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# Duality of Gaudin Models

We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\lc z_{k})$ and $\bar{\alpha}=(\alpha_{1}\lc\alpha_{n})$, respectively.

We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide.

To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.

One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians.

We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- Indianapolis

## Advisor/Supervisor/Committee Chair

Vitaly Tarasov## Additional Committee Member 2

Evgeny Mukhin## Additional Committee Member 3

Alexander Its## Additional Committee Member 4

Daniel Ramras## Usage metrics

## Categories

- Algebraic structures in mathematical physics
- Mathematical aspects of quantum and conformal field theory, quantum gravity and string theory
- Mathematical aspects of classical mechanics, quantum mechanics and quantum information theory
- Mathematical physics not elsewhere classified
- Pure mathematics not elsewhere classified

## Keywords

Bethe algebraGaudin modelBethe ansatzAlgebraic Structures in Mathematical PhysicsMathematical Aspects of Quantum and Conformal Field Theory, Quantum Gravity and String TheoryMathematical Aspects of Classical Mechanics, Quantum Mechanics and Quantum Information TheoryMathematical Physics not elsewhere classifiedPure Mathematics not elsewhere classified

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