Fermionic spectra in integrable systems

Kedem, Rinat

Fermionic spectra in integrable systems

Rinat Kedem

May 21, 2014

20 pages

e-Print:

1405.5585 [math-ph]

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Abstract: (arXiv)

This is a brief review of several algebraic constructions related to generalized fermionic spectra, of the type which appear in integrable quantum spin chains and integrable quantum field theories. We discuss the connection between fermionic formulas for the graded dimensions of the spaces of conformal blocks of WZW theories, quantum cluster algebras, discrete integrable noncommutative evolutions and difference equations.

Note:

20 pages, contribution to proceedings of ICM 2014

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[1]

Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas

Eddy Ardonne
(
- Santa Barbara, KITP and
- Caltech
)
,
Rinat Kedem
(
- Illinois U., Urbana, Math. Dept.
)

- J.Algebra 308 (2007) 270-294
•
e-Print:
- math/0602177
•
DOI:
- 10.1016/j.jalgebra.2006.08.024

[2]

Exactly solved models in statistical mechanics

R.J. Baxter
(
- Australian Natl. U., Canberra
)

DOI:
- 10.1142/9789814415255_0002

[3]

Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras

- Adv.Math. 195 (2005) 405-455

[4]

Theorie der metalle i. eigenwerte und eigenfunktionen der linearen atomkette. Zeitschrift fr Physik, (71):-6, 1931

H. Bethe

[5]

Vyjayanthi Chari. On the fermionic formula and the Kirillov-Reshetikhin conjecture. Internat. Math. Res. Notices, (12):629-654

[6]

Spectrum and completeness of the three state superintegrable chiral Potts model

Srinandan Dasmahapatra
(
- SUNY, Stony Brook
)
,
Rinat Kedem
(
- SUNY, Stony Brook
)
,
Barry M. McCoy
(
- SUNY, Stony Brook
)

- Nucl.Phys.B 396 (1993) 506-540
•
e-Print:
- hep-th/9204003
•
DOI:
- 10.1016/0550-3213(93)90662-9

[7]

Diagonalization of the XXZ Hamiltonian by vertex operators

Brian Davies
(
- Australian Natl. U., Canberra
)
,
Omar Foda
(
- Nijmegen U. and
- Melbourne U.
)
,
Michio Jimbo
(
- Kyoto U.
)
,
Tetsuji Miwa
(
- Kyoto U., RIMS
)
,
Atsushi Nakayashiki
(
- Kobe U.
)

- Commun.Math.Phys. 151 (1993) 89-153
•
e-Print:
- hep-th/9204064
•
DOI:
- 10.1007/BF02096750

[8]

Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property

- Lett.Math.Phys. 89 (2009) 183-216
•
e-Print:
- 0803.0362
•
DOI:
- 10.1007/s11005-009-0354-z

[9]

Philippe Di Francesco and Rinat Kedem. Q-system cluster algebras, paths and total positivity. SIGMA Symmetry Integrability Geom. Methods Appl., 6:Paper 014, 36

[10]

Philippe Di Francesco and Rinat Kedem. unpublished

[11]

On generalized Kostka polynomials and the quantum Verlinde rule. In Differential topology, infinite-dimensional Lie algebras, and applications, volume 194 of Amer. Math. Soc. Transl. Ser. 2, pages 61-79. Amer. Math. Soc., Providence, RI

B. Feigin
,
S. Loktev

[12]

Boris Feigin, Evgeny Feigin, Michio Jimbo, Tetsuji Miwa, and Evgeny Mukhin. Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian

- Lett.Math.Phys. 88 (2009) 39-77

[13]

Cluster X-varieties, amalgamation, and PoissonLie groups. In Algebraic geometry and number theory, volume 253 of Progr. Math., pages 27-68. Birkhuser Boston, Boston, MA

V.V. Fock
,
A.B. Goncharov

[14]

Sergey Fomin. Total positivity and cluster algebras. In Proceedings of the International Congress of Mathematicians. Volume II, pages 125-145. Hindustan Book Agency, New Delhi

[15]

Sergey Fomin and Andrei Zelevinsky. Cluster algebras (electronic)

I. Foundations

- J.Am.Math.Soc. 15 (2002) 497-529

[16]

Sergey Fomin and Andrei Zelevinsky. The Laurent phenomenon

- Adv.Appl.Math. 28 (2002) 119-144

[17]

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

G. Fourier
,
P. Littelmann

- Adv.Math. 211 (2007) 566-593

[18]

Philippe Di Francesco and Rinat Kedem. Proof of the combinatorial KirillovReshetikhin conjecture. Int. Math. Res. Not. IMRN, (7):Art. ID rnn006, 57

[19]

Philippe Di Francesco and Rinat Kedem. Quantum cluster algebras and fusion products. Int. Math. Res. Not. IMRN

•
DOI:
- 10.1093/imrn/rnt004)

[20]

Edward Frenkel and Nicolai Reshetikhin. The q-characters of representations of quantum affine algebras and deformations of w-algebras. In Recent developments in quantum affine algebras and related topics (Raleigh, NC,), volume 248 of Contemp. Math., pages 163-205. Amer. Math. Soc., Providence, RI, 1999

[21]

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Cluster algebras and Poisson geometry 1199,. {Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday}

- Moscow Math.J. 3 (2003) 899-934

[22]

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Cluster algebras and Poisson geometry, volume 167 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI

[23]

Remarks on fermionic formula. In Recent developments in quantum affine algebras and related topics (Raleigh, NC,), volume 248 of Contemp. Math., pages 243-291. Amer. Math. Soc., Providence, RI, 1999

G. Hatayama
,
A. Kuniba
,
M. Okado
,
T. Takagi
,
Y. Yamada

[24]

Michio Jimbo, Tetsuji Miwa, and Feodor Smirnov. Fermions acting on quasi-local operators in the XXZ model. In Symmetries, integrable systems and representations, volume 40 of Proc. Math. Stat., pages 243-261 Heidelberg

[25]

Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, volume 2 of Advanced Series in Mathematical Physics Publishing Co., Inc., Teaneck, NJ

V.G. Kac
,
A.K. Raina

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