Matrix Models and Superintegrability: Deformations and Integrals
Explore the concepts of superintegrability, W-operators, and matrix integrals in mathematical physics through deformations and generalizations. Learn about matrix models, Hermitian Gaussian model, W-representations, and more. Discover the significance of deformation of integration space, combining W-operators, and partition functions in this insightful study.
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SUPERINTEGRABILITY, W-OPERATORS AND MATRIX INTEGRALS THREE SIDES OF A COIN THROUGH THE DEFORMATIONS A.A. Oreshina ITEP, MIPT Problems of the Modern Mathematical Physics February, 2024
Plan Short review on matrix models Simple example: Hermitian Gaussian model Deformations and generalizations Central object of report: -WLZZ Helpful functions: Izikson-Zuber integral and Dunkle operator The main result: -WLZZ integral representation
Matrix models Matrix models special kind of matrix integrals ? = ?? ??? ?(?) Historically, they arose in the statistical physics and have found further application in string theory, combinatory, topological gravity. Three types of representation: Integral W-representation Polynomial representation (superintegrability)
Simple example: Hermitian Gaussian model ? = ?????[ 1 2?? ?2+ ??????] Integral representation ? 1 2 ? = ??? 2(?)? ?? ?? 2 ?? ? ? 1 2 ?+1 2(?)? ?? ?? 2 ?? 0 = ??? ?? ??? ? 1 2 ?) 2(?)? ?? ?? 2 ?? 0 = ??? ?( ?? ? ?= ? ?? ?? ?? ? ?? ?? ?? ??? ? 1 ?2 2= ??? = 0 ? ? ???+2) ??? 2(?)? ?? ?? 2 ?? ( ? ?? ???+?+ ?????? ? ???? 2? = ?0 ?2? = 0 ? = ??2 1 W-representation
Hermitian Gaussian model ?2SRpk = ??,? ?? (??) ? =?+ + 1 ?!(?2)?? pk ? = ??2 1 = ? ?(??) = ??????,2??(??) = ????(??) ? ? Superintegrability: ??? ????,1 ????,2 = ?? ??? ????,1 ??= = (? ? + ?) (?,?) ?
Deformations and generalizations 1) Deformation of integration space: ? 1 2 -deformation: ? = ??? 2?(?)? ?? ?? ( =1, , 2 for hermitian, orthogonal and simplectic models) q,t-deformation 2 ?? 2) Combining different W-operators (and respective models) to a class of models (WLZZ class) Object of our interest -WLZZ class
-WLZZ ??= ? ??/? 1, ?0= ??0 ???1 ??= ???/? ? ????, ? < 0 Partition function of positive, negative and zero branches: ? > 0 ?2= [?0,[?0,?1]] 1 ?![?2,? ?],? 2 Negative branch ? ? 1= ?2 ? +(1 ?) ? Zero branch ?0= ? (? + ?)???? + ? ? ??+? ? ? 1 ?? ???+? ?????? 2 ??? ? ?2= [?0,[?0, ??1]] Positive branch ??+1= 1 ?![?2,??],? 2
Auxillary objects Izikson-Zuber integral original definition angular matrix integral ?(?,?) = ?? exp[???+?] Harish-Chandra form: ? 1 ? ?,? =detexp[????] ?! (?) (?) ?=1 IZ integral as eigenfunction of Calogero-Mozer hamiltonian ? ??? Polynomial form: 2 1 (? ??? ? 2???,? ??????,? = ( + 2? ???))???,? = ?? ?? ?? ? ?<? ??(??,1) ??(?) ??? ??(?) ||??||2 1??? ??(?) ???,? = ? 1 = ?? ? ? Dunkle operator 1 ??,? ?? ?? ??= ??+ ? ? ?
-WLZZ integral representation Motivation: Ambiguity in the defining W-representation Integral representation lets study dividing into phases, that appears in the Kontsevich model and Brezin-Gross-Witten model Convenient for making further generalizations and deformations, that could bound the model with other knowing models. For this purpose in the integral representation one could add a potential or change the integrating space Virasoro constraints can not be restored from the W- representation, but there are some methods that produced it from the integral representation Some symmetries could be found only in the integral representation
-WLZZ integral representation ????,??, = ??? ??? 2 x 2 y exp ?? ?+ ?? ?? ?,? ? (?, ) ? ?? ? ?? Sketch of proof: Starting with =0. Notice, that for ? ? = ??? 2 x exp ?? ?+??+1 ? + 1 ?? ?+1? ?,? ? ?? ?+??)? ? = 0 for all j ??? 2 y exp ?? It is true that (?? ? ?? ?(?? ?+??)? ? = 0 ? ?? Leads to the Virasoro equation ?? (?)? = ?? ?+1 ???+1 (?)reproduces partition function of -WLZZ And the operator ?? ?+1
-WLZZ integral representation Use polynomial representation of the partition function ? = ??? ??? 2 x 2 y exp ?? ?+ ?? ?? ?,? ? ?? ? ?? ??? ????,1 = ??(??)??(??) ? Orthogonallity relation: ???? = ??,? |R| ????,??, = ??? ??? 2 x 2 y exp ?? ?+ ?? ?? ?,? ? ?, = ? ?? ? ?? ??? ????,1 ??? ????,1 ????,1 ??? = ??? ??? 2 x 2 y ? ?,? ??(??)??(?) ??(?)??(??) ?? ??? = ? ? ? ????,1 ??? ??? ????,1 ?|?| = ?,? ??????/????? ????,1 ??? ??? ????,1 ?|?|= ????,??, = ??????/????? ?,? = ??? ??? 2 x 2 y exp ?? ?+ ?? ?? ?,? ? (?, ) ? ?? ? ??
