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

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 Liquid 2 Gauge/Gravity Duality, MPI Munich Ingo Kirsch carbon nanotube (source: wiki) Coset approach to the Luttinger Liquid Vertically aligned Carbon Nanotubes by using a photolithography method (source: Dept. of Electronics U. York) Luttinger relation

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 SUSY in 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)? Luttinger Liquid and matrix cosets 3 Coset approach to the Luttinger Liquid Gauge/Gravity Duality, MPI Munich Ingo Kirsch

Outline: I.Motivation: Luttinger Liquid II.Matrix coset theories III.Partition function Z N (for higher N) IV.Spectrum of primary fields V.Chiral ring of chiral primaries Conclusions Overview ETH Zurich, 30 June 2010 Coset approach to the Luttinger Liquid 4 Gauge/Gravity Duality, MPI Munich Ingo Kirsch

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

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

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

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

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

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

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

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

A particular feature of superconformal field 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 Chiral Ring 13 Coset approach to the Luttinger Liquid Gauge/Gravity Duality, MPI Munich Ingo Kirsch

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. Chiral Ring 14 Coset approach to the Luttinger Liquid Gauge/Gravity Duality, MPI Munich Ingo Kirsch

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

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

I discussed diagonal coset theories 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 Z N =Z N (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 Z N in terms of characters N=4, 5: - we explicitly derived the q-expansion of Z N (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 + AdS dual description Conclusions Coset approach to the Luttinger Liquid 17 Gauge/Gravity Duality, MPI Munich Ingo Kirsch