Krzysztof Gawȩdzki

Problem Set for Lecture 1. Simple functional integrals

PROBLEM 1. \underline{Feynman-Kac formula for harmonic oscillator}

The Feynman Kac formula in the one-dimensional case (d = 0, Σ = [0, L]_{periodic}) asserts for the 2-point function that, for x_{1} ≤ x_{2} ,

( ∫_{Cper([0,L])}ϕ(x_{1}) ϕ(x_{2}) dμ_{G}(ϕ) =) G(x_{1}, x_{2}) = \frac{tr e^{−x_{1}H} φe^{(x_{1}−x_{2})H} φe^{(x_{2}−L)H}}{tr e^{−LH}} .  (1)

In the normalizations from the lecture, G = \frac{2π}{β} (−\frac{d^{2}}{dx^{2}} + m^{2})^{−1} with \frac{d^{2}}{dx^{2}} with periodic b.c. on [0, L] and H is the Hamiltonian of a harmonic oscillator,

H = −\frac{π}{β} \frac{d^{2}}{dφ^{2}} + \frac{βm^{2}}{4π^{2}} φ^{2} − \frac{m}{2} = ma^{∗}a

where

a^{∗} = −\sqrt{\frac{π}{βm}} \frac{d}{dφ} + \sqrt{\frac{βm}{4π}} φ, a = \sqrt{\frac{π}{βm}} \frac{d}{dφ} + \sqrt{\frac{βm}{4π}} φ, [a, a^{∗}] = 1,

are operators in L^{2}(R, dφ). Recall that the spectrum of H is m{0, 1, …}. (a). Use the Fourier transform to write the left hand side. Show that its limit G_{∞}(x_{1}, x_{2}) when L → ∞ exists. (b). Prove that for x_{1}, …, x_{n} > 0 and complex numbers λ_{1}, …, λ_{n}

∑_{k,l=1}^{n} λ̄_{k} λ_{l} G_{∞}(−x_{k}, x_{l}) ≥ 0.

(c). What is the limit of the right hand side of Eq. (1) when L → ∞? (d). Show that both sides of Eq. (1) coincide at L = ∞. Prove (b) using this result. (e). Prove the relation (1) for finite L.

PROBLEM 2. \underline{Zeta-regularized determinants} (for calculators)

(a). Using the identity λ^{−s} = Γ(s)^{−1} ∫_{0}^{∞} t^{s−1} e^{−λt} dt show that ζ(0) = −\frac{1}{2} where ζ is the Riemann zeta function ζ(s) = ∑_{n=1}^{∞} n^{−s} (for Res > 1, analytically continued elsewhere).

(b). For τ = τ_{1} + iτ_{2} with τ_{i} real, τ_{2} > 0 show using the identity from (a) and the Poisson resummation that for Res sufficiently large

∑_{n=−∞}^{∞} |τ + n|^{−2s} = \frac{\sqrt{π}}{Γ(s)} (∑_{n≠0} e^{2πinτ_{1}} ∫_{0}^{∞} t^{s−3/2} e^{−τ_{2}^{2}t−π^{2}n^{2}/t} dt + τ_{2}^{−2s+1} Γ(s − 1/2)) .

Note that the right hand side is analytic in s around s = 0. Using (easy) relations Γ(s)^{−1} = s + O(s^{2}), Γ(− \frac{1}{2} ) = −2\sqrt{π} and ∫_{0}^{∞} t^{−3/2} e^{−x(t+t^{−1})} dt = \sqrt{π} x^{−1/2} e^{−2x} obtain:

\frac{d}{ds} |_{s=0} ∑_{n} |τ + n|^{−2s} = − ln |1 − q|^{2} − 2πτ_{2}

with the standard notation q ≡ e^{2πiτ}. (c). By taking τ → 0 in the last formula show that ζ′(0) = − \frac{1}{2} ln(2π).

(d). Prove that for the periodic b.c. operator \frac{d^{2}}{dx^{2}} on [0, L] the zeta-regularized determinant

det′(− \frac{β}{2π} \frac{d^{2}}{dx^{2}} ) = 2πL^{2} .

(c). Show that the spectrum of the Laplacian Δ_{τ} on the torus C/(Z + τZ) in the metric |dz|^{2} is given by λ_{m,n} = −(\frac{2π}{τ_{2}})^{2} |τm + n|^{2} for n, m ∈ Z.

(d). Proceeding as in (b) decompose

∑_{(m,n)≠(0,0)} |τm + n|^{−2s} = ∑_{m≠0, n} |τm + n|^{−2s} + 2ζ(2s)

= \frac{\sqrt{π}}{Γ(s)} ( ∑_{m,n≠0} e^{2πimnτ_{1}} ∫_{0}^{∞} t^{s−3/2} e^{−m^{2}τ_{2}^{2}t−π^{2}n^{2}/t} dt + ∑_{m≠0} m^{−2s+1} τ_{2}^{−2s+1} Γ(s − 1/2)) + 2ζ(2s)

and show that (after analytic continuation)

ζ_{−Δ′_{τ}} (s) ≡ ∑_{(m,n)≠(0,0)}(−λ_{m,n})^{−s} = −1 − 2s ln | ∏_{m=1}^{∞}(1 − q^{m})|^{2} − 2s ln τ_{2} + \frac{1}{3} π τ_{2}s.

Infer that

ζ_{−Δ′_{τ}} (0) = −1, ζ′_{−Δ′_{τ}} (0) = −2 ln | ∏_{m=1}^{∞}(1 − q^{m})|^{2} − 2 ln τ_{2} + \frac{1}{3} π τ_{2} ,

and that

det′(−Δ_{τ}) = τ_{2}^{2} |η(τ)|^{4}

where the Dedekind eta function η(τ) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}).

PROBLEM 3. \underline{Rationality of the free field with values in S^{1}}

The toroidal partition function of the free field compactified on the circle of radius squared r^{2} is

Z_{r^{2}}(τ) = ∑_{n,m∈Z} e^{πiτ (mr+n/r)^{2}/2 − πiτ̄(mr−n/r)^{2}/2} |η(τ)|^{−2} .  (2)

Show that for rational r^{2} it is a finite sesqui-linear combination of functions analytic in τ.

PROBLEM 4. \underline{Toroidal compactification to a complex torus}

Let (Σ, γ) be a Riemann surface. Let R = R_{1} + iR_{2} and T_{1} + iT_{2} be two numbers in the upper half plane. Consider on the torus (R/2πZ)^{2} the complex structure given by the coordinate ψ = ϕ^{1} + T ϕ^{2} , the metric g = \frac{R_{2}}{T_{2}} |dψ|^{2} and the 2-form ω = i\frac{R_{1}}{T_{2}} dψ ∧ dψ̄. Show that the partition functions Z_{R,T} of the free field on Σ with values in (R/2πZ)^{2} with action S(ϕ) = \frac{1}{4π}(∥dϕ∥_{L^{2}}^{2} + i ∫_{Σ} ϕ^{∗}ω) satisfy the identities:

Z_{R,T} = Z_{R+1,T} = Z_{R,T +1} = Z_{R,−T^{−1}} = Z_{−R^{−1},−T^{−1}} .

additionally to the mirror symmetry Z_{R,T} = Z_{T,R} shown in the lecture.

PROBLEM 9. \underline{Conformal anomaly}

Using the short time expansion of the heat kernel of the Laplacian −Δ on a Riemann surface (Σ, γ)

e^{tΔ}(x, x) = \frac{1}{4πt} + \frac{1}{12π} r(x) + O(t)

where r stands for the scalar curvature, prove that under the infinitesimal local rescaling of the metric γ ↦ e^{σ}γ

\frac{δ}{δσ(x)} |_{σ=0} ln (\frac{det′(−Δ)}{^{vol^{Σ}}} ) = − \frac{1}{12π} r(x).  (3)

Show that this infinitesimal relation is equivalent to the global one

\frac{det′(−Δ)}{^{vol^{Σ}}} |_{e^{σ}γ} = e^{− \frac{1}{48π} (∥dσ∥_{L^{2}}^{2} + 4 ∫_{Σ} σr dv)} \frac{det′(−Δ)}{^{vol^{Σ}}} |_{γ} .

PROBLEM 10. \underline{Massless Fermions on Riemann surface}

Let (Σ, γ) be a Riemann surface. Spin structure on Σ may be identified with the square root L of the canonical bundle K = T^{∗}^{1,0}(Σ). A Dirac spinor Ψ = (ψ, ψ̃) is an element of Γ(L ⊕ L̄) where L̄ is the bundle complex conjugate to L. The conjugate spinor is Ψ̄ = (χ̃, χ) ∈ Γ(L̄ ⊕ L) and in the euclidean Dirac theory it should be treated as an independent field (χ = ψ, χ̃ = ψ̃ for Majorana fermions). Denote by ¯∂_{L} the ¯∂ operator of L and by ∂L̄ its complex conjugate which may be naturally identified with ¯∂_{L}^{∗}. The action is a function on the odd vector space Π(Γ(L ⊕ L̄) ⊕ Γ(L̄ ⊕ L)):

S(Ψ, Ψ̄) = \frac{1}{π} ∫_{Σ}(χ¯∂L̄ψ + χ̃∂_{L}ψ̃)

(note that the integrand is naturally a 2-form). Partition functions of the Dirac fermions are given by the formal Berezin integral

Z_{L} = ∫ e^{−S(Ψ,Ψ̄)} DΨ̄DΨ = det(∂L̄) det(¯∂_{L}) = det(¯∂_{L}^{∗} ¯∂_{L}).

The last determinant may be zeta-regularized giving a precise sense to the partition function Z_{L} of the Dirac field on Σ.

On the elliptic curve C/(Z + τZ) with τ in the upper half-plane, the canonical bundle K may be trivialized by the section dz and spin structures correspond to the choice of periodic or anti-periodic boundary conditions under z → z + 1 and z → z + τ:

L = pp, pa, ap, aa.

(a). Show that the eigenvalues of ¯∂_{L}^{∗} ¯∂_{L} are

λ_{m,n} = (\frac{π}{τ_{2}} )^{2}|τm + n|^{2}

with

m ∈ Z,  n ∈ Z  for L = pp,
m ∈ Z,  n ∈ Z + \frac{1}{2}  for L = pa,
m ∈ Z + \frac{1}{2} ,  n ∈ Z  for L = ap,
m ∈ Z + \frac{1}{2} ,  n ∈ Z + \frac{1}{2}  for L = aa.

(b). Infer that

Z_{pp}(τ) = 0.

(c). Show that

ζ ¯∂_{pa}^{∗} ¯∂_{pa}(s) = 2^{2s} (ζ_{−Δ′_{2τ}} (s) − ζ_{−Δ′_{τ}} (s)).

Infer from Eq. (2) that

Z_{pa}(τ) = 4|q^{1/24} ∏_{n=1}^{∞}(1 + q^{n})|^{4} .

In the Hilbert space picture

Z_{pa}(τ) = tr_{H_{R}⊗H̃_{R}} q^{L_{0}−1/24}q̄^{L̃_{0}−1/24} .  (4)

The “Ramond sector” Hilbert space is H_{R} ⊗ H̃_{R} with

H_{R} = C^{2} ⊗ (∧( ⊕_{n=1}^{∞} C))^{⊗2}  (5)

and H̃_{R} is another copy of H_{R}. L_{0} acts in the first copy. It has eigenvalue \frac{1}{8} on C^{2} (the “Ramond ground states”) and the occupied n^{th} mode in the fermionic Fock space adds n to it. The periodic partition function is

Z_{pp}(τ) = tr_{H_{R}⊗H̃_{R}} (−1)^{F +F̃} q^{L_{0}−1/24}q̄^{L̃_{0}−1/24} ≡ str_{H_{R}⊗H̃_{R}} q^{L_{0}−1/24}q̄^{L̃_{0}−1/24}  (6)

where (1, 0), (0, 1) ∈ C^{2} correspond to the eigenvalues +1, −1 of (−1)^{F} and each occupied fermionic Fock space mode adds 1 to F . Z_{pp}(τ) vanishes since modes with odd and even Fermi numbers are paired. (d). Show that

ζ ¯∂_{ap}^{∗} ¯∂_{ap}(s) = 2^{4s}ζ_{−Δ′_{τ/2}}(s) − 2^{2s}ζ_{−Δ′_{τ}} (s) and Z_{ap}(τ) = |q^{−1/48} ∏_{n=0}^{∞}(1 − q^{n+1/2})|^{4} .

The Hilbert space interpretation is

Z_{ap}(τ) = str_{H_{NS}⊗H̃_{NS}} q^{L_{0}−1/24}q̄^{L̃_{0}−1/24}  (7)

where the “Neveu-Schwarz sector” Hilbert space is

H_{NS} = (∧( ⊕_{n=0}^{∞} C))^{⊗2} .  (8)

The “Neveu-Schwarz ground state” has eigenvalue zero of L_{0} and the n^{th} occupied zero mode contributes (n+ \frac{1}{2}) to it. The fermion number of the NS-ground state vanishes and each occupied fermionic mode adds 1 to it. (e). Show that

ζ ¯∂_{aa}^{∗} ¯∂_{aa}(s) = 2^{2s}ζ ¯∂_{pa}^{∗} ¯∂_{pa}(s)|_{τ/2} − ζ ¯∂_{pa}^{∗} ¯∂_{pa}(s)|_{τ} and Z_{aa}(τ) = |q^{−1/48} ∏_{n=0}^{∞}(1 + q^{n+1/2})|^{4}

and that

Z_{aa}(τ) = tr_{H_{NS}⊗H̃_{NS}} q^{L_{0}−1/24}q̄^{L̃_{0}−1/24} .  (9)

(f). Prove the modular properties:

Z_{pa}(τ + 1) = Z_{pa}(τ),  Z_{ap}(τ + 1) = Z_{aa}(τ),  Z_{aa}(τ + 1) = Z_{ap}(τ),
Z_{pa}(−1/τ) = Z_{ap}(τ),  Z_{ap}(−1/τ) = Z_{pa}(τ),  Z_{aa}(−1/τ) = Z_{aa}(τ).

PROBLEM 10. \underline{Bosonization} The spin structure is called even (odd) if the dimension of the kernel of ¯∂_{L} is even (odd). Denote by σ(L) the parity of L. The bosonization formula asserts that

∑_{L}(−1)^{σ(L)} Z_{L} = C^{h−1} Z_{1/2}  (10)

where on the right hand side we have the partition function of the bosonic free field with values in the circle of radius squared \frac{1}{2}, C is a constant and h the genus of the Riemann surface Σ. These equalities extend to correlations. For example, the fermionic fields (ψψ̃)(x) correspond to bosonic fields : e^{iϕ(x)} : and (χ̃χ)(x) to : e^{−iϕ(x)} :. What are their conformal weights? Prove identity (10) for Σ = C/(Z + τZ) using the expression (2) for Z_{1/2}(τ) and the classical product expressions for the theta functions

ϑ(z|τ) ≡ ∑_{n∈Z} e^{πiτn^{2} + 2πinz} = ∏_{n=1}^{∞}(1 − q^{n})(1 + e^{2πiz} q^{n−1/2})(1 + e^{−2πiz}q^{n−1/2}).

What is the Hilbert space interpretation of the left hand side of Eq. (10) on the elliptic curve?