Scientists / Open questions

Edward Witten

String theory; quantum field theory; mathematical physics

1990 Fields Medal

83 questions

All-orders quantum consistency of the heterotic (2,0) supersymmetric sigma model with torsion: higher-loop anomaly cancellation, the b-field transformation law, and compatibility with conformal invariance
Tensionless-string fixed points and the global structure of the six-dimensional N=1 string landscape
First-principles worldsheet derivation of the AdS-deformed constraint equations and alpha-prime structure of vertex operators for the superstring on AdS3 x S3 x M with Ramond-Ramond flux
Quantum completion of the Hořava–Witten boundary construction: the $\delta(0)$ boundary singularities, their finite remainders, and higher-order corrections in eleven-dimensional supergravity on a manifold with boundary
Hamiltonian (phase-space) formulation of covariant open string field theory: symplectic structure, initial-value problem, and Hilbert-space interpretation
Completing the short-distance analysis of weak interactions: rigorous all-orders multi-current operator products in gauge theories and first-principles long-distance rare-kaon amplitudes
Open agenda of the axion topological field theory of 3d time-reversal-invariant topological superconductors: beyond-weak-pairing validity, chiral-vortex realization and fermion-parity pumping, and measurable gravitational (thermal) response in real materials
All-orders large-radius expansion and stable-bundle existence for heterotic superstring compactifications beyond the standard embedding
Non-perturbative foundations of the string duality web: exactness of type IIB $\mathrm{SL}(2,\mathbb{Z})$ self-duality, $U$ -duality groups $G(\mathbb{Z})$ , and the heterotic dualities
Purely three-dimensional Chern-Simons theory with supergroup gauge group: formulation and one-loop level shift
Foundations of fermionic localization for the Schwarzian theory: the SL(2,R)-invariant lattice regulator, a formal Duistermaat-Heckman theorem for symplectic supermanifolds, and the microscopic N=2 winding-saddle sum
The action–partition-function relation in background-independent open string field theory beyond matter–ghost decoupling
Exact solution and quantum consistency of the hybrid-formalism worldsheet CFT for the superstring on $AdS_3\times S^3\times T^4$ (or $K3$ ) with Ramond-Ramond flux away from the Wess-Zumino-Witten point
Background-independent integration on the field-antifield function space and a natural regularization of the Batalin–Vilkovisky Laplacian (the open problem posed in Witten's 1990 antibracket note)
A priori determination of the renormalization constant $\Delta v$ in Witten's two-dimensional quantum gauge theory and its Chern-Simons correspondence on $\Sigma\times S^1$
Intrinsic proof of perturbative unitarity (cutting rules) in covariant string perturbation theory via the Feynman iε integration cycle
An algebraic foundation for the generalized second law at out-of-equilibrium black hole horizon cuts
Rigorous nonperturbative establishment of the 2+1-dimensional boson–fermion duality (3d bosonization) and its duality web
Smooth extension of the superstring measure across the bad locus for super Riemann surfaces with two Ramond punctures
Completing the Lorentzian OPE inversion formula: higher-point correlators, low-spin data, canonical subtractions, and perturbative spin bounds
Completing the mathematical and physical foundations of the Vafa-Witten strong-coupling test of S-duality
Nonperturbative completion of two-dimensional quantum gravity: string-equation integration constants, Stokes data, and the matrix-model-continuum correspondence
Reproducing the torsion part of the heterotic anomaly-cancellation condition from Type IIA on K3: space-time global anomalies of the solitonic heterotic string
The M5-brane instanton one-loop determinant: the holomorphic prefactor of non-perturbative superpotentials in M-theory on Calabi-Yau four-folds

Back to open problems

All-orders quantum consistency of the heterotic (2,0) supersymmetric sigma model with torsion: higher-loop anomaly cancellation, the b-field transformation law, and compatibility with conformal invariance

Tensionless-string fixed points and the global structure of the six-dimensional N=1 string landscape

First-principles worldsheet derivation of the AdS-deformed constraint equations and alpha-prime structure of vertex operators for the superstring on AdS3 x S3 x M with Ramond-Ramond flux

Quantum completion of the Hořava–Witten boundary construction: the δ(0)\delta(0)δ(0) boundary singularities, their finite remainders, and higher-order corrections in eleven-dimensional supergravity on a manifold with boundary

Hamiltonian (phase-space) formulation of covariant open string field theory: symplectic structure, initial-value problem, and Hilbert-space interpretation

Completing the short-distance analysis of weak interactions: rigorous all-orders multi-current operator products in gauge theories and first-principles long-distance rare-kaon amplitudes

Open agenda of the axion topological field theory of 3d time-reversal-invariant topological superconductors: beyond-weak-pairing validity, chiral-vortex realization and fermion-parity pumping, and measurable gravitational (thermal) response in real materials

All-orders large-radius expansion and stable-bundle existence for heterotic superstring compactifications beyond the standard embedding

Non-perturbative foundations of the string duality web: exactness of type IIB SL(2,Z)\mathrm{SL}(2,\mathbb{Z})SL(2,Z) self-duality, UUU-duality groups G(Z)G(\mathbb{Z})G(Z), and the heterotic dualities

Purely three-dimensional Chern-Simons theory with supergroup gauge group: formulation and one-loop level shift

Foundations of fermionic localization for the Schwarzian theory: the SL(2,R)-invariant lattice regulator, a formal Duistermaat-Heckman theorem for symplectic supermanifolds, and the microscopic N=2 winding-saddle sum

The action–partition-function relation in background-independent open string field theory beyond matter–ghost decoupling

Exact solution and quantum consistency of the hybrid-formalism worldsheet CFT for the superstring on AdS3×S3×T4AdS_3\times S^3\times T^4AdS3​×S3×T4 (or K3K3K3) with Ramond-Ramond flux away from the Wess-Zumino-Witten point

Background-independent integration on the field-antifield function space and a natural regularization of the Batalin–Vilkovisky Laplacian (the open problem posed in Witten's 1990 antibracket note)

A priori determination of the renormalization constant Δv\Delta vΔv in Witten's two-dimensional quantum gauge theory and its Chern-Simons correspondence on Σ×S1\Sigma\times S^1Σ×S1

Intrinsic proof of perturbative unitarity (cutting rules) in covariant string perturbation theory via the Feynman iε integration cycle

An algebraic foundation for the generalized second law at out-of-equilibrium black hole horizon cuts

Rigorous nonperturbative establishment of the 2+1-dimensional boson–fermion duality (3d bosonization) and its duality web

Smooth extension of the superstring measure across the bad locus for super Riemann surfaces with two Ramond punctures

Completing the Lorentzian OPE inversion formula: higher-point correlators, low-spin data, canonical subtractions, and perturbative spin bounds

Completing the mathematical and physical foundations of the Vafa-Witten strong-coupling test of S-duality

Nonperturbative completion of two-dimensional quantum gravity: string-equation integration constants, Stokes data, and the matrix-model-continuum correspondence

Reproducing the torsion part of the heterotic anomaly-cancellation condition from Type IIA on K3: space-time global anomalies of the solitonic heterotic string

The M5-brane instanton one-loop determinant: the holomorphic prefactor of non-perturbative superpotentials in M-theory on Calabi-Yau four-folds

Quantum completion of the Hořava–Witten boundary construction: the $\delta(0)$ boundary singularities, their finite remainders, and higher-order corrections in eleven-dimensional supergravity on a manifold with boundary

Non-perturbative foundations of the string duality web: exactness of type IIB $\mathrm{SL}(2,\mathbb{Z})$ self-duality, $U$ -duality groups $G(\mathbb{Z})$ , and the heterotic dualities

Exact solution and quantum consistency of the hybrid-formalism worldsheet CFT for the superstring on $AdS_3\times S^3\times T^4$ (or $K3$ ) with Ramond-Ramond flux away from the Wess-Zumino-Witten point

A priori determination of the renormalization constant $\Delta v$ in Witten's two-dimensional quantum gauge theory and its Chern-Simons correspondence on $\Sigma\times S^1$