Non-abelian quantum field theories on an algebraic curve: explicit multiplicative Ward identities and uniqueness of the expectation-value functional

Witten's 1988 paper formulated two-dimensional chiral conformal field theory on an arbitrary smooth projective algebraic curve in algebraic (adelic and residue) terms, tied the free-fermion theory to the infinite Grassmannian and its determinant and Pfaffian line bundles, and introduced a multiplicative analogue of the additive Ward identities of current algebra, together with an influential analogy between conformal field theory and automorphic representations. The construction is complete only for free fermions (uniqueness of the expectation functional when the spin structure has no global sections) and, after Takhtajan's later work, for the abelian additive- and multiplicative-boson theories; Witten himself emphasized that for curves of genus at least one the Ward identities, even in the Lie-algebra case, do not determine the expectation-value functional uniquely. The present problem is the non-abelian tier of this program: construct, within Witten's formalism, the quantum field theories whose global symmetries are the rational maps of the curve into a semisimple Lie algebra (current algebras) or into a Lie group (loop groups) — deriving the explicit non-abelian multiplicative Ward identities with normalized operator insertions — and solve the associated uniqueness problem, determining the dimension of the space of admissible invariant functionals and the additional conditions that single out the expectation functional. No construction or uniqueness result for the non-abelian case is known; the problem remains open.

Background

Two-dimensional chiral conformal field theory (CFT) — quantum field theory of fields depending on a single complex coordinate, with holomorphic factorization — admits a purely algebraic reformulation on an arbitrary smooth projective algebraic curve $X$ over a field $k$. In this reformulation, which goes back to Serre's adelic description of the cohomology of curves and to Tate's general residue theorem for differentials on curves, one replaces Laurent expansions at one point by the ring of adèles $\mathbb{A}$ of $X$ — the restricted product of the fields $k((t_P))$ of formal Laurent series over all points $P \in X$ — together with its subring of integer adèles $\mathbb{O}$ built from the $k[[t_P]]$. A basic dictionary identifies isomorphism classes of rank-$n$ vector bundles on $X$ with the double quotient $GL_n(k(X)) \backslash GL_n(\mathbb{A}) / GL_n(\mathbb{O})$, where $k(X)$ is the field of rational functions on $X$.

Witten's 1988 paper, with its companion “Free fermions on an algebraic curve”, placed free chiral fermions on $X$ in this setting and connected them to the Sato–Segal–Wilson infinite Grassmannian $Gr$: the infinite-dimensional manifold (ind-scheme) of subspaces of a polarized Hilbert or Laurent-series space commensurable with a reference half-space. Krichever's construction — the algebraic-geometric construction of solutions of the soliton equations from the data of a pointed curve with a marked point and a local parameter — embeds that data into $Gr$, so that fermionic correlation functions on $X$ become holomorphic sections over $Gr$; $Gr$ carries natural determinant line bundles of the Dolbeault operators coupled to the data encoded by a point $W \in Gr$, and, in antisymmetric-pairing situations, Pfaffian line bundles — square-root-type analogues, with the Pfaffian $Pf(D_W)$ a section of the dual Pfaffian line bundle — whose holomorphic sections are the τ-functions of the Kadomtsev–Petviashvili (KP) integrable hierarchy, the soliton equations of mathematical physics. In the algebraic formulation, the free-fermion theory on $X$ is specified by a Fock space $F_X$ (a restricted tensor product of local Clifford-algebra modules over the points of $X$) and an expectation-value linear functional $\langle \cdot \rangle : F_X \to k$ generalizing vacuum correlation functions.

Invariance of the free theory is expressed by Witten's additive Ward identities: $\langle f \cdot v \rangle = 0$ for all meromorphic sections $f$ of the relevant bundle acting through the Clifford algebra — quantum conservation laws that encode the global residue theorem $\sum_{P \in X} \mathrm{Res}_P(f\, dg) = 0$. Witten proved that when the spin structure has no global holomorphic sections, the additive Ward identities determine $\langle \cdot \rangle$ uniquely. He then introduced the multiplicative analogue for group-valued symmetries: in the abelian case $G = GL(1)$, the identities for $G$-valued rational maps express A. Weil's reciprocity law via the tame symbol — the local two-function pairing whose product formula is Weil reciprocity — with operator insertions carrying explicit conformal-normalization factors; the non-abelian case was treated only in outline.

For non-free theories the symmetry is enlarged: for current algebras (the affine Kac–Moody algebras, central extensions of the loop algebras of a finite-dimensional semisimple Lie algebra $\mathfrak{g}$ — the chiral symmetry algebras of Wess–Zumino–Witten models) the global symmetries are the rational maps $X \to \mathfrak{g}$, and for the group version the rational maps $X \to G$ (loop groups). Witten emphasized (Sect. IV of the paper) that on a curve of genus at least one the Ward identities, even in the Lie-algebra case, do not determine the expectation functional uniquely: the space of invariant Ward functionals, in his notation the quotient $W/\mathscr{G}W$ dual to the space of admissible expectation functionals, is not shown to be one-dimensional, and no canonical selection principle was given. Takhtajan, who axiomatized and developed this program, called the search for additional conditions determining $\langle \cdot \rangle$ uniquely “the main problem of constructing quantum field theories on an algebraic curve”, and solved the simplest case: for the abelian Lie algebra $k$ and its group $k^*$ (the additive- and multiplicative-boson theories) he proved that extending the symmetries from rational functions to algebraic analogues of multi-valued functions (abelian integrals of the second and third kind with zero periods) determines the expectation functional uniquely. The non-abelian case — current algebras of semisimple $\mathfrak{g}$ and loop-group symmetries — is the subject of the present problem. Witten's paper also drew an influential analogy between this adelic CFT and the automorphic representations of function fields (realized later in the geometric Langlands program), and proposed a residue-based string field theory; those threads, and the rigorous Pfaffian geometry, are adjacent results outside this problem's completion boundary.

Problem Statement

Let $X$ be a smooth projective algebraic curve over an algebraically closed field $k$ of characteristic zero (analytically, a compact Riemann surface) of arbitrary genus $g \geq 0$. Within Witten's algebraic formalism for quantum field theory on $X$ — the Fock-type module $F_X$ built from local data at the points of $X$, acted on by meromorphic sections through a Clifford-type algebra, together with the expectation-value functional $\langle \cdot \rangle : F_X \to k$ — construct the non-abelian theories, in two linked parts.

(1) Transformation law. For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$ and its simply connected group $G$, derive the explicit action of the global symmetries of the theory — the rational maps $X \to \mathfrak{g}$ (current-algebra case) and the rational maps $X \to G$ (loop-group case) — on the Fock-type module, including the central-extension cocycles, and state the resulting Ward identities: the additive identities for $\mathfrak{g}$-valued rational maps and the multiplicative Ward identities for $G$-valued rational maps, with the normalization factors of operator insertions given in closed form, at the level of explicitness of Witten's abelian $GL(1)$ identities and consistent with them in that limit.

(2) State Selection. For $g \geq 1$, compute the dimension of the space of invariant Ward functionals — Witten's quotient $W/\mathscr{G}W$, dual to the space of admissible expectation functionals — for the current-algebra and loop-group theories, and identify additional structural conditions (the non-abelian analogue of Takhtajan's multi-valued additive and multiplicative functions) under which $\langle \cdot \rangle$ is uniquely determined, or prove that no such canonical choice exists within the formalism.

The construction is to carry a uniqueness theorem valid for all semisimple $G$ and all genera, or, per Part (2), a proof that no such canonical choice of $\langle \cdot \rangle$ exists within the formalism; it reduces for $G = GL(1)$ to Takhtajan's abelian theory and for the free-fermion sector to Witten's additive identities. Both parts serve the single objective of completing the non-free, non-abelian tier of the free → abelian → non-abelian ladder of algebraic quantum field theories on curves.

References

1. Edward Witten, “Quantum field theory, Grassmannians, and algebraic curves”, Communications in Mathematical Physics 113 (1988) 529–600, DOI: 10.1007/BF01223238, URL: https://doi.org/10.1007/BF01223238
2. Edward Witten, “Free fermions on an algebraic curve”, Proceedings of Symposia in Pure Mathematics 48 (1988) 329–344, DOI: 10.1090/pspum/048/974345, URL: https://doi.org/10.1090/pspum/048/974345
3. Leon A. Takhtajan, “Quantum field theories on an algebraic curve”, Letters in Mathematical Physics 52 (2000) 79–91, DOI: 10.1023/A:1007605903393, URL: https://doi.org/10.1023/A:1007605903393
4. Leon A. Takhtajan, “Quantum field theories on algebraic curves. I. Additive bosons”, Izvestiya: Mathematics 77 (2013) 378–406, arXiv: 0812.0169, DOI: 10.1070/IM2013v077n02ABEH002640, URL: https://arxiv.org/abs/0812.0169 (Theorem 5.4 on uniqueness via multi-valued functions appears in arXiv v1 (2008) and v2 (2009) of 0812.0169, posted under the earlier title “Quantum field theories on algebraic curves and A. Weil reciprocity law”; v3, the published version, removes the multiplicative section)
5. Siddhartha Sen, A. K. Raina, “Grassmannians, multiplicative Ward identities and theta-function identities”, Physics Letters B 203 (1988) 256–262, DOI: 10.1016/0370-2693(88)90548-5, URL: https://doi.org/10.1016/0370-2693(88)90548-5
6. T. R. Ramadas, “Rational and global forms of certain Chiral Conformal Field Theories; Vertex Algebras I”, arXiv: 1401.1969 (single version, posted 2014-01-09, labeled “Preliminary version”), URL: https://arxiv.org/abs/1401.1969
7. David Borthwick, “The Pfaffian line bundle”, Communications in Mathematical Physics 149 (1992) 463–493, DOI: 10.1007/BF02096939, URL: https://doi.org/10.1007/BF02096939
8. Arthur Jaffe, Andrzej Lesniewski, Jonathan Weitsman, “Pfaffians on Hilbert space”, Journal of Functional Analysis 83 (1989) 348–363, DOI: 10.1016/0022-1236(89)90024-4, URL: https://doi.org/10.1016/0022-1236(89)90024-4
9. John Palmer, “Pfaffian bundles and the Ising model”, Communications in Mathematical Physics 120 (1989) 547–574, DOI: 10.1007/BF01260387, URL: https://doi.org/10.1007/BF01260387
10. John Palmer, “Determinants of Cauchy–Riemann operators as τ-functions”, Acta Applicandae Mathematicae 18 (1990) 199–223, DOI: 10.1007/BF00049126, URL: https://doi.org/10.1007/BF00049126
11. Simon G. Scott, “Determinants of Dirac boundary value problems over odd-dimensional manifolds”, Communications in Mathematical Physics 173 (1995) 43–76, DOI: 10.1007/BF02100181, URL: https://doi.org/10.1007/BF02100181
12. Simon G. Scott, Krzysztof P. Wojciechowski, “Determinants, Grassmannians and elliptic boundary problems for the Dirac operator”, Letters in Mathematical Physics 40 (1997) 135–145, DOI: 10.1023/A:1007391620108, URL: https://doi.org/10.1023/A:1007391620108
13. Richard A. Wentworth, “Precise constants in bosonization formulas on Riemann surfaces. I”, Communications in Mathematical Physics 282 (2008) 339–355, DOI: 10.1007/s00220-008-0560-z, URL: https://doi.org/10.1007/s00220-008-0560-z
14. A. A. Beilinson, V. V. Schechtman, “Determinant bundles and Virasoro algebras”, Communications in Mathematical Physics 118 (1988) 651–701, DOI: 10.1007/BF01221114, URL: https://doi.org/10.1007/BF01221114
15. Yongchang Zhu, “Global vertex operators on Riemann surfaces”, Communications in Mathematical Physics 165 (1994) 485–531, DOI: 10.1007/BF02099421, URL: https://doi.org/10.1007/BF02099421
16. A. K. Raina, “Chiral fermions on a compact Riemann surface: a sheaf cohomology analysis”, Letters in Mathematical Physics 19 (1990) 1–5, DOI: 10.1007/BF00402254, URL: https://doi.org/10.1007/BF00402254
17. Toshiyuki Katsura, Yuji Shimizu, Kenji Ueno, “New bosonization and conformal field theory over Z”, Communications in Mathematical Physics 121 (1989) 603–627, DOI: 10.1007/BF01218158, URL: https://doi.org/10.1007/BF01218158
18. A. Álvarez Vázquez, J. M. Muñoz Porras, F. J. Plaza Martín, “The algebraic formalism of soliton equations over arbitrary base fields”, arXiv: alg-geom/9606009, URL: https://arxiv.org/abs/alg-geom/9606009
19. J. M. Muñoz Porras, F. J. Plaza Martín, “Equations of the moduli of pointed curves in the infinite Grassmannian”, Journal of Differential Geometry 51 (1999), no. 3, 431–469, arXiv: alg-geom/9711022, DOI: 10.4310/jdg/1214425138, URL: https://arxiv.org/abs/alg-geom/9711022
20. Francisco J. Plaza Martín, “Arithmetic infinite Grassmannians and the induced central extensions”, Collectanea Mathematica 61 (2010) 107–129, arXiv: 0810.0354, DOI: 10.1007/BF03191229, URL: https://arxiv.org/abs/0810.0354
21. José M. Muñoz Porras, Fernando Pablos Romo, “Generalized reciprocity laws”, Transactions of the American Mathematical Society 360 (2008) 3473–3493, DOI: 10.1090/S0002-9947-08-04554-6, URL: https://doi.org/10.1090/S0002-9947-08-04554-6
22. José M. Muñoz Porras, Fernando Pablos Romo, Francisco J. Plaza Martín, “An explicit reciprocity law associated to some finite coverings of algebraic curves”, Mediterranean Journal of Mathematics 15 (2018), no. 3, article 82, DOI: 10.1007/s00009-018-1125-9, URL: https://doi.org/10.1007/s00009-018-1125-9
23. Edward Frenkel, “Lectures on the Langlands program and conformal field theory”, arXiv: hep-th/0512172 (2005), URL: https://arxiv.org/abs/hep-th/0512172
24. Anton Kapustin, Edward Witten, “Electric–magnetic duality and the geometric Langlands program”, Communications in Number Theory and Physics 1 (2007) 1–236, arXiv: hep-th/0604151, DOI: 10.4310/CNTP.2007.v1.n1.a1, URL: https://arxiv.org/abs/hep-th/0604151
25. David Ben-Zvi, Thomas Nevins, “W-symmetry of the adèlic Grassmannian”, Communications in Mathematical Physics 293 (2010) 185–204, arXiv: 0807.4992, DOI: 10.1007/s00220-009-0925-y, URL: https://arxiv.org/abs/0807.4992
26. F. Balogh, J. Harnad, J. Hurtubise, “Isotropic Grassmannians, Plücker and Cartan maps”, Journal of Mathematical Physics 62 (2021) 021701, arXiv: 2007.03586, DOI: 10.1063/5.0021269, URL: https://arxiv.org/abs/2007.03586
27. I. M. Krichever, “Methods of algebraic geometry in the theory of non-linear equations”, Russian Mathematical Surveys 32 (1977), no. 6, 185–213, DOI: 10.1070/rm1977v032n06abeh003862, URL: https://doi.org/10.1070/rm1977v032n06abeh003862
28. Graeme Segal, George Wilson, “Loop groups and equations of KdV type”, Publications Mathématiques de l'IHÉS 61 (1985) 5–65, DOI: 10.1007/BF02698802, URL: https://doi.org/10.1007/BF02698802
29. Jean-Pierre Serre, “Algebraic Groups and Class Fields”, Graduate Texts in Mathematics 117, Springer, New York, 1988, DOI: 10.1007/978-1-4612-1035-1, URL: https://doi.org/10.1007/978-1-4612-1035-1
30. John Tate, “Residues of differentials on curves”, Annales scientifiques de l'École normale supérieure (4) 1 (1968) 149–159, DOI: 10.24033/asens.1162, URL: https://doi.org/10.24033/asens.1162