On an Equivalence of Divisors on $\overline {\text {M}}_{0,n}$ from Gromov-Witten Theory and Conformal Blocks
Abstract

We consider a conjecture that identifies two types of base point free divisors on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$to the same statement forn= 4. A reinterpretation leads to a proof of the conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

Authors:
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Publication Date:
NSF-PAR ID:
10369886
Journal Name:
Transformation Groups
ISSN:
1083-4362
Publisher:
National Science Foundation
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1. Abstract

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x1,…,xn) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum }_{i}{x}_{i}$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

#SAT algorithms fornk-size$\mathcal { C}$$C$-circuits running in 2n/nktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$\left(f\circ C\right)$-circuits of polynomial size.

#SAT algorithms for$2^{n^{{\varepsilon }}}$${2}^{{n}^{\epsilon }}$-size$\mathcal { C}$$C$-circuits running in$2^{n-n^{{\varepsilon }}}$${2}^{n-{n}^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$$\left(f\circ C\right)$-circuits of polynomial size.

Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJACC0THRcircuits of polynomialmore »

2. Abstract

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3. Abstract

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4. Abstract

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