Counting minimal surfaces in negatively curved 3-manifolds

Definition 2.1 .

[ 12 ] An algebra $J$ over a field $\rm{F}$ is a Jordan algebra satisfying for any $x,y\in J$ ,

(1)

$x\circ y=y\circ x$ ;
(2)

$(x^{2}\circ y)\circ x=x^{2}\circ(y\circ x)$ .

Definition 2.5 .

Suppose that $\mathcal{A}$ is a ternary Jordan algebra over $\rm{F}$ and $f:\mathcal{A}\times\mathcal{A}\rightarrow\rm{F}$ is a bilinear form on $\mathcal{A}$ . If for any $x,y,z,w\in\mathcal{A}$ , $f$ satisifies

f(\rm{[\![\it{x},\it{y},\it{z}]\!]},\it{w})=f(\it{x},\rm{[\![\it{w},\it{y},\it% {z}]\!]}),

then $f$ is called invariant.

Definition 2.9 .

Let $\mathcal{A}$ be a ternary Jordan algebra. The centroid of $\mathcal{A}$ is a vector space spanned by all elements $\it{f}\in\rm{End}(\mathcal{A})$ which satisfies for all $x,y,z\in\mathcal{A}$

\rm{[\![\it{f}(x),\it{y},\it{z}]\!]}=\rm{[\![\it{x},\it{f}(y),\it{z}]\!]}=\rm{% [\![\it{x},\it{y},\it{f}(z)]\!]}=\it{f}(\rm{[\![\it{x},\it{y},\it{z}]\!]}),

denoted by $\Gamma(\mathcal{A})$ .

Definition 2.10 .

Let $\mathcal{A}$ be a ternary Jordan algebra. The quasicentroid of $\mathcal{A}$ is a vector space spanned by all elements $\it{f}\in\rm{End}(\mathcal{A})$ which satisfies for all $x,y,z\in\mathcal{A}$

\rm{[\![\it{f}(x),\it{y},\it{z}]\!]}=\rm{[\![\it{x},\it{f}(y),\it{z}]\!]}=\rm{% [\![\it{x},\it{y},\it{f}(z)]\!]},

denoted by $Q\Gamma(\mathcal{A})$ .

Definition 3 .

[ 1 ] Let $(M,g)$ be a $m$ -dimensional generalized Riemannian manifold, $\nabla$ is an affine connection on $M$ , if

\nabla g=0

then $\nabla$ is admissible connection of generalized Riemannian manifold $(M,g)$ .

Definition 5.6 .

Let $(\xi=(E,p,N),s)$ be a definable $C^{r}$ bilinear spaces over a definable $C^{r}$ manifold $N$ . Consider a definable $C^{r}$ map $f:M\rightarrow N$ between definable $C^{r}$ manifolds. The definable $C^{r}$ bilinear space $f^{*}(\xi,s)$ induced by $f$ is a definable $C^{r}$ bilinear space whose vector bundle is $f^{*}\xi$ and whose definable $C^{r}$ bilinear form is $f^{*}s$ defined by

f^{*}s((x,v),(x,w))=s(v,w)

for $x\in M$ and $v,w\in E$ with $p(v)=p(w)=f(x)$ .

Definition 3.4.2 .

The boundary map $\partial:\mathbb{M}_{k}(\Gamma)\rightarrow\mathbb{B}_{k}(\Gamma)$ is the natural map extending linearly the map

\partial(v\otimes\{\alpha,\beta\})=v\otimes\alpha-v\otimes\beta

The space $\mathbb{S}_{k}(\Gamma)=\ker\partial$ is called the space of cuspidal modular symbols . As in subsection 3.3 , we can define $\mathbb{B}_{k}(\Gamma,\varepsilon)$ , a boundary map $\partial:\mathbb{M}_{k}(\Gamma,\varepsilon)\rightarrow\mathbb{B}_{k}(\Gamma,\varepsilon)$ and its kernel will be denoted by $\mathbb{S}_{k}(\Gamma,\varepsilon)$ .

In order to have a concrete realization of $\mathbb{S}_{k}(\Gamma,\varepsilon)$ , it remains to efficiently compute the boundary map.

Definition 2.5 .

Let $X$ be a finite dimensional free module over a commutative ring with identity and a non-degenerate anti-symmetric bilinear form $\langle,\rangle$ . Then ( $X$ , $\langle,\rangle$ ) is a quandle with the quandle operation

x\rhd y=x+\langle x,y\rangle y\quad\text{and}\quad x\rhd^{-1}y=x-\langle x,y% \rangle y.

This type of quandle is called a $symplectic$ $quandle$ . See [ 29 ] for more details.

Definition 3.2 .

Let $X$ be a complex manifold and let $f\colon X\to X$ be a holomorphic self-map. A semi-model for $f$ is a triple $(\Lambda,h,\varphi)$ where $\Lambda$ is a complex manifold called the base space , $h\colon X\to\Lambda$ is a holomorphic mapping, and $\varphi\colon\Lambda\to\Lambda$ is an automorphism such that

h\circ f=\varphi\circ h,

(3.2)

and

\bigcup_{n\geq 0}\varphi^{-n}(h(X))=\Lambda.

(3.3)

Let $(Z,\ell,\tau)$ and $(\Lambda,h,\varphi)$ be two semi-models for the map $f$ . A morphism of semi-models $\hat{\eta}\colon(Z,\ell,\tau)\to(\Lambda,h,\varphi)$ is given by a holomorphic map $\eta:Z\to\Lambda$ such that the following diagram commutes: \SelectTips xy12

\xymatrix{X\ar[rrr]^{h}\ar[rrd]^{\ell}\ar[dd]^{f}&&&\Lambda\ar[dd]^{\varphi}\\ &&Z\ar[ru]^{\eta}\ar[dd]^{(}.25)\tau\\ X\ar^{\prime}[rr]^{h}[rrr]\ar[rrd]^{\ell}&&&\Lambda\\ &&Z\ar[ru]^{\eta}}

If the mapping $\eta\colon Z\to\Lambda$ is a biholomorphism, then we say that $\hat{\eta}\colon(Z,\ell,\tau)\to(\Lambda,h,\varphi)$ is an isomorphism of semi-models . Notice that then $\eta^{-1}\colon\Lambda\to Z$ induces a morphism ${\hat{\eta}}^{-1}\colon(\Lambda,h,\varphi)\to(Z,\ell,\tau).$

Definition 5.2 .

Let $X$ be a complex manifold and let $f\colon X\to X$ be a holomorphic self-map. A pre-model for $f$ is a triple $(\Lambda,h,\varphi)$ where $\Lambda$ is a complex manifold called the base space , $h\colon\Lambda\to X$ is a holomorphic mapping, and $\varphi\colon\Lambda\to\Lambda$ is an automorphism such that

f\circ h=h\circ\varphi.

(5.1)

Let $(\Lambda,h,\varphi)$ and $(Z,\ell,\tau)$ be two pre-models for $f$ . A morphism of pre-models $\hat{\eta}\colon(\Lambda,h,\varphi)\to(Z,\ell,\tau)$ is given by a holomorphic mapping $\eta\colon\Lambda\to Z$ such that the following diagram commutes: \SelectTips xy12

\xymatrix{\Lambda\ar[rrr]^{h}\ar[rd]^{\eta}\ar[dd]^{\varphi}&&&X\ar[dd]^{f}\\ &Z\ar[rru]^{\ell}\ar[dd]^{(}.25)\tau\\ \Lambda\ar^{\prime}[r][rrr]^{(}.25)h\ar[rd]^{\eta}&&&X\\ &Z\ar[rru]^{\ell}}

If the mapping $\eta\colon\Lambda\to Z$ is a biholomorphism, then we say that $\hat{\eta}\colon(\Lambda,h,\varphi)\to(Z,\ell,\tau)$ is an isomorphism of pre-models . Notice then that $\eta^{-1}\colon Z\to\Lambda$ induces a morphism ${\hat{\eta}}^{-1}\colon(Z,\ell,\tau)\to(\Lambda,h,\varphi).$

Definition 2.1 .

An $\mathrm{SU}(3)$ structure $(\omega,\psi=\psi^{+}+i\psi^{-})$ on a $6$ -manifold $M$ is a pair of forms $\omega\in\Omega^{2}(M)$ and $\psi\in\Omega^{3}(M,\mathbb{C})$ satisfying

2\omega^{3}=3\psi^{+}\wedge\psi^{-},\qquad\omega\wedge\psi=0.

Definition 3.5 (Properly supported operators) .

A continuous linear operator $A:C^{\infty}_{0}(\Omega)\rightarrow C^{\infty}(\Omega)$ is said to be properly supported if, for any compact subset $K\subset\Omega$ , there exists a compact subset $K^{\prime}\subset\Omega$ with:

\operatorname{supp}u\subset K\Longrightarrow\operatorname{supp}Au\subset K^{% \prime}\text{ and }u=0\text{ on }K^{\prime}\Longrightarrow Au=0\text{ on }K

Definition 5.1 .

Let $(H,\beta)$ be a monoidal Hom-Hopf algebra. If there exists a convolution invertible bilinear form $\zeta:H\otimes H\longrightarrow\Bbbk$ , such that for any $h,g,k\in H$ ,

	$\displaystyle\zeta(\beta(h),\beta(g))=\zeta(h,g);$		(5.1)
	$\displaystyle\zeta(h_{1},g_{1})g_{2}h_{2}=h_{1}g_{1}\zeta(h_{2},g_{2});$		(5.2)
	$\displaystyle\zeta(\beta^{-1}(h),gk)=\zeta(h_{1},\beta(g))\zeta(h_{2},\beta(k));$		(5.3)
	$\displaystyle\zeta(hg,\beta^{-1}(k))=\zeta(\beta(h),k_{2})\zeta(\beta(g),k_{1}),$		(5.4)

then $\zeta$ is called a coquasitriangular form of $H$ , and $(H,\beta,\zeta)$ is called a coquasitriangular monoidal Hom-Hopf algebra.

Definition 5.4 .

Let $(H,\beta)$ be a monoidal Hom-Hopf algebra. If the linear map $\sigma:H\otimes H\longrightarrow\Bbbk$ such that the following conditions hold:

	$\displaystyle\sigma(\beta(h),\beta(g))=\sigma(h,g),$		(5.5)
	$\displaystyle\sigma(h_{1},g_{1})\sigma(h_{2}g_{2},k)=\sigma(g_{1},k_{1})\sigma% (h,g_{2}k_{2}),$		(5.6)

for any $h,g,k\in H$ , then $\sigma$ is called a left monoidal Hom- $2$ -cocycle.

Definition 2.8 .

Let $H$ and $K$ be two groups and $\theta:K\rightarrow$ Aut $\,H$ be a homomorphism. The semidirect product $H\rtimes K$ of $H$ by $K$ relative to $\theta$ is the group with underlying set $H\times K$ consisting of pairs $(h,k)$ for $h\in H$ and $k\in K$ , and operation

(h,k)(h^{\prime},k^{\prime})=(h(\theta(k)(h^{\prime})),kk^{\prime}).

The identity element is $(1_{H},1_{K})$ and the inverse of $(h,k)$ is $(h,k)^{-1}=(\theta(k^{-1})(h^{-1}),k^{-1})$ .

Definition 2.9

Let $\phi$ be a two-valued scf, and let $\{a,b\}$ be its range. We say that $\phi$ is compatible with the dominance relation $\sqsupset$ when

P\underset{\phi(P)}{\sqsupset}Q\,\Rightarrow\,\,\phi(Q)=\phi(P).

Definition 7.15 .

We define a function $u\colon T_{\Gamma}Y\to T_{\Gamma}Y$ by

u(\gamma,y,r)=(t(y)^{-1}\gamma,y,r).

Definition 4.1 .

$\mathbf{x}$ and $\mathbf{\alpha}$ are said to be a Complementary Pair if any one of those properties hold (It can be shown that they are all equivalent)

f^{\prime}(x)=\alpha,\,\,\,f^{*^{\prime}}(\alpha)=x,\,\,\,f(x)+f^{*}(\alpha)=x\alpha,

where $f^{*}(\alpha)$ is the concave conjugate defined in ( 4 ).

Definition 11 (Additivity and Subadditivity) .

Let $f_{\mathcal{H}}$ be a family of functions from $\mathcal{S}\left(\mathcal{H}\right)$ to $\mathbb{R}$ , where $\mathcal{H}$ is a quantum system. We may omit the subscript of $f_{\mathcal{H}}$ to write $f$ for brevity. Then, $f$ is said to be fully additive if it holds that

f\left(\psi\otimes\phi\right)=f\left(\psi\right)+f\left(\phi\right)

(111)

for any states $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and $\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$ . On the other hand, $f$ is said to be weakly additive if it holds that

f\left(\psi^{\otimes n}\right)=nf\left(\psi\right)

(112)

for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and for any positive integer $n$ . In this paper, we use the word “additivity” to refer to weak additivity for brevity.

Similarly, $f$ is said to be fully subadditive if it holds that

f\left(\psi\otimes\phi\right)\leqq f\left(\psi\right)+f\left(\phi\right)

(113)

for any states $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and $\phi\in\mathcal{S}\left(\mathcal{H}^{\prime}\right)$ . On the other hand, $f$ is said to be weakly subadditive if it holds that

f\left(\psi^{\otimes n}\right)\leqq nf\left(\psi\right)

(114)

for any state $\psi\in\mathcal{S}\left(\mathcal{H}\right)$ and for any positive integer $n$ .

Definition 5.1 .

Let us say that two plane Cremona maps $\varphi,\varphi^{\prime}\colon\mathbb{P}^{2}\dasharrow\mathbb{P}^{2}$ are equivalent if there exist two automorphisms $\alpha,\alpha^{\prime}\in\operatorname{Aut}(\mathbb{P}^{2})$ such that

\varphi^{\prime}=\alpha^{\prime}\circ\varphi\circ\alpha.

Definition 7.6 .

Let $F\colon\mathcal{L}^{n}\to\mathbb{R}$ be a European function (Definition 1.12 ) given by $u_{1},\dots,u_{r}\in U_{\ell\text{-pos}}$ and $s_{1},\dots,s_{r},C\geq 0$ . Let $P(b)$ be non-empty. For any $0\leq i\leq m$ set

\beta(0)=1\qquad\text{and}\qquad\beta(i)=b^{\prime\prime}(i)=\tfrac{1+b(i)}{2}.

For any $1\leq j\leq r$ and any $0\leq i\leq m$ set

\alpha_{j}(0)=u_{j}(\nu_{0})\qquad\text{and}\qquad\alpha_{j}(i)=u_{j}(\nu_{i})% -u_{j}(\nu_{0}).

The minimizer of $F$ on $P(b)$ is the function $G\colon\mathcal{T}^{*}\to\mathbb{R}$ defined as follows. Using the notation in 6.2 for $u_{j}$ and $\beta$ and $\alpha_{j}$ , for any $1\leq k\leq n$ and any $\omega\in\mathcal{T}_{n-k}$ set

G^{(k)}(\omega)=\sum_{\mathbf{i}\in\{0,\dots,m\}^{k}}\,\beta(\mathbf{i})\big{(% }\sum_{j=1}^{r}s_{j}\cdot\alpha_{j}(\mathbf{i})\cdot u_{j}(\omega)-C\big{)}^{+}.

Definition 2.7 (Symmetrized distance) .

Let $(X,d)$ be a quasi-metric space. The symmetrized distance $d^{\phi}$ is given by

d^{\phi}(x,y)=\phi(d(x,y),d(y,x)),

where $\phi:\mathbb{R}_{+}^{2}\to[0,\infty)$ is a symmetric (i.e. $\phi(a,b)=\phi(b,a)$ , for all $a,b\in\mathbb{R}_{+}$ ) continuous, coercive function satisfying $\phi(a,b)=0$ if and only if $a=b$ , for every $a,b\geq 0$ , and $\phi(a,b)\geq\max\{a,b\}$ .

Definition 2.11 (Abstract cone) .

A cone on $\mathbb{R}_{+}$ is a triple $(C,+,\cdot)$ such that $(C,+)$ is an abelian monoid, and $\cdot$ is a mapping from $\mathbb{R}_{+}\times X$ to $X$ such that for all $x,y\in C$ and $r,s\in\mathbb{R}_{+}$ :

(i)

$r\cdot(s\cdot x)=(rs)\cdot x$ ;
(ii)

$r\cdot(x+y)=(r\cdot x)+(r\cdot y)\quad\text{and}\quad(r+s)\cdot x=(r\cdot x)+(% s\cdot x)$ ;
(iii)

$1\cdot x=x\quad\text{and}\quad 0\cdot x=0.$

Definition 2.12 (Cancellative cone) .

A cone $(C,+,\cdot)$ is called cancellative if for any $x,y,z\in C$ ,

x+z=y+z\implies x=y.

Definition 3.9 .

We define the defect of a monomial in $\alpha,\beta,\gamma$ and $\eta$ via the assignment

\text{def}(\alpha)=0,\ \text{def}(\beta)=\text{def}(\gamma)=\text{def}(\eta)=2

and extending it by multiplicativity. We also define the defect of any polynomial in $\alpha,\beta,\gamma$ and $\eta$ to be the minimum of the defects of its monomials.

Definition 2.6 .

A divergence operator on $E$ is a first order differential operator $\text{div}\colon\Gamma(E)\to C^{\infty}(M)$ satisfying the Leibniz rule

\text{div}(fz)=\pi(z)f+f\,\text{div}(z),

$f\in C^{\infty}(M)$ , $z\in\Gamma(E)$ . Given a generalized connection $D$ one may define the associated divergence operator

\text{div}_{D}(z)=\text{tr}(Dz).

Definition 3.2 .

The pair $(\phi,\psi)\in H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$ is said to be a (weak) solution of ( 1.13 ) if

\begin{cases}\displaystyle\int\phi wdx+\int\phi^{\prime}w^{\prime}dx-\int f(% \phi,\psi)wdx=0,\\ \displaystyle\int\psi zdx+\int\psi^{\prime}z^{\prime}dx-\int g(\phi,\psi)zdx=0% ,\end{cases}

(3.4)

for any $(w,z)\in H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$ .

Definition 2.11 .

$L$ is monotone if there exists $\kappa>0$ such that for all $\beta\in\pi_{2}(X,L)$ ,

[\omega](\beta)=\kappa\cdot\mu(\beta).

(2.8)

Definition 2.1 .

An affine structure on a Lie algebra $\mathfrak{g}$ over $k$ is a left-symmetric product $\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying

(1)

[x,y]=x\cdot y-y\cdot x

for all $x,y,z\in\mathfrak{g}$ . If the product is Novikov, we say that $\mathfrak{g}$ admits a Novikov structure .

Definition 2.4 .

We define the secondary particles to be the sums of two consecutive primary particles in terms of $\succ_{\epsilon}$ . We denote by $\mathcal{S}_{\epsilon}=\mathbb{Z}\times\mathcal{C}^{2}$ the set of secondary particles, in such a way that the particle

(k,c,c^{\prime})=(k+\epsilon(c,c^{\prime}),c)+(k,c^{\prime})

(2.4)

has potential $2k+\epsilon(c,c^{\prime})$ and state $cc^{\prime}$ . In the following, we identify a secondary particle as $(k,c,c^{\prime})$ or $(2k+\epsilon(c,c^{\prime}))_{cc^{\prime}}$ . We denote by $\gamma(k,c,c^{\prime})$ and $\mu(k,c,c^{\prime})$ the primary particles

\gamma(k,c,c^{\prime})=(k+\epsilon(c,c^{\prime}),c)\quad\text{and}\quad\mu(k,c% ,c^{\prime})=(k,c^{\prime})\,,

respectively called upper and lower halves of the secondary particles $(k,c,c^{\prime})$ .

Definition 27 (Lie bracket) .

Let $G$ be a Lie group, and $\phi:G\to GL_{n}(\mathbb{R})$ be a homomorphism, namely

\phi(p\star q)=\phi(p)\phi(q)

Furthermore, let $\phi_{*}$ , the pushforward of $\phi$ , be a bijection at $e\in G$ , the identity element. The Lie bracket $[\cdot,\cdot]$ on $T_{e}(G)$ is a bilinear form, s.t.

[U,V]_{G}=\phi_{*}^{-1}\left(\phi_{*}(U)\phi_{*}(V)-\phi_{*}(V)\phi_{*}(U)\right)

Definition B.6 .

Let $f$ be a Laurent polynomial. Then, $f$ can be uniquely factorized as

f=gh,

where $g$ is a monic Laurent monomial and $h$ is a polynomial without any monomial factor. We shall call $h$ the polynomial part of $f$ .

Definition 2.3 .

A Lie algebra over a commutative ring $R$ is an $R$ -module $\mathfrak{g}$ equipped with an $R$ -bilinear map $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the following conditions:

•

For any $x\in\mathfrak{g},[x,x]=0$ .
•

For any $x,y,z\in\mathfrak{g},[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.$

Definition 4.3 .

A $3$ -multipede isafiniterelationalstructure $\mathbf{M}$ withthesignature $\{<,E,H\}$ ,where $<,E$ arebinarysymbolsand $H$ isaternarysymbol,suchthat $\mathbf{M}$ satisfiesthefollowingaxioms.Thedomainof $\mathbf{M}$ hasapartition $\{\mathit{Sg},\mathit{Ft}\}$ into segments and feet suchthat ${<}^{\mathbf{M}}$ isalinearorderon $\mathit{Sg}$ ,and $E^{\mathbf{M}}$ isthegraphofasurjectivefunction $\mathrm{seg}\colon\mathit{Ft}\rightarrow\mathit{Sg}$ with $|\mathrm{seg}^{-1}(x)|=2$ forevery $x\in\mathit{Sg}$ .Forevery ${\bm{t}}\in H^{\mathbf{M}}$ ,eithertheentriesof ${\bm{t}}$ arecontainedin $\mathit{Sg}$ andwecall ${\bm{t}}$ a hyperedge ,ortheyarecontainedin $\mathit{Ft}$ andwecall ${\bm{t}}$ a positivetriple .Therelation $H^{\mathbf{M}}$ istotallysymmetricandonlycontainstripleswithpairwisedistinctentries.Foreverypositivetriple ${\bm{t}}$ ,thetriple $(\mathrm{seg}({\bm{t}}\raisebox{0.5pt}{{{[$1$]}}}),\mathrm{seg}({\bm{t}}% \raisebox{0.5pt}{{{[$2$]}}}),\mathrm{seg}({\bm{t}}\raisebox{0.5pt}{{{[$3$]}}}))$ isahyperedge.If ${\bm{t}}\in H^{\mathbf{M}}$ isanhyperedgewith $\mathrm{seg}^{-1}({\bm{t}}\raisebox{0.5pt}{{{[$i$]}}})=\{x^{i}_{0},x^{i}_{1}\}$ ,thenwerequirethatexactlyfourelementsoftheset $\smash{\{(x^{1}_{i},x^{2}_{j},x^{3}_{k})\mid i,j,k\in\{0,1\}\}}$ arepositivetriples.Wealsorequirethatforeachtwotuples $\smash{(x^{1}_{i},x^{2}_{j},x^{3}_{k}),(x^{1}_{i^{\prime}},x^{2}_{j^{\prime}},% x^{3}_{k^{\prime}})}$ fromthissetwehave

(i-i^{\prime})+(j-j^{\prime})+(k-k^{\prime})=0\text{mod}2.

Definition 1 .

The expected average gap between primes $p\leq x$ in progression (P) is

a(q,x)~{}=~{}{x\varphi(q)\over\mathop{\mathrm{li}}x},

(5)

where $\varphi(q)$ is Euler’s totient function, and $\mathop{\mathrm{li}}x$ is the logarithmic integral.

Definition 4.1 .

Let $N$ be a simply connected nilpotent Lie group with left invariant metric and magnetic form. For any $\gamma\in N$ not equal to the identity, a magnetic geodesic $\sigma\left(t\right)$ is called $\gamma$ -periodic with period $\omega$ if $\omega\neq 0$ and for all $t\in\mathbb{R}$

(29)

\displaystyle\gamma\sigma\left(t\right)=\sigma\left(t+\omega\right).

We also say that $\gamma$ translates the magnetic geodesic $\sigma(t)$ by amount $\omega$ . The number $\omega$ is called a period of $\gamma$ .

Definition 2.1 .

The DIM algebra, which we denote by $\mathcal{U}=\mathcal{U}_{q,t}$ , is a unital associative algebra generated by the currents $x^{\pm}(z)=\sum_{n\in\mathbb{Z}}x^{\pm}_{n}z^{-n}$ , $\psi^{\pm}(z)=\sum_{\pm n\in\mathbb{Z}_{\geq 0}}\psi^{\pm}_{n}z^{-n}$ and the central elements $c^{\pm 1/2}$ . The defining relations are

	$\displaystyle\psi^{+}(z)x^{\pm}(w)=g(c^{\mp 1/2}w/z)^{\mp 1}x^{\pm}(w)\psi^{+}% (z),$		(2.1)
	$\displaystyle\psi^{-}(z)x^{\pm}(w)=g(c^{\mp 1/2}z/w)^{\pm 1}x^{\pm}(w)\psi^{-}% (z),$		(2.2)
	$\displaystyle\psi^{\pm}(z)\psi^{\pm}(w)=\psi^{\pm}(w)\psi^{\pm}(z),\qquad\psi^% {+}(z)\psi^{-}(w)=\dfrac{g(c^{+1}w/z)}{g(c^{-1}w/z)}\psi^{-}(w)\psi^{+}(z),$		(2.3)
	$\displaystyle[x^{+}(z),x^{-}(w)]=\dfrac{(1-q)(1-1/t)}{1-q/t}\bigg{(}\delta(c^{% -1}z/w)\psi^{+}(c^{1/2}w)-\delta(cz/w)\psi^{-}(c^{-1/2}w)\bigg{)},$		(2.4)
	$\displaystyle G^{\mp}(z/w)x^{\pm}(z)x^{\pm}(w)=G^{\pm}(z/w)x^{\pm}(w)x^{\pm}(z% )\,,$		(2.5)

where

g(z)=\dfrac{G^{+}(z)}{G^{-}(z)}\,,\qquad G^{\pm}(z)=(1-q^{\pm 1}z)(1-t^{\mp 1}% z)(1-q^{\mp 1}t^{\pm 1}z)\,,\qquad\delta(z)=\sum_{n\in\mathbb{Z}}z^{n}\,.

(2.6)