1. CERNTR, 16 February 2012 Non-perturbative Physics from String Theory Dr. Can Kozçaz, CERN

2. Outline ✤ Motivations ✤ Bosonic String Theory ✤ Geometric Engineering ✤ AdS/CFT Duality & Drag Force

3. Motivations

4. Motivations to study String Theory ✤ The strongest candidate to unify all observed fundamental forces of nature ✤ Theory of Quantum Gravity ✤ Enhances the non-perturbative understanding of gauge theories ✤ Introduces new ideas in Physics: holography, mirror symmetry, new dualities ✤ Inspires results in Mathematics

5. Two Important Questions ✤ Is the String Theory describing the Nature? ✤ We do not know yet! And we may not know in the near future! ✤ Is String Theory a well-defined and consistent theory? ✤ Yes, it is! And it teaches us a lot of new ‘physics’! The following two questions are distinct questions and should be distinguished precisely

6. String Theory

7. Unification of Fundamental Forces Electric Magnetic Weak Strong Gravity Special Relativity Quantum Electrodynamics Electroweak Theory Quantum Chromodynamics General Relativity

8. Unification of Fundamental Forces The running of coupling constants in Standard Model and Minimal Supersymmetric Standard Model (without the gravity)

9. String Theory The basic idea of string theory is to promote the point particles to vibrating strings! A particle sweeps a worldline A string sweeps a worldsheet

10. String Theory What is the action for the relativistic point particle? with the proper time along the worldline The square-root in the action makes it very hard to quantize! Let us introduce a new field and a new action: This new action is equivalent to the previous action, and it is quadratic, i.e., easier to quantize!!! In Feyman path integral approach, we sum over all possible worldlines connecting the initial and final points.

11. String Theory What is the action for the relativistic string? Since the string is an extended object, one more variable is needed to parametrize it: For the closed string, will be periodic and we require (*) Picture from David Tong’s lecture notes on string theory (*)

12. String Theory Motivated by the action of the point particle, the obvious generalization is that the string action is proportional to the area of the worldsheet (*) Picture from David Tong’s lecture notes on string theory The area is intrinsic to the worldsheet, and is therefore reparametrization invariant.

13. String Theory This heuristic argument can be more formalized, the are can be computed using the induced metric on the worldsheet which is the pull-back of the flat Minkowski metric The action is given by where T is the tension: for, the instantaneous kinetic energy vanishes, the action is proportional to potential energy Nambu-Goto action

14. String Theory We again encountered a nasty square-root, can we find an equivalent action which is quadratic? YES! where g is the dynamical worldsheet metric. From the worldsheet point of view, this is the action of a number of scalar (bosonic) fields coupled to 2d gravity! Polyakov action The action has Poincare invariance, reparametrization invariance and Weyl invariance (invariance under local rescaling of the metric, keeping the angles the same)! Need to be careful about anomalies when we are writing the quantum theory

15. String Theory Having written the action, we can look at the classical equations of motion. For the scalar fields These equations look scary. Is this as far as we can go with Polyakov action? Fortunately, the answer is ‘No!’. We can exploit the symmetries: 1) Using the two reparametrization, we can bring the metric to the form 2) Using the Weyl invariance, we can rescale the metric as well

16. String Theory After the gauge fixing of the worldsheet metric the action reduces to The equations of motion for the scalar fields are free wave equations Let us forget about the gauging for a while, and look at the equation of motion for the metric Stress-energy tensor vanishes and this will impose constraints

17. String Theory The free wave equations can be easily solved be mode expansions, in lightcone coordinates: The wave equation in lightcone coordinates is The most general solution will be of the form with Center of mass coordinate Center of mass momentum Vibrational modes

18. String Theory The solutions are constructed ignoring the constraints due to the equation of motion of the metric. Imposing them give the mass formula Level matching At this point we can quantize the strings by promoting the modes into operators A generic state is given by

19. String Theory ALERT!!! We have negative norm states!!! These states can be avoided by carefully choosing the physical states. Let us define The classical constraints can be converted to quantum requirements to find the physical states However, we are not still done: for we have ordering ambiguity

20. String Theory After solving the ordering ambiguity, the mass formula becomes Skipping some technical details let us look at the ground state and the first excited state of the string spectrum: 1) The ground state is tachyonic! This would imply the instability of the vacuum we picked. 2) The first excited states form a massless representation of the Lorentz group, and include the graviton. On the other hand, the mass is given by

21. String Theory We have looked at the closed strings by imposing periodic boundary conditions. Are these the only possible strings? No, we can also allow open strings. Let us look again at the Polyakov action and find the equations of motion This boundary term vanishes by fixing the initial and final states Neumann or Dirichlet boundary conditions can be chosen to vanish this boundary term

22. String Theory All the steps for the closed strings can be repeated. We end up with the same string theory and realize in addition to closed strings there are open strings as well. Moreover, string theory has dynamical higher dimensional branes. These branes source certain bosonic form fields in string theory. A stack of coinciding N branes support supersymmetric U(N) gauge theories.

23. String Theory So far, we have not included fermions. The fermions are introduced by worldsheet supersymmetry

24. Geometric Engineering

25. Identify these ‘edges’, i.e., compactify on a circle Imagine the cylinder as a line with a circle ‘living’ over each point Consider 10D compactification in String Theory The compact space will be a more complicated space We should keep in mind that at each space time point we have a 6D toric Calabi-Yau 3fold

26. Topological String Theory worldsheet of genus gtarget space M

27. Topological String Theory If the worldsheet is flat then the Lagrangian is invariant under supersymmetry! However, we are interested in formulating the theory on a curved Riemannian surface. The action is not invariant under SUSY transformations: unless we have covariantly constant spinors. We need to topologically twist the theory

28. Seiberg-Witten Solution A priory the coefficients appearing in the instanton part are unknown, Seiberg and Witten determined them in a self-consistent way classical part 1-loop perturbative (exact) instanton part The low energy dynamic of supersymmetric gauge theories is governed by an holomorphic function defined on the moduli space of the theory, called the prepotential,. In terms of the prepotential the low energy effective action, in the Wilsonian sense, has the following form The prepotential has a classical piece, 1-loop piece and contributions from the instanton sector

29. Geometric Engineering The gauge group Encoded in the K3: the Cartan matrix of the gauge group the intersection matrix of the 2-cycles Asymptotical freedom Further compactify on a Riemann surface with genus 0 or 1 Supersymmetry Gauge bosons D2-branes wrapping 2-cycles in K3, e.g. For SU(2) two different ways of wrapping gives rise to W-bosons and integrating the RR 3-form over the (only) 2-cycle to get a 1-form for Z! Matter multiplets and quivers Modifications on the 2d space, e.g. enhance over a single point of the base space the singularity, introduce intersecting spheres Toric geometries The low energy dynamics is determined by the prepotential

30. Topological Vertex The prepotential of is given by the genus g=0 amplitude,, of topological string theory All genus answer of the A-model topological string theory is solved by topological vertex formalism for the toric geometries Example:

31. Topological Vertex ✤ Divide the toric diagram into trivalent vertices ✤ Compute the amplitude of each vertex ✤ Glue the amplitudes with appropriate propagators to obtain the full amplitude

32. Refined Topological Vertex The motivation is based on the microscopic derivation of prepotential due to Losev-Moore- Nekrasov-Shatashvili, i.e. performed the integrals over the instanton moduli space! LMNS deform the space-time from 4d to 6d which admits some Lie group action that can be lifted to the moduli space of instantons, hence, allows to equivariantly integrate the following integrals equivariant parameters Localization formulas can be used to perform the integrals (Duistermaat-Heckman formula). The prepotential is obtain in the limit of

33. Refined Topological Vertex The implication of this computation to the topological string theory is best understood in terms of the target space interpretation due to Gopakumar&Vafa: In the limit, with Degeneracies of BPS particles in M-theory compactifications to 5d due to M2 branes wrapping 2-cycle which is charged under When, the refined topological string theory partition functions reads Our refined topological vertex is constructed to compute this free energy

34. AdS/CFT Duality

35. ‘t Hooft argued that in the large rank limit of a gauge theory corresponds to a string theory. He introduced a large rank expansion and showed in a particular limit the Feynman diagrams can be grouped such that the dual diagrams are triangulations of the worldsheet:

36. AdS/CFT Duality Maldacena gave a very explicit example for ‘t Hooft’s proposal: In the large N and large coupling limit, the 4D maximal supersymmetric SU(N) Yang-Mills theory is dual to the motion of a string in 5D AdS background (the field theory lives on the boundary of AdS space)

37. Drag Force The quark-gluon plasma is governed by strongly coupled, thermal QCD and is in a non- equilibrium state. We modeled this system using AdS/CFT correspondence and determined the energy loss rate Quark Energy flow Momentum flow

38. Thank you!