nLab twisted de Rham cohomology

Idea

Twisted de Rham cohomology is the twisted cohomology-version of de Rham cohomology, a simple example of twisted differential cohomology.

For degree-3 twists this is the codomain of the twisted Chern character on twisted K-theory, and in its orbifold cohomology-generalization it is the codomain of the twisted equivariant Chern character on twisted equivariant K-theory.

Definition

Plain

Definition

(1-twisted de Rham cohomology)
(…)

Definition

(3-twisted de Rham cohomology)
For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential

$$d + H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$$

Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.

Remark

There is also the cohomology of the chain complex whose maps are just multiplication by $H$.

$$H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$$

This is also called H-cohomology (Cavalcanti 03, p. 19).

Equivariant

We discuss notions of twisted de Rham cohomology on (global quotient) orbifolds, as they are used for the codomain of the twisted equivariant Chern character on twisted equivariant K-theory.

This combines the above twistings in degrees 1 and 3, the latter induced from the curvature 3-form on a twisting 3-class, the former (an “inner local system”) induced from the flat connection on the circle principal bundle (over the inertia orbifold) which is classified by the transgression of the 3-twist to a 2-class. This requires that this connection be flat, hence that the transgressed 2-class is torsion, which is guaranteed by a Lemma that we discuss first, in Transgression of the 3-twist to a 1-twist on Inertia.

In all of the the following:

• Let $G$ be a finite group.

  • For $g \in G$ any element, write $C_G(g) \,\subset\, G$ for its centralizer subgroup.

  • Write $C_G(g) \times \mathbb{Z}$ for the direct product group with the additive abelian group of integers.

• Let

  (1)$$X \in Proper G Actions(SmthMfds)$$

  be a proper smooth G-manifold, hence a smooth manifold equipped with a proper action (from the right, say) of $G$ by diffeomorphisms.

  • Notice that for any $g \in G$ the fixed locus

    (2)$$X^g \xhookrightarrow{\;i_g\;} X$$

    is a smooth submanifold (by this Prop.).

All twisting classes, in the following are in ordinary Borel-equivariant cohomology, hence in the ordinary cohomology of a Borel construction. The “3-twist” and “torsion 2-twist” are in integral cohomology (of the Borel construction) and the “1-twist” has coefficients a finite cyclic group.

Transgression of 3-twist to 1-twist on inertia

We discuss the transgression of a 3-twist on a (global quotient) orbifold to a 1-twist on its inertia orbifold, namely to a torsion 2-class:

1. via the Künneth theorem

   as argued in Becerra & Uribe 2009, Section 3.2;

2. via a Serre spectral sequence

   as sketched in Freed, Hopkins & Teleman 07, (3.5).

There is also a corresponding argument in terms of bundle gerbes, given in Tu & Xu 2006, Prop. 2.6 & 3.6.

Via Künneth theorem

Throughout, fix an element $g \in G$.

Consider the right group action of the direct product group $C_G(g) \times \mathbb{Z}$ (of the centralizer subgroup with the integers) on the fixed locus $X^g$ (2) given by

(3)$$\array{ X^g \times C_G(g) \times \mathbb{Z} &\xrightarrow{\;\;\;\;}& X^g \\ \big( x, \, (h, n) \big) &\mapsto& x \cdot h \cdot g^n \mathrlap{\,.} }$$

and notice that the following function is a group homomorphism (by the fact that all elements of $C_G(g)$ commute with $g$ in $G$):

(4)$$\array{ C_G(g) \times \mathbb{Z} &\xrightarrow{\;\;\phi_g\;\;}& G \\ (h,n) &\mapsto& h \cdot g^n \mathrlap{\,.} }$$

In view of the action (3), the homomorphism (4) induces a map of Borel constructions

$$\big( X^g \sslash C_G(g) \times B \mathbb{Z} \big) \;\simeq\; X^g \sslash \big( C_G(g) \times \mathbb{Z} \big) \xrightarrow{ \;\; i_g \sslash \phi_g \;\; } X \sslash G \,,$$

(where the homotopy equivalence shown on the left follows since the group action of $\mathbb{Z}$ on $X^g$ is the trivial action, by definition (3), and using that the Borel construction on a point is the classifying space $\ast \sslash \mathbb{Z} \,\simeq\, B \mathbb{Z} \simeq S^1$, hence the circle, in the present case)

and hence the corresponding pullback in integral cohomology:

(5)$$\begin{array}{rrl} H^\bullet \big( X \sslash G \; \, \mathbb{Z} \big) & \xrightarrow{ \; (i_g \sslash \phi_g)^\ast \; } & H^\bullet \big( ( X^g \sslash C_G(g) ) \times B \mathbb{Z} \; \, \mathbb{Z} \big) \\ & \,\simeq\, & H^\bullet \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \otimes_{\mathbb{Z}} \underset{ \mathclap{ \simeq \left\{ \array{ \mathbb{Z} &\vert& \bullet \in \{0,1\} \\ 0 &\vert& else } \right. } }{ \underbrace{ H^\bullet \big( B \mathbb{Z} ;\, \mathbb{Z} \big) } } \\ &\simeq & H^\bullet \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \,\oplus\, H^{\bullet-1} \big( ( X^g \sslash C_G(g) ) \; \, \mathbb{Z} \big) \,, \end{array}$$

where the second line uses the Künneth theorem.

(The following definition becomes a proposition if one uses conceptualization of transgression as laid out in transgression in group cohomology.)

Definition

For $g \in G$ write

$$\tau_g \;\colon\; H^\bullet \big( X \sslash G ;\, \mathbb{Z} \big) \xrightarrow{\;\;} H^{\bullet-1} \big( ( X^g \sslash C_G(g) ) ;\, \mathbb{Z} \big)$$

for the composite of (5) with projection onto the second direct summand.

Proposition

The image of the transgression map $\tau_g$ (Def. ) is in the torsion subgroup.

(Becerra & Uribe 2009, Lemma 3.2)

Proof

The group homomorphism (4) factors through the cyclic group $\mathbb{Z} \to \langle g\rangle \to G$ which is generated by $g \in G$. By the assumption that $G$ is a finite group, so that its integral cohomology is entirely torsion. Since the transgression map factors through a tensor product with pullback along this factorization, by definition, the claim follows.

Via the Serre spectral sequence

Lemma

For each $g \in G$ and each point in the Borel construction $X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle}$, there is a homotopy fiber sequence of the form

(6)$$\array{ B \langle{g}\rangle &\longrightarrow& X^g \!\sslash\! \langle{g}\rangle \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \ast &\longrightarrow& X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} }$$

Proof

First notice that the short exact sequence

$$1 \to \langle{g}\rangle \hookrightarrow C_G(g) \twoheadrightarrow G/\langle{g}\rangle \to 1$$

deloops to a homotopy fiber-sequence of the form

(7)$$\array{ B \langle{g}\rangle &\longrightarrow& B C_G(g) \\ && \big\downarrow \\ && B \frac{C_G(g)}{\langle{g}\rangle} }$$

(using this Prop.).

Then notice that we have a pasting diagram of homotopy pullbacks as follows:

$$\array{ B \langle{g}\rangle &\longrightarrow& \ast \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ X^g \!\sslash\! C_G(g) &\xrightarrow{\;}& X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} \\ \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ B G &\xrightarrow{\;}& B \frac{C_G(g)}{\langle{g}\rangle} }$$

Here:

• the bottom square being a homotopy pullback expresses that the group action of $\langle{g}\rangle$ on $X^g$ is trivial, so that the group action of $C_G(g)$ on $X^g$ is induced by that of $C_G(g)/\langle{g}\rangle$;

• the total rectangle is a homotopy pullback by (7);

• hence the top square is a homotopy pullback by the pasting law.

It follows – once we know (do we?) that the action of the fundamental group of $X^g \sslash \frac{C_G(g)}{\langle{g}\rangle}$ on the integral cohomology of $B\langle{g}\rangle$ is trivial – that we have a Serre spectral sequence

(8)$$H^p \left( X^g \!\sslash\! \frac{C_G(g)}{\langle{g}\rangle} ;\, H^q \big( B\langle{g}\rangle{g} ;\, \mathbb{Z} \big) \right) \;\; \overset{\;\;\;\;}{\Rightarrow} \;\; H^{p+q} \big( X^g \!\sslash\! C_G(g) ;\, \mathbb{Z} \big) \,.$$

But since $\langle{g}\rangle \simeq \mathbb{Z}/(ord(g))$ is a cyclic group (by the assumption that $G$ is a finite group) it follows (by this Prop. or Lem. 4.51 in BSST 07) that its group cohomology is concentrated in even degrees:

$$H^n \big( B \langle{g}\rangle ;\, \mathbb{Z} \big) \;=\; H^n_{grp} \big( \langle{g}\rangle ;\, \mathbb{Z} \big) \;\simeq\; \left\{ \begin{array}{ccl} \mathbb{Z} &\vert& n = 0 \\ \mathbb{Z}/ord(g) &\vert& n = \,\text{positive multiple of}\, 2 \\ 0 &\vert& \text{otherwise} \end{array} \right.$$

Therefore the spectral sequence (8) associates with any 3-twist $\alpha \mapsto \alpha_{\vert X^g} \in H^3\big( X^g \sslash C_G(g);\, \mathbb{Z} \big)$ a “transgressed” degree-1 class

(9)$$\tau_g(\alpha) \;\in\; H^1 \left( X^g ;\, \mathbb{Z}/ord(g) \right) \xrightarrow{\;\;\beta\;\;} H^2 \left( X^g ;\, \mathbb{Z} \right) \,,$$

which the Bockstein homomorphism identifies with a torsion 2-class in integral cohomology.

Essentially this conclusion is claimed as FHT 07, (3.5).

Definition

Now we can indicate the definition of the twisted equivariant de Rham cohomology. In outline:

Write $\Lambda (\prec (X \sslash G))$ for the inertia orbifold of the global quotient orbifold of $X$ (1).

For $\alpha \in H^3\big( X \sslash G; \, \mathbb{Z} \big)$ a “3-twist” in the degree-3 integral cohomology of the homotopy quotient (Borel construction) of $X$ by $G$ (1)

• let $H_3 \in \Omega^3\big( \Lambda (\prec (X \sslash G)) \big)$ be a de Rham image of $\alpha$ pulled back to the inertia orbifold,

• let $\nabla$ be a connection on the transgression (Def. ) of $\alpha$ to the inertia orbifold, which is flat by Prop. or, alternatively, by (9).

Then equivariant twisted de Rham cohomology of $X$ is the de Rham cohomology of $\Lambda (\prec (X \sslash G))$ which is both

• 1-twisted by $\nabla$ and

• 3-twisted by $H_3$

(Tu & Xu 2006, Def. 3.10, Freed, Hopkins & Teleman 2007, (3.19), Bunke, Spitzweck & Schick 08, Def. 3.15).

Properties

Of 1-Twisted de Rham cohomology

Equivalence with $\pi_1$-invariant dR cohomology

In the following, let $\mathrm{X}$ be a smooth manifold, which we assume, without real restriction of generality, to be connected. Therefore we write $\pi_1(\mathrm{X})$ for its fundamental group, for any fixed choice of basepoint $x_0 \,\in\, \mathrm{X}$.

We write $\widehat{\mathrm{X}} \xrightarrow{\;} \mathrm{X}$ for its universal cover (also canonically a smooth manifold). This is a $\pi_1(\mathrm{X})$-principal bundle, in particular it has a canonical $\pi_1(\mathrm{X})$-action by deck transformations.

For $\mathcal{L}$ a complex line bundle over $\mathrm{X}$ with flat connection $\nabla$, write

(10)$$hol_\nabla \,\colon\, \pi_1(\mathrm{X}) \xrightarrow{\;\;} \mathbb{C}^\times$$

for the group homomorphism from the fundamental group to the multiplicative group of units $\mathbb{C}^\times \,=\, \mathbb{C} \setminus \{0\}$ which is given by sending a smooth curve $\lambda \colon [0,1] \to \mathrm{X}$ (with $\lambda(0) = \lambda(1)$) to its holonomy under the parallel transport with respect to $\nabla$.

Proposition

(1-Twisted dR cohomology equivalent to $\pi_1$-invariant dR cohomology on universal cover)
For $\mathcal{L}$ a complex line bundle over $\mathrm{X}$ with flat connection $\nabla$, there is a natural isomorphism between

1. the $\nabla$-twisted de Rham cohomology on $\mathrm{X}$:

   $$H^\bullet \Big( \Omega_{dR}^\bullet \big( \mathrm{X} ;\, \mathcal{L} \big) ,\, \nabla \Big)$$

2. the untwisted but $\pi_1$-invariant complex-valued de Rham cohomology on the universal cover $\widehat{\mathrm{X}}$

   $$H^\bullet \bigg( \Big( \Omega^\bullet_{dR} \big( \widehat{\mathrm{X}} ;\, \mathbb{C} \big) \Big)^{\pi_1(\mathrm{X})} ,\, \mathrm{d} \bigg)$$

   where $\pi_1(\mathrm{X})$ acts on differential forms by pullback along deck transformations combined with multiplication by the holonomy (10) of $\nabla$:

   $$\array{ \pi_1(\mathrm{X}) \times \Omega^\bullet_{\mathrm{dR}} \big( \widehat{\mathrm{X}} ;\, \mathbb{C} \big) &\xrightarrow{\phantom{--}}& \Omega^\bullet_{\mathrm{dR}} \big( \widehat{\mathrm{X}} ;\, \mathbb{C} \big) \\ \big( [\lambda] ,\, A \big) &\mapsto& hol_\nabla(\lambda) \cdot [\lambda]^\ast(A) }$$

A proof is spelled out in this pdf. It proceeds by observing that the bundle and its connection trivialize after pullback to the universal cover $\widehat{\mathrm{X}}$, in fact that the local connection form $\widehat{\omega}$ on $\widehat{\mathrm{X}}$ becomes exact (by the Poincaré lemma for simply connected domains, here):

(11)$$\underset{ \mathclap{\phantom{\vert^{\vert}}} { \ell \in } \atop { C^\infty\big( \widehat{\mathrm{X}} ;\, \mathbb{C}^\times \big) } }{\exists} \;\;\; \widehat{\omega} \;=\; - \mathrm{d} log \ell \,, \;\;\;\; \in \; \Omega^1_{dR} \big( \widehat{\mathrm{X}} ,\, \mathbb{C} \big) \,,$$

and checking that multiplication by this potential $\ell$ (11) constitutes a isomorphic (bijective) chain map between the cochain complexes in question.

Example

(Hypergeometric integral solutions of KZ-equation)
For the special case that the complex line bundle $\mathcal{L}$ is trivial (so that the flat connection $\nabla$ is represented by a globally defined differential 1-form already on the base manifold $\mathrm{X}$) the statement of Prop. (or rather its holomorphic version) plays a central role in the discussion of the “hypergeometric integral construction” of solutions to the Knizhnik-Zamolodchikov equation, where it is applied to the case that $\mathrm{X}$ is an $n$-punctured Riemann sphere (e.g. a trinion). In fact it is so central to this construction that the function $\ell$ (11) which trivializes the connection form on the universal cover and thereby induces the isomorphism in Prop. came to be called the “master function”, in this context (Slinkin & Varchenko 2019, §2.1).

On the other hand, none of the many references listed there really make the Proposition explicit.

Of 3-Twisted de Rham cohomology

Proposition

If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology (Def. ) of $X$ coincides with its H-cohomology, for any closed 3-form $H$.

(Cavalcanti 03, theorem 1.6).

Related concepts

• twisted Chern character, twisted equivariant Chern character

• inner local system

• Hanany-Witten effect

References

Twist in degree 1

The classical case of twisted de Rham cohomology with twists in degree 1, given by flat connections on flat line bundles and more generally on flat vector bundles), and its equivalence to sheaf cohomology with coefficients in abelian sheaves of flat sections (local systems):

• Claire Voisin (translated by Leila Schneps), Section II 5.1.1 of: Hodge theory and Complex algebraic geometry II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3 (doi:10.1017/CBO9780511615177)

• Alexandru Dimca, Section 2.5 of: Sheaves in Topology, Universitext, Springer (2004) $[$doi:10.1007/978-3-642-18868-8$]$

Review:

• Anatoly Libgober, Sergey Yuzvinsky, Cohomology of local systems, Advanced Studies in Pure Mathematics 27, Mathematics Society of Japan (2000) 169-184 [pdf, doi:10.2969/aspm/02710169]

• Cailan Li, Cohomology of Local Systems on $X_\Gamma$ (pdf, pdf)

• Youming Chen, Song Yang, Section 2.1 in: On the blow-up formula of twisted de Rham cohomology. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)

For extensive application, see also the “hypergeometric integral construction” of solutions to the Knizhnik-Zamolodchikov equation.

In this context there is also:

• Alexey Slinkin, Alexander Varchenko, Twisted de Rham Complex on Line and Singular Vectors in $\widehat{\mathfrak{sl}}_2$ Verma Modules, SIGMA 15 (2019) 075 [arXiv:1812.09791, doi:10.3842/SIGMA.2019.075]

See also at supersymmetric quantum mechanics and see

• Albert Schwarz, Ilya Shapiro, Twisted de Rham cohomology, homological definition of the integral and “Physics over a ring”, Nucl. Phys. B 809 (2009) 547-560 [arXiv:0809.0086, doi:10.1016/j.nuclphysb.2008.10.005]

Twist in degree 3

Plain

The concept of $H_3$-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory), in:

• Ryan Rohm, Edward Witten, around (23) and appendix of: The antisymmetric tensor field in superstring theory, Annals of Physics Volume 170, Issue 2, September 1986, Pages 454-489 (doi:10.1016/0003-4916(86)90099-0)

Further discussion (often as the codomain of the twisted Chern character on twisted K-theory):

• Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, Danny Stevenson, Section 9.3 of: Twisted K-theory and K-theory of bundle gerbes , Commun Math Phys, 228 (2002) 17-49 (arXiv:hep-th/0106194, doi:10.1007/s002200200646)

• Varghese Mathai, Danny Stevenson, Section 3 of: Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236:161-186, 2003 (arXiv:hep-th/0201010)

• Daniel Freed, Michael Hopkins, Constantin Teleman, Section 2 of: Twisted equivariant K-theory with complex coefficients, Journal of Topology, Volume 1, Issue 1, 2007 (arXiv:math/0206257, doi:10.1112/jtopol/jtm001)

• Constantin Teleman, around Prop. 3.7 in: K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra, in: Ulrike Tillmann (ed.) Topology, geometry and quantum field theory, Cambridge 2004 (arXiv:math/0306347, spire:660158)

• Gil Cavalcanti, Section I.4 of: New aspects of the $d d^c$-lemma, Oxford 2005 (arXiv:math/0501406)

Equivariant

The generalization of 3-twisted de Rham cohomology to orbifolds (often as the codomain of the twisted equivariant Chern character on twisted equivariant K-theory):

• Varghese Mathai, Danny Stevenson, p. 18 of: Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236 (2003) 161-186 (arXiv:hep-th/0201010, doi:10.1007/s00220-003-0807-7)

  (via equivariant de Rham cohomology)

• Jean-Louis Tu, Ping Xu, Def. 3.10 in: Chern character for twisted K-theory of orbifolds, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 (arXiv:math/0505267, doi:10.1016/j.aim.2005.12.001)

• Ulrich Bunke, Markus Spitzweck, Thomas Schick, Section 3.2 of: Inertia and delocalized twisted cohomology, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) (arXiv:math/0609576, doi:10.4310/HHA.2008.v10.n1.a6)

See also

• Freed, Hopkins & Teleman 07, Paragraph (3.8)

Twist in higher degrees

Discussion of higher-degree twisted de Rham cohomology (often as the Chern character-like image of higher twisted K-theory):

• Teleman 04, Sec. 3

• Hisham Sati, A Higher Twist in String Theory, J. Geom. Phys. 59:369-373, 2009 (arXiv:hep-th/0701232)

• Varghese Mathai, Siye Wu, Analytic torsion for twisted de Rham complexes, J. Diff. Geom. 88:297-332, 2011 (arXiv:0810.4204)

  (relation to analytic torsion)

A spectral sequence for higher twisted de Rham cohomology (analgous to the Atiyah-Hirzebruch spectral sequence for twisted K-theory from Atiyah & Segal 2005):

• Weiping Li, Xiugui Liu, He Wang, On a spectral sequence for twisted cohomologies, Chin. Ann. Math. Ser. B 35, 633–658 (2014) (arXiv:0911.1417, doi:10.1007/s11401-014-0842-z).

In the broader context of twisted cohomology theory and the Chern-Dold character map:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3.3 of: The character map in (twisted differential) non-abelian cohomology (arXiv:2009.11909)

Braid representations via twisted cohomology of configuration spaces

The “hypergeometric integral” construction of conformal blocks for affine Lie algebra/WZW model-2d CFTs and of more general solutions to the Knizhnik-Zamolodchikov equation, via twisted de Rham cohomology of configuration spaces of points, originates with:

• Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$

• Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 $[$doi:10.1142/S0217979290000528$]$

  (for $\mathfrak{sl}(2,\mathbb{C})$)

• Atsushi Matsuo, An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 $[$doi:10.1007/BF02102089$]$

• Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$

• Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$

following precursor observations due to:

• Vladimir S. Dotsenko, Vladimir A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

• Philippe Christe, Rainald Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B

  282 (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

The proof that for rational levels this construction indeed yields conformal blocks is due to:

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$

  (for $\mathfrak{sl}(2, \mathbb{C})$)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 [euclid:cmp/1104272957]

Review:

• Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)

• Ivan Cherednik, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$

• Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$

• Toshitake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 $[$doi:10.5427/jsing.2012.5g, pdf$]$

• Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 $[$doi:10.1007%2Fs40306-014-0088-6, pdf$]$

• Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 $[$pdf, pdf$]$

  (motivation from braid representations)

See also:

• Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$

• Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 $[$ISBN:978-1-4704-1315-6, web$]$

This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:

• Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 $[$doi:10.1007/BF01392549$]$

• Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 $[$doi:10.2969/jmsj/03920191$]$

• Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 $[$pdf$]$

• Vadim Schechtman, H. Terao, Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 $[$arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N$]$

also:

• Peter Orlik, Hypergeometric integrals and arrangements, Journal of Computational and Applied Mathematics 105 (1999) 417–424 $[$doi:10.1016/S0377-0427(99)00036-9, pdf$]$

• Daniel C. Cohen, Peter Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000) 299-316 $[$arXiv:math/9907117, doi:10.4310/MRL.2000.v7.n3.a5$]$

reviewed in:

• Yukihito Kawahara, The twisted de Rham cohomology for basic constructions of hyperplane arrangements and its applications, Hokkaido Math. J. 34 2 (2005) 489-505 $[$doi:10.14492/hokmj/1285766233$]$

Discussion for the special case of level$=0$ (cf. at logarithmic CFT – Examples):

• Fedor A. Smirnov, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, $[$arXiv:hep-th/9210051$]$

• Fedor A. Smirnov, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 $[$doi:10.1007/BF02096723, arXiv:hep-th/9210052$]$

• S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 $[$doi:10.1142/S0217732394001647$]$

Interpretation of the hypergeometric construction as happening in twisted equivariant differential K-theory, showing that the K-theory classification of D-brane charge and the K-theory classification of topological phases of matter both reflect braid group representations as expected for defect branes and for anyons/topological order, respectively:

• Hisham Sati, Urs Schreiber, Anyonic defect branes in TED-K-theory $[$arXiv:2203.11838$]$