Quantum Hitchin Systems via beta-deformed Matrix Models
aa r X i v : . [ h e p - t h ] M a y SISSA 18/2011/EP-FM
Quantum Hitchin Systems via β -deformed Matrix Models Giulio Bonelli, Kazunobu Maruyoshi and Alessandro TanziniInternational School of Advanced Studies (SISSA)via Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di TriesteAbstractWe study the quantization of Hitchin systems in terms of β -deformations of generalized matrixmodels related to conformal blocks of Liouville theory on punctured Riemann surfaces. We showthat in a suitable limit, corresponding to the Nekrasov-Shatashvili one, the loop equations ofthe matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphereand the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the ǫ -deformation of the chiral observables of the corresponding N = 2 four dimensional gaugetheory. Moreover, we find the exact wave-functions in terms of the matrix model representationof the conformal blocks with degenerate field insertions. Introduction N = 2 supersymmetric gauge theories in four dimensions display very interesting mathemat-ical structures in their supersymmetrically saturated sectors. These structures allow an exactcharacterization of several important physical aspects, such as their low energy behavior andstable spectra. These data are encoded in the celebrated Seiberg-Witten solution [1]. It wassoon realized that the Seiberg-Witten data can be recovered from integrable systems in termsof their spectral curves [2]. In this context the Hitchin integrable system has emerged asthe fundamental geometric structure underlying the M-theory description of N = 2 theories[3, 4, 5].On the other hand the Seiberg-Witten solution can also be recovered, at least in the case oflinear and elliptic quiver N = 2 theories, via equivariant localization on the instanton modulispace [6, 7]. Indeed, this approach contains further information encoded in the expansion in theequivariant parameters of the Ω-background. This opens the issue of relating the full Nekrasovpartition function to a suitable quantization of the Hitchin system.A crucial result in this context is provided by the AGT correspondence [8] relating theNekrasov partition function to conformal blocks of Liouville/Toda field theories in two dimen-sions. In [9, 10] it was proposed that this correspondence should be regarded as a two parameterquantization of the Hitchin system itself, or, in field theory language, as its second quantization.Here we will address these issues in the particular limiting case in which one of the two equiv-ariant parameters is vanishing. This was identified by Nekrasov and Shatashvili [11] to providethe first quantization of the integrable system ∗ . In the context of AGT correspondence theinstanton partition function in the Nekrasov-Shatashvili limit can be related to the insertion ofdegenerate fields in the Liouville theory [28, 29], which corresponds to the insertion of surfaceoperators in the gauge theory side [30, 31, 32, 33, 34, 35, 36].In our approach, we will make use of the matrix model perspective on AGT correspondencedeveloped in [37]. This was further elaborated for Liouville theory on the sphere in [38, 39,40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], on the torus in [54, 55] and in [56] atall genera. In this context the equivariant parameters of the Nekrasov partition function areencoded in the β -deformation of the standard Van-der-Monde measure [37]. Notice that for β -deformed matrix models the algebraic equation defining the spectral curve gets deformed intoa differential equation which can be interpreted as a Schr¨odinger equation [57, 58, 59]. Ourproposal identifies this differential equation, in the Nekrasov-Shatashvili limit, as providing thequantum Hamiltonians of the associated Hitchin integrable system. Moreover the associated ∗ See also [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] for the relation between the gaugetheory and the quantized integrable system. β -deformed generalized matrix model correspondingto degenerate field insertions in the Liouville theory [30, 31, 29].Let us notice also that the quantization of Hitchin systems plays a vital rˆole in the contextof Langlands duality [60].The organization of this paper is as follows. In section 2, we review the M-theory perspectiveon the Hitchin system and some basic facts on its quantization at low genera, namely the sphereand the torus with marked points. We derive the loop equation for the generalized β -deformedmatrix model both in the sphere and torus case in sections 3 and 4 respectively and we show thatin the Nekrasov-Shatashvili limit these reproduces the Hamiltonians of the quantized Hitchinsystem. Moreover, we provide a description of the associated wave-functions in terms of the β -deformed generalized matrix model describing degenerate field insertions in the Liouville theory[30, 31, 29]. In section 5 we present some final comments and further directions. in nuce In the M-theory framework, the Hitchin system arises by considering the geometry of a systemof N M5-branes wrapped on a manifold Y with topology C × R × { pt } in T ∗ C × R × R where C is a Riemann surface. This [37] should be equipped with a non trivial fibration of R over C which specifies the Ω-background of Nekrasov [6]. The geometry of the M5-branes bound stateis described by an N -fold branched covering of C given by the algebraic equation x N = N X j =2 φ j ( z ) x N − j (2.1)where x is a section of T ∗ C and φ j are ( j, C whose singularitystructure at the punctures identifies the matter content of the gauge theory. The quantummechanics of this latter structure passes by interpreting (2.1) as the spectral curve of theassociated classical system encoding the Seiberg-Witten solution (2.1), and then quantizing viaa suitable deformation.In what follows, we focus on the case with two M5-branes and the corresponding Hitchinsystems. In this case, the M-theory curve is specified by the quadratic differential: x = W ( z ).We further focus on the situation where the singularities of the quadratic differential are all of regular type meaning the poles of at most degree 2. There is one kind of regular punctures inthis two M5-branes case, so the Riemann surface C is just specified by genus and the number ofpunctures. Thus, we denote this by C g,n . Under the particular marking of C g,n , the worldvolumelow energy theory is the weakly coupled SU (2) n +3 g − superconformal quiver gauge theory [61, 4].The Hitchin system is associated with this gauge theory.2n order to have explicit parametrization of the Hamiltonians, let’s review and specifyHitchin systems and their quantization at low genera, that is sphere and torus with an arbitrarynumber of punctures, following the approach of [62].Let us start from some general features of the system. The Higgs field satisfies the condition¯ ∂ A Φ = X k λ k δ w k which can be simplified by ¯ A = h − ¯ ∂h and Φ = h − ˜Φ h to¯ ∂ ˜Φ = X k ν k δ w k , (2.2)where ν k = h − ( w k ) λ k h ( w k ). Eq.(2.2) admits a unique solution iff P k ν k = 0, given by˜Φ = X k ν k ω w k ,z ∗ + Φ (2.3)where ω P Q is the unique normalized abelian differential of third kind, i.e. holomorphic onΣ \ {
P, Q } with simple poles at P and Q with residues respectively +1 and − = φ I ω I is a Lie algebra valued holomorphic differential and ω I is a basisof normalized holomorphic differentials on Σ.As explained in [62], the Poisson brackets are induced by the Lie algebra of the complexifiedgauge group. At every puncture the residues are expanded as ν k = P a ν ak t a , where t a is a basisof the Lie algebra and the Poisson brackets are { ν ak , ν bl } P B = iδ kl f abc ν ck .The Hamiltonians are the Chern polynomials of the Higgs field, namely the coefficients ofthe expansion of the spectral curve (2.1) as det (Φ − x · ) = 0 . The quantization of the integrable system, as proposed in [62], is induced by the quantizationof the Poisson brackets above.The case of our interest is a projection to the Cartan degrees of freedom of the generalintegrable system specified to SL (2 , C ).In the sphere case, there are no holomorphic differentials and the Higgs field reads (see also[10]) ˜Φ = X k ν k z − w k dz πi where ν k is the only Cartan element. The corresponding relevant Hamiltonians are generatedby Tr Φ = X k J k ( z − w k ) + H (0) k z − w k ! (cid:18) dz πi (cid:19) (2.4)3here J k = Tr ν k H (0) k = 2 X l = k w k − w l Tr ν l ν k (2.5)According to the general discussion above, the quantization of these operators is provided byreplacing ν k at each puncture with the corresponding spin operators.Analogously, in the torus case, the Higgs field is˜Φ = X k ν k ϑ ′ ( z − w k ) ϑ ( z − w k ) + 2 πip ! dz πi (2.6)from which it follows thatTr Φ = X k (cid:18) P ( z − w k ) J k + ϑ ′ ( z − w k ) ϑ ( z − w k ) H (1) k + H (1)0 (cid:19) (cid:18) dz πi (cid:19) , (2.7)where † H (1) k = 2 X l = k Tr ν k ν l ϑ ′ ( w k − w l ) ϑ ( w k − w l ) + 4 πi Tr ν k p, H (1)0 = − π Tr p − η X k J k + 12 X k,l ; k = l Tr ν k ν l ϑ ′′ ( w k − w l ) ϑ ( w k − w l ) . (2.8)See Appendix A for the definition of the theta functions. In (2.7), P is the Weierstrass P -function. To obtain (2.7) we used the identity (A.8) relating the Weierstrass P -function andits primitive ζ ′ ( z ) = −P ( z ).In what follows, we will see the appearing of the above Hitchin Hamiltonians via the gen-eralized beta-deformed matrix model. The AGT relation associated with a sphere is the one between the Nekrasov partition functionof N = 2 superconformal SU (2) n − linear quiver gauge theory and the n -point conformalblock on the sphere. Both of them are specified by the marking of C ,n We can obtain thebeta-deformed matrix model starting from the Dotsenko-Fateev integral representation of theconformal block [63] as follows. (See [37, 47]). In terms of the free field φ ( z ) (whose OPE is † Notice that with respect to (4.10) in [62], in H (1)0 we find also a term proportional to η , which will revealto be crucial in comparing with the quantum Seiberg-Witten curve. This term was invisible to the authors of[62], being the absolute normalization of the differential which they admittedly do not check. ( z ) φ ( ω ) ∼ − log( z − ω )), the n -point conformal block is described by inserting the screeningoperators Z C ,n = *(cid:18)Z dλ I : e bφ ( λ I ) : (cid:19) N n − Y k =0 V m k ( w k ) + free on C , (3.1)where the vertex operator V m k ( w k ) is given by : e m k φ ( w k ) :. The momentum conservationcondition relates the external momenta and the number of integrals as P n − k =0 m k = bN − Q .By evaluating the OPEs, it is easy to obtain Z C ,n = C ( m k , w k ) e Z C ,n ≡ e F C ,n /g s , (3.2)where e Z C ,n is the beta-deformation of one matrix model e Z C ,n = Z N Y I =1 dλ I Y I 3) as follows: w = q , w = q q , . . . , w n − = q q . . . q n − . (3.5)While the dependence on m n − disappeared in the potential, this is recovered by the momentumconservation condition n − X k =0 m k + m n − = bg s N − g s Q. (3.6)We will refer to F m as free energy.The identification of the parameter b with the Nekrasov’s deformation parameters is givenby ǫ = bg s , ǫ = g s b . (3.7)Note that, in the case of b = i ( c = 1), this reduces to the usual hermitian matrix model andthis case corresponds to the self-dual background ǫ = − ǫ .Here we define the resolvent of the matrix model as R ( z , . . . , z k ) = ( bg s ) k X I z − λ I .... X I k z k − λ I k . (3.8)For k = 1, this reduces to the usual resolvent. 5 .1 Wave-function and conformal block In the following, we mainly concentrate on the limit where ǫ → ǫ and the otherparameters keeping fixed. In other words, the limit is b → ∞ and g s → bg s and N keeping finite. This is the limit by Nekrasov and Shatashvili [11]. In [10, 29], it was shownthat the the conformal blocks on a sphere with the additional insertion of the degenerate fields V b ( z ) = e − φ ( z ) b capture the quantization of the integrable systems.In this limit, the beta-deformed partition function can be written as R Q NI =1 dλ I exp( − ǫ f W )where f W = X I W ( λ I ) + 2 ǫ X I 0. In this limit, the terms with k > k = 1 terms since the connected part of theexpectation value can be ignored in this limit. Thus, we obtainlog Z C ,n + ℓ deg Z C ,n → ǫ X i Z z i x ( z ′ ) dz ′ , (3.18)7here we have used (3.12). Thus, by setting ℓ = n , we have obtained (3.11). This indicates thatthe properties of the conformal block with degenerate field insertions are build in the resolventof the matrix model in the ǫ → The argument in the previous section shows that the relation with the integrable system canbe seen by analyzing the resolvent, in particular, the loop equations. Thus, we derive it herewith finite β . First of all, we keep the potential arbitrary and obtain0 = 1 e Z C ,n Z N Y I =1 dλ I X K ∂∂λ K " z − λ K Y I 2. First of all, due to the equations of motion: h P I W ′ ( λ I ) i = 0, the sum of c k is constrained to vanish P n − k =0 c k = 0. In order to find another8onstraint, we consider the asymptotic at large z of the loop equation. The asymptotic of theresolvent is h R ( z ) i ∼ bg s Nz , so that the leading terms at large z in the loop equations satisfy − ( bg s N ) + ( ǫ + ǫ ) bg s N + bg s N n − X k =0 m k − n − X k =0 w k c k = 0 . (3.24)The leading term of order 1 /z in f ( z ) vanishes via the first constraint. Thus, we obtain n − X k =0 w k c k = n − X k =0 m k + m n − + ng s b + g s Q ! n − X k =0 m k − m n − + ng s b ! ≡ M , (3.25)where we have used the momentum conservation ‡ (3.14) with ℓ = n . Therefore, c and c canbe written in terms of c k (3.23). These constraints are related to the Virasoro constraints [64]. ǫ → limit As above, in the ǫ → h R ( z, z ) i → h R ( z ) i . Taking this into account, the loop equation (3.21) becomes0 = −h ˜ R ( z ) i − ǫ h ˜ R ( z ) ′ i + h ˜ R ( z ) i W ′ ( z ) − ˜ f ( z ) , (3.26)where ˜ R and ˜ f are R | ǫ → and f | ǫ → respectively. In the following, we will omit the tildes of R and f . Then, in terms of x = h R ( z ) i − W ′ ( z )2 , the equation becomes0 = − x − ǫ x ′ + U ( z ) , (3.27)where U ( z ) = W ′ ( z ) − ǫ W ′′ ( z ) − f ( z ) . (3.28)This equation is similar to the one obtained in [57]. It is easy to see that this can be writtenas the Schr¨odinger-type equation:0 = − ǫ ∂ ∂z Ψ( z ) + U ( z )Ψ( z ) , (3.29)where Ψ( z ) is defined in (3.11).The above argument is applicable for an arbitrary potential W ( z ). Here we return to thePenner-type one (3.4) and see the relation with the Gaudin Hamiltonian § . (3.28) becomes inthis case U ( z ) = n − X k =0 m k ( m k + ǫ )( z − w k ) + X k H k z − w k − n − X k =0 c k z − w k (3.30) ‡ We are using the modified momentum conservation to apply this to the argument in the previous section.However, the difference will disappear in the ǫ → § A relation with Gaudin system at finite ǫ has been noticed also in [57] H k = X ℓ ( = k ) m k m ℓ w k − w ℓ . (3.31) U ( z ) is the vacuum expectation value of (2.4). In particular, notice that the residue of thequadratic pole in (3.30) corresponds to the eigenvalue of J k quantized in ǫ units and that H k − c k are the vacuum energies of the quantum Hamiltonians H (0) k (2.5).Let us rewrite c k in terms of the gauge theory variables. Since the moduli of the sphereare related with the gauge coupling constants of the linear quiver gauge theory as in (3.5), thederivatives with respect to w k can be written as 2 πiw k ∂∂w k = ∂∂τ k − − ∂∂τ k , where the second termvanished when k = n − 2. Therefore, for k = 2 , . . . , n − 2, by using (3.23), we obtain c k = 12 πiw k ∂ ˜ F C ,n ∂τ k − − ∂ ˜ F C ,n ∂τ k ! = 12 πiw k ( u k − − u k ) . (3.32)where u k are closely related with the gauge theory variables h tr φ k i , φ k being the vector multipletscalar of the k -th gauge group. Indeed, supposing that the free energy F C ,n is identified withthe prepotential (with ǫ and ǫ ) of the gauge theory (this has indeed been checked for n = 4in [55, 44, 45, 46] in some orders in the moduli), we can use the ǫ -deformed version [65, 66, 9]of the Matone relation [67, 68, 69] to relate u k with h tr φ k i . Note that there is still differencebetween ˜ F C ,n and F C ,n , we will explicitly consider this in an example below. Instead, for c and c , we can use the two constraints derived above and obtain c = n − X k =2 ( w k − c k − M = 12 πi n − X k =2 w k − w k ( u k − − u k ) − M ,c = − n − X k =2 w k c k + M = − πi n − X k =2 ( u k − − u k ) + M . (3.33)We have obtained the differential equations for Ψ = Q i Ψ i ( z i ) each of which is (3.29) satisfiedby Ψ i ( z i ). These are similar to the differential equations satisfied by the ( n + n )-point Virasoroconformal block where n vertex operators are chosen to be degenerate V ( z ) = e − φ ( z ) b , as in[10]. As an example, we will explicitly see in the subsequent section the differential equationfor n = 4 is the same as that obtained from the Virasoro conformal block. We now consider the case corresponding to a sphere with four puncture where the matrix modelpotential is given by W ( z ) = X k =0 m k log( z − w k ) , w = 0 , w = 1 , w = q. (3.34)10his corresponds to SU (2) gauge theory with four flavors and the Gaudin model on a spherewith four punctures where the number of commuting Hamiltonian is just one. Thus we simplyconsider the wave-function Ψ( z ) = e ǫ R z xdz ′ and the differential equation satisfied by it.In this case, U ( z ) can be evaluated in terms of c as U ( z ) = X k =0 m k + ǫ m k ( z − w k ) + m − P i =0 m i + ǫ ( m − P k =0 m k ) z ( z − − q ( q − c + 2( qm m + ( q − m m ) z ( z − z − q ) , (3.35)Let us relate this with the one obtained from the Virasoro conformal block. We consider thelast line of U ( z ). Let us recall the definition of the free energy (3.2) and (3.3). Since thedifference of them is expressed by C = q − m m g s (1 − q ) − m m g s in this case, the free energies arerelated by F C , = e F C , − m m log(1 − q ) − m m log q. (3.36)Therefore, its derivative is q (1 − q ) ∂F C , ∂q = q (1 − q ) ∂ e F C , ∂q + 2 qm m − − q ) m m . (3.37)We notice that the right hand side is the numerator of the last line of U ( z ).Here let us redefine the mass parameters as˜ m = m + ǫ , ˜ m = m − ǫ . (3.38)In this notation, U can be written as U ( z ) = ˜ m − ǫ z + m ( m + ǫ )( z − + m ( m + ǫ )( z − q ) − − ˜ m + ˜ m + m ( m + ǫ ) + m ( m + ǫ ) z ( z − 1) + q (1 − q ) z ( z − z − q ) ∂F∂q . (3.39)Note here that the first four terms are exactly the potential which considered in [29] V ( z ).Also, the last term might correspond to the “eigenvalue” in [29]. The Schr¨odinger-like equationbecomes − ǫ ∂ ∂z Ψ( z ) + V ( z )Ψ( z ) = − q (1 − q ) z ( z − z − q ) ∂F∂q Ψ( z ) . (3.40)Note that this differential equation has also been derived in [20] from the free field expression,the first line of (3.13). 11 Matrix model: genus one In this section, we consider the matrix model corresponding to the conformal block on a toruswith punctures [37, 54, 55, 56]. We will derive the loop equations of the matrix model andrelate it with the differential equations of the corresponding Hitchin system.We consider the n -point conformal block on a torus whose integral description is Z C ,n = e F C ,n /g s = Z Y I dλ I Y I 1, where we have defined u k = ∂ e F C ,n ∂τ k and u ≡ u n . For k = n , we calculate ∂ e F C ,n ∂w n = − n − X k =1 ∂ e F C ,n ∂w k + 4 πp n X k =1 m k = − u n − + u n + 4 πp n X k =1 m k (4.31)where we have used the equations of motion and the momentum conservation. (We also ignoredthe term depending on ǫ .) Also, for the derivative with respect to q we obtain − ∂ e F C ,n ∂ ln q = − πi u n . (4.32)As in the sphere case, these u k are related with h tr φ k i . Note however that there could be adifference of them since u k here are the derivatives of e F , which will be seen below.17or completeness, let us rewrite the above equation in terms of the original partition function Z C ,n . The difference between Z C ,n and e Z C ,n is given by C (4.2), which gives rise to4 ∂ e F C ,n ∂ ln q = 4 ∂F C ,n ∂ ln q + 3 X k m k ( m k + ǫ ) η − X k<ℓ m k m ℓ ϑ ′′ ( w k − w ℓ ) ϑ ( w k − w ℓ ) , − ∂ e F C ,n ∂w k = − ∂F C ,n ∂w k − X ℓ ( = k ) m k m ℓ ϑ ′ ( w k − w ℓ ) ϑ ( w k − w ℓ ) − πpm k , ( k = 1 , . . . , n − − ∂ e F C ,n ∂w n = n − X k =1 ∂F C ,n ∂w k + X ℓ ( = n ) m n m ℓ ϑ ′ ( w ℓ ) ϑ ( w ℓ ) − πpm n , (4.33)where we have used (4.31). Thus, we obtain from (4.27)0 = − x − ǫ x ′ + X k m k ( m k + ǫ ) P ( z − w k ) + 2 X k m k ( m k + ǫ ) η (4.34) − n − X k =1 ϑ ′ ( z − w k ) ϑ ( z − w k ) ∂F C ,n ∂w k + ϑ ′ ( z ) ϑ ( z ) n − X k =1 ∂F C ,n ∂w k + X ℓ ( = n ) m n m ℓ ϑ ′ ( w ℓ ) ϑ ( w ℓ ) + 4 π p + 4 ∂F C ,n ∂ ln q . In the n = 1 case, we can see the relation with the elliptic Calogero-Moser model. We will take w = 0. The potential is W ( z ) = 2 m log ϑ ( z ) + 4 πpz. (4.35)In this case, it is easy to calculate the loop equation (4.34):0 = − x ( z ) − ǫ x ′ ( z ) + m ( m + ǫ ) P ( z ) − u ( ǫ ) , (4.36)where u ( ǫ ) = − π p − g s ∂ ln Z C , ∂ ln q − m ( m + ǫ ) η − π p + ∂∂ ln q (cid:0) F C , − m ( m + ǫ ) ln η (cid:1) , (4.37)where we have used that η = 4 ∂ ln η∂ ln q (See Appendix A) and defined the free energy as F C , =lim ǫ → F C , . Note that the free energy is the one evaluated in the ǫ → ǫ -deformed version of the relation between h tr φ i and the Coulomb modulus u [66] whichcoincides with the one found in [24].By introducing the “wave-function” Ψ = e ǫ R z dz ′ x ( z ′ ) , we finally obtain (cid:20) − ǫ ∂ ∂z + m ( m + ǫ ) P ( z ) (cid:21) Ψ = 4 u Ψ . (4.38)18he left hand side is the Calogero-Moser Hamiltonian and u in the right hand side can beconsidered as the eigenvalue of the Hamiltonian. Note that similar equations have been derivedfrom the Virasoro conformal block [29] and affine s ˆ l conformal block [28]. Indeed, by assumingthe equivalence of the partition function of the matrix model Z C , with the one-point conformalblock on a torus, (4.38) becomes the exactly same equation as the one obtained from theconformal block with the degenerate field. (See Section 3.1.2 in [29]. The identification of theparameter is a = iπp .)We also emphasize that the differential xdz in the wave-function satisfies the special ge-ometry relation (4.17). This is equivalent to the proposal in [12] stating that the ǫ -deformedprepotential can be obtained from the ǫ -deformed special geometry relation for the N = 2 ∗ theory, by using the same argument as in [29]. N limit and prepotential Before going to next, let us consider the loop equation in the large N limit which can be obtainby taking ǫ → x = m P ( z ) − u ( ǫ = 0) , (4.39)which is the Seiberg-Witten curve of the N = 2 ∗ gauge theory [3]. Indeed, the parameter u can be written as u ( ǫ = 0) = − π p + ∂∂ ln q (cid:16) F C , − m ln η (cid:17) . (4.40)where F C , is the leading contribution of the full free energy in the limit where ǫ , → 0. Thefirst term corresponds the classical contribution to the prepotential. The last term denotes theshift of the Coulomb moduli parameter from the value of the physical expectation value h Tr φ i [1, 76, 66].Indeed, we can be more precise. Under the identification iπp with the vev of the vectormultiplet scalar a , it is easy to show from (4.13) that a = iπp = 12 πi Z π xdz. (4.41)This and the fact that the form of x here is the same as the Seiberg-Witten differential ofthe N = 2 ∗ gauge theory where F C , is changed to the prepotential (see [77, 29]) lead to theconclusion that the free energy in the large N limit of this matrix model is exactly the same asthe prepotential of the gauge theory (under the identification above).19 Conclusions In this paper we proposed that the β -deformation of matrix models provides, in a suitable limit,the quantization of the associated integrable system. In particular we have shown that the loopequations for the β -deformed generalized matrix models [54, 56] reproduce in the Nekrasov-Shatashvili limit the Hamiltonians of the quantum Hitchin system associated to the sphere andtorus with marked points. Moreover, we have shown how to obtain the wave function fromdegenerate field insertions.It would be interesting to understand if this procedure could provide a general quantizationprescription of integrable systems which can be linked to specific matrix models. To this endit would be very useful to provide further evidence and examples. For instance, it would beinteresting to investigate the β -deformed Chern-Simons matrix model [49] in this direction.Furthermore, the extension of our approach to q-deformed conformal blocks, along the lines of[78], would be worth to be analyzed with the aim of connecting our results with topologicalstrings.On a more specific side, a natural extension of our analysis concerns Hitchin systems oncurves of higher genera, the point being a generalization of the identity (A.8). A furtherexplorable direction would be the extension to higher rank gauge groups with a multi-matrixmodel approach.The problem of understanding the proper quantization of the Seiberg-Witten geometry hasbeen explored recently also from a different view point consisting in a saddle point analysis ofthe instanton partition in the Nekrasov-Shatashvili limit [19, 24, 27]. In Section 4 we have seenthat for the N = 2 ∗ theory the eigenvalues of the Hamiltonian match. It would be interestingto further explore the relation between the two quantizations. Acknowledgements We thank A. Brini, B. Dubrovin, H. Itoyama, S. Pasquetti, F. Yagi, for interesting discussionsand comments.G.B. and K.M. are partially supported by the INFN project TV12. A.T. is partially sup-ported by PRIN “Geometria delle variet´a algebriche e loro spazi di moduli” and the INFNproject PI14 “Nonperturbative dynamics of gauge theories”.20 ppendixA Elliptic functions The elliptic theta function is defined by ϑ ( z | τ ) = 2 q / sin z ∞ Y n =1 (1 − q n )(1 − q n cos 2 z + q n ) (A.1)which has pseudo periodicity ϑ ( z + π | τ ) = − ϑ ( z | τ ) , ϑ ( z + πτ | τ ) = e − i (2 z + πτ ) ϑ ( z | τ ) . (A.2)This function satisfies ϑ ′′ ( z | τ ) = − ∂∂ ln q ϑ ( z | τ ) where ϑ ′ ( z | τ ) = ∂∂z ϑ ( z | τ ).The Weierstrass elliptic function P is double periodic with periods π and πτ and is expressedas P ( z ) = − ζ ′ ( z ) , ζ ( z ) = ϑ ′ ( z | τ ) ϑ ( z | τ ) + 2 η z, η = − ϑ ′′′ ( z | τ ) | z =0 ϑ ′ ( z | τ ) | z =0 , (A.3)where ϑ ( z | τ ) is elliptic theta function. The Weierstrass function satisfies P ( z ) ′ = 4 P ( z ) − g P ( z ) − g , (A.4)where g = 43 ∞ X n =1 n q n − q n ! ,g = 827 − ∞ X n =1 n q n − q n ! , (A.5)We also define g whose expansion is g = − − ∞ X n =1 nq n − q n ! , (A.6)which is related with η as g = − η : η = 16 − ∞ X n =1 nq n − q n ! = 4 ∂∂ ln q ln η, (A.7)where η = q / Q ∞ n =1 (1 − q n ) is the Dedekind eta function.The Weierstrass and zeta functions satisfies the identity P ( a + b ) + P ( a ) + P ( b ) = ( ζ ( a + b ) − ζ ( a ) − ζ ( b )) . 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