Progress in the synthesis of integrabilities arisingfrom gauge-string duality 2014 @ Lake Biwa

ODE/IM correpondence and modified
affine Toda field equations: IIChristopher Locke
Tokyo Institute of Technology

Based on: [K.Ito-CL arXiv:1312.6759]

Introduction●●●

The ODE/IM correspondence is a relation between spectral
analysis of ODEs, and the ˝functional relations˝ approach to2d quantum integrable models (IM)Dr. Ito just gave an excellent introduction on this topic, so wewill focus on only a specific part of this correspondence forthe modified affine Toda field equation
We will
1. First, look at asymptotic behavior of solutions
2. Then introduce the so called ψ-system associated with the algebra
3. Use ψ-system to derive Q-functions and Bethe ansatz equations

Affine Toda equation●●

We will use complex coordinates
Motivated by recent papers [ : Lukyanov-Zamolodchikov 2010 and: Dorey-Faldella-Negro-Tateo 2012], we definewith h the Coxeter number, and M a positive real parameter●

Using this, the modified affine Toda field equation isAsymptotic behavior of●●In the large |z| limit, we can simply assume that theasymptotic solution to the modified affine Toda equation is
For small z near 0, we assume logarithmic divergence, withexpansion

Here, g is a vector that controls the small-z behavior●Substituting this expansion into the modified affine Toda
equation, one can show that

Note about representations– In our paper [1312.6759], we focused on specific a specificmatrix representation, the fundamental representation;– however, in this presentation we will present the same basic

results using a general highest weight representation, andarrive at the Bethe ansatz equations much fasterLinear equation formulation
●

●

●

●

The modified affine Toda equation can be written as a linear
problem as well for an appropriate one-form connection

A can exist in any highest weight representation
Denote fundamental modules with highest weightwith set of weight vectors

asIn this notation, Ψ is a wavefunction in this representation
space,

Linear equation formulation●

The small z limit of the linear problem is●This has a basis of non-trivial solutions given by●This basis will be used to expand a preferred solution thatcomes from looking at the large z behavior

Linear equation formulation●

In the large z limit, it is convenient to gauge transform theconnection into the form
where

●Denoting the unique eigenvalue of Λ+ in representation V(a)with largest real part as λ(a), we have the unique solution
which decays fastest along the positive real axis [Sun-2012]

Linear equation formulation●●●

The previous asymptotically decaying unique solution forlarge z can now be expanded in terms of our basis ofsolutions defined by their behavior around small z:

These Q-coefficients do not depend on the coordinates z
As per previous work with sinh-Gordon and Bullough-Dodd
cases, we expect that in the ODE/IM correspondence, thatthese will coincide with the vacuum expectation values of theQ-functions of a 2D massive QFT

Conformal limit
●●

●

●

To study the massless limit, we can define the following
The conformal limit is defined as the limit where we take tozero, then take to 0 and to infinity such that x and E
remain finite values
This converts the affine Toda field equation to the massless
Toda field equation
To handle the massive case, we need the full connection, butwhen looking at the massless case just the holomorphic
connection is enoughSymanzik rotation●●●

Before proceeding, we need to define a Symanzik rotation
Our linear problem is symmetric under the transformation forintegers k

We denote a Symanzik rotated function with a subscript real
number, asψ-system●

To obtain the Bethe ansatz equations (BAE) for this, we needto determine the so-called ψ-system, which are relations
between the different ψ(a)– Focus on just the case of A-type Lie algebra●●It can be shown that the following highest weights areequivalent by showing they have the same eigenvalues withrespect to H, and are annihilated by each raising operator
This can be written down as
where the Symanzik rotated ψ on the left ensures that thewedge product is not zeroψ-system●●●

By applying
to both sides, one can see that eachside is a solution to the linear problem
Furthermore, each side has the same large z behavior if k=½,
so the A-type ψ-system isNow, substituting in the expansion
on both sides, and extracting just the coefficient of the most
singular term near x = 0, we get the relationψ-system●●●

Denote the zeros of

asThen evaluating the previous equation at this point after
Symanzik rotating the whole equation by ±½, then dividing
the two resulting equations gives

This is the BAE for A-type Lie algebras, and we can arrive at
it without the need for complex Plücker relations ofdeterminants

Matrix representation and ODE●

Quick aside on (pseudo-)ODEs:– these highest weight representations can be written out as a matrix– by looking at each component equation of the linear problem, one

can arrive at a (pseudo-)differential equation that characterizes thebehavior of solutions– for instance in A-type case we have in the vector representation,– in previous papers, these (pseudo-)differential equations were theobjects of study, but our work shows that one can focus entirely
on just the linear problem and get the same information(asymptotic behaviors, Q-coefficients, T-Q relations, BAE, etc.)

Other algebras●●The above analysis was carried out for all but the E- andF-type affine Lie algebras
In the case of non-simply laced algebras, we foundsomething different than previous papers:– Our results were in agreement only if we took the Langlands dual●

Our calculated (pseudo-)ODEs had the following relation

Other algebras●●

Why is it that the Langlands dual for our modified affine
Toda equation correspond to the equations of Dorey et al.?
Would like to check if there are differences for the dual pairs
in terms of, for instance, BAE, or T-Q relations

Conclusion
1) Looked at asymptotic behavior of modified affine Toda
equation, and linear problem solutions
2) Found unique asympotically decaying solution to linear
problem that leads naturally to ψ-system
3) Showed how ψ-system can give BAE

Future directions
1) Calculate Bethe ansatz equations for non-simply laced cases
to see if there is any difference
2) Work on E- and F-type cases as well3) Investigate effects of using
in the Dorey et al. paper
4) Look into the massive ODE/IM correspondence for theseToda equations5) Supersymmetric Toda equations?

as