arXiv subjects

Alexander Givental

Publications and source records attributed to Alexander Givental.

At least 19 recordsLinked to original sources

arXiv2026

Intelligent qubits

We introduce the method of ``intelligent qubits,'' which replace live observers in Wigner's friend and Frauchiger--Renner thought experiments, in order to expose the source of the paradoxes: tacit substitution of Boolean hidden variables for non-commuting quantum propositions. The fallacy traces back to London and Bauer's attempt (1939) to legitimize a self-referential interpretation of observers' ``introspection,'' and thus intertwines the measurement and mind-body problems. We cut the Gordian knot by abandoning reductionism, and describe an alternative which fits, we argue, equally naturally with both physics and anthropology takes on the problems.

quant-ph↗

arXiv2024

Virasoro Constraints for Toric Bundles

We show that the Virasoro conjecture in Gromov--Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base $B$. The main steps are: (i) we establish a localization formula that expresses Gromov--Witten invariants of $E$, equivariant with respect to the fiberwise torus action, in terms of genus-zero invariants of the toric fiber and all-genus invariants of $B$; and (ii) we pass to the non-equivariant limit in this formula, using Brown's mirror theorem for toric bundles.

math.AG↗

arXiv2024

On merit and equity in math

In recent paper "Quantifying Inequities and Documenting Elitism in PhD-granting Mathematical Sciences Departments in the United States" (arXiv:2308.13750) by a group of accomplished and/or aspiring mathematicians, the authors use data to challenge ``the idea that the mathematical sciences in the United States is a meritocracy''. We show that, regardless of the validity of the challenged idea, the arguments in the paper are invalid. Namely, among other flaws, they rely on a logical trick of asserting the validity of the conclusion derived from an assumption which the authors neglect to test (and which actually contradicts their own data).

math.HO↗

arXiv2023

On quantum measurement

We propose a solution to the quantum measurement paradox by first identifying its classical counterpart.

physics.gen-ph↗

arXiv2021

Quantum K-theory of grassmannians and non-abelian localization

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

math.AG↗

arXiv2020

Permutation-equivariant quantum K-theory X. Quantum Hirzebruch-Riemann-Roch in genus 0

We extract genus $0$ consequences of the all genera quantum HRR formula proved in Part IX. This includes re-proving and generalizing the adelic characterization of genus $0$ quantum K-theory found in [Givental A., Tonita V., in Symplectic, Poisson, and Noncommutative Geometry, Math. Sci. Res. Inst. Publ. , Vol. 62, Cambridge University Press, New York, 2014, 43-91]. Extending some results of Part VIII, we derive the invariance of a certain variety (the "big J-function"), constructed from the genus $0$ descendant potential of permutation-equivariant quantum K-theory, under the action of certain finite difference operators in Novikov's variables, apply this to reconstructing the whole variety from one point on it, and give an explicit description of it in the case of the point target space.

math.AG↗

arXiv2017

Permutation-equivariant quantum K-theory XI. Quantum Adams-Riemann-Roch

We introduce twisted permutation-equivariant GW-invariants, and compute them in terms of untwisted ones. The computation is based on Grothendieck-like RR formula corresponding to Adams' operations from K-theory to itself, and the result can be understood as a "quantum" version of such Adams-RR. As in the case of cohomological quantum RR theorem [3], the result is applied to express the invariants of bundle and super-bundle spaces in terms of those of the base. The bonus feature of permutation-equivariant K-theory is that the twisting classes can be understood as the simpler kappa-classes of Kabanov--Kimura [9].

math.AG↗

arXiv2017

Permutation-equivariant quantum K-theory IX. Quantum Hirzebruch-Riemann-Roch in all genera

We introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic characterization from the paper [7] by Givental-Tonita, and is based on the application of Kawasaki's Riemann-Roch formula to moduli spaces of stable maps.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory VIII. Explicit reconstruction

In Part VII, we proved that the range of the big J-function in permutation-equivariant genus-0 quantum K-theory is an overruled cone, and gave its adelic characterization. Here we show that the ruling spaces are D_q-modules in Novikov's variables, and moreover, that the whole cone is invariant under a large group of symmetries defined in terms of q-difference operators. We employ this for the explicit reconstruction of the cone from one point on it, and apply the result to toric target spaces, when such a point is given by the q-hypergeometric function.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory VII. General theory

We introduce K-theoretic GW-invariants of mixed nature: permutation-equivariant in some of the inputs and ordinary in the others, and prove the ancestor-descendant correspondence formula. In genus 0, combining this with adelic characterization, we derive that the range ${\mathcal L}_X$ of the big J-function in permutation-equivariant theory of a target space $X$ is overruled.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory VI. Mirrors

We present here the K-theoretic version of mirror models of toric manifold. First, we recall the construction of cohomological mirrors for toric manifolds, i.e. representations of the toric hypergeometric functions from quantum cohomology theory by complex oscillating integrals. Then we repeat the construction in the K-theoretic situation, and obtain complex oscillating integrals representing q-hypergeometric functions from quantum K-theory of toric manifolds, bundles, and super-bundles. Finally, we examine the Lagrangian varieties parameterized by critical points of the phase functions of these oscillating integrals.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory V. Toric $q$-hypergeometric functions

We first retell in the K-theoretic context the heuristics of $S^1$-equivariant Floer theory on loop spaces which gives rise to $D_q$-module structures, and in the case of toric manifolds, vector bundles, or super-bundles to their explicit $q$-hypergeometric solutions. Then, using the fixed point localization technique developed in Parts II--IV, we prove that these $q$-hypergeometric solutions represent K-theoretic Gromov-Witten invariants.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory IV. $D_q$-modules

In Part II, we saw how genus-0 permutation-equivariant quantum K-theory of a manifold with isolated fixed points of a torus action can be reduced via fixed point localization to permutation-equivariant quantum K-theory of the point. In Part III, we gave a complete description of genus-0 permutation-equivariant quantum K-theory of the point by means of adelic characterization. Here we apply the adelic characterization to introduce the action on this theory of a certain group of $q$-difference operators. This action will enable us to prove that toric $q$-hypergeometric functions represent K-theoretic GW-invariants of toric manifolds.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory III. Lefschetz' formula on $\overline{M}_{0,n}/S_n$ and adelic characterization

We continue our study of the genus-$0$ permutation-equivariant quantum K-theory of the target $X=pt$, and completely determine the "big J-function" of this theory. The computation is based on the application of Lefschetz' fixed point formula to the action of $S_n$ on $\overline{M}_{0,n+1}$. It is an instance of the general "adelic characterization" (which we state at the end with reference to arXiv:1106.3136) of quantum K-theory for any target $X$ in terms of quantum cohomology theory. Yet, some simplifications of non-conceptual nature occur in this example, making it a lucid illustration to the general theory.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory II. Fixed point localization

Using projective spaces as examples of toric manifolds, we examine K-theoretic fixed point localization. On the one hand, we will see how the permutation-equivariant theory of the point target space emerges as a necessary ingredient. On the other hand, we will completely characterize the genus-0 permutation-equivariant quantum K-theory of the given toric manifold in terms of such theory for the point, and a certain recursion relation.

math.AG↗

arXiv2015

Permutation-equivariant quantum K-theory I. Definitions. Elementary K-theory of $\overline{\mathcal M}_{0,n}/S_n$

K-theoretic Gromov-Witten invariants of a compact Kahler manifold $X$ are defined as super-dimensions of sheaf cohomology of interesting bundles over moduli spaces of n-pointed holomorphic curves in X. With this article, we begin a series of publications on K-theoretic Gromov-Witten invariants, cognizant of the $S_n$-module structure on the sheaf cohomology, induced by renumbering of the marked points. In the opening paper, we introduce such invariants, explain how the representation-theoretic information with varying $S_n$ is incorporated into generating functions of Gromov-Witten theory, and compute one of them, the small J-function for $X=pt$, by using Kapranov's description of Deligne-Mumford spaces $\overline{\mathcal M}_{0,n}$.

math.AG↗

arXiv2015

Explicit reconstruction in quantum cohomology and K-theory

Cohomological genus-0 Gromov-Witten invariants of a given target space can be encoded by the "descendant potential," a generating function defined on the space of power series in one variable with coefficients in the cohomology space of the target. Replacing the coefficient space with the subspace multiplicatively generated by degree-2 classes, we explicitly reconstruct the graph of the differential of the restricted generating function from one point on it. Using the Quantum Hirzebruch--Riemann--Roch Theorem from our joint work with Valentin Tonita, we derive a similar reconstruction formula in genus-0 quantum K-theory. The results amplify the role of the divisor equations, and the structures of $D$-modules and $D_q$-modules in quantum cohomology and quantum K-theory with respect to Novikov's variables.

math.AG↗

arXiv2011

The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory

We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by applying (a virtual version of) the Kawasaki-Hirzebruch-Riemann-Roch formula for expressing holomorphic Euler characteristics of orbibundles on moduli spaces of genus-0 stable maps, analyzing the sophisticated combinatorial structure of inertia stacks of such moduli spaces, and employing various quantum Riemann--Roch formulas from "fake" (i.e. orbifold-ignorant) quantum K-theory of manifold and orbifolds (formulas, either previously known from works of Coates-Givental, Tseng, and Coates-Corti-Iritani-Tseng, or newly developed for this purpose in separate papers by Tonita). The ultimate formulation combines properties of overruled Lagrangian cones in symplectic loop spaces (the language, that has become traditional in description of generating functions of genus-0 Gromov-Witten theory) with a novel framework of "adelic characterization" of such cones. As an application, we prove that tangent spaces of the overruled Lagrangian cones of quantum K-theory carry a natural structure of modules over the algebra of finite-difference operators in Novikov's variables. As another application, we compute one of such tangent spaces for each of the complete intersections given by equations of degrees $l_1,...,l_k$ in a complex projective space of dimension $\geq l_1^2+...+l_k^2-1$.

math.AG↗