A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

Abstract

Abstract. In Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these invariants. A purpose of this note is to show how the conjecture arises. Because of the integral identity's nature we shall give a quick tour on theories of motivic integration, from which lead to a proof of the conjecture for ground fields algebraically closed of characteristic zero. IntroductionHistorically, Thomas in his thesis and his paper [31] introduced an invariant for a 3-dimensional Calabi-Yau manifold M as a counting invariant of coherent sheaves on M , which analogizes the Casson invariant on a real 3-dimensional manifold. This kind of invariant was then named after him and his advisor Donaldson. According to [16], the moduli space M of coherent sheaves on M can be locally presented as the critical locus of a holomorphic Chern-Simons functional f . By this, a Donaldson-Thomas invariant for M can be described as an integral of the Behrend function over the moduli space M. In view of [1], the value of the Behrend function is defined in terms of the Euler characteristic of the Milnor fiber F f of the Chern-Simons functional f , from which the Donaldson-Thomas invariant arises. In [17], replacing the Milnor fiber by a so-called motivic Milnor fiber (defined by Denef-Loeser [8] using motivic integration) Kontsevich and Soibelman study motivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yau manifolds. The theory of Kontsevich and Soibelman allows, under appropriate realizations (e.g., a cohomology functor), to obtain refinements of (classical) Donaldson-Thomas invariants. The motivic Donaldson-Thomas invariants are also realized physically as "BPS invariants". Among 3-dimensional Calabi-Yau categories, the derived category of coherent sheaves on a compact (or local) 3-dimensional Calabi-Yau manifolds is a central object.Let us now give a brief review due to [17] and [18] on the direct elements in KontsevichSoibelman's theory concerning the integral identity conjecture. Let C be an ind-constructible triangulated A ∞ -category over a field k. For any strict sector V in R 2 , we consider a collection of full subcategories C V of C. Then one can construct the motivic Hall algebra H(C) as a graded associative algebra admitting for any strict sector V an element A Hall V invertible in a completion H(C V ) of H(C V ). More precisely, in terms of a countable decomposition Ob(C) = i∈I Y i into 2010 Mathematics Subject Classification. Primary 03C60, 14B20, 14E18, 14G22, 32S45, 11S80. Key words and phrases. motivic integration, formal schemes, rigid varieties, volume Poincaré series, resolution of singularity, integral identity conjecture, definable sets. where 1 S is the identity function interpreted as a counting measure (see [17, Section 6.1]). One requires that A Hall V must satisfy the factorization property that A Hall V = A Hall V 1 · A Hall V 2 , where V = V 1 V 2 and th...