Rigorous construction of the two-dimensional supersymmetric nonlinear sigma model: the loop-space Hamiltonian $H_s$ and the $1/s$ expansion

The two-dimensional supersymmetric nonlinear sigma model with compact curved target still lacks a rigorous mathematical foundation. Its canonical quantization on a circle formally lives on the infinite-dimensional loop space of the target, with deformed supercharges $d_s = d + s\,i(K)$ and Hamiltonian $H_s$, and its index computations rest on an unjustified asymptotic expansion in $1/s$ at large $s$ that localizes onto constant loops. Constructing this interacting, asymptotically free quantum field theory rigorously — as a self-adjoint loop-space Hamiltonian or as a Euclidean constructive field theory satisfying the supersymmetry identities — and justifying the $1/s$ expansion remains open, beyond the finite-dimensional analogs (Simon; Helffer and Sjöstrand; Bismut), the Floer-theoretic special cases, the perturbative construction (Nguyen), and the supersymmetric lattice regularizations (Catterall and Ghadab).

Background

Supersymmetry is a proposed symmetry of physics that pairs bosonic degrees of freedom (integer-spin fields, for example scalar particles) with fermionic ones (half-integer-spin fields, for example the electron). Its simplest laboratory is supersymmetric quantum mechanics on a compact Riemannian manifold $M$: the Hilbert space is the space of square-integrable differential forms on $M$ (an antisymmetric covariant tensor field, on which the exterior derivative $d$ acts, sending $k$-forms to $(k+1)$-forms with $d^2=0$), the supercharges are $d$ and its adjoint $d^*$, and the Hamiltonian is $H = dd^* + d^*d$ (a multiple of the Laplacian). The zero-energy states are then precisely the harmonic forms, which represent the de Rham cohomology of $M$ (the quotient of the kernel of $d$ by the image of $d$), so the graded dimension of the space of supersymmetric ground states equals the Euler characteristic $\chi(M)$ of $M$.

Witten's 1982 paper deformed $d$ by a Morse function $h$ — a smooth real-valued function on $M$ whose critical points (where the derivative vanishes) are all nondegenerate, meaning the Hessian, the matrix of second derivatives, is invertible there — by setting $d_t = e^{-th}\,d\,e^{th}$ and $H_t = d_td_t^* + d_t^*d_t$. As $t\to\infty$, the low-lying eigenfunctions of $H_t$ concentrate near the critical points of $h$, which recovers the Morse inequalities (the number of critical points of each Morse index — the number of negative eigenvalues of the Hessian at the critical point — bounds the corresponding Betti number), and the first-order corrections — matrix elements $n(a,b)$ defined as signed counts of gradient-flow lines of $h$ between critical points, with signs from orientations of the flow lines — define a coboundary operator $\delta$ with $\delta^2=0$ whose cohomology computes the de Rham cohomology of $M$. This recovered the classical Morse-theoretic complex from quantum-mechanics arguments. Witten also treated the analogous deformation $d_s = d + s\,i(K)$, where $K$ is a Killing field (the vector field generating a one-parameter group of isometries) on $M$ and $i(K)$ denotes contraction of a form with $K$, showing that index-type quantities localize as $s\to\infty$ onto the fixed-point set of $K$ — a supersymmetric analogue of equivariant localization in geometry.

A nonlinear sigma model is a quantum field theory whose basic field is a map $\phi$ from spacetime into a fixed Riemannian target manifold $B$, with action the Dirichlet energy of $\phi$ measured with the target metric; the field equations are nonlinear whenever $B$ is curved. Witten's Section 4 promoted the supersymmetric quantum mechanics above to the supersymmetric nonlinear sigma model in one space and one time dimension. Quantizing such a theory on a spatial circle of circumference $L$ makes the configuration space the loop space $\Omega(B)$ — the infinite-dimensional manifold of smooth maps from the circle into $B$ — and formally the quantum states are differential forms on $\Omega(B)$, the supercharges are the exterior derivative $d$ along the loop space and its adjoint, and the Hamiltonian is the corresponding Laplacian. Rotation of the spatial circle acts on $\Omega(B)$ as a Killing field $K$, and the deformation $d_s = d + s\,i(K)$ produces the deformed Hamiltonian $H_s = d_sd_s^* + d_s^*d_s$. Supersymmetry forces every nonzero-energy state to occur in a boson-fermion pair that cancels in the trace weighted by fermion parity, the operator $(-1)^F$ that equals $+1$ on bosonic states and $-1$ on fermionic states; this trace is the Witten index; its value is the number of bosonic ground states minus the number of fermionic ones. For the sigma model Witten computed it to equal the Euler characteristic $\chi(B)$ of the target, the computation being an asymptotic expansion in $1/s$ at large $s$ in which the loop-space trace localizes onto the constant loops — the fixed points of the rotation, that is, onto $B$ itself. The same section treats a supersymmetric scalar field theory (one whose fields are a real scalar and a fermion on a spatial circle) with polynomial superpotential $W$ — the function whose derivatives determine the interactions and the supersymmetry transformations — whose index is $1$ or $0$ according to the parity of the degree of $W$.

Each step of the Section 4 argument is formal. The two-dimensional sigma model with curved compact target is an interacting quantum field theory: its short-distance (ultraviolet) behavior requires renormalization — defining the theory as a limit in which a microscopic cutoff is removed while finitely many observables stay finite — and the model is asymptotically free, meaning its effective coupling tends to zero at short distances but grows with distance, so the theory is strongly coupled in the infrared and expected to have a mass gap (a strictly positive lowest nonzero energy level). The Hilbert space of 'forms on loop space' is not a mathematical object as it stands: quantized fields in two dimensions are distributions rather than smooth maps, and there is no rotation-invariant Lebesgue-type measure on an infinite-dimensional space. Constructive field theory is the program that defines such models rigorously, typically as probability measures on spaces of distributions satisfying the Osterwalder–Schrader axioms — Euclidean invariance, reflection positivity, and regularity conditions whose reconstruction theorem yields a unitary relativistic theory. It has succeeded in two dimensions for super-renormalizable interactions (interactions with only finitely many divergent diagrams, such as polynomial potentials), but no interacting asymptotically free scalar model at a finite number of fields has ever been constructed. Witten's same-year paper 'Constraints on supersymmetry breaking' introduced the Witten index as the diagnostic for spontaneous supersymmetry breaking — the absence of any zero-energy ground state, not merely the pairing of positive-energy states — and analyzed scalar models with polynomial superpotentials; the concluding discussion of the present paper takes up the cubic example $W(\phi) = \phi^3 + b\phi$.

The formal loop-space ideas of the paper nonetheless seeded rigorous mathematics: Floer built 'Witten complexes' for infinite-dimensional variational problems satisfying special compactness conditions, founding Floer homology (a homology theory for infinite-dimensional manifolds built from the critical points and gradient-flow lines of a functional), and topological field theory grew out of the same localization philosophy. What remains missing is the object these formal manipulations presuppose: the sigma model itself as a rigorous quantum theory, and with it a proof of the asymptotic expansion in $1/s$.

Problem Statement

This is the first of the unresolved issues in the concluding discussion of Witten's 'Supersymmetry and Morse theory' (1982), Section 4: the supersymmetric nonlinear sigma model in two dimensions requires a rigorous mathematical foundation — the construction of the Hamiltonian $H_s = d_sd_s^* + d_s^*d_s$ on the infinite-dimensional loop space $\Omega(B)$ (where $d_s = d + s\,i(K)$, $K$ is the Killing field generated by rotations of the circle, and $i(K)$ is contraction), and the justification of the asymptotic expansion in $1/s$ on which the paper's loop-space index computations, such as $\operatorname{Tr}(-1)^F = \chi(B)$, rest. In modern terms the task is the constructive quantization of the two-dimensional supersymmetric nonlinear sigma model. A solution must produce, for a general compact curved target $B$ (the genuinely interacting case; a flat target reduces to free compact bosons and is exactly solvable), either (a) a Hamiltonian construction — a separable Hilbert space carrying self-adjoint operators that satisfy the supersymmetry algebra of the model, realize the paper's deformation family $d_s$ and $H_s$ for all $s \geq 0$, with the undeformed supersymmetric sigma model realized within the family, and for which the Witten index is well defined and equals $\chi(B)$, together with a proof of the large-$s$ expansion: a controlled asymptotic formula in $1/s$, with explicitly bounded remainder, for the relevant spectral or trace quantities, exhibiting localization onto the constant loops; or (b) an equivalent Euclidean constructive construction — Osterwalder-Schrader-positive Schwinger functions (Euclidean correlation functions) for the model on the plane or torus, satisfying the supersymmetric Ward identities, which are the identities among correlation functions that follow from the supersymmetry of the action, from which the loop-space Hamiltonian picture and its $1/s$ expansion follow. Perturbative constructions (formal power series in the coupling), lattice regularizations without a proven continuum and infinite-volume limit, the finite-dimensional quantum-mechanical analogs, and special exactly solvable or large-$N$ (number of fields tending to infinity) targets do not by themselves resolve the problem.

References

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