Precise constraints on a $tau$ function in 2D quantum gravity

Quantum gravity is an interesting object in the current research of mathematical physics. In 2D quantum gravity, Kontsevich [1] proved Witten's conjecture [2] that two different approaches to 2D quantum gravity coincide. That is, a partition function for the intersection theory of moduli space is the logarithm of some τ function [3] which satisfies the string equation and the 2-reduced KP hierarchy. Meanwhile, using Kontsevich's matrix integral representation of the partition function, Witten [4] showed the exponent of the partition function is a vacuum vector for the Virasoro algebra. Together with the conclusion [5,6] that a τ function which satisfies the string equation and the 2-reduced KP hierarchy is equivalent to a vacuum vector for the Virasoro algebra, he also obtained the equivalence of the two approaches. Since integrable system has close connection with the string theory and the intersection theory [7,8], this conclusion has been wildly researched with various methods in this field [9,10,11,12,13]. An interesting problem is to extend the conclusion in [5,6] from 2 to an arbitrary p, which is to obtain the equivalence between a τ function constrained by the string equation and the p-reduced KP hierarchy and a vacuum vector of some algebra which include the Virasoro algebra as a subalgebra. When p = 3, Goeree [14] showed that it is true. And the case for bigger p had also been researched in [6,7,13].

In order to obtain the above equivalence for an arbitrary p, it need to obtain the precise constraints which the KP hierarchy and the string equation impose on tau functions. When we calculate them, it creates a lots of constants in the obtained constraints whose values are unknown. When p = 2, the constrains constitute the Virasoro algebra and we could use the commutation relations of the Virasoro algebra to calculate the values of the constants. But when p ≥ 4, although there are some classical conclusions about W algebra [6,13], it is so hard to calculate the commutation relations of the constraints that the constants in the higher order constraints are still unknown. As far as we have known, there is not an effective computable method to determine them when p ≥ 4. Due to the uncertainty of the constants, we also could not obtain the precise algebraic structure of these constraints. In this study, for an arbitrary p, we propose a new computable method which can determine values of the constants. It is a recursive process and we can directly calculate them step by step. It is usually that assign the value of 0 to all the constants; but here, by this method, we know that all of them are not equal 0. And the none zero constants are closely related with the centers of the algebra that the constrains constitute. When p = 2, our conclusion coincide with the current conclusion, that is, the constants determined through our method being the same as those determined through commutation relations of Virasoro. Consequently, with these values we obtain the precise constraints. And further we obtain the precise algebra which the constraints constitute. The algebra does not include the redundant variables of t mp , and we find the connection between the algebra and the W 1+∞ algebra which include all the variables of t mp . Based on this connection, we can calculate the commutation relations of one algebra from those of the other algebra. And the calculation is much simpler than a straightforward calculation. Furthermore, the obtained algebra include the Virasoro algebra as a subalgebra. So the above conclusion in [5,6] is also included in our conclusion. In addition, we mainly use the tool of pseudo-differential operators to prove the conclusions, which is introduced by Dickey [15] and greatly simplifies the proof.

The organization of the paper is as follows. In section 2, for self-contained we give a brief description of the KP hierarchy. In section 3, we prove the connection between W 1+∞ algebra and the algebra of W = {W (m) n | tmp=0 }. In section 4, we show the approach to calculate unknown constants, which is our main theorem. Meanwhile, we give some examples for our theorems. Section 5 is devoted to conclusions. §2. KP hierarchy To be self-contained, we give a brief introduction to the KP hierarchy based on a detailed research in [15].

Let F be an associative ring of functions which include infinite time variables t i ∈ R:

Denote ∂ t 1 by ∂, which is the common differential operator on the first variable t 1 . Its actions on f (t) are

Here the symbol " • " denote the multiplication between operators. If we consider a function f (t) as a operator whose action on g(t) ∈ F is f (t)g(t), we can infer the following identity about multiplication of function operators and differential operators. That is for any j ∈ Z

If the function operators are located on the the left-hand side we omit " • “. So with (2.2) we could obtain an associative ring F (∂) of formal pseudo differential operators, which includes

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