Quantum Information Geometry and the Connes–Araki–Haagerup Cones

Abstract: The profound interplay between von Neumann algebras and quantum field theory has increasingly highlighted their importance in higher category theory and topology. A central insight emerges from Tomita–Takesaki theory, which studies the modular automorphism groups of von Neumann algebras. Using techniques from affine differential geometry, we establish an explicit connection between the Connes–Araki–Haagerup cones—objects invariant under modular operators—and geometric structures intrinsic to the axiomatization of 2D topological quantum field theory (TQFT).

We demonstrate that these strictly convex symmetric cones possess a pre-Frobenius structure and contain a submanifold satisfying the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation, thereby forming a Frobenius submanifold. This result reveals new and concrete relationships between objects invariant under modular operators and low-dimensional TQFTs, with additional implications for quantum information geometry.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

---

Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

***** We're transitioning to a new platform google meet. Please bear with us and we apologize for the inconvenience! ****

---