Holmstrom Quantum sheaf cohomology

Quantum sheaf cohomology

Talk by Prof. Eric Sharpe of Virginia Tech

TITLE:

An introduction to quantum sheaf cohomology

ABS:

In this talk, I will give an introduction to `quantum sheaf cohomology,'

an analogue of ordinary quantum cohomology determined by a space $X$

plus a holomorphic vector bundle ${\cal E} \rightarrow X$ satisfying

certain consistency conditions. Physically, this structure encodes

alpha’-nonperturbative corrections to charged matter couplings in

heterotic strings, such as (27)^3 couplings. Just as ordinary quantum

cohomology gives nonperturbative corrections to classical cohomology

rings, quantum sheaf cohomology gives nonperturbative corrections to

sheaf cohomology rings $H^*(X, \wedge^* {\cal E}^*)$, and reduces to

ordinary quantum cohomology in the special case ${\cal E} = TX$. Just as

ordinary quantum cohomology arises in the study of the A model

topological field theory, quantum sheaf cohomology arises in the A/2

model holomorphic field theory, and plays a role in a generalization of

mirror symmetry known as (0,2) mirror symmetry. After giving a brief

introduction to general aspects of (0,2) mirrors and formal aspects of

quantum sheaf cohomology, we will explain general results for $X$ a

toric variety and ${\cal E}$ a deformation of the tangent bundle, and

give a detailed derivation for an example on $\mathbb{P}^1 \times

\mathbb{P}^1$.

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Quantum sheaf cohomology

A Mathematical Theory of Quantum Sheaf Cohomology: http://front.math.ucdavis.edu/1110.3751

nLab page on Quantum sheaf cohomology