Gauge/Gravity Duality 2013 Max Planck Institute for Physics, 29 July to 2 Aug 2013

1. ` M. Isachenkov, I.K., V. Schomerus, arXiv: 1308.XXXX Coset approach to the Luttinger Liquid Ingo Kirsch DESY Hamburg, Germany Gauge/Gravity Duality2013 Max Planck Institute for Physics, 29 July to 2 Aug 2013

2. Coset approach to the Luttinger Liquid LuttingerLiquid • Electron transport in conductors is usually well-described by Fermi-liquid theory (d>1) • BUT d=1: Electrons in a one-dimensional system form a quantum liquid which can be described as a Luttinger liquid rather than by Landau's Fermi-liquid theory • Fermi surface but no weakly-coupled quasi-particles above FS • Experimentally realized e.g. in quantum wires/carbon nanotubes Luttinger relation carbon nanotube (source: wiki) Vertically aligned Carbon Nanotubes by using a photolithography method (source: Dept. of Electronics U. York) Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

3. Coset approach to the Luttinger Liquid LuttingerLiquid and matrix cosets • Gopakumar-Hashimoto-Klebanov-Sachdev-Schoutens (2012): • UV: 2d SU(N) gauge theory coupled to Dirac fermions • IR: effective low-energy theory flows to 2d coset CFT: • emergent SUSYin the IR - not present in the UV theory! • coset studied only for N=2, 3: equivalence to minimal models: • What can we do (no relation to minimal models)? Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

4. Cosetapproach to the Luttinger Liquid Overview • Outline: • Motivation: Luttinger Liquid • Matrix coset theories • Partition function ZN (for higher N) • Spectrum of primary fields • Chiral ring of chiral primaries • Conclusions ETH Zurich, 30 June 2010 Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

5. Coset approach to the Luttinger Liquid Example: The partition function Z3 (N=3) Numerator partition function: Denominatorpartition function: D-type modular invariant of the type Coset partition function: Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

6. Coset approach to the Luttinger Liquid Example: The partition function Z3 (N=3) It is possible to write Two successive decompositions: i) decomposition of SO(16)1characters into characters of SO(8)1 x SO(8)1: ii) decomposition of SO(8)1characters into characters of SU(3)3 branching the characters of SO(16)1 into those of SU(3)6: Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

7. Coset approach to the Luttinger Liquid Example: The partition function Z3 (N=3) Result for partition function Z3 : where and Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

8. Coset approach to the Luttinger Liquid Partition function ZN (general N) We constructed an expression for the coset partition function ZN : are branching functions: Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

9. Coset approach to the Luttinger Liquid Bouwknegt-McCarthy-Pilch formula The branching functions are computed using a formula for diagonal cosets. This gives the q-expansion of . Spectrum: Bouwknegt-McCarthy-Pilch(1991) Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

10. Coset approach to the Luttinger Liquid N=2,3: Partition function ZN and supersymmetry Goal: Compute the cosetpartition function ZN in terms of the branching functions and then rewrite it in terms of characters. Example: N=2 cf. w/ characters Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

11. Coset approach to the Luttinger Liquid N=4, 5: The partition function ZN(q) For N>3 it is difficult to rewrite ZN in terms of characters. BUT: Still possible to write ZN =ZN (q) using BMP formula Example: N=3 similarly Likewise, we found the q-expansions of Z4 =Z4 (q), Z5 =Z5 (q) by computing 700 (N=4) and 10292 (N=5) branching functions… Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

12. Coset approach to the Luttinger Liquid Spectrum: cosetelements and their conformal weights Parallel computing on DESY’s theory and HPC clusters we also have N=5 ... N=2 N=3 N=4 Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

13. Coset approach to the Luttinger Liquid Chiral Ring A particular feature of superconformalfield theories is the chiral ring of NS sector chiral primary fields. These fields form a closed algebra under fusion. Let’s identify the chiral primary fields (h=Q) by introducing charge into the branching functions (i.e. make them z-dependent) Example: N=3 Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

14. Coset approach to the Luttinger Liquid Chiral Ring Do the chiral primaries form a closed algebra under fusion? - Yes. For instance, for N=3: Generator of the chiral ring (h=Q=1/6): Claim: Repeatedly act with x on the identity. This generates the chiral ring of NS chiral primary fields. Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

15. Coset approach to the Luttinger Liquid Chiral Ring Visualization of the chiral ring by tree diagrams: An arrow represents the action of x on a field, e.g. OPE (N=3) Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

16. Coset approach to the Luttinger Liquid Chiral Ring N=4, 5: In the large N limit, the number of chiral primaries is governed by the partition function p(6h). Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch

17. Coset approach to the Luttinger Liquid Conclusions • I discussed diagonal cosettheories of the type • Gopakumar et al. studied this space for N=2, 3 (by relating it to minimal models) • Our method works in principle for general N: • general N: we derived the partition function ZN=ZN (b(q)), • the branching functions b(q) can be computed using the BPM formula • (needs a lot of computer power for higher N though) • N=2, 3: we rewrote ZN in terms of characters • N=4, 5: - we explicitly derivedthe q-expansion of ZN (up to some order) • - we identified the chiral primary fields and • - showed that they form a chiral ring under fusion • Outlook (work in progress): • Large N limit + AdSdual description Gauge/Gravity Duality, MPI Munich 2013 -- Ingo Kirsch