many models with low degree of supersym-metry but they become more and more restricted withincreasing N , and unique for maximal supersymmetry.

Distinguish between two types of maximal theories:

• Supersymmetric matter interactions (no gravity)

N = 4 multiplet: 1×[1] ⊕ 4×[

12

]

⊕ 6×[0]

• Supersymmetric gravity = supergravity

N = 8 multiplet: 1×[2]⊕ 8×[

32

]

⊕ 28×[1]⊕ 56×[

12

]

⊕ 70×[0]

N = 8 Supergravity (I)

Unique theory (modulo ‘gauging’) based on multiplet

1×[2] ⊕ 8×[

32

]

⊕ 28×[1] ⊕ 56×[

12

]

⊕ 70×[0]

This theory incorporates all the known and expectedsymmetries ∗. In addition it has an unexpected ‘hid-den’ exceptional E7 duality symmetry. [Cremmer,Julia (1979)]

It is thus the most symmetric known field theoreticextension of Einstein’s General Relativity!———————————————————————∗ General covariance and local Lorentz symmetry, N = 8 local

supersymmetry and SU(8) R-symmetry (local or rigid).

N = 8 Supergravity (II)

N = 8 Supergravity (III)

In the late 1970s this theory was thought to be apromising candidate for a unified theory of quantumgravity and matter interactions. However,

• Existence of supersymmetric counterterms suggestedthe appearance of non-renormalizable infinities fromthree loops onwards;

• Its properties [absence of chiral fermions, huge neg-ative cosmological constant for gauged theory] seemto be in obvious conflict with observations.

And: in both regards superstring theory seemed to domuch better!

BUT: remarkable recent progress on UV finiteness...

Finiteness: to be or not to be?

Einstein gravity is perturbatively non-renormalizable

Γ(2)div =

1

ε

209

2880

1

(16π2)2

∫

dV CµνρσCρσλτCλτ

µν

where Cµνρσ = Weyl tensor. [Goroff, Sagnotti(1986); van de Ven(1992)]

NB: calculation of the coefficient 2092880 requires consid-

eration of O(100 000) Feynman diagrams!

Although this 2-loop counterterm does not allow for asupersymmetric extension, supersymmetry is not enoughto rule out UV infinities from three loops onwards!

However, for N = 8 supergravity computation of 3-loopcounterterm coefficient would require consideration ofO(10xxx) Feynman diagrams → hopeless??? ......

Finiteness: to be or not to be?

Einstein gravity is perturbatively non-renormalizable

Γ(2)div =

1

ε

209

2880

1

(16π2)2

∫

dV CµνρσCρσλτCλτ

µν

where Cµνρσ = Weyl tensor. [Goroff, Sagnotti(1986); van de Ven(1992)]

NB: calculation of the coefficient 2092880 requires consid-

eration of O(100 000) Feynman diagrams!

Although this 2-loop counterterm does not allow for asupersymmetric extension, supersymmetry is not enoughto rule out UV infinities from three loops onwards!

However, for N = 8 supergravity computation of 3-loopcounterterm coefficient would require consideration ofO(10xxx) Feynman diagrams → hopeless???...... NO!

3& 4-loop finiteness of N = 8 supergravity

[Bern,Carrasco,Dixon,Johansson,Roiban, PRL103,081301(2009)]

• Exploit Gravity = (Yang Mills)2.

• Use unitarity based arguments to reduce all ampli-tudes to integrals over products of tree amplitudes.

• All particles are on-shell → only 3-point vertices.

• Instead of O(10xxx) Feynman diagrams need only cal-culate O(50) ‘Mondrian-like’ diagrams!

Beyond L = 4 loops

There is now mounting evidence from different sourcesthat N = 8 supergravity is UV finite at L < 7 loops:

• spectacular computational advances (see above...)

• exploiting nonlinear symmetries (SUSY, E7(7),...)

... and could thus be UV finite to all orders!

BUT L = 7 requires 10xxxx Feynman diagrams! AND:

• Even if N = 8 supergravity is finite, (non-linear) E7

and SUSY may not be enough to prove it.

• Even if it is finite we still do not understand what‘happens’ to space and time at the Planck scale.

What about ‘real’ physics?

Even if finite it is usually taken for granted that

• N = 8 supergravity does not match with particle physics

This seems hard to argue with but ....

... after complete breaking of N = 8 supersymmetry we are leftwith 48 = 3 × 16 basic spin-12 fermions – equal to the numberof quarks and leptons in three families!

This could be either a deep truth or a mirage...

... but again: what you see in your Lagrangian maybe very different from what you see in the detector!

Famous example: QCD would have been considered‘obviously wrong’ if it had been proposed in 1950.

Exceptional Mathematics and Supergravity

Continuing on the road towards more symmetry!

• N = 8 supergravity has more symmetry than meetsthe eye (SUSY, E7 duality), but still not enough!

• Also true for D 6= 4: hidden En for maximal super-gravity in D space-time dimensions with n = 11−D!

Below D = 3 symmetries become infinite-dimensional:

• E9 ≡ E(1)8 for maximal supergravity in D = 2.

• ... suggests E10 for D = 1: no space, only time?!?

But why should we care about physics in one dimen-sion if we live in D = 4 space-time dimensions???

Physics near the Big Bang

For T → 0 causal decoupling of spatial points −→

Physics becomes effectively one-dimensional in vicinityof cosmological singularity! [Belinski,Khalatnikov,Lifshitz (1972)]

IDEA: Full symmetry only visible at the singularity!

What is E10?

(No one knows, really....)

E10 is the ‘group’ associated with the Kac-Moody Liealgebra g ≡ e10 defined via the Dynkin diagram [e.g. Kac]

1 2 3 4 5 6 7 8 9

0

y y y y y y y y y

y

y

Defined by generators {ei, fi, hi} and relations via Car-tan matrix Aij (‘Chevalley-Serre presentation’)

[hi, hj] = 0, [ei, fj] = δijhi,

[hi, ej] = Aijej, [hi, fj] = −Aijfj,

(ad ei)1−Aijej = 0 (ad fi)

1−Aijfj = 0.

e10 is the free Lie algebra generated by {ei, fi, hi}modulothese relations → infinite dimensional as Aij is indefi-nite → Lie algebra of exponential growth !

Infinite Complexity from simple recursion

A Mandelbrot set generated from zn+1 = fc(zn).

Vistas into E10...

[from: Teake Nutma (formerly University of Groningen, now at AEI)]

What is so special about E10?

• E10 occupies a uniquely distinguished place amongall infinite-dimensional Lie groups (much like E8

among the finite-dimensional Lie groups).

But it is monstrously complicated .... [see hep-th/0301017]

• Near the initial singularity (for 0 < T < TPlanck) thedynamics of maximal supergravity ‘asymptotes’ toa ‘cosmological billiards’ system based on E10.

• E10 may provide an algebraic mechanism for the‘de-emergence’ of space and time near the big bang.

• E10 ‘knows all’ about maximal supersymmetry.

Only for the specialists: this also works for the fermions!

E10 Versatility

y y y y y y y y y

y

y

z

sl(10) ⊆ e10

D = 11 SUGRA

y y y y y y y y y

y

z

y

so(9, 9) ⊆ e10

mIIA D = 10 SUGRA

y y y y y y y y y

y

z z

sl(9)⊕ sl(2) ⊆ e10

IIB D = 10 SUGRA

y y y y y y y y

y

z

sl(3)⊕ e7 ⊆ e10

N = 8, D = 4 SUGRA

Main Conjecture

For 0 < T < TP space-time ‘de-emerges’, and space-timebased (quantum) field theory is replaced by purely al-gebraic description in terms of E10. [Cf. DN, 0705.2643]

Outlook

• Symmetry by no means exhausted as a guiding prin-ciple of physics but many open questions remain.

• Both N = 8 supergravity and E10 are uniquely dis-tinguished by their symmetry properties.

• E10 ‘knows all’ about maximal supersymmetry andunifies many known (S, T, U, ...) string dualities.

[so supersymmetry may not be as fundamental as we thought...]

• Exponentially increasing complexity of E10 algebra→ an element of non-computability for T → 0 ?

•→ what is the role of INFINITY in Physics?

Outlook

• Symmetry by no means exhausted as a guiding prin-ciple of physics but many open questions remain.

• Both N = 8 supergravity and E10 are uniquely dis-tinguished by their symmetry properties.

• E10 ‘knows all’ about maximal supersymmetry andunifies many known (S, T, U, ...) string dualities.

[so supersymmetry may not be as fundamental as we thought...]

• Exponentially increasing complexity of E10 algebra→ an element of non-computability for T → 0 ?

•→ what is the role of INFINITY in Physics?

THANK YOU