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The Drinfel’d double and twisting in stringy orbifold theory. (English) Zbl 1174.14048

The paper under review deals with the central role that the Drinfel’d double (and its twists) of the group ring of a finite group \(G\) has in stringy orbifold theories (and their twists) as cohomology, Chow, Grothendieck \(K_{0}\) and topological \(K\)-theory.
In the first part of the paper, the authors recall the definition of the twisted Drinfel’d double \(D^{\beta}(k[G])\) of the group ring \(k[G]\) of a finite group \(G\), where \(\beta\in Z^{3}(G,k^{*})\) is a \(3\)-cocycle. It is a quasi-triangular quasi-Hopf \(k\)-algebra which as \(k\)-vector space has a basis given by elements \(g\llcorner_{x}\) for every \(x,g\in G\) (giving a natural \(G\times G\)-grading), and whose algebra and co-algebra structures, Drinfel’d associator, \(R\) matrix and antipode are defined by means of \(\beta\) (see Definition 2.1). If \(\beta=1\), then it is the Drinfel’d double \(D(k[G])\). Moreover, as \(D^{\beta}(k[G])\) is a quasi-triangular quasi-Hopf algebra, the category \(D^{\beta}(k[G])\)-Mod of (left) \(D^{\beta}(k[G])\)-modules has a natural braided monoidal structure, and every left \(D^{\beta}(k[G])\)-module \(A\) has a \(G\)-grading.
In section 3 the authors introduce the notion of \(G\)-Frobenius algebra over a field \(k\) of characteristic 0 with character \(\chi\in \operatorname{Hom}(G,k^{*})\), which is a Frobenius algebra with a \(G\)-grading and an action of \(G\) verifying some compatibility axioms (see Definition 3.12). Any \(G\)-Frobenius algebra is shown to be a \(D(k[G])\)-module by R. M. Kaufmann [J. Algebra 282, 232–259 (2004; Zbl 1106.16040)]. In the paper the authors show that a \(G\)-Frobenius algebra with character \(\chi\) is a unital, associative, commutative algebra object in \(D(k[G])\)-Mod and defines a \(k_{\chi}\)-twisted Frobenius algebra object (see Proposition 3.15). Moreover, the main result of section 3 is Theorem 3.16, where it is shown that a \(G\)-Frobenius algebra with character \(\chi\) is a (non-degenerate) \(\mathbb{I}_{\chi}\)-twisted Frobenius algebra object in \(D(k[G])\)-Mod verifying two axioms (S) and (T), where \(\mathbb{I}_{\chi}\) is an even invertible element in \(D(k[G])\)-Mod associated to \(\chi\), i. e. it is a \(G\)-Frobenius algebra object in \(D(k[G])\)-Mod (see Definition 3.17).
In section 4 the authors relate the (twisted) Drinfel’d double to stringy cohomology theories. Consider a smooth projective variety \(X\) with an action of a finite group \(G\), and let \(I(X,G)\) be the inertia variety, i. e. \(I(X,G)=\coprod_{g\in G}X^{g}\), where \(X^{g}\) is the subset of \(X\) of the points fixed by \(g\). If \(\mathcal{F}\) is any functor among \(H^{*}\) (cohomology), \(K_{0}^{*}\) (Grothendieck \(K_{0}\)), \(A^{*}\) (Chow) and \(K^{\mathrm{top}}\) (topological \(K\)-theory) with coefficients in \(\mathbb{Q}\), then one defines \(\mathcal{F}_{\mathrm{stringy}}(X,G):=\mathcal{F}(I(X,G))\), which as vector space is isomorphic to \(\coprod_{g\in G}\mathcal{F}(X^{g})\). The stringy product giving to \(\mathcal{F}_{\mathrm{stringy}}(X,G)\) the structure of an algebra is defined by means of the obstruction bundle \(\mathcal{R}\). T. Jarvis, R. Kaufmann and T. Kimura [Invent. Math. 168, 23–81 (2007; Zbl 1132.14047)] showed that the cases where \(\mathcal{F}\) is \(H^{*}\) or \(K^{\mathrm{top}}\) yield \(G\)-Frobenius algebras. In Theorem 4.3, the authors show that if \(\mathcal{F}\) is \(A^{*}\) or \(K_{0}\), the stringy functors are \(G\)-Frobenius algebra objects.
Moreover, the authors study the stack case, namely when one considers the stack \([X/G]\). In this case we have three notions of stringy \(K\)-theory: the global stringy \(K\)-theory \(K_{\mathrm{global}}((X,G)):=K(I(X,G))\) (isomorphic as vector space to \(\bigoplus_{g\in G}K(X^{g})\)), the small stringy \(K\)-theory \(K_{\mathrm{small}}([X,G]):=K_{\mathrm{global}}((X,G))^{G}\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K(X^{g})^{Z(g)}\), where \(C(G)\) is the set of conjugacy classes of \(G\) and \(Z(g)\) is the group of elements of \(G\) commuting with \(g\)) and the full stringy \(K\)-theory \(K_{\mathrm{full}}([X,G]):=K(\mathcal{I}[X/G])\), the \(K\)-theory of the inertia stack of \([X/G]\) (isomorphic as vector space to \(\bigoplus_{[g]\in C(G)}K([X^{g}/Z(g)])\)). In the case where \(X=\mathrm{pt}\) the authors show that (as rings) \(K_{\mathrm{global}}(I(\mathrm{pt},G),G)=D(k[G])^{\mathrm{comm}}\) (the subalgebra of \(D(k[G])\) generated by the elements \(g\llcorner x\) such that \([g,x]=e\)) and that \(K_{\mathrm{full}}([\mathrm{pt}/G])\simeq \mathrm{Rep}(D(k[G]))\) (see Proposition 4.10 and Theorem 4.13). This last result can be obtained, at least as vector spaces, by D. S. Freed, M. J. Hopkins and C. Teleman [Twisted \(K\)-theory and loop-group representations, arXiv:math.AT/0312155] and [Loop-groups and twisted \(K\)-theory II, arXiv:math.AT/0511232].
The last part of the paper deals with twistings by \(0\)-, \(1\)- or \(2\)-gerbes: when considering a global quotient \((X,G)\), these are simply elements in \(Z^{i}(G,k^{*})\) for \(i=1,2,3\) respectively. The choice of such an element \(\beta\), i. e. the choice of a gerbe \(\mathcal{G}\), induces a twisting of the algebra structure on \(K_{\mathrm{global}}\). If \(\beta\in Z^{3}(G,k^{*})\) one gets \(K_{\mathrm{global}}^{\beta}(I(\mathrm{pt},G),G)=D^{\beta}(k[G])^{comm}\), getting a geometrical interpretation of the twisted Drinfel’d double. When considering the stack case, in Theorem 5.8 the authors show that \(K_{\mathrm{full}}^{\beta}([\mathrm{pt}/G])=D^{\beta}(k[G])\). The final sections deal with a purely algebraic version of the twisting and equivariant \(K\)-theory using the language of modules.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19L47 Equivariant \(K\)-theory
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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