KYUNGPOOK Math. J. 2019; 59(3): 403-414
Super Theta Vectors and Super Quantum Theta Operators
Hoil Kim
Department of Mathematics and Institute for Mathematical Convergence, Kyungpook National University, Daegu 41566, Korea
e-mail : hikim@knu.ac.kr
Received: March 20, 2019; Accepted: June 5, 2019; Published online: September 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Theta functions are the sections of line bundles on a complex torus. Noncommutative versions of theta functions have appeared as theta vectors and quantum theta operators. In this paper we describe a super version of theta vectors and quantum theta operators. This is the natural unification of Manin’s result on bosonic operators, and the author’s previous result on fermionic operators.

Keywords: super theta vectors, quantum theta operators, super Heisenberg group.
Introduction
roduction

In physics, one considers two related notions: observables and states. In classical theory, the observables are real valued functions on a phase space (position with momentum), and the states are probability measures on the phase space. In quantum theory, the observables are self-adjoint (Hermitian) operators on a Hilbert space, and the pure states are vectors in the Hilbert space with length one, mixed states are a mixture of pure states. Note that pure states correspond to Dirac measures on the phase space and mixed states are correspond to probability measures. Roughly speaking, in both classical and quantum theories, observables contain states.

Theta functions are functions on complex spaces, but more precisely, are sections of line bundles on the complex torus. The Noncommutative torus was introduced in [7]; however, the concept of the noncommutative torus had already been developed in terms of the Heisenberg group and Schrödinger representation, in [6]. Noncommutative tori are used in physics in the toroidal compactification by Connes, Douglas and Schwarz in [2]. Later the concept of theta vectors was introduced by Schwarz in [8]. Finally, Manin studied an operator version of theta functions, called quantum theta functions in [4]. Classically, theta functions play the role of observables (and states) or a Hilbert space in the geometric quantization. Theta vectors are the vacuum states which have the minimum energy and from which the other states are constructed, in the Hilbert space and quantum theta functions are observables. For more explanation and interpretation, see [10].

A classical theta function of z ∈ ℂn is $θ(z,T)=∑l∈ℤneπiltTl+2πiltz,$where T is a symmetric complex matrix of size n with Im T > 0. This function satisfies $θ(z+k, T)=θ(z,T)θ(z+Tk,T)=e−πiktTk−2πiktzθ(z,T)$for all k ∈ ℤn, and $θ(T−1z,−T−1)=(det(T/i))12eπiztT−1zθ(z,t).$The corresponding theta vectors are defined as fT (x) = eπixtTx with the same T which is considered as a vacuum vector in L2(ℝn). A quantum theta function is an element of a noncommutative algebra C(D, χ) which consists of $∑h∈DaheD,χ(h)$ over a lattice D in ℝ2n with ah ∈ ℂ satisfying the Schwarz condition, where eD,χ(h)’s are generators satisfying $eD,χ(h)eD,χ(g)=χ(h,g)eD,χ(h+g)$and χ is a skew symmetric bilinear operator on D with value in U (1) = S1. This C(D, χ) is called a quantum torus or a noncommutative torus.

Two questions were raised by Schwarz in [8]. The first one was of the connection between quantum theta functions and theta vectors, and the second one was of the existence of a quantum analogue of the classical functional equation for thetas. Manin answered both of these questions in [5].

Let f, gL2(ℝn) and π be the Heisenberg group representation of Heis(ℝ2n) on L2(ℝn), where $(π(t,x,y)f)(s)=e2πi(t+sty)+πi(xty)⋅f(s+x).$Rieffel [7] defined C(D, χ) valued inner product on L2(ℝn) as $〈〈f,g〉〉=∑h∈D〈f,πhg〉 eD,χ(h).$

Manin showed the follwing in [5].

### Theorem 1.1

For fT(x) = eπixtTx, with Tt = T, Im(T) > 0, $〈〈fT,fT〉〉=12ndet ImT∑h∈De−π2H(h_,h_) eD,χ(h).$Moreover,$ΘD:=∑h∈De−π2H(h_,h_)eD,χ(h)$is a quantum theta function in the ring C(D, χ) satisfying the following functional equations: $∀ g ∈D,cgeD,χ(g)sg*(ΘD)=ΘD,$where$cg=e−π2H(g_,g_),sg*(eD,χ(h))=e−πH(g_,h_)eD,χ(h),$and$H(g_,h_)=(Tg1+g2)tT2−1(Th1+h2)*$with g = (g1, g2), h = (h1, h2) and T = T1 + iT2. Here (g1, g2) = (0, g1, g2) and (h1, h2) = (0, h1, h2) are in Heis(ℝ2n), so that g1, g2, h1, h2 ∈ ℝn, and T1 = Re(T), T2 = Im(T) for TMn×n (ℂ).

Also he showed that $∑h∈De−πH(h_,h_)−πH(s_,h_)=∑g∈D!e−πH(g_,g_)−πH(s_,g_)$as functions of variable s, where $D!={x∈R2n|2ψ(x,y)=x1ty2−y1tx2∈Z,∀y∈D}$.

What the author obtained in [3] on ℝ0|2m, as an analogue of the theorem by Manin is as follows. For a lattice D in odd space ℝ0|2m, gR (η) = eπiξt with Rt = −R, where R2 is nondegenerate, we have $〈〈gR(η),gR(η)〉〉=∑δ∈D〈gR(η),πδgR(η)〉eD,χ(δ)=∑δ∈D2m2Pf (R2)e−π2K(δ_,δ_)eD,χ(δ)$with R = R1 + iR2, δ = (δ1, δ2) and $K(δ_,∊_)=(Rδ1+δ2)tR2−1(R∊1+∊2)*$, without knowing the meaning of the super theta vector defined. Note that if the dimension is not even, Pf(R2) = 0 (see the second section.) In this paper we find a general construction from the super theta vector naturally defined.

Let L2(ℝn|m) = L2(ℝn) ⊗ Λ•(ℝm), which is the completion of the Schwarz space S (ℝn) ⊗ Λ•(ℝm). Here Λ•(ℝm) is the (Grassmann algebra spanned by {η1 ∧· · · ∧ ηl | ηi ∈ ℝm, lm}. For with and , where is in L2(Rn|m). Here means can be expressed as $Z2=Wst(IJ)W$ for some nondegenerate W, where $J=(I−I)$, so that m must be even for nondegenerate. We call as a super theta vector generalizing the theta vector fT (x) = eπixtTx with Tt = T, Im(T) > 0.

As in the bosonic case we define C(D, χ) valued inner product on L2(ℝn|m) as $〈〈f,g〉〉=∑(h,δ)∈ D〈f,π(h,δ)g〉 eD,χ(h,δ).$

Then we get our main theorem.

### Theorem 1.2.

(1) $〈〈FZ(s,η),FZ(s,η)〉〉=1sdet (2Z2)∑(h,δ)∈De−π2H((h,δ)_,(h,δ)_)eD,χ(h,δ),$where, , and * is the complex conjugation.

(2) Let $ΘD=∑(h,δ)∈De−π2H((h,δ)_,(h,δ)_)eD,χ(h,δ).$

Then ∀ (g, µ) ∈ D,$c(g,μ)eD,χ(g,μ)(s,η)(g,μ)*ΘD=ΘD,$where$c(g,μ)=e−π2H((g,μ)_,(g,μ)_)(s,η)g,μ*eD,χ(h,δ)=e−πH((g,μ)_,(h,δ)_)eD,χ(h,δ).$(3) $∑(h,δ)∈De−πH((h,δ)_,(h,δ)_)−πH((s,η)_,(h,δ)_) =∑(g,μ)∈D!e−πH((g,μ)_,(g,μ)_)−πH((s,η)_,(g,μ)_),$as functions of variables (s, η).

There are several different approaches to define super theta functions [1, 9]. However their approaches and the concepts are different from ours in the sense that they deal with classical super theta functions while our quantum super theta functions are operators (observables) coming from super theta vectors which are vacuum states.

The contents of this paper are as follows. In section 2, we describe the necessary materials on superspaces for this paper. In section 3, we discuss the super Heisenberg groups and the super theta vector which is a vacuum state generalizing the classical theta vector. In section 4, we construct the super quantum theta functions on a superspace ℝn|2m coming from the super theta vector constructed in the previous section, which generalizes Manin’s result on even spaces and our previous result on odd spaces.

Superspaces
perspaces

In this section, we explain necessary materials for later sections including superspace, superlinear algebra, integration on superspace.

On ℝn, also denoted by ℝn|0, we define polynomials $ℝ[x1,…,xn]=ℝ〈x1,…,xn〉/(xixj−xjxi),1≤i,j≤n$as the quotient of the free algebra generated by x1, . . . , xn by the ideals generated by (xixjxjxi) for 1 ≤ i,jn. It is a commutative algebra. On odd ℝm, denoted by ℝ0|m, we can similarly define the algebra as the quotient of ℝ〉ξ1, . . . , ξm〈 by the ideal generated by (ξαξβ +ξβξα) for 1 ≤ α, βm. It is an anticommutative algebra, called a Grassmann algebra. Combining these two concepts, we can define ℝn|m, where the polynomial functions are ℝ[x1, . . . , xn, ξ1, . . . , ξm] = ℝ〉x1, . . . , xn, ξ1, . . . , ξm〈/I, where I is the ideal generated by (xixjxjxi), (ξαξβ + ξβξα), (xiξαξαxi) for 1 ≤ i,jn, 1 ≤ α, βm, which is supercommutative, in the sense that xixj = xjxi, ξαξβ = −ξβξα, xiξα = ξαxi.

Let A ={ai,j} be a 2m × 2m skew-symmetric matrix. The Pfaffian of A is defined by the equation $Pf(A)=12mm!∑σ∈S2msgn(σ)∏i=1maσ(2i−1),σ(2i)$where S2m is the symmetric group of the dimension (2m)! and sgn(σ) is the signature of σ. Then Pf(A)2 = det(A). The Pfaffian of a m × m skew-symmetric matrix for m odd is defined to be zero, as the determinant of an odd skew-symmetric matrix is zero.

For $X=(ABCD)$, the supermatrix representing a linear map from ℝn|m to ℝn|m, where A of size n × n and D of size m × m have even entries and B of size n × m and C of size m × n have odd entries, sdet(X) is defined by det(ABD−1C) det(D)−1 or, equivalently, by det(A) det(DCA−1B)−1, where A and D are invertible. This is the generalization of the determinant of even matrices and the Pfaffian of odd matrices.

For any supermatrix $X=(ABCD)$, we define the supertrace $Xst=(ABCD)st=(AtCt−BtDt).$

Here we introduce how to integrate on odd space called Berezin integral. Let Λm be the exterior algebra of polynomials in anticommuting elements θ1, . . . , θm over the field of complex numbers. (The ordering of the generators θ1, . . . , θm is fixed and defines the orientation of the exterior algebra.) The Berezin integral on Λm is the linear functional $∫Λm⋅dθ$ with the following properties: $∫Λmθm…θ1 dθ=1,∫Λm∂f∂θidθ=0, i=1,…,m$for any f ∈ Λm, where ∂/∂θi means the left or the right partial derivative. These properties define the integral uniquely. The formula $∫Λmf(θ)dθ=∫Λ1(…∫Λ1(∫Λ1f(θ)dθ1)dθ2…) dθn$expresses the Fubini law.

Super Theta Vectors
a Vectors

In this section, we want to define a super theta vector which is a vacuum state. First we want to define the super Heisenberg group sHeis(ℝ2n|2m, ψ) as follows, generalizing the Heisenberg group Heis(ℝ2n, ψ). For t, t′ ∈ ℝ, and (x, α), (y, β), (x′, α′), (y′, β′) ∈ ℝn|m, we define the multiplication of (t, x, y, α, β), (t′, x′, y′, α′, β′)∈ ℝ × ℝ2n|2m by $(t,x,y,α,β)⋅(t′,x′,y′,α′,β′) =(t+t′+ψ(x,y,α,β;x′,y′ ,α′,β′),x+x′,y+y′,α+α′,β+β′),$where ψ : ℝ2n|2m × ℝ2n|2m → ℝ, satisfies the cocycle condition $ψ(x,y,α,β;x′,y′,α′,β′)+ψ(x+x′,y+y′,α+α′,β+β′;x″,y″,α″,β″)=ψ(x,y,α,β;x′+x″,y′+y″,α′+α″,β′+β″)+ψ(x′,y′,α′,β′;x″,y″,α″,β″),$a necessary and sufficient for the associative multiplication. Then there is a central extension $0→ℝ→isHeis(ℝ2n|2m,ψ)→jℝ2n|2m→0,$which is an exact sequence, with the inclusion i (t) = (t, 0), the projection j (t, z) = z, for z ∈ ℝ2n|2m, where i(ℝ) is the center in sHeis(ℝ2n|2m, ψ). As in the bosonic case, we can introduce the unitary representation of the super Heisenberg group.

Let $ψ(x,y,α,β;x′,y′,α′,β′)=12(xty′−ytx′−αtβ′−βtα′).$Then ψ satisfies the cocycle condition. We define $(π(t,x,y,α,β)f)(s,η)=e2πi(t+sty−ηtβ)+πi(xty−αtβ)⋅f(s+x, η+α).$Then $π(t1,x1,y1,α1,β1)π(t2,x2,y2,α2,β2) =eπi(x1ty2−y1tx2−α1tβ2−β1tα2)π(t1+t2,x1+x2,y1+y2,α1+α2,β1+β2),$so that $π(t1,x1,y1,α1,β1)π(t2,x2,y2,α2,β2) =e2πi(x1ty2−y1tx2−α1tβ2−β1tα2)π(t2,x2,y2,α2,β2)π(t1,x1,y1,α1,β1).$

Let D be a lattice in ℝ2n|2m. Let ψ be a ℝ valued bililnear form on D. We define C(D, χ) of infinite series $∑(h,δ)∈Da(h,δ)eD,χ(h,δ),$where $eD,χ(g,μ)eD,χ(h,δ)=χ((g,μ),(h,δ))eD,χ(g+h,μ+δ),$with $χ((g,μ),(h,δ))=e2πiψ(g,μ;h,δ).$This C(D, χ) is a super quantum torus, generalizing the notion of a quantum torus by Rieffel.

We choose such that and . We define as the space of super holomorphic functions $F((s,η)_)$ on ℂn|m such that $‖F‖2=∫ℂn|m|F((s,η_))|2e−πH((s,η_),(s,η_))d(s,η_)<∞,$where , , and * is the complex conjugation. Here super holomorphic on ℂn|m means holomorphic on ℂn and integration on ℂn|m is the Berezin integral exaplained in the second section. We define the Heisenberg group representation as $(Uλ,(h,δ)F)((s,η_))=λ−1e−πH((s,η_),(h,δ_))−π2H((h,δ_), (h,δ_))F((s,η_)+(h,δ_)).$

Now we determine the representation of the Heisenberg algebra associated to the above Heisenberg group representation. Let Ai, Aµ, Bi, Bµ, C denote the basis of the Heisenberg Lie algebra 1 ≤ in, 1 ≤ µm, such that $exp(∑si(1)Ai)=(1,(si(1)0))exp(∑ημ(1)Aμ)=(1,(ημ(1)0))exp(∑si(2)Bi)=(1,(0si(2)))exp(∑ημ(2)Bμ)=(1,(0ημ(2)))exp (tC)=(e2πit,0).$In realization, we have $δUAi(F)(s,η)=∂F∂siδUAμ(F)(s,η)=∂F∂ημδUBi(F)(s,η)=2π isiF(s,η)δUBμ(F)(s,η)=2π iημF(s,η),$so that $δ(UA)F((s,η_))=−πZ¯Z2−st((s,η_))F((s,η_))+Z∂F∂(s,η_)δ(UB)F((s,η_))=−πZ2−st((s,η_))F((s,η_))+∂F∂(s,η_).$This implies that defined as the span of is equal to the span of ${F→(s,η)_F}$, defined as the span of is equal to the span of ${F→∂F∂(s,η)_}$.

In the Heisenberg representation of sHeis(ℝ2n|2m, ψ), there is an element , unique up to scalar, such that is defined and equal to 0 for all . In $ℋϕ2(ℂn|m)$, there is a unique killed by and it is , the vacuum state. Hence $ℋϕ2$ is irreducible and in the conjugate linear isomorphism with L2(ℝn|m), 1 corresponds to $FZ(s,η)=eπi(st,ηt)Z(sη),$where $Z=(TΔ∇S)$ with Tt = T, ∆t = −∇, St = −S and , with the above condition.

### Definition 3.1

(Super Theta Vector) We define this as the super theta vector.

Super theta operator
operator

In this section, we generalize Manin’s result on quantum theta function for ℝn|0 and our result on quantum theta function for ℝ0|2m. We construct the super quantum theta operators for a superspace ℝn|2m coming from the super theta vector constructed in the previous section.

### Definition 4.1

(Super Quantum Theta Operator) Let , the super theta vector. As in the bosonic case we define C(D, χ) valued inner product on L2(ℝn|2m) as $〈〈f,g〉〉=∑(h,δ)∈D〈f,π(h,δ)g〉 eD,χ(h,δ).$We define as the super quantum theta operator.

The following is the proof of our main theorem (Theorem 1.2).

Proof of Theorem 1.2

The proof of (1) is obtained by the computation of $〈FZ(x,η),πh,δFZ(x,η)〉 =∫FZ(x,η)πh,δFZ(x,η)¯ dxdη =∫eπi(xt,ηt)Z(xη)e−πi((x+h1)t,(η+δ1)t)Z¯(x+h1η+δ1) .e−2πi(xth2−ηtδ2)e−πi(h1th2−δ1tδ2) dxdη.$If we compute the exponent, $πi(xt,ηt)Z(xη)−πi(xt,ηt)Z¯(xη) −2πi(xt,ηt)(Z¯(h1δ1)+(h2−δ2))−πi(h1t,δ1t)(Z¯(h1δ1)+(h2−δ2))=−2π(xt,ηt)Z2(xη) −2πi(xt,ηt)(Z¯(h1δ1)+(h2−δ2))−πi(h1t,δ1t)(Z¯(h1δ1)+(h2−δ2))=−2π((xt,ηt)+i2(h,δ)_*tZ2−st)Z2((xη)+i2Z2−1(h,δ)_*) −π2(h,δ)_*Z2−stZ2Z2−1(h,δ)_*−πi(h1t,δ1t)(h,δ)_* =−2π((xt,ηt)+i2(h,δ)_*tZ2−st)Z2((xη)+i2Z2−1(h,δ)_*) −π2(h,δ)_tZ2−st(((h,δ)_*)).$Here the first term gives $1sdet2Z2$ after integration.

The proof of (2) comes from the following observation. $Im H((g,μ_), (h,δ_)) =(g1t,μ1t)(Z1(h1δ1)+(h2−δ2))+((g2t,−μ2t)+(g1t,μ1t)Z2st)Z2−st(−Z2)(h1δ1) =g1th2−g2th1−μ1tδ2−μ2tδ1 =2Ψ((g,μ),(h, δ)),$by using $Zist=Zi(1−1)$$ReH((g,μ)_,(h,δ)_)=12(H((g,μ)_,(h,δ)_)+H((h,δ)_,(g,μ)_)),$with the property $H((g,μ)_,(h,δ)_)=(H((h,δ)_,(g,μ)_))*$.

The proof of (3) is a generalization of Manin, by using the Fourier transform of $Fs,η(h,δ)=e−πH((h,δ)_,(h,δ)_)−πH((s,η)_,(h,δ)_)$and checking that $F^s,η(g,μ)=e−πH((g,μ)_,(g,μ)_)−πH((s,η)_,(g,μ)_)$and using the Poisson summation formula. $F^s,η(g,μ)=∫e−πH((h,δ)_,(h,δ)_)−πH((s,η)_,(h,δ)_)−2πiA˜((g,μ)_,(h,δ)_)d(h,δ)_,$where $A˜((g,μ),(h,δ))=(g1,μ1,g2,−μ2)(11−11)(h1δ1h2−δ2).$In fact, Ã = 2Ψ. $H((s,η)_,(h,δ)_)+H((h,δ)_,(h,δ)_)+2iA˜((g,μ),(h,δ)) =(s1t,η1t,s2t,−η2t)B(h1δ1h2−δ2)+i(s1t,η1t,s2t,−η2t)A(h1δ1h2−δ2) +2i(g1t,μ1t,g2t,−μ2t)A(h1δ1h2−δ2)+(h1t,δ1t,h2t,−δ2t)B(h1δ1h2−δ2),$where $A=(11−11) and B=(Z1stZ2−stZ1+Z2Z1stZ2−stZ2−stZ1Z2st).$Then $B−1=(Z2−1−Z2−1Z1st−Z1Z2−1Z2st+Z1Z2−1Z1st)$and $B−st=(Z2−st−Z2−stZ1st−(1−1)Z1Z2−1(1−1)Z2(1−1)+(1−1)Z1Z2−1Z1(1−1)).$Let $(a1α1a2α2)=12[(s1η1s2−η2)+iB−1A[(s1η1s2−η2)+2(g1μ1g2−μ2)]].$Then we can show that $H((s,η)_,(h,δ)_)+H((h,δ)_,(h,δ)_)+2iA˜((g,μ).(h,δ)) =(h1t+a1t,δ1t+α1t,h2t+a2t,−(δ2t+α2t))B(h1+a1δ1+α1h2+a2−(δ2+α2)) −(a1t,α1t,a2t,−α2t)B(a1α1a2−α2).$By using −AstBstB = A and AstBstA = B, we have $−(a1t,α1t,a2t,−α2t)B(a1α1a2−α2)=H((g,μ)_,(g,μ)_)+H((s,η)_,(g,μ)_).$For $B=(B11B12B21B22) =(IB12B22−10I)(B11−B12B22−1B2100B22)(I0B22−1B21I),sdet B=sdet(B11−B12B22−1B2100B22)=sdet(Z2Z2−st)=1.$Then after integration on (h, δ) space we get $F^s,η(g,μ)=e−πH((g,μ)_,(g,μ)_)−πH((s,η)_,(g,μ)_).$

References
1. M. Bergvelt, and J. Rabin. Supercurves, their Jacobians, and super KP equations. Duke Math. J., 98(1)(1999), 1-57.
2. A. Connes, M. Douglas, and A. Schwarz. Noncommutative geometry and matrix theory. J. High Energy Phys., JHEP9802(1998) 003, 35.
3. H. Kim. Quantum super theta vectors and theta functions. Kyungpook Math. J., 56(1)(2016), 249-256.
4. Y. Manin. Theta functions, quantum tori and Heisenberg Groups. Lett. Math. Phys., 56(3)(2001), 295-320.
5. Y. Manin. Functional equations for quantum theta functions. Publ. Res. Inst. Math. Sci., 40(3)(2004), 605-624.
6. D. Mumford. Tata lectures on theta III. Progress in Math, 97, Birkhäuser, Boston, 1991. (with M Nori and P Norman),.
7. M. Rieffel. Projective modules over higher dimensional noncommutative tori. Canad. J. Math., 40(2)(1988), 257-338.
8. A. Schwarz. Theta functions on noncommutative tori. Lett Math Phys., 58(2001), 81-90.
9. Y. Tsuchimoto. On super theta functions. J. Math. Kyoto Univ., 34(3)(1994), 641-694.
10. Tyurin. A. Quantization and “theta functions”.