What is String Theory? David Tong University of Cambridge

Transcription

1 What is String Theory? David Tong University of Cambridge The Structure of Gravity and Space=me, Oxford, February 2014

2 The Faces of String Theory D- Branes Perturba=ve String Theory Gravity Gauge Theory AdS/CFT

3 Quantum Gravity Z = topologies Dg Dφ exp i d 4 x gr + L matter [φ] Can we make sense of this?

4 A Simpler Toy Model Z = topologies Dg Dφ exp 1 d 2 x gr + L matter [φ] Quantum field theory in d=1+1 is osen more tractable. We have also gone to Euclidean space to make life easier. M d 2 x gr =4πχ(M)

5 An Even Simpler Toy Model Z = topologies Dg Dφ exp 1 d 2 x gr + L matter [φ] L matter [φ] = gg αβ α φ β φ i.e. a massless scalar field. This is special because it has a new symmetry

6 The Symmetries L matter [φ] = gg αβ α φ β φ Diffeomorphisms (in two dimensions) Weyl Symmetry: g αβ Ω 2 (x) g αβ Both diffeomorphisms and Weyl symmetry are gauge symmetries. i.e. they are redundancies of our descrip=on This wouldn t work if we included a poten=al for the scalar And it doesn t work in higher dimensions.

7 Conformal Field Theories L matter = L CFT Any conformal field theory (CFT) coupled to 2d gravity has classical Weyl symmetry The central charge, c: This characterises the number of degrees of freedom of a CFT Perturba=ve string theory is the study of quantum field theories with Weyl symmetry.

8 Anomalies Anomalies are symmetries of classical field theories that do not survive quan=sa=on. Anomalies in global symmetries are interes=ng Most of the mass in the Universe Chiral anomaly in QCD (pion decay, eta- prime mass) Anomalies in gauge symmetries are fatal Cancella=on of gauge (and gravity!) anomalies in Standard Model

9 Weyl Anomaly Polyakov The Weyl symmetry suffers an anomaly. This is bad. In fact, it suffers two anomalies Trace of stress tensor should vanish in a conformal field theory But it doesn t typically vanish in a curved background T α α = c R c is the central charge of the CFT. It counts the number of degrees of freedom. But you get an extra term coming from the dynamical metric. (Strictly speaking, from ghosts)

10 The Punchline Quantum field theories with Weyl symmetry only make sense if c = 26 e.g. 26 free scalars 52 copies of the Ising model Many more interes=ng possibilies

11 An Example of a CFT with c=26 L CFT = gg αβ α φ µ β φ ν G µν (φ) This has the interpreta=on of a two- dimensional object (i.e. string) moving in a d=26 dimensional space with metric G. This is where the extra dimensions of string theory come from.

12 An Example of a CFT with c=26 L CFT = gg αβ α φ µ β φ ν G µν (φ) But.this isn t a CFT for any choice of background metric G. It needs to obey R µν [G] =0 Friedan These are the vacuum Einstein equa=ons in d=26!

13 Other Examples of CFTs with c=26 L CFT = g g αβ α φ µ β φ ν G µν (φ)+ αβ α φ µ β φ ν B µν (φ)+α Φ(φ)R The theory is a CFT only if the func=ons G, B and D obey the equa=ons of mo=on arising from the ac=on S = d 26 X Ge 2D (R[G] H µνρ H µνρ +4 µ D µ D) H = db

14 Some Comments c=26 is also special for another reason: the whole set- up works in Lorentzian signature. e.g. L CFT = gg αβ α φ µ β φ ν G µν (φ) Include fermions in d=1+1 dimensional theory and you find that the cri=cal dimension is d=10. Now CFT gives you equa=ons of d=10 supergravity. Main Lesson: Geometry is replaced by CFT

15 Finding Gravitons Vacuum state of theory = Einstein equa=ons Excited states in d=1+1 = Par=cles in higher dimensions Correla=on func=ons in d=1+1 = S- Matrix in higher dimension = Note: In this way, can change the background space=me. It s not fixed! Summing over topologies gives quantum correc=ons to gravity

16 Open Problems It s a first quan=sed theory. What is the more general formula=on? Note: first quan=sed but not fixed par=cle number! Time dependent backgrounds poorly understood Finite perturba=on theory?

17 Thank you for your afen=on