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 f_T (x) = e^πixtTx 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 a_h ∈ ℂ satisfying the Schwarz condition, where e_D,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, g ∈ L²(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 f_T(x) = e^πixtTx, with T^t = 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,wherecg=e−π2H(g_,g_),sg*(eD,χ(h))=e−πH(g_,h_)eD,χ(h),andH(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 T ∈ M_n×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 ℝ^0|2m, as an analogue of the theorem by Manin is as follows. For a lattice D in odd space ℝ^0|2m, g_R (η) = e^−πiξtRξ with R^t = −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(ℝ^n|m) = L²(ℝⁿ) ⊗ Λ•(ℝ^m), which is the completion of the Schwarz space S (ℝⁿ) ⊗ Λ•(ℝ^m). Here Λ•(ℝ^m) is the (Grassmann algebra spanned by {η₁ ∧· · · ∧ η_l | η_i ∈ ℝ^m, l ≤ m}. For with and , where is in L²(R^n|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 f_T (x) = e^πixtTx with T^t = 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,wherec(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 ℝ^n|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.

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

On ℝⁿ, also denoted by ℝ^n|0, we define polynomials ℝ[x1,…,xn]=ℝ⟨x1,…,xn⟩/(xixj−xjxi),1≤i,j≤nas the quotient of the free algebra generated by x₁, . . . , x_n by the ideals generated by (x_ix_j − x_jx_i) for 1 ≤ i,j ≤ n. It is a commutative algebra. On odd ℝ^m, denoted by ℝ^0|m, we can similarly define the algebra as the quotient of ℝ⟩ξ₁, . . . , ξ_m⟨ by the ideal generated by (ξ_αξ_β +ξ_βξ_α) for 1 ≤ α, β ≤ m. It is an anticommutative algebra, called a Grassmann algebra. Combining these two concepts, we can define ℝ^n|m, where the polynomial functions are ℝ[x₁, . . . , x_n, ξ₁, . . . , ξ_m] = ℝ⟩x₁, . . . , x_n, ξ₁, . . . , ξ_m⟨/I, where I is the ideal generated by (x_ix_j − x_jx_i), (ξ_αξ_β + ξ_βξ_α), (x_iξ_α − ξ_αx_i) for 1 ≤ i,j ≤ n, 1 ≤ α, β ≤ m, which is supercommutative, in the sense that x_ix_j = x_jx_i, ξ_αξ_β = −ξ_βξ_α, x_iξ_α = ξ_αx_i.

Let A ={a_i,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 S_2m 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 ℝ^n|m to ℝ^n|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(A − BD⁻¹C) det(D)⁻¹ or, equivalently, by det(A) det(D − CA⁻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 Λ^m be the exterior algebra of polynomials in anticommuting elements θ₁, . . . , θ_m over the field of complex numbers. (The ordering of the generators θ₁, . . . , θ_m is fixed and defines the orientation of the exterior algebra.) The Berezin integral on Λ^m is the linear functional ∫Λm⋅dθ with the following properties: ∫Λmθm…θ1 dθ=1,∫Λm∂f∂θidθ=0, i=1,…,mfor any f ∈ Λ^m, where ∂/∂θ_i 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…) dθnexpresses the Fubini law.

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(ℝ^2n|2m, ψ) as follows, generalizing the Heisenberg group Heis(ℝ²ⁿ, ψ). For t, t′ ∈ ℝ, and (x, α), (y, β), (x′, α′), (y′, β′) ∈ ℝ^n|m, we define the multiplication of (t, x, y, α, β), (t′, x′, y′, α′, β′)∈ ℝ × ℝ^2n|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 : ℝ^2n|2m × ℝ^2n|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 ∈ ℝ^2n|2m, where i(ℝ) is the center in sHeis(ℝ^2n|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 ℝ^2n|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 ℂ^n|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 ℂ^n|m means holomorphic on ℂⁿ and integration on ℂ^n|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 A_i, A_µ, B_i, B_µ, C denote the basis of the Heisenberg Lie algebra 1 ≤ i ≤ n, 1 ≤ µ ≤ m, 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 ℋ of sHeis(ℝ^2n|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(ℝ^n|m), 1 corresponds to FZ(s,η)=eπi(st,ηt)Z(sη),where Z=(TΔ∇S) with T^t = T, ∆^t = −∇, S^t = −S and , with the above condition.

Definition 3.1

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

In this section, we generalize Manin’s result on quantum theta function for ℝ^n|0 and our result on quantum theta function for ℝ^0|2m. We construct the super quantum theta operators for a superspace ℝ^n|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(ℝ^n|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).