Integrable Field Theories
from4d Chern-Simons Theory
Masahito Yamazaki
Strings 2018

Based on collaboration
with Kevin Costello and Edward WittenBased on collaboration
with Kevin Costello and Edward Witten

Part I arXiv:1709.09993
Part II arXiv:1802.01579

Based on collaboration
with Kevin Costello and Edward Witten

Part I arXiv:1709.09993
Part II arXiv:1802.01579
Part III to appear
Part IV to appear

Based on collaboration
with Kevin Costello and Edward Witten

Part I arXiv:1709.09993
Part II arXiv:1802.01579
Part III to appear
Part IV to appear

Integrable Lattice Models(Part I and II)integrability: characterized by Yang-Baxter equationintegrability: characterized by Yang-Baxter equationintegrability: characterized by Yang-Baxter equation
with spectral parametersintegrability as topological invariance?

integrability as topological invariance?

4d = 2d (topological) + 2d (holomorphic)

spectral curve

topologicalholomorphic

4d = 2d (topological) + 2d (holomorphic)

spectral curve

topologicalholomorphic

“4d Chern-Simons” by [Costello] (‘13)

“4d Chern-Simons” by [Costello] (‘13)

“4d Chern-Simons” by [Costello] (‘13)“T-dual” to ordinary 3d Chern-Simons[Vafa-Y] (to appear)“4d Chern-Simons” by [Costello] (‘13)

Perturbative expansion
around isolated classical solution

statistical lattice from Wilson linesstatistical lattice from Wilson lines

Integrable Field Theories(Part III and IV)thermodynamic limit

lattice model fromWilson linesthermodynamic limit

lattice model fromWilson linesthermodynamic limit

2d field theory from surface defects
coupled 4d-2d system

two defects: vertical and horizontal

two defects: vertical and horizontal

two defects: chiral and anti-chiraltwo defects: chiral and anti-chiralWhy Integrable?
(Part III)Lax operator

Lax operator

Flat connection

Lax operator

Flat connection

infinitely-many conserved charges

Lax operator = 4d Wilson line!

Effective 2d Theory(Part III)

4d-2d systemeffective 2d system

No 4d zero modes: we have perturbative
expansion around an isolated solution ofequation of motion

All zero modes comes from 2d surface defects

The interaction comes from exchange of 4dgauge bosons

The interaction comes from exchange of 4dgauge bosons

The interaction comes from exchange of 4dgauge bosons

only this diagram on the left contributes attree-level

For example, no such diagram:

Let’s now compute this diagram

The computation is the same as in thecomputation of leading-order term ofR-matrix in Part I

The computation is the same as in thecomputation of leading-order term ofR-matrix in Part I

We thus have the classical r-matrix

We obtained the effective 2d theory:

Similarly, we can compute Lax matrix for theeffective 2d theory:

For the rational case

we haveand we reproduce the standard formula

Examples and Generalizations

Simple example: chiral/anti-chiral free fermionsReproduce Gross-Neveu and Thirring models

The framework generalizes in several directions:
1. trigonometric/elliptic cases2. more general defects
e.g. curved beta-gamma systemfrom which we obtain sigma models
Also non-chiral defects, e.g. free boson

3. multiple defects

Quantum Integrability(Part IV)

Let’s assume for now that anomalies
cancel for the coupled 4d-2d system

Recall: Lax operator = 4d Wilson lineRTT relation: definition of the Yangian(and their trigonometric/elliptic counterparts),
and ensures quantum integrability

This can be thought of the “continuum limit” of
the RTT relation for discrete lattice models,
discussed in Part II

Our 4d framework says more, about e.g.
l Local conserved charges
l Renormalization group flowl S-matrix factorizationl Higher genus spectral curves

SummaryThank you
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