[Paper Review] On quasimaps to quadrics

M. V. Movshev|arXiv (Cornell University)|Aug 4, 2010

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper rigorously establishes the acyclicity of the mini-BRST complex for $etaeta$-systems on quadrics using Drinfeld's quasimaps to approximate loop spaces. By constructing finite-dimensional quasimap models, it proves that cohomology is concentrated in degree zero, confirming a key technical assumption in earlier work on pure spinor string theory and providing a framework for analyzing infinite-dimensional quantum field theories via algebraic geometry.

ABSTRACT

We put on a rigorous basis results of Aisaka and Aldo Arroyo on curved βγ systems on a quadric. Drinfeld's quasimaps turned to be indispensable in accurate computation of mini-BRST cohomology.

Motivation & Objective

• To provide a mathematically rigorous foundation for the mini-BRST cohomology computation in curved $etaeta$-systems on quadrics, as proposed in prior work.
• To resolve the ambiguity in infinite-dimensional cohomology by replacing loop spaces with finite-dimensional quasimap models via Drinfeld's construction.
• To prove that the Koszul complex associated with the quadric constraint has cohomology only in degree zero, confirming a central assumption in the physics literature.
• To establish the Koszul property and compute the Poincaré series of the algebra of quasimap functions, enabling further algebraic analysis.
• To lay the groundwork for extending this framework to the more complex case of pure spinor targets in string theory.

Proposed method

• Approximate the space of loops on a quadric using finite-dimensional Drinfeld quasimaps, defined by Laurent polynomials with bounded Fourier modes: $ ho(z) = igoplus_{-N_1 o N_2} ho^i[k] e_i z^k $.
• Define the algebra $ AQ_{-N_1}^{N_2}(n) $ as the quotient of polynomial functions on Fourier coefficients modulo the relations $ f[k] = igsum_{t+s=k} igsum_i ho^i[t] ho^i[s] = 0 $.
• Construct the Koszul complex $ C_{-N_1}^{N_2}(n) $ with generators $ ho^i[k] $ (even) and $ c[k] $ (odd), and differential $ d = igsum_k f[k] rac{ au}{ au c[k]} $.
• Use Gröbner basis techniques and the Buchberger algorithm to compute the Poincaré series and verify the Koszul property of $ AQ $, relying on straightening laws and monomial orderings.
• Apply the canonical reduction algorithm to compute $ ext{con}(f, I) $, ensuring unambiguous reduction via a well-ordered basis of monomials.
• Prove that the direct limit of $ ext{Spec}(AQ_{-N_1}^{N_2}(n)) $ captures essential geometric and cohomological features of the full loop space, avoiding infinite-dimensional analysis.

Experimental results

Research questions

• RQ1 Does the mini-BRST complex for $etaeta$-systems on quadrics have cohomology only in degree zero, as assumed in physics literature?
• RQ2 Can Drinfeld's quasimaps provide a mathematically rigorous and finite-dimensional approximation to the infinite-dimensional space of loops on a quadric?
• RQ3 What is the structure of the algebra $ AQ $ of quasimap functions, and does it satisfy the Koszul property necessary for acyclicity?
• RQ4 How can the Poincaré series of $ AQ $ be computed algebraically, and what does it reveal about the Hilbert series of the quantum cohomology?
• RQ5 Can this framework be extended to the more complex case of pure spinor targets in the pure spinor formalism of superstring theory?

Key findings

• The zeroth cohomology of the Koszul complex $ C_{-N_1}^{N_2}(n) $ is isomorphic to $ AQ_{-N_1}^{N_2}(n) $, and all higher cohomology groups vanish for $ n o 3 $, confirming the acyclicity of the complex.
• The algebra $ AQ $ of quasimap functions satisfies the Koszul property, which is essential for proving the acyclicity of the resolution and enables the use of Poincaré series techniques.
• The Poincaré series of $ AQ $ is computed via Gröbner basis methods, and the result is consistent with the expected Hilbert series of the coordinate ring of the quadric.
• The canonical reduction algorithm applied to $ S $-polynomials ensures unambiguous computation of normal forms, validating the use of Buchberger’s algorithm in this infinite-dimensional context.
• The direct limit of $ ext{Spec}(AQ_{-N_1}^{N_2}(n)) $ as $ N_1, N_2 o iginfty $ captures the essential geometric and cohomological structure of the loop space, justifying the finite-dimensional approximation.
• The method provides a viable algebraic alternative to infinite-dimensional analysis, enabling the study of quantum cohomology and BRST structures in a finite, computable setting.